Dynamics of
quasi-two-dimensional
turbulent jets
Julien Re´my Dominique Ge´rard Landel
Churchill College
A dissertation submitted for the degree of Doctor of Philosophy
Department of Applied Mathematics and Theoretical Physics
and
BP Institute
The University of Cambridge
May 2012
Dynamics of quasi-two-dimensional turbulent jets
Julien Re´my Dominique Ge´rard Landel
Abstract
The study of quasi-two-dimensional turbulent jets is relevant to chemical reactors,
the coking process in oil refinement, as well as rivers flowing into lakes or oceans. In
the event of a spillage of pollutants into a river, it is critical to understand how these
agents disperse with the flow in order to assess damage to the environment.
For such flows, characteristic streamwise and cross-stream dimensions can be much
larger than the fluid-layer thickness, and so the flow develops in a confined environment.
When the distance away from the discharge location is larger than ten times the fluid-
layer thickness, the flow is referred to as a quasi-two-dimensional jet.
From experimental observations using dyed jets and particle image velocimetry, we
find that the structure of a quasi-two-dimensional jet consists of a high-speed mean-
dering core with large counter-rotating eddies developing on alternate sides of the core.
The core and eddy structure is self-similar with distance from the discharge location.
The Gaussianity of the cross-stream distribution of the time-averaged velocity is due,
in part, to the sinuous instability of the core.
To understand the transport and dispersion properties of quasi-two-dimensional jets
we use a time-dependent advection–diffusion equation, with a mixing length hypothesis
accounting for the turbulent eddy diffusivity. The model is supported by experimental
releases of dye in jets or numerical releases of virtual passive tracers in experimentally-
measured jet velocity fields.
We consider the statistical properties of this flow by releasing and then tracking large
clusters of virtual particles in the jet velocity field. The probability distributions of two-
point properties (such as the distance between two particles) reveal large streamwise
dispersion. Owing to this streamwise dispersive effect, a significant amount of tracers
can be transported faster than the speed predicted by a simple advection model.
Using potential theory, we determine the flow induced by a quasi-two-dimensional
jet confined in a rectangular domain. The streamlines of the induced flow predicted by
the theory agree with experimental measurements away from the jet boundary.
Finally, we investigate the case of a quasi-two-dimensional particle-laden jet. De-
pending on the bulk concentration of dense particles, we identify different flow regimes.
At low concentrations, the jet features the same core and eddy structure observed with-
out the particles, and thus quasi-two-dimensional jet theory can apply to some extent.
At larger concentrations, we observe an oscillating instability of the particle-laden jet.
Preface
This thesis, which is submitted for the degree of Doctor of Philosophy at Churchill
College, University of Cambridge, describes work carried out from January 2009
to May 2012 in the Department of Applied Mathematics and Theoretical Physics
and the BP Institute, University of Cambridge. This dissertation is the result
of my own work and includes nothing which is the outcome of work done in
collaboration, except where specifically indicated in the text. No part of this
work has been, or is being submitted for any other qualification at this or any
other university.
Acknowledgements
First of all, I would like to express my deep appreciation to my supervisor Dr
Colm Caulfield and my advisor Prof. Andy Woods for giving me the opportunity
to pursue my PhD research on this fascinating project, here at the University
of Cambridge. I am particularly grateful for the constant support and useful
advice I have received from Dr Colm Caulfield. He has taught me the rigour of
mathematical modelling, and how it can be applied to physical problems. I wish
to thank Prof. Andy Woods for all the fruitful and enthusiastic discussions. His
unique way of simplifying any complex problem into its essential components will
continue to serve me as a guide throughout my professional career.
I am very grateful to Dr Stuart Dalziel for his invaluable help in experimental
and technical matters, and in the use of DigiFlow. He has always made himself
available at any time during my PhD. His knowledge and expertise in experimental
Fluid Dynamics is a wonderful source of inspiration.
I would like to extend my gratitude to my friend Samuel Rabin for helping me
resolve numerous mathematical problems and equations throughout my PhD. I
would like to thank my friends Megan Davies Wykes, Alan Jamieson, Antoine
Julia, Hugh Lund and Dr Ben Maurer for proofreading parts of this document. I
am also grateful to all my colleagues and friends at the G.K. Batchelor lab and
at the BPI lab for useful chats as well as many memorable events and dinners
together.
Thanks are due to the technician of the BPI, Andrew Pluck, who built my
experimental apparatus. I am also grateful to the DAMTP technicians, David
Page-Croft, Colin Hitch, John Milton and Neil Price, for their technical support.
I would like to thank BP, the Engineering and Physical Sciences Research Coun-
cil, the Cambridge Philosophical Society, DAMTP, the BPI and Churchill College
for their financial support.
I would like to express my love and gratitude to my parents, sister, brothers
and other family members. This work would not have been possible without their
continuous encouragement, support and love.
Finally, I am eternally grateful to the Creator of all things. I am simply fasci-
nated by the wonders of His work.
Contents
1 Overview 1
2 Meandering due to large eddies and the statistically self-similar
dynamics of quasi-two-dimensional jets 5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Dye tracking experiments . . . . . . . . . . . . . . . . . . 8
2.2.2 Particle image velocimetry experiments . . . . . . . . . . . 9
2.3 Qualitative observations . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Time-averaged mean flow field . . . . . . . . . . . . . . . . . . . . 12
2.5 Quantitative analysis of the time-dependent core and eddy structure 20
2.5.1 Time-dependent eddy structure . . . . . . . . . . . . . . . 20
2.5.2 Time-dependent core structure . . . . . . . . . . . . . . . 26
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 Advection–diffusion model for the streamwise transport, disper-
sion and mixing in quasi-two-dimensional jets 33
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Turbulent model hypothesis . . . . . . . . . . . . . . . . . . . . . 37
i
Contents
3.3 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.1 Similarity transformation . . . . . . . . . . . . . . . . . . . 42
3.3.2 Constant-flux release: concentration . . . . . . . . . . . . . 43
3.3.3 Constant-flux release: concentration flux . . . . . . . . . . 49
3.3.4 Finite-volume release: instantaneous release fundamental
solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3.5 Finite-volume release: time-dependent release general solu-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4 Streamwise transport, dispersion and mixing in quasi-two-dimen-
sional jets: experimental results 67
4.1 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . 67
4.1.1 Constant-flux releases of dye . . . . . . . . . . . . . . . . . 68
4.1.2 Instantaneous finite-volume releases of clusters of virtual
particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.1.3 Finite-volume releases of dye . . . . . . . . . . . . . . . . . 71
4.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2.1 Qualitative assessment . . . . . . . . . . . . . . . . . . . . 74
4.2.2 Constant-flux releases of dye . . . . . . . . . . . . . . . . . 80
4.2.3 Instantaneous finite-volume releases of clusters of virtual
particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.2.4 Finite-volume releases of dye . . . . . . . . . . . . . . . . . 93
4.3 Statistical significance of the experimental results . . . . . . . . . 99
4.3.1 Constant-flux release of dye . . . . . . . . . . . . . . . . . 100
4.3.2 Instantaneous finite-volume release of virtual particles . . . 102
4.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5 Two-point statistics for turbulent relative dispersion in quasi-
two-dimensional jets 111
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.2 Mathematical definitions of two-point probability distributions . . 115
5.2.1 Continuous formulation . . . . . . . . . . . . . . . . . . . . 115
5.2.2 Discrete formulation . . . . . . . . . . . . . . . . . . . . . 117
ii
Contents
5.3 Test studies in diverging velocity fields . . . . . . . . . . . . . . . 117
5.3.1 Circular domain in an axisymmetric diverging velocity field 118
5.3.2 Elliptical domain in a non-axisymmetric diverging velocity
field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.3.3 Square cluster of virtual particles in a diffusive velocity field 126
5.3.4 Conclusion of the test studies . . . . . . . . . . . . . . . . 129
5.4 Analysis of the virtual particles in the jet structures . . . . . . . . 131
5.4.1 Virtual particles: time-dependent versus time-averaged ve-
locity fields . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.4.2 Two-point statistics: time-dependent versus time-averaged
velocity fields . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.5 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . 139
6 Flow induced by a quasi-two-dimensional jet in a confined rect-
angular domain 145
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.2 Potential flow model . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.2.1 Description of the entrainment problem . . . . . . . . . . . 149
6.2.2 Decomposition of the problem . . . . . . . . . . . . . . . . 151
6.2.3 Solution to the uniform problem ϕ˜u . . . . . . . . . . . . . 152
6.2.4 Solution to the perturbation problem ϕ˜p . . . . . . . . . . 155
6.2.5 Solution to the entrainment problem . . . . . . . . . . . . 161
6.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.3.1 Experimental procedure . . . . . . . . . . . . . . . . . . . 165
6.3.2 Qualitative observations . . . . . . . . . . . . . . . . . . . 166
6.3.3 Quantitative results . . . . . . . . . . . . . . . . . . . . . . 168
6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7 Dynamics of particle-laden jets in quasi-two-dimensional envi-
ronments 179
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
7.2 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . 181
7.3 Phenomenological description . . . . . . . . . . . . . . . . . . . . 182
7.3.1 Regime diagram . . . . . . . . . . . . . . . . . . . . . . . . 182
7.3.2 Regime I: fluidized bed . . . . . . . . . . . . . . . . . . . . 184
iii
Contents
7.3.3 Regime II: oscillatory flow . . . . . . . . . . . . . . . . . . 184
7.3.4 Regime III: core and eddy flow . . . . . . . . . . . . . . . 187
7.4 Core and eddy flow model . . . . . . . . . . . . . . . . . . . . . . 187
7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
7.5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
7.5.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . 192
8 Conclusion and future work 197
8.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
8.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
A Advection–diffusion model for quasi-two-dimensional jets 203
A.1 Proof of equation (3.89) . . . . . . . . . . . . . . . . . . . . . . . 203
B Two-point statistics in circular distributions 205
B.1 Conditional probability for the x-distance between two points in a
disc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
B.2 Value at the origin for the p.d.f. of the lateral distance between
two points in a disc . . . . . . . . . . . . . . . . . . . . . . . . . . 207
B.3 Value at 2R(t) for the p.d.f. of the lateral distance between two
points in a disc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
B.4 P.d.f of the x-distance between two points in a square domain . . 208
B.5 Conditional probability for the Euclidean distance between two
points in a disc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
B.6 Conditional probability for the ratio of the lateral distance to the
streamwise distance between two points in a disc . . . . . . . . . . 211
Bibliography 212
iv
List of Figures
1.1 Photographs of quasi-two-dimensional jets in the laboratory and
in nature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Schematic diagram of the experimental apparatus. . . . . . . . . . 8
2.2 Sequence of grey-scale pictures of a dyed jet. . . . . . . . . . . . . 12
2.3 The eddy and core structure of quasi-two-dimensional jets. . . . . 13
2.4 Average dye edge and average velocity spread rate. . . . . . . . . 16
2.5 Maximum time-averaged streamwise velocity versus height. . . . . 17
2.6 Time-averaged momentum flux versus height. . . . . . . . . . . . 18
2.7 Time-averaged streamwise velocity at various heights. . . . . . . . 19
2.8 Identification of the eddy and core structures. . . . . . . . . . . . 22
2.9 Eddy locations in quasi-two-dimensional jets. . . . . . . . . . . . . 23
2.10 Eddy z-coordinate versus time. . . . . . . . . . . . . . . . . . . . 24
2.11 Eddy frequency and Strouhal number versus height. . . . . . . . . 25
2.12 Time-averaged mean core structure. . . . . . . . . . . . . . . . . . 28
2.13 Instantaneous streamwise velocity at different heights. . . . . . . . 29
2.14 Mean streamwise velocity and mean core–eddy structure. . . . . . 32
3.1 Concentration similarity solution for the constant-flux release. . . 47
v
List of Figures
3.2 Centroid and standard deviation of the concentration solution for
the constant-flux release. . . . . . . . . . . . . . . . . . . . . . . . 48
3.3 Volume of tracers ahead of the advective front for the constant-flux
release. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4 Concentration-flux similarity solution for the constant-flux release. 52
3.5 Centroid and standard deviation of the concentration-flux solution
for the constant-flux release. . . . . . . . . . . . . . . . . . . . . . 54
3.6 Concentration flux of tracers ahead of the advective front for the
constant-flux release. . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.7 Similarity solution for the instantaneous finite-volume release. . . 58
3.8 Centroid and standard deviation of the concentration solution for
the instantaneous finite-volume release. . . . . . . . . . . . . . . . 60
3.9 Volume of tracers ahead of the advective front for the instantaneous
finite-volume release. . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.10 Deviation between the general solution and the fundamental solu-
tion for the finite-volume release. . . . . . . . . . . . . . . . . . . 64
4.1 Schematic diagram of the experimental apparatus. . . . . . . . . . 68
4.2 Core and eddy structures in quasi-two-dimensional jets revealed by
dye and passive tracers. . . . . . . . . . . . . . . . . . . . . . . . 75
4.3 Typical trajectories of single virtual particles in an eddy, in the
core and at the interface between the two structures. . . . . . . . 78
4.4 Trajectories of virtual particle clusters seeded in an eddy, in the
core and at the interface between the two structures. . . . . . . . 79
4.5 Constant-flux and finite-time finite-volume dye releases in quasi-
two-dimensional jets. . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.6 Time evolution of the experimental dye concentration for constant-
flux releases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.7 Comparison between experimental data and theory for the concen-
tration of constant-flux releases. . . . . . . . . . . . . . . . . . . . 85
4.8 Comparison between experimental data and theory for the concen-
tration flux of constant-flux releases. . . . . . . . . . . . . . . . . 88
4.9 Instantaneous finite-volume releases of virtual particles. . . . . . . 90
4.10 Comparison between experimental data and theory for the concen-
tration of instantaneous finite-volume releases. . . . . . . . . . . . 92
vi
List of Figures
4.11 Experimental dye data for finite-time finite-volume releases. . . . 95
4.12 Comparison between experimental data and theory for the concen-
tration of finite-time finite-volume releases. . . . . . . . . . . . . . 98
4.13 Evolution in time of the theoretical prediction of the concentration
for a finite-time finite-volume release. . . . . . . . . . . . . . . . . 99
4.14 Probability density function and critical probability of the experi-
mental dye concentration in the case of constant-flux releases. . . 101
4.15 Probability density function and critical probability of the concen-
tration of virtual particles in the case of instantaneous finite-volume
releases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.1 Probability distributions of two-point properties for a uniformly
distributed disc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2 Time evolution of a uniform distribution of particles in an ellipse. 125
5.3 Probability density functions of two-point properties for a uni-
formly distributed ellipse. . . . . . . . . . . . . . . . . . . . . . . 127
5.4 Time evolution of an initially square distribution of particles fol-
lowing random walks. . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.5 Probability density functions of two-point properties for a diffusing
cluster of virtual particles. . . . . . . . . . . . . . . . . . . . . . . 129
5.6 Time evolution of three virtual-particle clusters seeded in an eddy,
in the core and at the interface between the two structures in both
a time-averaged and a time-dependent velocity field. . . . . . . . . 133
5.7 Probability distributions of two-point properties for a cluster of
virtual particles seeded at the location of an eddy for the time-
dependent and the time-averaged velocity fields. . . . . . . . . . . 136
5.8 Probability distributions of two-point properties for a cluster of
virtual particles seeded at the interface between an eddy and the
core for the time-dependent and the time-averaged velocity fields. 138
5.9 Probability distributions of two-point properties for a cluster of
virtual particles seeded in the core for the time-dependent and the
time-averaged velocity fields. . . . . . . . . . . . . . . . . . . . . . 140
6.1 Description of the entrainment problem. . . . . . . . . . . . . . . 152
vii
List of Figures
6.2 Decomposition of the entrainment problem into a uniform problem
and a perturbation problem. . . . . . . . . . . . . . . . . . . . . . 153
6.3 Potential and stream function of the uniform problem. . . . . . . 154
6.4 Velocity field of the uniform problem. . . . . . . . . . . . . . . . . 154
6.5 Potential and stream function of the perturbation problem. . . . . 157
6.6 Velocity field of the perturbation problem. . . . . . . . . . . . . . 158
6.7 Distribution of the flux at the jet boundary for the perturbation
problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.8 Convergence of the numerical truncated series for the flux at the
jet boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.9 Potential and stream function of the flow induced by a quasi-two-
dimensional jet modelled as a varying line sink. . . . . . . . . . . 164
6.10 Velocity field of the flow induced by a quasi-two-dimensional jet
modelled as a varying line sink. . . . . . . . . . . . . . . . . . . . 165
6.11 Schematic diagram of the experimental apparatus. . . . . . . . . . 166
6.12 Pictures of the flow induced by quasi-two-dimensional jets. . . . . 169
6.13 Experimental and theoretical distributions of the streamlines of the
flow induced by quasi-two-dimensional jets. . . . . . . . . . . . . . 170
6.14 Experimental and theoretical distributions of the velocity field in-
duced by quasi-two-dimensional jets. . . . . . . . . . . . . . . . . 171
6.15 Experimental and theoretical distributions of the volume flux and
momentum flux of the flow induced by quasi-two-dimensional jets. 174
7.1 Experimental apparatus to study quasi-two-dimensional particle-
laden jets (Q2DPL jets). . . . . . . . . . . . . . . . . . . . . . . . 182
7.2 Regime diagram. Three phenomenological regimes are observed
during the Q2DPL jet experiment. . . . . . . . . . . . . . . . . . 183
7.3 Illustration of the fluidization regime. . . . . . . . . . . . . . . . . 185
7.4 Illustration of the oscillatory flow regime. . . . . . . . . . . . . . . 186
7.5 Illustration of a large vortical structure in the oscillatory flow regime.186
7.6 Illustration of the core and eddy flow regime. . . . . . . . . . . . . 188
7.7 Particle maximum height and bed thickness versus flow rate. . . . 191
viii
Chapter 1
Overview
In turbulent jets, fluid is driven by momentum from an orifice into an environ-
ment filled with similar fluid. The complexity of this flow, which has been studied
for more than 80 years (see e.g. List, 1982, for a detailed review), resides in its
turbulent nature. Turbulence develops due to a shear instability at the boundary
between the jet fluid and the ambient fluid. The transition of the flow from lami-
nar to turbulent typically occurs at a Reynolds number Re = bU/ν (where b is the
characteristic width of the jet, U is the jet characteristic streamwise velocity and
ν is the kinematic viscosity of the fluid) of the order of 3000. From the equations
of motion, the momentum flux is approximately conserved (see e.g. Kotsovinos,
1978, for a discussion on the conservation of momentum in turbulent jets), while
its mean kinetic energy is dissipated by turbulence. Momentum spreads laterally
due to entrainment of ambient fluid in the jet. The entrainment process is gov-
erned by the large-scale turbulent structures in the flow and is self-similar in the
streamwise direction.
1
1 Overview
The capacity of turbulent jets to entrain ambient fluid and mix it efficiently
with jet fluid accounts in large part for the attention this flow has received in
both the scientific community and the industrial world. Also, behind the appar-
ent simplicity of the jet mean motion lies the fascination for the elusive underlying
physics of turbulence. Whether for their dilution properties, their efficient mixing
properties or the thrust they can provide, jets have been used in various indus-
trial applications, such as waste water disposal (Yannopoulos, 2006), chemical
reactors (Jirka & Harleman, 1979), or as a means of propulsion (Stanley, Sarkar
& Mellado, 2002). In geophysical flows, turbulent jets are, for instance, relevant
to the study of explosive volcanic eruptions, where a mixture of gas, fluid lava
and solid particles is initially driven by momentum out of the crater (Woods &
Caulfield, 1992).
In this study we are interested in a particular type of turbulent jet called a quasi-
two-dimensional steady turbulent jet (which we refer to, hereafter, as a quasi-two-
dimensional jet). Giger, Dracos & Jirka (1991) gave the first description of quasi-
two-dimensional jets (earlier studies of bounded plane jets include Foss & Jones,
1968; Holdeman & Foss, 1975, who focused on the near field of the flow). They
observed that, in the far field of a plane turbulent jet confined between two close
boundaries separated by a gap widthW , the flow develops into a meandering core
with large counter-rotating eddies growing on alternate sides of the core. A qua-
si-two-dimensional jet designates the region of the flow (starting from z ≈ 10W ,
with z the streamwise distance from the source) where the meandering core and
the large growing eddies appear. The sinuous instability of the jet is due to lateral
transverse shear (Jirka & Uijttewaal, 2004). Dracos, Giger & Jirka (1992) found
an inverse cascade of quasi-two-dimensional turbulence, which affects not only the
structure of the flow but also transport, dispersion and mixing properties. The
aim of this thesis is to investigate experimentally and theoretically the transport,
dispersion and mixing properties of quasi-two-dimensional jets.
In figure 1.1(a), we show a picture of a typical quasi-two-dimensional dyed
jet (Re ≈ 4000) produced in our experimental apparatus (whose gap width is
W = 1 cm). As we can see, for z > 10 cm, the jet meanders and large eddies
form on alternate sides of the core. The same core and eddy structure has been
observed in geophysical flows, such as rivers discharging into lakes or oceans. At
the discharge location, the depth of a river is often much smaller than the other
2
1 Overview
(a)
10
cm
(b)
(c)
Figure 1.1: Meandering quasi-two-dimensional jets in the laboratory and in nature:
(a) grey-scale picture of a dyed steady quasi-two-dimensional turbulent jet (Re ≈ 4000)
rising in our experimental apparatus; (b) photograph of a channel (Re ≈ 107) discharg-
ing from the Lower Mississippi River (near Baton Rouge, LA, USA) into an oxbow lake,
Image Source: 1998 US Geological Survey Digital Ortho-Quarter Quadrangle; (c) pho-
tograph of a river (Re ≈ 107) flowing into Balaton Lake, Hungary (Jirka & Uijttewaal,
2004). In (b) and (c), the meanders are made visible by the sediment transported by
the flow.
two characteristic dimensions of the environment in the horizontal plane. Thus,
as depicted in figures 1.1(b) and 1.1(c), a river flow can develop into a quasi-two-
dimensional jet flow. In figures 1.1(b) and 1.1(c), the core and eddy structures,
displayed by the two rivers discharging into lakes (Re ≈ 107), are revealed by the
sediment transported by the flow.
The study of river flows is relevant to coastal engineering problems, such as
sediment transport and coastal erosion (Joshi & Taylor, 1983), as well as environ-
mental pollution. In the event of a spillage of pollutants in rivers, the prediction
and monitoring of the transport and dispersion of the pollutants is crucial. Ac-
curate models of the flow, tested against experimental evidence, are therefore
needed to control this type of environmental pollution. The main objective of
this thesis is to address these issues.
We compare the flow of a river discharging into a large basin with a (laboratory)
3
1 Overview
quasi-two-dimensional jet. In Chapter 2, we analyze and model the time-averaged
velocity field of quasi-two-dimensional jets. We present a quantitative description
of the characteristic core and eddy structure. We discuss the implications of this
core and eddy structure on the velocity field and the entrainment mechanism
of the flow. Based on this analysis, we propose a model, in Chapter 3, for the
transport and dispersion of passive tracers in the flow. This model is derived from
a general effective advection–diffusion equation, using a mixing length hypothesis
to model the turbulent eddy diffusivity. The theoretical predictions are then
compared with experimental data in Chapter 4. We also study the statistical
significance of the experimental data, and describe a method, based on these
data, to assess pollution risks in quasi-two-dimensional jet flows. In Chapter 5,
we explore further the turbulent relative dispersion mechanisms of the core and
eddy structures using two-point statistical analysis. Then, Chapter 6 presents a
potential model for the flow induced by quasi-two-dimensional jets in a rectangular
domain. We study the impact of the induced flow on the jet. In Chapter 7, we
investigate particle-laden jets confined in a quasi-two-dimensional environment.
We compare the case of a dilute particle-laden jet (i.e. a particle-laden jet with
a small bulk concentration of particles) with a particle-free quasi-two-dimensio-
nal jet. Finally, we summarize the main findings of this thesis in Chapter 8 and
discuss future work.
The results presented in Chapter 2 have been published in Landel, Caulfield
& Woods (2012a). Most of the results described in Chapters 3 and 4 have been
submitted for publication in Journal of Fluid Mechanics, in an article by Landel,
Caulfield & Woods (2012b, sub judice). We adopt a similar structure in every
chapter, except in Chapters 3 and 4 which have a combined structure. The prob-
lem studied in the chapter is introduced in the first section, which also includes a
detailed review of past studies, and the last section is a conclusion of the chapter.
4
Chapter 2
Meandering due to large eddies and
the statistically self-similar dynamics
of quasi-two-dimensional jets
2.1 Introduction
The study of turbulent plane jets is relevant to a wide variety of problems where
both qualitative and quantitative knowledge of the concentration in time and
space of tracers transported by the jet is needed (Kotsovinos, 1975). In many
industrial applications, effluents, waste or even pollutants are released into large
basins such as lakes or oceans. The source of the discharge can be rivers (see
e.g. Rowland, Stacey & Dietrich, 2009, and references therein) or multiport dif-
fusers (for an extensive study, see Jirka, 2006). In both situations, characteristic
horizontal dimensions are much larger than the fluid-layer thickness and the flow
5
2 Meandering and self-similarity of quasi-two-dimensional jets
develops in a confined environment. Early experimental studies of bounded plane
jets by Foss & Jones (1968) and Holdeman & Foss (1975) showed the influence
of secondary flows on the mean flow. However, as Giger et al. (1991) and Dracos
et al. (1992) pointed out, these secondary flows disappear beyond a distance of
10 flow thicknesses. The present work focuses on this far-field region (z/W ≥ 10,
where z is in the flow direction and W is the fluid-layer thickness), where the jet
has been observed to meander due to the development of large eddies that grow
on its sides. In this far-field region, the initially planar two-dimensional jet is
referred to as a quasi-two-dimensional jet because of the influence of the spanwise
restriction on the flow. The key characteristic of quasi-two-dimensional jets is the
development of an instability (see Chen & Jirka, 1998, for a linear stability anal-
ysis of shallow-water jets) featuring large planar counter-rotating eddies. Dracos
et al. (1992) noted that the spanwise distribution of the velocity was approxi-
mately uniform. Moreover, they found that in the far field the mean properties
of the jet remained unchanged and turbulent energy was transferred to large
scales thus indicating an inverse cascade characteristic of quasi-two-dimensional
turbulence. Dracos et al. (1992) observed and studied the significance of large co-
herent eddy structures in the jet. However, using only point measurements, they
could not provide a complete dynamical study of these structures. Recently, Shin-
neeb, Bugg & Balachandar (2011) conducted a statistical analysis of large vortical
structures in shallow-water jets using particle image velocimetry. However, their
layer thickness (W ∼ 5–15 d) was such that the flow evolution was inherently
three-dimensional (albeit confined), and they did not focus on the far-field region
because their measurements were taken only up to z/W ≤ 16. Their study was
also uncorrelated in time, and so they were unable to identify the inherent time
dependence of the flow quantitatively.
We believe that a study of quasi-two-dimensional jets in the regime identified
by Dracos et al. (1992) is necessary to assess the impact of the characteristic flow
structures on the mixing, dispersion and diffusion of tracers in shallow jets, as sug-
gested by Jirka (2001). For instance, undiluted patches of pollutants carried by a
river discharging into the ocean can be disastrous for the local ecology. Informa-
tion about the size, speed and typical travel distances of these patches is therefore
crucial. To address this problem, we analyse the far field of a confined plane jet
using particle image velocimetry. With a fully resolved velocity field in time and
6
2.2 Experimental procedure
space, we can characterize the jet structure phenomenologically. We are par-
ticularly interested in understanding quantitatively the relationship between the
large-scale, and inevitably transient, flow structures and the long-time-averaged
mean properties of the plane jet.
The rest of this chapter is organized as follows. In § 2.2 we describe the ex-
perimental procedure. In § 2.3 we then provide a qualitative overview of the flow
structures observed from dyed-jet experiments and instantaneous velocity fields,
while in § 2.4 we compare measurements of the time-averaged velocity field with
classical theories for two-dimensional plane jets. In § 2.5 we present a quantita-
tive study of the flow structures, in particular by tracking the large eddies as they
interact with the high-speed core. We discuss how the frequency of occurrence of
the eddies changes with distance due to eddy merger. The study of the probabil-
ity density function of the core shows that the time-averaged mean distribution
of the velocity is due to the large-scale dynamics of the core and eddy structure.
Finally in § 2.6 we draw our conclusions.
2.2 Experimental procedure
The experimental apparatus is shown schematically in figure 2.1. Water jets were
discharged vertically upwards in a 1 m (L) × 0.01 m (W ) × 1 m (H) tank made
of 10 mm thick Perspex sheets. An aluminium structure, made of two vertical
beams located 0.4 m apart on each side of the jet axis and one horizontal beam
located 0.8m above the nozzle, was added on each side of the tank to increase the
rigidity of the walls and ensure a uniform gap width. Two overflows on the side
of the tank maintained a constant water depth at 0.915 m. The flow was driven
by a constant-head tank and discharged via a 0.1 m circular rigid tube of aspect
ratio 20, leading to a 5mm (d)×10mm (W )×20mm chamber and finally into the
tank. The aspect ratio of the tube was deemed sufficient to suppress any swirl
in the flow. The flow rate was controlled through a valve and measured with a
precision balance and a stopwatch for each experiment. The flow rate was found
to be consistent in time with an accuracy of approximately 1 %. We conducted
two distinct sets of experiments using two qualitatively different techniques: dye
tracking and particle image velocimetry (PIV).
7
2 Meandering and self-similarity of quasi-two-dimensional jets
x
z
d = 5 mm
Constant-head tank
Overflow
H
=
1
m
L = 1 m
W = 0.01 m
u
w
2b(z)
CCD camera
Red filter
(dyed jets)
Study area
2 (PIV)
Study area
1 (PIV)
Figure 2.1: Schematic diagram of the experimental apparatus. The two PIV study
areas are shown with overlapping dashed lines.
2.2.1 Dye tracking experiments
For the dye tracking experiments, we filled the tank with fresh tap water. We
injected dark blue food dye through a needle placed 0.2 m upstream of the noz-
zle. Simultaneously, we pumped the same volume of fluid out to minimize the
disturbance introduced into the flow. Also, we injected the dye after the flow
reached a steady state in the tank. We used diffuse ambient lighting for these
experiments. A red filter was placed between the objective of the camera and
the tank, as shown in figure 2.1, to increase the contrast between the jets and
the background. The flow motion was recorded with a high-speed 8 bit grey-scale
camera (Fastcam SA1.1 – Photron), mounted with a 62mm focal-length lens. We
analysed 40 dyed jets with jet Reynolds number 2280 ≤ Rej = dws/ν ≤ 4030,
where ws is the source velocity and ν is the kinematic viscosity of water, using
the software code DigiFlow (Sveen & Dalziel, 2005). We determined the location
of the edge of each dyed jet through an intensity criterion. Since the contrast
between the dyed surface and the background was very strong but not saturated,
the edge of the jet was very sharp.
8
2.2 Experimental procedure
2.2.2 Particle image velocimetry experiments
For PIV experiments the tank was filled with water mixed with Pliolite VTAC
particles of average diameter 0.23mm, which served as passive fluid tracers for the
PIV. Approximately 2 mL of rinsing agent (Finish R© rinse aid) was added to the
mixture to prevent aggregation of Pliolite particles. The small change in surface
tension had no influence on the measurements. The choice of this particle size
depended on both hydrodynamic and optical criteria (see e.g. Drayton, 1993).
We find that the particle diameter is of the order of the smallest Kolmogorov
length scale found in the flow, ηK ≈ 0.2 mm. Although this size is not optimal
to study small-scale turbulence, it was the minimum size that could be detected
by the image software while also capturing the largest length scales in the flow.
The particle Stokes number based on the Kolmogorov time scale was StkK ≤ 10−1
(see Xu & Bodenschatz, 2008), which guaranteed that these particles followed the
fluid motion closely. The particle concentration was kept relatively uniform at
approximately 1.7 × 10−5 by volume due to the turbulence in the tank. Since
the particle concentration was smaller than 10−3 by volume, particle–particle
interactions and any changes in the fluid viscosity were insignificant (see Fung,
1990, for more discussion). We adjusted the water density to match the particle
density of 1.03 g cm−3 by adding 35g of salt per litre of water. At rest, the particle
distribution remained unchanged over 18 hours, thus confirming that the particles
were neutrally buoyant. The mixture of salt water and particles recirculated in the
experimental set-up in order to have identical conditions (particle concentration,
water density and water temperature) for each experiment.
We performed the PIV experiments in a dark room. Two 1kW filament photo-
graphic lamps, each mounted with a long focal-length spherical lens to focus the
filament into a sheet, illuminated the tank from above through a 5mm slit centred
on the mid-plane (y = 0). Every effort was made to keep the width of the light
sheet constant and smaller than the gap width in order to attenuate reflection
issues with the tank walls. This also meant that we could not make any measure-
ments away from the mid-plane (y = 0) because as we moved the light sheet closer
to the wall in the narrow gap, reflection at the wall perturbed the measurements.
From image inspection, the number of particles that appeared much slower than
the rest, probably because they were trapped in the boundary layers, was suffi-
ciently small (of the order of 10%) not to affect the imaging analysis and corrupt
9
2 Meandering and self-similarity of quasi-two-dimensional jets
the computation of the velocity field. We recorded the flow motion using the same
high-speed camera as described above. The camera filmed two 0.4m×0.4m study
areas centred on the jet axis (as shown in figure 2.1). The frequency of image
acquisition was set at 500 frames per second for a duration of 10.9 s for study
area 1 and at 250 frames per second for a duration of 21.8 s for study area 2. The
acquisition frequency was much higher than the largest Kolmogorov frequency
scale. Moreover, the length of the video was long enough to compute meaningful
temporal averages. Study area 1 covered a height from z = 0 to 0.4m, while study
area 2 covered a height from z = 0.2 to 0.6m. Hence, the jet was studied from its
source up to a distance of 120 d. The width of the study area is larger than the
length scale of the jet at every height. The 1024×1024pixel images were analysed
using DigiFlow (see Sveen & Dalziel, 2005, and references therein for more detail
about the PIV algorithm used by DigiFlow). The spatial velocity resolution was
at 6.6 mm based on interrogation areas of 17 × 17 pixels with 75 % overlapping.
This resolution proved to be sufficiently small from z = 20 d upwards. Six steady
turbulent jets of flow rates 33.2, 37.0 and 40.3 cm3 s−1 were investigated in both
study areas. The jet Reynolds number was in the range 3320 ≤ Rej ≤ 4030.
2.3 Qualitative observations
A sequence of grey-scale pictures of a typical injection of dye in a steady-state
jet with Rej = 3850 is presented in figure 2.2 as the dye front rises through
the full depth of the quasi-two-dimensional tank. These pictures reveal many
interesting features of quasi-two-dimensional jets. The saturated dye clearly shows
the maximum lateral extent of the turbulent jet. The dye gradually fills a triangle
(plotted in black lines) which suggests that entrainment is self-similar with height,
at least when averaged over sufficiently long times. Before filling the full triangle
width, we can observe (especially in figures 2.2d and 2.2e) an oscillation of the jet,
as the dye path is clearly sinuous. Large round structures corresponding to eddies
can also be identified on either side of the centreline. Dracos et al. (1992) observed
similar structures for a range of distances 10 ≤ z/W ≤ 120. The curvy edge of
the jet suggests a characteristic scale, typically half the width of the triangle
(approximately 10 cm at mid-height). These eddies result from the instability
of the shear layer at the border between the jet and the ambient fluid (Jirka,
10
2.3 Qualitative observations
2001). Furthermore, tongues of ambient fluid (in white or light grey) appear at
the rear of the largest eddies (see arrow in figure 2.2e). This phenomenon was also
observed by Dimotakis, Miake-Lye & Papantoniou (1983) in the far field of round
turbulent jets, and by Thomas & Brehob (1986) for two-dimensional turbulent
jets. The role played by the eddies in the entrainment, by means of engulfment
mechanisms at their rear, was modelled by De Young (1997) in an attempt to
determine quantitatively the mass inflow contribution of large-scale structures in
two-dimensional mixing layers.
Although the eddies observed in quasi-two-dimensional jets, such as the jet
presented in figure 2.2, have some similarities with eddies in planar two-dimensio-
nal jets, it is important to note that the latter are genuine three-dimensional eddies
while the former should be referred to as quasi-two-dimensional eddies because
of the restriction imposed on the flow in the spanwise direction. The growth
dynamics of quasi-two-dimensional eddies is governed by an inverse cascade of
turbulence, while three-dimensional eddies tend to grow with mean-flow length
scales. On the other hand, quasi-two-dimensional eddies also differ from purely
two-dimensional eddies because friction at the boundaries, although relatively
weak, restrains the maximum size of the eddies (Jirka, 2001) and eventually leads
to their disintegration (Dracos et al., 1992). Finally, it is worth noting that at the
leading edge the dye concentration attenuates suggesting that diffusion occurs in
a steady jet. Diffusion in quasi-two-dimensional jets is likely to be the result of a
complex interaction between the eddies and the sinuous turbulent core of the jet.
We return to detailed investigation of this issue in Chapters 3 and 4.
The second batch of experiments involved quantitative measurements of the
velocity field using the PIV technique. Typical results for a jet at Reynolds
number 4030 analysed in study area 2 are depicted in figure 2.3. In figure 2.3(a),
a superposition of 40 images of the filming of the experiment shows the tracers
as streaks to help visualize Eulerian structures in the flow. The corresponding
velocity field is presented in figure 2.3(b), and it is clear that the main structures
of the jet have been captured by the PIV. A high-speed core undulates along the
centreline and is bordered by alternating counter-rotating eddies on the sides. The
eddies are responsible for the entrainment and detrainment of fluid to and from
the central core in a time-dependent fashion. Owing to the particular geometry
of the tank, the turbulence cannot develop isotropically and we observe rather
11
2 Meandering and self-similarity of quasi-two-dimensional jets
(a) (b) (c) (d) (e)
Figure 2.2: Sequence of grey-scale pictures of a dyed jet (Rej = 3850) rising in the
tank, at: (a) 1 s; (b) 2 s; (c) 3 s; (d) 4 s; and (e) 5 s. The average dye edge is plotted
with black lines. The arrow in (e) points at the engulfment mechanism occurring at
the rear of an eddy.
an inverse turbulent cascade in which quasi-two-dimensional eddies grow with
height (De Young, 1997). This mechanism is confirmed in the experiment, as
flow structures increase in size as they are advected upwards. The schematic
cartoon displayed in figure 2.3(c) summarizes these ideas. The time-averaged
mean picture of quasi-two-dimensional jets is associated with a triangular shape
encapsulating all the flow structures, while the time-dependent picture shows a
sinuous core flanked by large growing eddies. This two-part structure remains self-
similar with height and its dynamics is responsible for the Gaussian distribution
of the mean velocity, as we will discuss in § 2.5.
2.4 Time-averaged mean flow field
To characterize the mean behaviour of quasi-two-dimensional jets, we consider
the ideal model of a turbulent momentum jet in a two-dimensional semi-infinite
environment. Adopting the same conventions as Jirka & Harleman (1979), the
flow is considered incompressible and steady in the mean. The x-direction is
the lateral, cross-jet direction, the y-direction is the spanwise direction and the
z-direction is the streamwise, axial direction. The velocity components are desig-
nated by (u, v, w) for the Cartesian system (x, y, z) with the origin at the nozzle
exit. We assume a plane flow in the domain: the velocity field and any other
12
2.4 Time-averaged mean flow field
(a) (b) (c)
Figure 2.3: (a) Passive tracers (Pliolite particles) shown as streaks in a typical jet
(Rej = 4030) filmed in study area 2. (b) Velocity field (arrows) and vorticity field
(background) of the same jet. (c) Schematic diagram describing the structure of qua-
si-two-dimensional jets.
quantities are invariant with y, and v = 0 everywhere. This hypothesis can be
justified in three ways: the velocity profile across the gap must be self-similar
in the core and the influence of the boundary layers is of second order at high
Reynolds number; the v-component of the velocity is negligible compared to the
other two components; and ambient fluid can only be entrained from the sides of
the jet, i.e. in the x-direction. We also use the common hypothesis of a Gaussian
profile (see, for instance, List, 1982) for the time-averaged streamwise velocity,
w(x, z) = wm(z) exp
[
−
( x
b(z)
)2]
, (2.1)
where the over-bar represents an appropriate average in time, wm(z) is the max-
imum streamwise velocity at distance z from the source and b(z) is a measure
of the local lateral spread of the jet velocity. We derive briefly the governing
equations for plane jets, based upon the conservation of volume and momentum
(see, for instance, Kotsovinos & List, 1977, for more details). The time-averaged
volume flux and the time-averaged momentum flux are expressed respectively as
Q(z) =
∫ ∞
−∞
w(x, z) dx and M(z) =
∫ ∞
−∞
(w)2 (x, z) dx. (2.2a,b)
Solving the first-order integrated equations of motions
dM
dz = 0,
dQ
dz = 2αwm, (2.3a,b)
13
2 Meandering and self-similarity of quasi-two-dimensional jets
we find
M = M0, Q = Q0
(
4
√
2αM0zQ02
+ 1
)1/2
, (2.4a,b)
where we assume in equation (2.3b) that the entrainment velocity is proportional
to the maximum time-averaged streamwise velocity, with α the entrainment co-
efficient (Morton, Taylor & Turner, 1956), and Q0 and M0 are values at the
origin for the volume flux and momentum flux, in (2.4a) and (2.4b) respectively.
The e-folding value of the maximum time-averaged streamwise velocity and the
maximum time-averaged streamwise velocity are, respectively,
b(z) = Q0
2
√
2πM0
(
4
√
2αM0zQ02
+ 1
)
and wm(z) =
√
2M0
Q0
(
4
√
2αM0zQ02
+ 1
)−1/2
.
(2.5a,b)
We can infer the theoretical virtual origin of the jet
z0 = −Q02/(4
√
2αM0), (2.6)
which results from the choice of the boundary conditions (i.e. the distributions
of the volume flux and momentum flux at z = 0).
Alternatively, solving the plane jet equations assuming momentum-flux conser-
vation and similarity (see e.g. Pope, 2000) also leads to the same power laws for the
e-folding value of the maximum time-averaged streamwise velocity, b ∝ (z − z0),
and the maximum time-averaged streamwise velocity, wm ∝ (z − z0)−1/2. The
constants of proportionality and the virtual origin can differ because of the as-
sumptions we make for the x-distribution of wm (essentially due to ‘shape factors’)
and for the boundary conditions. As a direct comparison with the ‘velocity spread
rate’ S defined as dx1/2/dz = S (where x1/2 is the velocity half-width defined by
wm(z)/2 ≡ w(x1/2, z)), we can remark that S = 4(ln 2/π)1/2α (see Pope, 2000,
for further details about S).
Equations (2.5a,b) suggest that the natural scalings for length and time scales
in our problem are d, the source width, and τ = d2/Q0, respectively. Therefore,
when considering our experimental data, we will always scale quantities with these
scalings, i.e.
z˜ = zd, x˜ =
x
d , b˜ =
b
d, t˜ =
t
τ , w˜ =
τ
dw, (2.7a–e)
where tildes denote non-dimensional variables.
14
2.4 Time-averaged mean flow field
For comparison with the theoretical model, we time-averaged the velocity field
measured with PIV. We plot the lateral spread, the evolution with height and
the lateral distribution of the time-averaged streamwise velocity. We also discuss
the influence of the free surface at the top boundary, the impact of the lateral
confinement and possible three-dimensional effects on the flow, such as friction at
the walls constraining the flow.
In figure 2.4, we show the ensemble average of the edges of 40 dyed jets (plotted
with dots). The evolution of the dye edge with height clearly indicates that above
z/d = 120 the influence of the free surface becomes non-negligible. This height
serves as a lower bound for the ‘impingement region’ (see Jirka & Harleman,
1979, for a detailed study). The zone of established flow is found to start at
approximately z/d = 20, a value at which the streamwise velocity becomes self-
similar. This value is commonly reported in the literature (see e.g. Kuang, Hsu
& Qiu, 2001). A linear fit of the non-dimensional average dye edge (plotted with
a thin line in figure 2.4) calculated for 20 ≤ z/d ≤ 120 gives a slope of 0.22± 0.08
for the half-spreading angle. We can observe that the non-dimensional e-folding
value of the maximum time-averaged streamwise velocity b˜ (plotted with crosses)
is much narrower. We discuss this difference further below. We also compute the
quantity b˜ from the ensemble average of the 12 jets studied with PIV. A linear fit
(plotted with a thick line) calculated between 20 ≤ z/d ≤ 120 gives the rate of
change, db/dz = 0.154±0.016, which is slightly above the value of 0.135 reported
by Ramaprian & Chandrasekhara (1985) and very similar to the value reported
by Albertson et al. (1950). Using (2.5a) the corresponding entrainment coefficient
(determined to best-fit the streamwise variation of b) is αb = 0.068±0.007 (which
is equivalent to Sb = 0.125± 0.015), and we find that αb is almost constant in the
zone of established flow, thus confirming the entrainment assumption (Morton
et al., 1956).
In figure 2.5, we plot the non-dimensional maximum time-averaged streamwise
velocity wm/(Q0/d) against height. The crosses are plots of an ensemble average
over all the jets studied with PIV. Although the agreement is good, they lie
slightly but systematically above the theoretical curve (plotted with a solid line)
for z/d ≤ 100. We compute the theoretical curve from (2.5b) and using α = αb.
The value of Q0 was measured for each jet as described in the experimental
procedure. On the other hand, since M0 could not be measured directly, it was
15
2 Meandering and self-similarity of quasi-two-dimensional jets
0 5 10 15 20 25 30
x/d
0
20
40
60
80
100
120
140
160
z/
d
Average dye edge
Fit of dye edge
Average b˜
Fit of b˜
b˜ from best fit of wm/(Q0/d)
Figure 2.4: Non-dimensional average dye edge (dots) with a linear fit (thin line);
non-dimensional e-folding value of the maximum time-averaged streamwise velocity b˜
(crosses) with a linear fit (thick line); and a non-dimensional average velocity spread rate
(dashed line) using α = αwm computed from the best fit of wm/(Q0/d) (see figure 2.5).
replaced by M by virtue of (2.4a). As shown in figure 2.6, the momentum flux
M (plotted with pluses) computed from the time-averaged streamwise velocity
field using (2.2b) (the boundaries of the integral are chosen as the positions where
wm = 0) is found to be approximately constant for 34 ≤ z/d ≤ 110. For z/d < 34,
the data do not seem accurate, probably because the frame rate is not high enough
for the large velocity at that distance, and the resolution of the PIV could also
not be optimal where the jet is very narrow. For z/d > 110, the influence of the
impingement region as the jet approaches the free surface at the top seems to start
affecting the momentum flux. The mean non-dimensional value of the momentum
flux is < M > /
(
Q02/d
)
= 0.55 ± 0.03 (plotted with a solid line in figure 2.6).
Giger et al. (1991) reported and discussed the wide range of values for the non-
dimensional momentum flux measured in plane jets from the literature: from 0.52
(Cervantes de Gortari & Goldschmidt, 1981) to 1.77 (Antonia, Satyaprakash &
Hussain, 1980). We analysed the influence of friction at the rigid boundaries on
the momentum flux and found a Fanning friction factor of f ≈ 0.007 assuming
16
2.4 Time-averaged mean flow field
0 20 40 60 80 100 120
z/d
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
w
m
/(
Q
0
/d
)
Data
Theory
Best fit
Figure 2.5: Non-dimensional maximum time-averaged streamwise velocity (pluses)
versus height, theoretical curve (solid line, and using αb = 0.068) and best least-squares
fit (dashed line) optimising with respect to α (using αwm = 0.052).
a wall stress of the form τw = fρ < w >2 /2, where ρ is the water density
and < w > is the spatial averaged velocity in the y-direction (Bird, Stewart &
Lightfoot, 2007). The influence of friction is relatively small compared to the
mean value of the momentum flux (of the order of 10 %) and therefore has not
been included in our constant momentum-flux model (see Giger et al., 1991, for a
detailed study). In figure 2.5, a least-squares fit of the data (plotted with a dashed
line) assuming equation (2.5b) and optimising with respect to the entrainment
coefficient yields an optimal choice for α from the z dependence of the maximum
velocity αwm = 0.052 (which is equivalent to a velocity spread rate Swm = 0.098).
The fact that αwm (also plotted with a dashed line in figure 2.4) is slightly smaller
than αb means that some assumptions of the model underlying (2.5a,b) (which
should yield identical estimates for α using b(z) and wm(z)) are not perfect. In
particular, we believe that the Gaussian distribution hypothesis is not ideal, as
slight deviations from Gaussianity could explain the mismatch.
In figure 2.7(a), we show the lateral distribution of the normalized time-ave-
raged streamwise velocity w/wm. The x-axis is centred on the position of the
17
2 Meandering and self-similarity of quasi-two-dimensional jets
0 20 40 60 80 100 120
z/d
0
0.2
0.4
0.6
0.8
1.0
M
/(
Q
0
2
/d
)
Data
Average (34 < z/d < 110)
Figure 2.6: Non-dimensional time-averaged momentum flux (pluses) versus height and
average value < M > /
(
Q02/d
)
= 0.55 (solid line).
maximum time-averaged streamwise velocity. All the curves result from an en-
semble average of six or 12 jets, depending on where the z position of the curve
lies with respect to the two study areas for the PIV. The experimental data (plot-
ted with different colours) are in very good agreement with the theoretical curve
(plotted with a thick red line) computed from equation (2.1) using α = αb and
neglecting any consideration of virtual origin (Kotsovinos, 1976). Nevertheless,
the experimental curves are somewhat narrower than the theoretical Gaussian
velocity profile. This discrepancy is consistent with a smaller entrainment coeffi-
cient, as suggested by the best fit of wm/(Q0/d) in figure 2.5. The mismatch is
probably caused by the return flow in the tank which is not accounted for by the
model, where an infinitely wide domain is assumed.
The problem of the return flow in a domain of finite lateral extent is more promi-
nent in plane jets than in (fully unconfined non-planar) three-dimensional jets. In
plane jets, the entrainment velocity remains constant outside the jet, whereas it
decreases with distance in the three-dimensional case. As a consequence, we can
observe a negative shift in the lateral distribution of the time-averaged streamwise
velocity (see figure 2.7a), which denotes the presence of the return flow. The flux
18
2.4 Time-averaged mean flow field
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
x/z
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
w
/w
m
z/d =20
z/d =40
z/d =60
z/d =80
z/d =100
z/d =120
Theory −0.6 −0.2 0.2 0.6x/z
−0.2
0.2
0.6
1.0
(w
+
|w
r
|)/
(w
m
+
|w
r
|)(a) (b)
Figure 2.7: (a) Lateral distribution of the normalized time-averaged streamwise veloc-
ity at various heights (plotted with different colours) and theoretical prediction (plotted
with a thick red line). (b) Lateral distribution of the normalized sum of the time-
averaged streamwise velocity and the absolute value of the estimated time-averaged
streamwise velocity of the return flow (defined by equation 2.8) at the same heights as
in (a) (plotted with different colours) and theoretical prediction (plotted with a thick
red line).
of the return flow, Qr, increases with height, as it matches the jet volume flux
Q owing to conservation of volume at every height across the width of the tank.
We can estimate the time-averaged streamwise velocity distribution of the return
flow by applying volume conservation at each height. For all z, the total volume
flux on both sides of the jet is Qr = Q − Q0. We assume that the return veloc-
ity is distributed uniformly along −L/2 ≤ x ≤ −x0 and x0 ≤ x ≤ L/2, where
x0 ≈ 0.25z is defined as the location where w = 0. Therefore, using equations
(2.4b) and (2.5b) we find that the time-averaged return velocity is
wr
wm
= − Q0
2
2
√
2M0(L/2− x0)
[(
4
√
2αM0Q02
z + 1
)
−
(
4
√
2αM0Q02
z + 1
)1/2]
,
(2.8)
where we use α = αb = 0.068 and M0 =< M >= 0.55Q02/d to plot figure 2.7(b).
19
2 Meandering and self-similarity of quasi-two-dimensional jets
As can be seen in figure 2.7(b), adding this simple estimate of the return-flow ve-
locity wr to the jet velocity w has corrected the negative shift in the experimental
data. At every height, except z/d = 20, the velocity tends to a zero asymptotic
value for large |x/z|.
From comparison with similar experiments that we conducted in a smaller do-
main (0.5 m × 0.01 m × 0.5 m) and with experimental results reported in the
literature and obtained in larger tanks of various aspect ratios and with porous or
non-porous lateral boundaries (Giger et al. (1991); Dracos et al. (1992); Rowland
et al. (2009)), we believe that the impact of the return flow is limited and affects
principally the distribution of the time-averaged streamwise velocity in the man-
ner described above. From direct measurements we also find that the momentum
flux associated with the return flow is small compared with the momentum flux in
the jet (from 0 to 15% for z/d = 0 to 110). We have not observed any qualitative
or quantitative influence of the return flow on the time-dependent core and eddy
structure described in § 2.3. We discuss the spatial structure of this return flow
in more detail in Chapter 6.
The experimentally measured streamwise velocity field follows closely the pre-
dictions given by the derivation of the momentum and continuity equations for
two-dimensional turbulent jets. The small difference due to the lateral confine-
ment of the experimental jets leads us to the conclusion that the entrainment
coefficient is within the range 0.052 ≤ α ≤ 0.068. The purpose of the study of
the mean flow is not to understand all the details of this flow but rather to give
us some insight about the flow field in this particular geometry. More refined
models for the plane jet can be found in the literature (see e.g. Giger et al., 1991;
Hussein et al., 1994; Wang & Law, 2002).
2.5 Quantitative analysis of the time-dependent
core and eddy structure
2.5.1 Time-dependent eddy structure
We now analyse the core and eddy structure of the flow using the experimental
results given by the PIV. We identify large vortical structures or ‘eddies’ in in-
dividual frames of the instantaneous velocity field using DigiFlow, as shown in
20
2.5 Quantitative analysis of the time-dependent core and eddy structure
figure 2.8(a). Considering a specific frame, we find regions of the instantaneous
flow field where streamlines form a complete loop. This technique is similar to the
eddy identification method proposed by Robinson (1991). We plot the streamlines
forming a complete loop in figure 2.8(a) with grey curves. We then analyse each
patch, or eddy, to obtain statistical measurements such as the centroid (identified
by the location of the black crosses) and the standard deviations in the lateral
and streamwise directions (shown by the size of the crosses). The coordinates of
the centroid (xc,k, zc,k)(t) of eddy ‘k’ at time t are computed numerically as
(xc,k, zc,k)(t) =
1
Lx∑
x=0
Lz∑
z=0
∆k(x, z, t)
Lx∑
x=0
Lz∑
z=0
(x, z)(t)∆k(x, z, t), (2.9)
where Lx and Lz are the lateral and streamwise dimensions of the velocity field
and ∆k(x, z, t) is 1 if the point (x, z)(t) belongs to a streamline identified as part of
eddy k at time t and 0 otherwise. Similarly, the lateral and streamwise standard
deviations (xs,k, zs,k)(t) of eddy k at time t are computed numerically as
(xs,k, zs,k)(t) =


1
Lx∑
x=0
Lz∑
z=0
∆k(x, z, t)
×
Lx∑
x=0
Lz∑
z=0
(
(x, z)(t)− (xc,k, zc,k)(t)
)2
∆k(x, z, t)


1/2
. (2.10)
We applied the algorithm every 10 frames to the six experimental velocity fields.
The eddies are thus tracked in time at a frequency of 25 frames per second. As a
quality control of the technique, we conducted a visual inspection of all the eddies
identified by the algorithm. This showed that the algorithm was very robust. It
did not appear to be subject to ‘false positives’, i.e. the misidentification of a non-
eddy feature of the flow as an eddy. The algorithm only occasionally failed to
detect eddies (i.e. there were very few ‘false negatives’) when the eddies partially
21
2 Meandering and self-similarity of quasi-two-dimensional jets
appeared at the edges of the frame.
(a) (b)
Figure 2.8: (a) Identification of the vortical structures in the instantaneous velocity
field (plotted with black arrows). The streamlines identified within an eddy are plotted
with grey curves. The black crosses designate the centroid of the eddies, and the size of
the crosses represents their standard deviations in the lateral and streamwise directions.
(b) Identification of the core structure in the same instantaneous velocity field (plotted
with black arrows). The streamlines identified as part of the core structure are plotted
with grey curves.
The trajectories of 48 eddies are shown in figure 2.9 (plotted with black dots).
A linear fit (shown with a solid line) gives an average slope of 0.22 from the z-axis.
We also plot the linear fits of the ensemble-averaged lateral standard deviation of
the eddies, a measure of the average eddy width, with dashed lines. The lateral
and streamwise standard deviations were found to be almost identical, showing
that the eddies are close to circular in shape. They both have a linear trend
increasing with height at a rate of 0.07.
The non-dimensional location of each eddy in time has been plotted in fig-
ure 2.10 with dots. We can see a general trend, which follows the power law
z˜ ∝ t˜ 2/3 (plotted with a solid curve) derived from the maximum time-averaged
streamwise velocity formula (2.5b). The large scatter is due to the complex dy-
namics of individual eddies. We found that not all eddies travelled through the
depth of the PIV window completely unperturbed. We observed merging of close
successive eddies, with the first eddy slowing down considerably, sometimes even
halting, and the following eddy accelerating substantially. Similarly to the ob-
servations made by Dracos et al. (1992), we did not see eddies rotating around
22
2.5 Quantitative analysis of the time-dependent core and eddy structure
−40 −30 −20 −10 0 10 20 30 40
x/d
50
60
70
80
90
100
110
120
130
140
z/
d
Data
Average eddy path
Average eddy width
Figure 2.9: Eddy locations in PIV study area 2 (dots), linear fits of eddy locations
(solid lines) and average eddy lateral standard deviations (dashed lines).
a common axis before merging. We also noticed some small eddies disappearing
in the vicinity of the core. From the best fit of the eddy streamwise position in
time, we find that the average eddy speed is 0.28 times the theoretical wm and
0.24 times the best fit of wm. The fastest identified eddy rises approximately at
the same speed as the centreline time-averaged streamwise velocity, whereas the
slowest eddy rises at less than 15 % of wm.
To investigate the eddy frequency, we counted the number of eddies (identified
by the algorithm described above) passing at a given height on either side of the
core of the jet. We measured this number for the six PIV experiments performed
in study area 2 and then divided it by the duration of the experiment, i.e. 21.8 s.
The resulting non-dimensional eddy frequency f˜ = fd2/Q0 is plotted with thin
lines in figure 2.11(a). Dracos et al. (1992) found an empirical law for the eddy
frequency, f˜ = 176z˜−3/2 (plotted in non-dimensional form with a dashed curve),
and explained that f decreased as a result of the decrease of the eddy trans-
port velocity and merging mechanisms. However, if we assume that eddies form
periodically, then the eddy frequency should remain constant with height, since
eddies travel on average at the same velocity. The frequency can only decrease
23
2 Meandering and self-similarity of quasi-two-dimensional jets
0 200 400 600 800 1000 1200 1400
t/(d2/Q0)
50
60
70
80
90
100
110
120
z/
d
Data
Best fit
Figure 2.10: Eddy z-coordinate versus time (dots) and best least-squares fit (solid
curve) assuming z˜ ∝ t˜ 2/3.
if eddies merge (or, to a lesser extent, disappear). Merging occurs when the dis-
tance between two successive eddies is smaller than a critical value. The distance
between two eddies decreases because their transport velocity decreases like z˜−1/2
and because eddies grow approximately linearly with height due to entrainment
of ambient fluid.
The punctuated decrease in frequency can actually be observed in figure 2.11(a)
as we follow individual experiments (see the values of f for a typical experiment
plotted with red crosses). The frequency f is constant over a certain distance
and then drops by a discrete value in a stepwise way. This is also clearly shown
by the evolution of the Strouhal number, St = fb/wm, plotted with dots in
figure 2.11(b) and with red crosses for the same typical experiment. The Strouhal
number increases like St ∝ z˜ 3/2 from a minimum value of St = 0.07 consistent
with the value reported by Dracos et al. (1992) (plotted with a dashed curve)
and then drops, somewhat chaotically but consistently, to this minimum value
as merging occurs. Because merging becomes less frequent as z/d increases, the
length of time over which f is constant (and hence St increases) increases with
z/d. This leads to the increase in both typical values of the Strouhal number and
24
2.5 Quantitative analysis of the time-dependent core and eddy structure
50 60 70 80 90 100 110 120
z/d
0
0.05
0.10
0.15
0.20
0.25
0.30
S
t
Data
Individual experiment
Dracos et al. (1992)
(b)
60 80 100 120
z/d
0
2
4
10
00
f
d2
/Q
0
Data
Individual experiment
Dracos et al. (1992)
(a)
Figure 2.11: (a) Data for the non-dimensional eddy frequency fd2/Q0 versus height
(thin lines) and best fit of Dracos et al. (1992) (dashed curve). The values of fd2/Q0
for a typical individual experiment have been highlighted with red crosses. (b) Data for
the Strouhal number St = fb/wm versus height (dots) and Strouhal number reported
by Dracos et al. (1992) (dashed line). The values of St for the same experiment have
also been highlighted with red crosses.
its variance, as is apparent in figure 2.11(b). The actual value of the minimum
Strouhal number appears to depend on the eddy formation frequency, the travel
speed of the eddies, the growth of the eddies due to entrainment and the dynamics
of merging, in ways that are not as yet fully understood.
To summarize, the eddies have on average a linear trajectory, a constant growth
with height and a velocity similar to the time-averaged mean streamwise velocity
of the jet. All these findings lead to the conclusion that the dynamics of the
eddies is essentially self-similar with height, at least within the region of the flow
that we have studied. From the analysis of the time evolution of the streamlines
leading to the eddies, we can also attribute the growth of the eddies mainly to the
entrainment of ambient flow. Eddy merging occurs irregularly and is responsible
for the decrease of the long-time-averaged eddy frequency, with an apparently
well-defined minimum Strouhal number St ≥ 0.07.
25
2 Meandering and self-similarity of quasi-two-dimensional jets
2.5.2 Time-dependent core structure
Similarly, we identify the core of the jet by plotting all the streamlines that exit
through the top of a specific velocity field. Effectively, the algorithm follows the
streamlines backwards starting from the points at the top horizontal boundary of
the velocity field. However, in the following discussion, we consider the stream-
lines in the forward direction with their endpoint at the top of the velocity field.
The identification of the streamlines of the core is repeated every 10 frames for
each PIV velocity field, thus giving a dynamical picture of the core at a frequency
of 25 frames per second. It can be seen in figure 2.8(b) that some streamlines
(plotted with grey curves) start at the bottom boundary of the window while oth-
ers come laterally inwards. The streamlines coming from the bottom of the frame
reveal the volume flux brought by the jet itself into the frame. The streamlines
coming from the sides of the jet show the entrained volume flux. They actually
reveal how entrainment of ambient fluid occurs as they wrap around eddies and
then are incorporated into the core. It is clear that eddies constitute an essen-
tial entrainment mechanism by engulfing ambient fluid at their rear. The starting
point of entrained streamlines (i.e. the location at which we consider them as part
of the core) is chosen where the streamwise component of their gradient changes
sign.
This choice raises the more fundamental question about the boundaries of the
core. The boundary between the core and the eddy is clearly defined since the
algorithms used to identify both structures ensure mutual exclusion. However,
at the top and bottom of the window, this boundary can be ambiguous if large
eddies are not entirely seen in the image frame. At the top of the frame, the
error zone is actually restricted to z > 118 d, which is approximately where the
self-similarity region of the jet ends. At the bottom of the frame, the error zone
is insignificant since the eddies are much smaller. Moreover, we found that the
starting point chosen for entrained streamlines has no effect on the time-averaged
distribution of the core and negligible impact on time-dependent distributions.
Therefore, although somewhat arbitrary, we believe that our criterion determining
the boundary between the core and the ambient flow reflects the diffusion of
momentum from the jet to the ambient flow.
We present the lateral (or x-) distribution of the probability Pcore(x, z) of be-
ing in the core in time (plotted with thick solid curves) at different heights in
26
2.5 Quantitative analysis of the time-dependent core and eddy structure
figure 2.12. The discrete formulation of the probability Pcore is
Pcore(x, z) =
1
N
N∑
n=0
∆n(x, z), (2.11)
where N is the total number of frames for a given experiment, n designates the
nth frame and ∆n(x, z) is 1 if the point (x, z) belongs to a streamline identi-
fied as part of the core of the jet in the nth frame and 0 otherwise. Its shape
is Gaussian-like on the edges and flatter in the middle. The flat portion where
Pcore(x, z) = 1 corresponds to the section of the jet always occupied by the core in
time. The width of this section grows linearly with height on average, as shown
by the standard deviation measurement xstd (plotted with thin solid curves for
the experimental data and dashed lines for the linear fits), at a rate of 0.12. Fur-
thermore, the momentum flux of this portion remains constant with height at a
value of 78% of the total momentum of the jet. The edges of the probability Pcore
correspond to the lateral excursions of the core through time. It is interesting to
note a similarity between the distribution of the probability Pcore, as presented in
figure 2.12, and a typical distribution of the intermittency function measured in
quasi-two-dimensional jets (see e.g. Dracos et al., 1992). Both display a plateau
equal to 1 in the interior of the jet and a Gaussian-like decrease tending towards
0 as |x/z| increases. Nevertheless, the intermittency function and the probabil-
ity Pcore are different, both in the way they are computed and in their meaning.
The probability Pcore is a measure of the likelihood of being in the core in time
(which is identified by the algorithm described above). On the other hand, the
intermittency criterion measures the probability of being in a turbulent region
in time. The similarity observed between these two functions is probably due
to the fact that the core is a region where the amplitude of the turbulent fluc-
tuations increases towards the jet centreline. However, the lateral spreading of
the two functions should differ because, contrary to the intermittency function,
the probability Pcore excludes the eddies, which are also regions of large velocity
fluctuations.
A typical standard deviation xstd(t) of the distribution of the core streamlines
at the time instant corresponding to the jet shown in figure 2.8(b) is plotted with
dashed curves in figure 2.13. The undulations of the jet, which we already ob-
served on dyed jet pictures, are primarily a feature of the edges of the core. The
27
2 Meandering and self-similarity of quasi-two-dimensional jets
−40 −30 −20 −10 0 10 20 30 40
x/d
40
60
80
100
120
140
z/
d
0
1
P
co
r
e
Pcore
xstd
Fit of xstd
Figure 2.12: Time-averaged mean core structure in PIV study area 2. Lateral distribu-
tion of the probability Pcore of being in the core in time (thick solid curves) at different
heights, and data (thin solid curves) and linear fits (dashed lines) of the time-averaged
standard deviation xstd of the probability Pcore.
distribution of the instantaneous streamwise velocity w(t) corresponding to the
same time instant is plotted with solid curves at different heights in figure 2.13.
We normalize w(t) with the maximum instantaneous streamwise velocity wn mea-
sured at the lowest height in the frame, z˜n = 42.4. We can observe that the
instantaneous velocity decreases with height and spreads laterally as expected
from the self-similar theoretical model. Furthermore, the velocity distribution is
not centred on the z-axis but follows the undulations of the core described by
xstd(t). The velocity within the core is much larger than the velocity outside,
thus underlying the presence of this high-speed core in the jet. It is also very
interesting to note that the lateral decrease of the velocity is slower in the inte-
riors of the undulations than in the exteriors. This is due to the presence of the
eddies (shown as crosses, with the size of the crosses representing the lateral and
streamwise eddy standard deviations) located in the curves of the core structure
and which carry some upwards momentum flux (slightly less than a quarter of
the total momentum flux on average).
28
2.5 Quantitative analysis of the time-dependent core and eddy structure
−40 −30 −20 −10 0 10 20 30 40 50
x/d
40
50
60
70
80
90
100
110
120
130
z/
d
0
25
w
(c
m
s−
1
)
w(t)/wn xstd(t)
Typical eddies
Figure 2.13: Distribution of the instantaneous normalized streamwise velocity
w(t)/wn (solid curves) at different heights and corresponding to the jet presented in
figure 2.8(b). Instantaneous standard deviation xstd(t) (dashed curves) of the core of
the jet presented in figure 2.8(b) with its eddies (crosses, with the size representing the
lateral and streamwise eddy standard deviations).
The linear growth of the core shows that it is self-similar with height within
the flow region studied, as we found for the eddies. The spatial statistical dis-
tribution of the location of the core is due to its particular wave-like dynamics.
The undulations along the centreline of this high-velocity core are characterized
by an essentially self-similar spatial probability distribution Pcore. The standard
deviation of the probability Pcore increases with height at a rate of 0.12, which
is quite close to the rate of change with height of the mean velocity spread rate
db/dz = 0.15. The spatial Gaussian distribution of the time-averaged mean
streamwise velocity is therefore the result of the statistical spatial distribution
of the undulating core. It is difficult to assess whether the eddies have a di-
rect contribution to this statistical process. However, their role in the large-scale
dynamics of the core is essential.
29
2 Meandering and self-similarity of quasi-two-dimensional jets
2.6 Conclusion
In this experimental study of quasi-two-dimensional turbulent jets (and similarly
to Giger et al. (1991) and Dracos et al. (1992)), we have observed that the flow
organizes into a very interesting structure with a sinuous core of high streamwise
velocity oscillating about the centreline and eddies rising and growing along the
undulations. As predicted by the theoretical model, we find that: the mean ve-
locity field measured with PIV is self-similar with height (see figure 2.14); the
normalized time-averaged streamwise velocity profile w/wn (plotted with thick
solid curves, where wn is the maximum time-averaged streamwise velocity at the
lowest height, z˜n = 42.4) is close to a Gaussian distribution; and the velocity peak
decreases as z˜−1/2 with height. The return flow due to the lateral confinement
of the jet could explain the small mismatch between the theory and the experi-
mental results. Friction at the bounding walls has only a second-order effect on
the momentum flux (of the order of 10 % compared to the average value of the
momentum flux) and thus on the velocity field. Within the flow region studied,
we also find that both the eddies (average eddy paths plotted with dashed lines)
and the core (time-averaged standard deviation plotted with thin solid lines) are
on average self-similar with height, which is not described by the theory and is
fundamentally different from either a (fully unconfined planar) two-dimensional
jet or a (fully unconfined non-planar) three-dimensional jet, where the turbulence
is unconfined and three-dimensional.
The confinement of the jet in a narrow gap undoubtedly changes the struc-
ture of the turbulence in the flow with a quasi-two-dimensional inverse cascade
allowing large eddies to grow. This persistent growth of eddies is contrary to
three-dimensional turbulence. The eddies form within the intense shear layer at
the boundary between the jet and the ambient flow when the width b of the jet
is larger than the thickness W of the flow (Dracos et al., 1992). Then, the eddy
structures appear periodically at a given height. The eddy frequency decreases
with height due to merging and we find a well-defined minimum Strouhal number
St ≥ 0.07. The dynamics of these eddies is strongly coupled with the dynamics of
the core. The core, which moves on average four times faster and carries approx-
imately 75 % of the momentum flux, flows round the eddies. The consequence of
these lateral excursions is seen in the mean velocity field. We believe that the un-
stable dynamics of the core characterized by its probability density distribution
30
2.6 Conclusion
is principally responsible for the Gaussian profile of the time-averaged stream-
wise velocity. In this flow, it is the two-dimensional macrostructure and not the
three-dimensional small-scale turbulence that produces the Gaussian distribution.
Therefore, analysing the instantaneous flow field is key to understanding how
entrainment, mixing and dispersion occur in the jet. The eddies play a leading
role in the entrainment by engulfing ambient fluid at their rear, as we noticed from
the study of the streamlines in the core and eddy structure. This entrainment
mechanism ensures the linear growth of both the core and the eddies, therefore
explaining the self-similarity of these structures. The exchange of fluid between
the core and the eddies is permanent and in both directions as streamlines evolve
in time from being closed within an eddy to being open and stretched in the core.
It is perhaps surprising that the entrainment assumption of Morton et al. (1956),
modelling entrainment due to three-dimensional turbulent mechanisms, can also
describe the fundamentally different two-dimensional case. We find that the en-
trainment coefficient is 0.052 ≤ α ≤ 0.068, depending on how it is calculated.
This range of values for the entrainment coefficient is very similar to the values
reported in the literature for (fully unconfined planar) two-dimensional jets: for
example, 0.060 (Ramaprian & Chandrasekhara, 1985), 0.069 (Albertson et al.,
1950). The dyed jet experiments revealed the vigorous mixing effect of the ed-
dies. It is also worth noting that the average dye edge (shown with dotted lines
in figure 2.14) coincides with the average outer boundaries of the eddies, which
is the physical maximum lateral extent of the jet. Mixing is apparently not as
strong in the core, but intense stretching leading to large streamwise dispersion
occurs at the interface with the eddies. This region is delimited between the thin
solid lines and the dashed lines shown in figure 2.14.
In conclusion, a probabilistic description of the core–eddy structure of quasi-
two-dimensional jets leads to a self-similar Gaussian description of the time-
averaged flow. The instantaneous flow has a very different character from either
(fully unconfined planar) two-dimensional flows or (fully unconfined non-planar)
three-dimensional flows. Bulk long-time-averaged properties are consistent with
conventional theoretical models, but the mixing and dispersion cannot be ac-
counted for by these time-averaged models. We present a model for this mixing
and dispersion in the next chapter.
31
2 Meandering and self-similarity of quasi-two-dimensional jets
−40 −30 −20 −10 0 10 20 30 40
x/d
40
50
60
70
80
90
100
110
120
130
z/
d
w/wn
Core xstd
Average eddy path
Average dye edge
Figure 2.14: Distribution at different heights of: normalized time-averaged streamwise
velocity w/wn (thick solid curves); time-averaged standard deviation of the mean core
xstd (thin solid lines); ensemble-averaged mean trajectory of eddies (dashed lines); and
average dye edge (dotted lines).
32
Chapter 3
Advection–diffusion model for the
streamwise transport, dispersion and
mixing in quasi-two-dimensional jets
3.1 Introduction
In the event of a spill of pollutants, waste or any other tracers into a river, it is
crucial to predict how the tracers are advected and dispersed by the flow after
they reach a relatively shallow basin, such as a lake or the sea shelf. Such pre-
dictions can be used to monitor the spread of the tracers, control their impact
on the environment and assess any potential damage. One of the most impor-
tant aspects of these shallow river flows, and one which has raised the interest of
scientists for more than 20 years, is the emergence of large-scale eddy structures
and meanders at some distance away from the river mouth. These eddies and
33
3 Model for the streamwise transport, dispersion and mixing
meanders have been visualized in nature on several occasions due to sediments
transported by the flow (see e.g. Giger et al., 1991; Jirka & Uijttewaal, 2004;
Rowland et al., 2009). Giger et al. (1991) were interested in the entrainment and
mixing in shallow water flows, whose characteristic streamwise dimensions were
much larger than the fluid-layer thickness and where the flow developed in a con-
fined environment. They showed that these geophysical flows could be reproduced
in laboratory experiments by confining plane turbulent jets in the spanwise direc-
tion (i.e. the direction parallel to the line source of the jet). Giger et al. (1991)
observed that in the far field, or for z/W > 10 where z is the spatial coordinate
in the streamwise direction and W is the fluid-layer thickness in the spanwise
direction (i.e. W corresponded to the depth of the basin), the jet produced sim-
ilar large eddies and meanders as observed in shallow river flows. In Chapter 2,
we referred to turbulent plane jets in such a confined geometry in the far field
as quasi-two-dimensional jets and considered in detail the meandering flow due
to the large-scale eddy structures. The present chapter focuses on the advection
and dispersion properties of such quasi-two-dimensional jets, particularly when
considering the transport of a passive scalar.
The essential characteristics of quasi-two-dimensional jets have been described
previously. Dracos et al. (1992) showed that the large planar counter-rotating ed-
dies observed in quasi-two-dimensional jets developed due to an inverse cascade
of quasi-two-dimensional turbulence. Chen & Jirka (1998) proved through linear
stability analysis that the meanders of the jet were the result of a sinuous insta-
bility. According to Jirka & Uijttewaal (2004) the sinuous instability of the jet
originated from internal transverse shear across the jet. In Chapter 2, we showed
that the time-averaged velocity field of quasi-two-dimensional jets could be mod-
elled using two-dimensional plane jet theory. We also studied the instantaneous
velocity field and revealed the interactions between the high-speed meandering
core of the jet and the large eddies alternating on its sides. We showed that
these core and eddy structures were self-similar with distance and continuously
exchanged fluid between themselves, as well as with the ambient fluid surrounding
the jet. In particular, the eddies played a key role in the entrainment of ambient
fluid by means of engulfment at their rear. Entrained fluid could either be trapped
for a brief period in an eddy, where it experienced strong mixing, or be directly
incorporated in the core of the jet, where it was advected downstream much more
34
3.1 Introduction
rapidly. We further hypothesized that because of the difference in advection speed
between the core and the eddies (we measured that on average eddies travelled at
approximately 1/4 of the speed of the core), initially relatively close fluid parcels
entrained by the jet should experience large streamwise dispersion depending on
whether they were drawn into the eddies or the core.
In order to study and model the transport, mixing and dispersion of tracers in
shallow river flows, we investigate in this chapter the temporal and spatial evo-
lution of the concentration of tracers released in quasi-two-dimensional jets. The
mixing properties of turbulent jets have been studied experimentally many times.
Uberoi & Singh (1975) measured instantaneous temperature profiles in plane jets
and found that they differed from typical time-averaged Gaussian profiles. They
reported a relatively well-mixed interior while most of the mixing was performed
at the turbulent–non-turbulent interface of the jet. Schefer et al. (1994) also
noted a difference between the instantaneous distribution and the time-averaged
distribution of tracers in the case of three-dimensional round turbulent jets. They
attributed this discrepancy to the development of large-scale vortical structures.
Arguably, the dynamics of large-scale vortical structures is different in quasi-two-
dimensional jets from the case of three-dimensional round or plane jets due to the
confinement of the flow in one direction (see Jirka, 2001, for a discussion on large-
scale flow structures in shallow flows, or Chapter 2 for quasi-two-dimensional
jets specifically). Nevertheless, large-scale vortical structures do have an influ-
ence on the mixing and dilution properties of quasi-two-dimensional jets. Giger
et al. (1991) reported that mixing efficiency and dilution in quasi-two-dimensio-
nal jets tended to diminish with distance. From turbulence spectral analysis and
intermittency analysis, Dracos et al. (1992) argued that the decrease of mixing
efficiency was due to the development of quasi-two-dimensional turbulence. Us-
ing laser-induced fluorescence in quasi-two-dimensional jets, Chen & Jirka (1999)
showed that quasi-two-dimensional turbulence induced patchiness in the time-
dependent distribution of the tracer concentration. They found distinct regions
of large concentration which corresponded to the large-scale turbulent structures.
Jirka (2001) reflected upon the impact of large vortical structures in shallow river
flows and emphasized their ability to transport relatively unmixed fluid over large
distances.
Despite the large number of experimental studies, there appear to have been
35
3 Model for the streamwise transport, dispersion and mixing
relatively few attempts to provide a comprehensive model of the advection and
dispersion processes in quasi-two-dimensional jets. Moreover, most models as-
sume a steady state. Paranthoe¨n et al. (1988) suggested a limited model for the
initial phase of the dispersion process in a turbulent plane jet. Then, from con-
servation of mass in a classical plane jet, Chen & Jirka (1999) showed that the
decay of the time-averaged concentration of passive tracers C along the centreline
of quasi-two-dimensional jets followed C ∝ z−1/2. Using conservation of mass
and the Reynolds-averaged Navier–Stokes equation with the boundary-layer ap-
proximation for three-dimensional round and plane jets, Law (2006) proposed an
analytic solution for the time-averaged concentration distribution across the jet.
To close the problem, he used the common assumption that the turbulent dif-
fusive term was proportional to the gradient of the time-averaged concentration
across the jet. He also assumed that the coefficient of proportionality between
these two quantities (i.e. the turbulent diffusivity) was constant across the jet
and depended only on the eddy diffusivity and the turbulent Schmidt number
(see e.g. Mathieu & Scott, 2000, for more details).
Previous models often assume purely lateral entrainment, and then simple time-
averaged streamwise motion. Owing to the cross-stream variation in along-stream
velocity (due to the time-dependent core–eddy interaction and the time-averaged
Gaussian streamwise velocity distribution) quasi-two-dimensional jets inevitably
have significant along-stream dispersion. We want to investigate the implications
of this along-stream dispersion for tracer transport and how it affects advection
in quasi-two-dimensional jets.
In this chapter, we propose a new one-dimensional model solving the time-
dependent effective advection–diffusion equation along the direction of the flow,
based on mixing-length theory. Mixing-length theory is appropriate because of
the central role of large eddies (scaling with the local jet width) on the dis-
persion within the flow. We find analytical solutions in similarity form for the
case of a constant-flux release and the case of a finite-volume release of trac-
ers, which appear to describe correctly some new experimental measurements of
tracer transport. We are able to formulate the general solution for any time-
dependent release in integral form, effectively by means of a Green’s-function-like
solution. We also show the importance of along-stream dispersion mechanisms
in quasi-two-dimensional jets, by comparing our full effective advection–diffusion
36
3.2 Turbulent model hypothesis
model with a simple advection model. In § 3.2, we present our model hypothesis
starting from the advection–diffusion equation, where the diffusive term models
the dispersion by the turbulent flow field of quasi-two-dimensional jets. In § 3.3,
we derive analytical solutions for both a constant volume-flux release and an in-
stantaneous finite-volume release. We also show how to generalize the analytical
solution for an instantaneous finite-volume release into a solution for an arbitrary
time-dependent release. In the next chapter, we compare the theoretical results
obtained in this chapter with experimental data. In § 4.1, we describe our experi-
mental procedure. In § 4.2, we first provide a qualitative assessment of our model
hypothesis, then we compare our theoretical predictions with experimental data
obtained using dye tracking experiments and virtual particle tracking experiments
in both the constant-flux and the finite-volume cases. In § 4.3, we analyse the
statistical significance of the experimental measurements presented in § 4.2 for
the cases of constant-flux releases of dye and instantaneous finite-volume releases
of virtual particles. Finally, in § 4.4 we draw our conclusions for both Chapters 3
and 4.
3.2 Turbulent model hypothesis
To characterize the evolution of the concentration of tracers released in quasi-
two-dimensional jets, we consider the ideal model of a turbulent momentum jet
in a two-dimensional semi-infinite environment. Adopting the same conventions
to those used in § 2.4, the flow is considered incompressible and statistically
steady. The x-direction is the lateral, cross-jet direction and the z-direction is
the streamwise, axial direction. Assuming a plane flow in the domain, the velocity
is labelled u = (u, w) in a Cartesian coordinate system (x, z) with the origin at
the nozzle exit. The temporal and spatial evolution of the concentration of tracers
C(x, z, t) (where t is time) in a two-dimensional steady turbulent jet satisfies (see
e.g. Itoˆ, 1992)
∂tC +∇ · (uC) = κ∆C, (3.1)
where ∇ is the gradient operator, κ is the molecular diffusivity and ∆ is the
Laplacian in two dimensions. We take a point-wise ensemble average (i.e. an
37
3 Model for the streamwise transport, dispersion and mixing
ensemble average at each point (x, z, t) in space and time) of equation (3.1)
∂tCE +∇ · (uECE) +∇ · ([uFCF]E) = κ∆CE, (3.2)
where the subscript in XE denotes the ensemble average of a quantity X and
XF denotes the fluctuations such that X = XE + XF, [XF]E = 0. Thus, the
ensemble-averaged concentration is defined as
CE(x, z, t) =
1
N
N∑
n=1
Cn(x, z, t), (3.3)
where N is the total number of realisations of an experiment and n designates the
nth realisation. We then make the modelling assumptions that uECE behaves as
an advective contribution and is equal to λ1uCE (where the overbar represents
an appropriate average in time and λ1 is a constant), while [uFCF]E effectively
acts diffusively so that [uFCF]E = −D · ∇CE (with D a turbulent eddy diffusive
tensor). We expect that advection is governed by the mean flow and dispersion
by eddy processes. The term [uFCF]E can be seen as a turbulent flux, which is
usually defined as u′C ′ with u′ = C ′ = 0 (Mathieu & Scott, 2000), where u′ and
C ′ designate the temporal fluctuations of the velocity field and the temporal fluc-
tuations of the concentration field, respectively. In other words, we assume that
the statistical diffusive effect of the turbulent fluctuations is equivalent whether
averaged in time or over many realisations. The diffusive effect of [uFCF]E de-
scribes and parameterizes physically the interaction between the high-speed core
and the growing eddies described in Chapter 2. Therefore, neglecting molecu-
lar diffusion under the assumption that it is less significant than eddy diffusion
processes (Mathieu & Scott, 2000), equation (3.2) becomes
∂tCE + λ1∇ · (uCE) = ∇ · (D · ∇CE) , (3.4)
We believe that the interaction between the high-speed core and the growing
eddies has a strong streamwise dispersive effect. On the other hand, the cross-
jet distribution of the concentration remains confined laterally by two linearly-
expanding straight-sided boundaries (as observed in Chapter 2). As we already
mentioned, the transport and dispersion of tracers in quasi-two-dimensional jets
is more critical along the streamwise direction (Jirka, 2001). Therefore, we choose
38
3.2 Turbulent model hypothesis
to integrate equation (3.4) across the jet
∂tφ+ λ1
∫ ∞
−∞
(
∂x (uCE) + ∂z (wCE)
)
dx =
∫ ∞
−∞
∇ · (D · ∇CE) dx, (3.5)
where
φ(z, t) =
∫ ∞
−∞
CE(x, z, t) dx. (3.6)
Since CE vanishes as x→ ±∞ and u remains finite we have
∂tφ+ λ1∂z
(∫ ∞
−∞
wCE dx
)
=
∫ ∞
−∞
∇ · (D · ∇CE) dx. (3.7)
We assume that the eddy diffusive coefficient is independent of x and that, in
the streamwise direction, it scales like the local characteristic velocity wm(z) (the
maximum time-averaged streamwise velocity in the jet at height z) and the local
characteristic size b(z) (the velocity spread rate or e-folding distance of the time-
averaged streamwise velocity at height z) of this core and eddy structure, such
that
Dzz(z) ∝ b(z)wm(z). (3.8)
This is essentially a ‘mixing-length’ model (Prandtl, 1925), where the mixing
length is the local characteristic width of the jet, and where streamwise transport
and dispersion are dominant. Therefore, since ∂xCE and ∂zCE vanish as x → ∞
and D remains finite equation (3.7) becomes
∂tφ+ λ1∂z
(∫ ∞
−∞
wCE dx
)
∝ ∂z (wmb ∂zφ) . (3.9)
We found in (2.5a,b)
b(z) = Q0
2
√
2πM0
(
4
√
2αM0zQ02
+ 1
)
and wm(z) =
√
2M0
Q0
(
4
√
2αM0zQ02
+ 1
)−1/2
,
(3.10a,b)
where α is the entrainment coefficient (Morton et al., 1956), Q0 is the initial
volume flux of the jet, and M0 is the initial momentum flux, which is conserved
with distance in the z-direction (see figure 2.6). The time-averaged streamwise
velocity can be further decomposed into a spatial-averaged part and a fluctuating
39
3 Model for the streamwise transport, dispersion and mixing
part:
w =< w > +wˆ, (3.11)
where
< w >= 1
2b
(∫ ∞
−∞
w dx
)
. (3.12)
Therefore, we obtain
∂tφ+ λ1∂z
(
< w > φ+
∫ ∞
−∞
wˆCE dx
)
∝ ∂z (wmb ∂zφ) . (3.13)
We again face a closure problem with the third term on the left-hand side of
(3.13), which we address by assuming that this term has an advective effect of
the form < w > φ. Therefore, considering that < w >∝ wm, we can introduce
two constants ka and kd to obtain
∂tφ+ ka ∂z (wmφ) = kd ∂z (wmb ∂zφ) . (3.14)
We can rewrite the quantities b and wm using the power laws (2.5a) and (2.5b)
(neglecting the virtual origins) respectively, to obtain the effective advection–
diffusion equation for the laterally-integrated ensemble-averaged concentration φ
∂tφ+KaM01/2 ∂z
( φ
z1/2
)
= KdM01/2 ∂z
(
z1/2∂zφ
)
, (3.15)
where the constants Ka and Kd are a dimensionless advection parameter and a
dimensionless dispersion parameter, respectively, which we will determine exper-
imentally. The parameters Ka and Kd can be related to ka and kd using (2.5a)
and (2.5b) (and, again, neglecting the virtual origins) in the following manner
Ka =
ka
(
2α
√
2
)1/2 and Kd = 2kd
(
α
√
2
π
)1/2
, (3.16a,b)
with α ≈ 0.068 (as calculated in Chapter 2). It is interesting to note that in
(3.15) the dispersion term increases with distance like z1/2, whereas the advection
term decreases with distance like z−1/2.
40
3.3 Mathematical model
3.3 Mathematical model
In order to test our turbulent model hypothesis, we impose different, appropriate
initial, boundary and integral conditions on solutions to the general effective
advection–diffusion equation (3.15), for example,
φ(z, 0) = 0, z > 0, φ(z, t) → 0 as z →∞, and
∫ ∞
0
φ(z, t) dz ∝ tϑ, t > 0.
(3.17a–c)
Equation (3.17a) imposes that the concentration is 0 everywhere initially; equa-
tion (3.17b) imposes that, at all time, the concentration vanishes at infinity;
and equation (3.17c) imposes that, for t > 0, the total integrated concentration
evolves as a power law of time. The integral condition (3.17c) effectively controls
the release of the passive tracers in the jet.
In this theoretical section, we solve analytically equation (3.15) for three dif-
ferent sets of initial boundary and integral conditions. We consider the simple
case of a constant-flux release of passive tracers (i.e. we impose ϑ = 1 in (3.17c)),
which we solve by analysing either the concentration (see § 3.3.2) or the concen-
tration flux (see § 3.3.3). In the second case, presented in § 3.3.4, we consider an
instantaneous release of a finite volume of passive tracers at the origin of the jet
(i.e. we impose ϑ = 0 in (3.17c)). Then, based on the solution for the instan-
taneous finite-volume release, we show in § 3.3.5 how to formulate, in integral
form, the solution for a general and more realistic time-dependent release of trac-
ers governed by an arbitrary source function (i.e. not limited to a power law
of time). We give an analytical solution in the case where the source function
models a constant-flux release over a finite period of time T0. We further show
that the solutions for the first two simpler cases of a constant-flux release and
an instantaneous finite-volume release are the two asymptotic limits of the more
general solution when T0 →∞ and t≫ T0, respectively.
We choose to solve the problems of a constant-flux release and a finite-volume
release because we can reproduce them experimentally, and thus test our turbulent
model hypothesis and the various associated assumptions, stated in § 3.2, against
experimental measurements (presented in § 4.2). Before deriving the solutions of
the three cases, we use below a similarity transformation to simplify the partial
differential equation (3.15) into an ordinary differential equation (ODE), which
we can then solve.
41
3 Model for the streamwise transport, dispersion and mixing
3.3.1 Similarity transformation
We introduce the dilation transformation
zˇ = εaz, tˇ = εbt, φˇ = εcφ(ε−azˇ, ε−btˇ), (3.18)
and so equation (3.15) becomes
εb−c∂tˇφˇ+ ε
3
2a−cKaM01/2 ∂zˇ
( φˇ
zˇ1/2
)
= ε 32a−cKdM01/2 ∂zˇ
(
zˇ1/2∂zˇφˇ
)
. (3.19)
If b = 3a/2, then equation (3.15) is invariant under this transformation. This
suggests that we look for a solution for (3.15) of the form
φ(z, t) = t2c/3a y(η) with η = z
t2/3M01/3
. (3.20)
Thus (3.15) becomes
(
2c
3a −
Ka
2η3/2
)
y +
(
(2Ka −Kd)
2η1/2 −
2η
3
)
y′ −Kdη1/2y′′ = 0. (3.21)
The general effective advection–diffusion problem has thus been simplified to the
ODE (3.21). This second-order ODE, written in similarity form, captures both
the temporal and spatial streamwise evolution of the concentration of tracers
in quasi-two-dimensional steady turbulent jet. Most importantly, (3.21) allows
not only for streamwise advection transport, but also for streamwise turbulent
dispersion (based on a mixing-length assumption). Furthermore, we can note
that (3.21) depends on the ratio of two dilation constants, c/a. This ratio can be
determined using the integral condition (3.17c), which becomes, using (3.20), for
t > 0,
∫ ∞
0
φ(z, t) dz = t2c/3a
∫ ∞
0
y
( z
t2/3
)
t2/3M01/3dη = M01/3t(
2c
3a+
2
3)
∫ ∞
0
y(η) dη ∝ tϑ.
(3.22)
Therefore, this condition can hold for all t > 0 if and only if
c
a =
3ϑ− 2
2
. (3.23)
42
3.3 Mathematical model
3.3.2 Constant-flux release: concentration
In the case of a release of tracers at a constant source flux F , if the general
effective advection–diffusion equation (3.15) is satisfied for z > 0, t > 0 and if, in
addition, φ(z, t) satisfies (following (3.17a–c) with ϑ = 1)
φ(z, 0) = 0, z > 0, φ(z, t) → 0 as z →∞, and
∫ ∞
0
φ(z, t) dz = Ft, t > 0,
(3.24a–c)
then the condition (3.24c) can hold for all t > 0 if and only if a = 2c according
to (3.23) with ϑ = 1. Thus, (3.20) becomes
φ(z, t) = t1/3y(η) with η = z
t2/3M01/3
. (3.25)
In this case, the initial boundary value problem for φ(z, t), defined by (3.21) with
a = 2c, (3.24a–c) and (3.25), reduces to
(
1
3
− Ka
2η3/2
)
y +
(
(2Ka −Kd)
2η1/2 −
2η
3
)
y′ −Kd η1/2y′′ = 0, (3.26)
subject to the conditions
y(η) → 0 as η →∞,
∫ ∞
0
y(η) dη = F
M01/3
, t > 0. (3.27a,b)
Equation (3.26) can then be rewritten using
y(η) = s
1
3
(
Ka
Kd
−1
)
p(s), with s = 4η
3/2
9Kd
, (3.28)
to obtain
p′′ + p′ +


1
3
(
Ka
Kd
− 2
)
s +
1
4 −
(
1
3
(
Ka
Kd
− 12
))2
s2

 p = 0. (3.29)
Making the change of variable p = e−s/2W , we obtain the Whittaker differen-
tial equation (Gradshteyn & Ryzhik, 2007). The Whittaker functions Wk,m[s]
and Mk,m[s] are two linearly independent solutions of the Whittaker differential
43
3 Model for the streamwise transport, dispersion and mixing
equation where
k = 1
3
(Ka
Kd
− 2
)
, m = 1
3
(Ka
Kd
− 1
2
)
. (3.30a,b)
Therefore, the solution of (3.29) is
p(s) = e−s/2 (JWWk,m + JMMk,m) [s], (3.31)
where JW and JM are constants of integration which will be determined using the
boundary conditions (3.27a,b). We can rewrite equation (3.31) in the similarity
form
y(η) =
(
4η3/2
9Kd
) 1
3
(
Ka
Kd
−1
)
e−
2η3/2
9Kd (JWWk,m + JMMk,m)
[
4η3/2
9Kd
]
= JWW + JMM,
(3.32)
defining two linearly independent solutions: W (involving Wk,m), and M (in-
volving Mk,m) of the underlying equation (3.31). Since m− k − 1/2 = 0, we can
actually simplify the Whittaker functionsWk,m andMk,m (see equations (13.18.5)
and (13.18.4) for Wm−1/2,m and Mm−1/2,m, respectively, in National Institute of
Standards and Technology, 2011-08-29) to find
W(η) =
(
4η3/2
9Kd
)1/3
Γ
[
2
3
(Ka
Kd
− 1
2
)
, 4η
3/2
9Kd
]
, (3.33)
M(η) = 2
3
(Ka
Kd
− 1
2
)(
4η3/2
9Kd
)1/3
γ
[
2
3
(Ka
Kd
− 1
2
)
, 4η
3/2
9Kd
]
, (3.34)
where Γ[g, ι] =
∫∞
ι hg−1e−h dh is the upper incomplete Gamma function and
γ[g, ι] =
∫ ι
0 hg−1e−h dh is the lower incomplete Gamma function. We can prove
that, as η →∞,
W ∼ e−η3/2η
(
Ka
Kd
− 32
)
, M∼ η1/2, (3.35a,b)
(see equation (8.11.2) in National Institute of Standards and Technology, 2011-
08-29, for the asymptotic expansion of the upper incomplete Gamma function)
for Ka > Kd/2 (we will find later that for our experimental data, Ka appears to
be substantially greater than Kd). So, in order to satisfy the far-field boundary
condition (3.27a) requiring decay of y, we must have JM = 0 with the solution
depending on W alone. JW can then be determined using the boundary condition
44
3.3 Mathematical model
(3.27b):
JW =
F
M01/3
∫ ∞
0
W(η) dη
. (3.36)
Therefore, the general solution of the effective advection–diffusion problem for the
case of a constant flux release at the source is, in similarity form, for Ka > Kd/2
yF (η) =
2F
3KdM01/3Γ
[
2
3
(
Ka
Kd
+ 1
)]η1/2Γ
[
2
3
(Ka
Kd
− 1
2
)
, 4η
3/2
9Kd
]
, (3.37)
where Γ[g] =
∫∞
0 hg−1e−h dh is the Gamma function. We can note that the
laterally-integrated concentration φF (z, t) = t1/3yF (η) (according to equation
(3.25) with yF described in (3.37)) tends towards a simple asymptotic distribution
φF ∝ z1/2 as t2/3M01/3 ≫ z (or η ≪ 1). In the limit t2/3M01/3 ≫ z, it appears
that the laterally-integrated concentration φF depends only on z and increases
with distance like z1/2. On the other hand, we will see in the next chapter that
the ensemble-averaged concentration CE,F (see (3.6)) should actually decrease like
z−1/2, because the experimental cross-jet distribution of φF spreads linearly with
distance (see figure 4.5a). Since the asymptotic distribution of the concentration
CE,F is independent of time in the limit t2/3M01/3 ≫ z, this asymptotic distri-
bution represents the steady state solution. This finding is in agreement with
Chen & Jirka (1999), who also showed that the time-averaged concentration of
passive tracers in quasi-two-dimensional jets decays like C ∝ z−1/2 along the jet
axis. Note that in the steady-state case, the ensemble average is equivalent to the
time average. Mathematically, the concentration CE,F is singular at the origin
z = 0 and tends to infinity. However, this is not the case in practice because
the concentration of tracers must be finite at the source and the jet has a virtual
origin z0.
Interestingly, in the purely advective limit where Kd → 0 (corresponding to a
so-called ‘top-hat’ velocity profile, see e.g. Turner, 1986) equation (3.26) becomes
(
1
3
− Ka
2η3/2
)
y +
( Ka
η1/2 −
2η
3
)
y′ = 0, (3.38)
45
3 Model for the streamwise transport, dispersion and mixing
which integrates to
yF,a(η) =
{
J1η1/2, 0 ≤ η < ηa
J2η1/2, ηa < η
, (3.39)
where J1 and J2 are integration constants, and
ηa =
(
3Ka
2
)2/3
(3.40)
is the location of the advective front considering ‘top-hat’ velocity profiles in the
jet. Using the boundary condition at infinity (3.27a), we obtain J2 = 0. J1
can be determined using the integral condition (3.27b). Therefore, the similarity
solution of the purely advective problem for the case of a constant-flux release at
the source is
yF,a(η) =



F
KaM01/3
η1/2, 0 ≤ η < ηa
0, ηa < η
. (3.41)
We have plotted in figure 3.1 the non-dimensional quantities yF/
(
F/M01/3
)
and yF,a/
(
F/M01/3
)
. The five different curves show the concentration profile
in similarity form for different values of Ka and Kd. As we increase Ka (deter-
mining the advection strength), the maximum of the curve is displaced upwards,
further away from the origin, while if we increase Kd (determining the dispersion
strength), the front drops less rapidly, and there is still asymmetry about the
maximum. As expected, without dispersion (i.e. in the ‘top-hat’ limit Kd → 0)
the distribution of tracers yF,a/
(
F/M01/3
)
has a discontinuity at ηa, the location
of the advective front (defined in (3.40)), where it vanishes.
To study the distribution of yF , we can compute the location of its centroid
normalized with the advective front ηa
µF =
∫ ∞
0
yF (η)η dη
ηa
∫ ∞
0
yF (η) dη
(3.42)
=
3
5
(
3Kd
2Ka
)2/3 Γ
[
2Ka
3Kd
+ 43
]
Γ
[
2Ka
3Kd
+ 23
] , (3.43)
46
3.3 Mathematical model
η
yF /
(
F/M01/3
)
Ka = 1, Kd = 0.1
Ka = 1, Kd = 0
Ka = 1, Kd = 1
Ka = 10, Kd = 1
Ka = 10, Kd = 0
0 0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
7
8
9
10
Figure 3.1: Variation of the non-dimensional similarity solution yF /
(
F/M01/3
)
, de-
fined in (3.37), against the similarity variable η = z/
(
t2/3M01/3
)
for the problem of
advection–dispersion in the case of a constant flux at the source and for different values
of the advection and dispersion parameters, Ka and Kd respectively. In the ‘top-hat’
limit Kd → 0, we use the non-dimensional similarity solution yF,a/
(
F/M01/3
)
defined
in (3.41).
and its standard deviation normalized with the advective front ηa
σF =


∫ ∞
0
yF (η)η2 dη
ηa2
∫ ∞
0
yF (η) dη
− µF 2


1/2
(3.44)
=
(
3Kd
2Ka
)2/3


3Γ
[
2Ka
3Kd
+ 2
]
7Γ
[
2Ka
3Kd
+ 23
] −


3Γ
[
2Ka
3Kd
+ 43
]
5Γ
[
2Ka
3Kd
+ 23
]


2

1/2
. (3.45)
We plot µF in figure 3.2(a). We can see that µF decreases when Ka/Kd increases.
We can prove that µF → 3/5 as Ka/Kd → ∞ (using equation (5.11.7) in Na-
tional Institute of Standards and Technology, 2011-08-29), thus meaning that the
centroid of yF recedes behind the advective front at a fixed relative distance. The
47
3 Model for the streamwise transport, dispersion and mixing
Ka/Kd
µ
F
Theory
Data
µF = 3/5
0 5 10 15 20 25
0
0.5
1
1.5
2
(a)
Ka/Kd
σ F
√
Theory
Data
σF =
√
12/175
0 5 10 15 20 25
0
0.5
1
1.5
2
(b)
Figure 3.2: Constant-flux case for the tracer concentration: (a) plot of the theoreti-
cally predicted variation of µF (defined in (3.43)), the centroid of yF (defined in (3.37))
normalized with the advective front ηa (defined in (3.40)), as a function of Ka/Kd (plot-
ted with a solid line), with the experimentally determined value (obtained from the best
fit of the constant-flux case shown in figure 4.7) marked with a cross, the asymptotic
value of µF is plotted with a dashed line; (b) plot of the theoretically predicted variation
of σF (defined in (3.45)), the standard deviation of yF normalized with the advective
front ηa, as a function of Ka/Kd (plotted with a solid line), with the experimentally de-
termined value (obtained from the best fit of the constant-flux case shown in figure 4.7)
marked with a cross, the asymptotic value of σF is plotted with a dashed line.
normalized standard deviation σF is plotted in figure 3.2(b). σF also decreases
whenKa/Kd increases. We can prove that σF →
√
12/175 asKa/Kd →∞ (using
equation (5.11.7) in National Institute of Standards and Technology, 2011-08-29).
Moreover, we can observe in figure 3.1 that for the solution yF of the general
effective advection–diffusion problem a non-negligible portion of the volume of
tracers is transported faster than the advective speed due to the combined effects
of advection and dispersion processes. We can compute the portion of the total
volume of tracers βF which travels ahead of the advective front
βF =
∫ ∞
ηa
yF dη
∫ ∞
0
yF dη
, (3.46)
using equation (3.37), we obtain
βF =
Γ
[
2
3
(
Ka
Kd
+ 1
)
, 2Ka3Kd
]
−
(
2Ka
3Kd
)
Γ
[
2
3
(
Ka
Kd
− 12
)
, 2Ka3Kd
]
Γ
[
2
3
(
Ka
Kd
+ 1
)] . (3.47)
48
3.3 Mathematical model
The ratio βF remains constant in time and space because (3.47) does not depend
on η. Moreover, βF depends only on the ratioKa/Kd. We have plotted βF against
Ka/Kd in figure 3.3(a). We can prove that βF tends asymptotically towards 0
at large Ka/Kd (see equation (8.11.10) in National Institute of Standards and
Technology, 2011-08-29), thus meaning that the portion of tracers in the dispersive
front becomes smaller as Ka/Kd increases (see figure 3.1 for the change in the
distribution of yF with various Ka and Kd). We can also compute the normalized
distance between the average location of the volume of tracers travelling ahead of
the advective front and the location of the advective front ηa
ξF =
1
ηa


∫ ∞
ηa
yF η dη
∫ ∞
ηa
yF dη
− ηa

 , (3.48)
which yields
ξF =
3
5
(
3Kd
2Ka
)2/3 Γ
[
2
3
(
Ka
Kd
+ 2
)
, 2Ka3Kd
]
−
(
2Ka
3Kd
)5/3
Γ
[
2
3
(
Ka
Kd
− 12
)
, 2Ka3Kd
]
Γ
[
2
3
(
Ka
Kd
+ 1
)
, 2Ka3Kd
]
−
(
2Ka
3Kd
)
Γ
[
2
3
(
Ka
Kd
− 12
)
, 2Ka3Kd
] − 1.
(3.49)
We plot ξF against Ka/Kd in figure 3.3(b). The distance ξF can be considered as
the normalized distance between the dispersive front (average location of the par-
ticles travelling ahead of the advective front) and the advective front ηa (defined
in (3.40)). In time and space coordinates, the distance between the dispersive
front zF and the advective front za is zF − za = ξFηat2/3. So the distance between
the dispersive front and the advective front increases with time like t2/3. We can
also see in figure 3.3(b) that ξF → 0 as Ka/Kd →∞, thus meaning that the front
becomes sharper as Ka/Kd increases (see also figure 3.1).
3.3.3 Constant-flux release: concentration flux
A somewhat more physically relevant quantity, which we can now study in space
and time for the case of a constant-flux release at the source, is the streamwise
49
3 Model for the streamwise transport, dispersion and mixing
Ka/Kd
β F
Theory
Data
0 5 10 15 20 25
0
0.2
0.4
0.6
0.8
1
(a)
Ka/Kd
ξ F
Theory
Data
0 5 10 15 20 25
0
1
2
3
(b)
Figure 3.3: Constant-flux case for the tracer concentration: (a) plot of the theoret-
ically predicted variation of βF (defined in (3.47)), the portion of the total volume of
tracers released which travels ahead of the advective front ηa (defined in (3.40)), as a
function of Ka/Kd (plotted with a solid line), with the experimentally determined value
(obtained from the best fit of the constant-flux case shown in figure 4.7) marked with
a cross; (b) plot of the theoretically predicted variation of ξF (defined in (3.49)), the
normalized distance between the average location of the volume of tracers travelling
ahead of the advective front and the location of the advective front ηa, as a function of
Ka/Kd (plotted with a solid line), with the experimentally determined value (obtained
from the best fit of the constant-flux case shown in figure 4.7) marked with a cross.
concentration flux of tracers in a steady quasi-two-dimensional jet, defined as:
Mφ =
∫ ∞
−∞
wC dx. (3.50)
We can take the point-wise ensemble average (as defined in (3.3) for the concen-
tration) of (3.50) and, neglecting the second-order turbulent contribution to the
flux (Wang & Law, 2002, found that the turbulent mass flux for round turbulent
jets was approximately 7.6% of the mean mass flux, so can be ignored to leading
order), we find
MφE =
∫ ∞
−∞
wECE dx. (3.51)
Using the same modelling assumptions we made in § 3.2, (3.51) becomes
MφE = λ1
∫ ∞
−∞
wCE dx, (3.52)
where we assume that the ensemble-averaged streamwise velocity is proportional
to the time-averaged streamwise velocity. Then, the time-averaged streamwise
50
3.3 Mathematical model
velocity can be further decomposed into a spatially-averaged part < w >, defined
in (3.12), and a fluctuating part wˆ, so that
MφE = λ1 < w > φ+ λ1
∫ ∞
−∞
wˆCE dx, (3.53)
Again, if we assume that the term
∫∞
−∞ wˆCE dx in (3.53) has an advective effect
similar to < w > φ, Mφ can be related to Ka and φ as (hereafter omitting the
subscript E for simplicity)
Mφ = KaM01/2
φ
z1/2 . (3.54)
Therefore, the solution of the concentration flux of tracers for the case of a con-
stant source flux is, for Ka > Kd/2,
Mφ(z, t) = yM(η) =
F
1− Kd2Ka
Γ
[
2
3
(
Ka
Kd
− 12
)
, 4η3/29Kd
]
Γ
[
2
3
(
Ka
Kd
− 12
)] , with η = z
t2/3M01/3
, (3.55)
where we use the solution for the laterally-integrated concentration φ = φF =
t1/3yF , with yF defined in (3.37). In the limit t2/3M01/3 ≫ z (or η ≪ 1), the
concentration flux is independent of time or space and tends towards a constant
Mφ → F (1−Kd/(2Ka)).
For comparison with a purely advective flow, in the limit Kd → 0 (relevant, as
already noted, to ‘top-hat’ velocity profiles) the concentration flux is
yM,a(η) =
{
F, 0 ≤ η < ηa
0, ηa < η
, (3.56)
according to yF,a, defined in (3.41), and (3.54) with φ = φF,a = t1/3yF,a.
We have plotted the normalized tracer flux yM/F as well as yM,a/F in figure 3.4.
The five different curves show the concentration profile in similarity form for
different values of Ka and Kd. As we increase the advection parameter the flux
of tracers extends from the origin into a plateau before dropping smoothly at the
front and eventually vanishing at large η. In the purely advective case (i.e. in the
‘top-hat’ limit Kd → 0), the solution yM,a/F has a discontinuity at the location
of the advective front ηa (defined in (3.40)). The steepness of the front tends
51
3 Model for the streamwise transport, dispersion and mixing
η
yM/F
Ka = 1, Kd = 0.1
Ka = 1, Kd = 0
Ka = 1, Kd = 1
Ka = 10, Kd = 1
Ka = 10, Kd = 0
0 0.5 1 1.5
0
1
2
3
4
5
6
7
8
9
10
Figure 3.4: Plot of the variation of the normalized similarity solution yM/F , defined
in (3.55), against the similarity variable η = z/
(
t2/3M01/3
)
for the concentration flux
of tracers in the case of a constant flux at the source F and for different values of the
advection and dispersion parameters, Ka and Kd respectively. In the ‘top-hat’ limit
Kd → 0, we use the normalized piecewise-constant similarity solution yM,a/F , defined
in (3.56).
to decrease with increasing dispersion parameter. Moreover, we can see that the
value at the origin yM(η = 0)/F decreases with Ka/Kd, from yM(0)/F → ∞ as
Ka/Kd → 0 to yM(0)/F → 1 as Ka/Kd →∞.
Similarly to the previous section, we can compute the centroid of the distribu-
tion of yM normalized with the advective front ηa
µM =
∫ ∞
0
yM(η)η dη
ηa
∫ ∞
0
yM(η) dη
(3.57)
=
1
2
(
3Kd
2Ka
)2/3 Γ
[
2Ka
3Kd
+ 1
]
Γ
[
2Ka
3Kd
+ 13
] , (3.58)
52
3.3 Mathematical model
and its standard deviation normalized with the advective front ηa
σM =


∫ ∞
0
yM(η)η2 dη
ηa2
∫ ∞
0
yM(η) dη
− µM 2


1/2
(3.59)
=
(
3Kd
2Ka
)2/3


Γ
[
2Ka
3Kd
+ 53
]
3Γ
[
2Ka
3Kd
+ 13
] −


Γ
[
2Ka
3Kd
+ 1
]
2Γ
[
2Ka
3Kd
+ 13
]


2

1/2
. (3.60)
We plot µM against Ka/Kd in figure 3.5(a). We can see that, similarly to µF
(defined in (3.43)), µM decreases when Ka/Kd increases. We can prove that
µM → 1/2 as Ka/Kd → ∞ (using equation (5.11.7) in National Institute of
Standards and Technology, 2011-08-29), thus meaning that the centroid of yM
recedes behind the advective front at a fixed relative distance. The normalized
standard deviation σM is plotted in figure 3.5(b) against Ka/Kd. Similarly to
σF (defined in (3.45)), σM decreases when Ka/Kd increases. We can prove that
σM →
√
3/6 as Ka/Kd → ∞ (using equation (5.11.7) in National Institute of
Standards and Technology, 2011-08-29).
In a similar fashion to the previous subsection (cf. (3.46) and (3.48)), we can
compute the portion of the total concentration flux of tracers βM which is ahead
of the advective front ηa
βM =
∫ ∞
ηa
yM dη
∫ ∞
0
yM dη
, (3.61)
using equation (3.55), we obtain
βM =
Γ
[
2
3
(
Ka
Kd
+ 12
)
, 2Ka3Kd
]
−
(
2Ka
3Kd
)2/3
Γ
[
2
3
(
Ka
Kd
− 12
)
, 2Ka3Kd
]
Γ
[
2
3
(
Ka
Kd
+ 12
)] . (3.62)
As before, the ratio βM remains constant in time and space because (3.62) does
not depend on η; and βM depends only on the ratio Ka/Kd. We have plotted
βM against Ka/Kd in figure 3.6(a). Similarly to βF , βM appears to vanish at
large Ka/Kd, thus meaning that the portion of the tracer flux in the dispersive
front becomes smaller as Ka/Kd increases (see figure 3.4 for the change in the
53
3 Model for the streamwise transport, dispersion and mixing
Ka/Kd
µ
M
Theory
Data
µM = 1/2
0 5 10 15 20 25
0
0.5
1
1.5
2
(a)
Ka/Kd
σ M
Theory
Data
σM =
√
3/6
0 5 10 15 20 25
0
0.5
1
1.5
2
(b)
Figure 3.5: Constant-flux case for the tracer concentration flux: (a) plot of the the-
oretically predicted variation of µM (defined in (3.58)), the centroid of yM (defined
in (3.55)) normalized with the advective front ηa (defined in (3.40)), as a function of
Ka/Kd (plotted with a solid line), with the experimentally determined value (obtained
from the best fit of the constant-flux case shown in figure 4.7) marked with a cross,
the asymptotic value of µM is plotted with a dashed line; (b) plot of the theoretically
predicted variation of σM (defined in (3.60)), the standard deviation of yM normalized
with the advective front ηa, as a function of Ka/Kd (plotted with a solid line), with the
experimentally determined value (obtained from the best fit of the constant-flux case
shown in figure 4.7) marked with a cross, the asymptotic value of σM is plotted with a
dashed line.
distribution of yM with various Ka and Kd). We can also compute the normalized
distance between the average location of the tracer flux ahead of the advective
front and the location of the advective front ηa
ξM =
1
ηa


∫ ∞
ηa
yM η dη
∫ ∞
ηa
yM dη
− ηa

 , (3.63)
which yields
ξM =
1
2
(
3Kd
2Ka
)2/3 Γ
[
2
3
(
Ka
Kd
+ 32
)
, 2Ka3Kd
]
−
(
2Ka
3Kd
)4/3
Γ
[
2
3
(
Ka
Kd
− 12
)
, 2Ka3Kd
]
Γ
[
2
3
(
Ka
Kd
+ 12
)
, 2Ka3Kd
]
−
(
2Ka
3Kd
)2/3
Γ
[
2
3
(
Ka
Kd
− 12
)
, 2Ka3Kd
] − 1.
(3.64)
We plot ξM against Ka/Kd in figure 3.6(b). Similarly to ξF (see figure 3.3b), we
can also see in figure 3.6(b) that ξM → 0 as Ka/Kd →∞, thus meaning that the
front becomes sharper as Ka/Kd increases.
54
3.3 Mathematical model
Ka/Kd
β M
Theory
Data
0 5 10 15 20 25
0
0.2
0.4
0.6
0.8
1
(a)
Ka/Kd
ξ M
Theory
Data
0 5 10 15 20 25
0
1
2
3
(b)
Figure 3.6: Constant-flux case for the tracer flux: (a) plot of the theoretically pre-
dicted variation of βM (defined in (3.62)), the portion of the total concentration flux
of tracers ahead of the advective front ηa (defined in (3.40)), as a function of Ka/Kd
(plotted with a solid line), with the experimentally determined value (obtained from the
best fit of the constant-flux case shown in figure 4.7) marked with a cross; (b) plot of
the theoretically predicted variation of ξM (defined in (3.64)), the normalized distance
between the average location of the concentration flux of tracers ahead of the advective
front and the location of the advective front ηa, as a function of Ka/Kd (plotted with a
solid line), with the experimentally determined value (obtained from the best fit of the
constant-flux case shown in figure 4.7) marked with a cross.
3.3.4 Finite-volume release: instantaneous release funda-
mental solution
We can also consider an instantaneous finite-volume release localized at the source
of a quasi-two-dimensional steady turbulent jet. If the general equation (3.15) is
satisfied for z > 0, t > 0 and if, in addition, φ(z, t) satisfies (following (3.17a–c)
with ϑ = 0)
φ(z, 0) = Bδ(z), φ(z, t) → 0 as z →∞,
∫ ∞
0
φ(z, t) dz = B, t > 0,
(3.65a–c)
where B is a constant representing the total volume of tracers released and δ(z)
is a Dirac delta function, then the condition (3.65c) can hold for all t > 0 if and
only if c = −a according to (3.23) with ϑ = 0. Thus, (3.20) becomes
φ(z, t) = t−2/3y(η) with η = z
t2/3M01/3
. (3.66)
55
3 Model for the streamwise transport, dispersion and mixing
In this case, the initial boundary value problem for φ(z, t), defined by (3.21) with
c = −a, (3.65a–c) and (3.66), reduces to
(
−2
3
− Ka
2η3/2
)
y +
(
(2Ka −Kd)
2η1/2 −
2η
3
)
y′ −Kd η1/2y′′ = 0, (3.67)
subject to the conditions
y(η) → 0 as η →∞,
∫ ∞
0
y(η) dη = B
M01/3
, t > 0. (3.68a,b)
Equation (3.67) can be rearranged
−2
3
(ηy)′ +Ka
( y
η1/2
)′
−Kd
(
η1/2y′
)′
= 0, (3.69)
and thus integrated twice to obtain
y(η) = ηKa/Kd exp
[
− 4
9Kd
η3/2
](
J4
+
2J3
3
(
− 4
9Kd
) 2
3
(
Ka
Kd
−1
)
γ
[
2
3
(
1− KaKd
)
,− 4
9Kd
η3/2
])
, (3.70)
where J3 and J4 are two integration constants and γ[g, ι] =
∫ ι
0 hg−1e−h dh is the
lower incomplete gamma function. Since η > 0 and the function γ[g, ι] is complex
for ι < 0, J3 must equal 0. J4 can be determined by integrating equation (3.70)
∫ ∞
0
y(η) dη = J4
(
3
2
) 4
3
(
Ka
Kd
+ 14
)
K
2
3
(
Ka
Kd
+1
)
d Γ
[
2
3
(Ka
Kd
+ 1
)]
, (3.71)
and applying the integral condition (3.68b) to obtain
J4 =
B
(
3
2
) 4
3
(
Ka
Kd
+ 14
)
K
2
3
(
Ka
Kd
+1
)
d Γ
[
2
3
(
Ka
Kd
+ 1
)]
M01/3
. (3.72)
Therefore, the ‘fundamental’ solution of the effective advection–diffusion problem
for the case of an instantaneous finite-volume release initially localized as a delta
56
3.3 Mathematical model
function at z = 0 is, in similarity form,
yδ(η) =
B
(
3
2
) 4
3
(
Ka
Kd
+ 14
)
K
2
3
(
Ka
Kd
+1
)
d Γ
[
2
3
(
Ka
Kd
+ 1
)]
M01/3
ηKa/Kd exp
[
− 4
9Kd
η3/2
]
.
(3.73)
We can note that the concentration φδ = t−2/3yδ(η) (from (3.66) with yδ described
in (3.73)) vanishes in time for all values of η, because of the streamwise dispersion
(i.e. for Kd > 0). Furthermore, we expect the actual concentration Cδ to vanish
even more rapidly due to the cross-jet dispersion as the flow transports the finite
volume of tracers (see the experimental results in figure 4.5a for finite-volume
releases in quasi-two-dimensional jets).
We have plotted the non-dimensional quantity yδ/
(
B/M01/3
)
in figure 3.7.
The three different curves show the concentration profile in similarity form for
different values of Ka and Kd. Unsurprisingly, we find that the location of the
peak, ηmax = (3Ka/2)2/3, only depends on Ka. Thus, increasing Ka shifts the
peak upwards (away from the origin), while increasing Kd spreads the width of
the distribution. There is always to a greater or lesser extent asymmetry, with
the leading edge being more diffuse than the rear.
Interestingly, in the ‘top-hat’, purely advective limit Kd → 0 equation (3.69)
integrates to ( Ka
η1/2 −
2η
3
)
y = J5, (3.74)
where J5 is a constant of integration. In order to satisfy the boundary condition
at infinity (3.68a) as well as the integral condition (3.68b) we must have J5 = 0 for
all 0 ≤ η < ηa and ηa < η, where ηa = (3Ka/2)2/3 is the location of the advective
front as defined in (3.40) (note that ηa is the same in both the constant-flux
case and the finite-volume case). Therefore, the similarity solution of the purely
advective problem for the case of an instantaneous finite-volume release initially
localized as a delta function at z = 0 is
yδ,a = Bδ (η − ηa) . (3.75)
As expected, without dispersion (i.e. in the ‘top-hat’ limit Kd → 0) the dis-
tribution of tracers remains the same in time (i.e. distributed as the initial
Dirac delta function). The delta function is located in the similarity domain at
57
3 Model for the streamwise transport, dispersion and mixing
η
yδ/
(
B/M01/3
)
Ka = 1, Kd = 0.1
Ka = 1, Kd = 1
Ka = 10, Kd = 1
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
Figure 3.7: Plot of the variation of the non-dimensional fundamental similar-
ity solution yδ/
(
B/M01/3
)
, defined in (3.73), against the similarity variable η =
z/
(
t2/3M01/3
)
for the problem of advection–dispersion in the case of an instantaneous
finite-volume release at the source and for different values of the advection and disper-
sion parameters, Ka and Kd respectively.
ηa = (3Ka/2)2/3, the location of the (purely) advective front. In time and space
coordinates, it means that the volume of tracers is located at za = (3Ka/2)2/3 t2/3
and travels at the speed wa = KaM01/2z−1/2 in the streamwise direction. We can
notice that the location of the advective front ηa is the same as the location of
the peak of the tracer concentration in the general effective advection–diffusion
problem: ηa = ηmax = (3Ka/2)2/3.
Similarly to the previous section, we can compute the centroid of the distribu-
tion of yδ normalized with the advective front ηa
µB =
∫ ∞
0
yδ(η)η dη
ηa
∫ ∞
0
yδ(η) dη
(3.76)
58
3.3 Mathematical model
µB =
(
3Kd
2Ka
)2/3 Γ
[
2Ka
3Kd
+ 43
]
Γ
[
2Ka
3Kd
+ 23
] , (3.77)
and its standard deviation normalized with the advective front ηa
σB =


∫ ∞
0
yδ(η)η2 dη
ηa2
∫ ∞
0
yδ(η) dη
− µB2


1/2
(3.78)
=
(
3Kd
2Ka
)2/3


Γ
[
2Ka
3Kd
+ 2
]
Γ
[
2Ka
3Kd
+ 23
] −


Γ
[
2Ka
3Kd
+ 43
]
Γ
[
2Ka
3Kd
+ 23
]


2

1/2
. (3.79)
We plot µB against Ka/Kd in figure 3.8(a). We can see that, similarly to µF
(defined in (3.43)) and µM (defined in (3.58)), µB decreases when Ka/Kd in-
creases. We can prove that µB → 1 as Ka/Kd → ∞ (using equation (5.11.7) in
National Institute of Standards and Technology, 2011-08-29), thus meaning that
the centroid of yB recedes precisely to the location of the advective front, which
is also the location of the peak. It is also important to note that µB does not only
depend on Ka but actually on the ratio Ka/Kd. Since the distribution is not sym-
metric with respect to its centroid, then both advection and dispersion processes
can affect the centroid. We believe that the underlying physical interpretation
of this asymmetry can be related to the asymmetry between the advective and
the dispersive terms in the general effective advection–diffusion equation (3.15).
The advection term decreases with distance like z−1/2, whereas the diffusion term
increases with distance like z1/2. The normalized standard deviation σB is plotted
in figure 3.8(b) against Ka/Kd. Similarly to σF (defined in (3.45)) and σM (de-
fined in (3.60)), σB decreases when Ka/Kd increases. We can prove that σB → 0
as Ka/Kd → ∞ (using equation (5.11.7) in National Institute of Standards and
Technology, 2011-08-29). σB vanishes at large Ka/Kd because, as we mentioned
previously, the concentration becomes distributed spatially according to a Dirac
delta function δ(z).
Similarly to the constant-flux case, in the general effective advection–diffusion
problem a non-negligible portion of the volume of tracers is transported faster
than the advective speed due to the combined effects of advection and dispersion
59
3 Model for the streamwise transport, dispersion and mixing
Ka/Kd
µ
B
Theory
Data
µB = 1
0 5 10 15 20 25
0
1
2
3
(a)
Ka/Kd
σ B
Theory
Data
0 5 10 15 20 25
0
1
2
3
(b)
Figure 3.8: Instantaneous finite-volume case for the tracer concentration: (a) plot of
the theoretically predicted variation of µB (defined in (3.77)), the centroid of yδ (defined
in (3.73)) normalized with the advective front ηa (defined in (3.40)), as a function of
Ka/Kd (plotted with a solid line), with the experimentally determined value (obtained
from the best fit of the constant-flux case shown in figure 4.7) marked with a cross,
the asymptotic value of µB is plotted with a dashed line; (b) plot of the theoretically
predicted variation of σB (defined in (3.79)), the standard deviation of yδ normalized
with the advective front ηa, as a function of Ka/Kd (plotted with a solid line), with the
experimentally determined value (obtained from the best fit of the constant-flux case
shown in figure 4.7) marked with a cross.
processes. We can compute the portion of the total volume of tracers βB which
travels ahead of the advective front
βB =
∫ ∞
ηa
yδ dη
∫ ∞
0
yδ dη
, (3.80)
using equation (3.73), we obtain
βB =
Γ
[
2
3
(
Ka
Kd
+ 1
)
, 2Ka3Kd
]
Γ
[
2
3
(
Ka
Kd
+ 1
)] , (3.81)
where, once again, Γ[g, ι] =
∫∞
ι hg−1e−h dh is the upper incomplete Gamma func-
tion. As in the constant-flux release case βF defined in (3.47), the ratio βB remains
constant in time and space because (3.81) does not depend on η. Moreover, βB
depends only on the ratio Ka/Kd. We have plotted βB against Ka/Kd in figure
3.9(a). However, in contrast to βF , we can prove that βB → 1/2 (plotted with a
60
3.3 Mathematical model
dashed line) as Ka/Kd →∞ (see equation (8.11.10) in National Institute of Stan-
dards and Technology, 2011-08-29), thus meaning that the distribution of tracers
yδ becomes more symmetrical with respect to the peak value as Ka/Kd increases
(see figure 3.7 for the change in the distribution of yδ with various Ka and Kd).
We can also compute the normalized distance between the average location of the
volume of tracers travelling ahead of the advective front and the location of the
advective front ηa
ξB =
1
ηa


∫ ∞
ηa
yδ η dη
∫ ∞
ηa
yδ dη
− ηa

 , (3.82)
which yields
ξB =
(
3Kd
2Ka
)2/3 Γ
[
2
3
(
Ka
Kd
+ 2
)
, 2Ka3Kd
]
Γ
[
2
3
(
Ka
Kd
+ 1
)
, 2Ka3Kd
] − 1. (3.83)
We plot ξB against Ka/Kd in figure 3.9(b). Similarly to the constant-flux case ξF
defined in (3.49), the normalized distance ξB can also be considered as the distance
between the dispersive front (average location of the particles travelling ahead of
the advective front) and the advective front ηa. In time and space coordinates, the
distance between the dispersive front zB and the advective front za is zB − za =
ξBηat2/3. This distance increases with time as t2/3, as we observed in the constant-
flux case. We can also see in figure 3.9(b) that ξB → 0 as Ka/Kd → ∞, thus
meaning that the spreading of the tracer distribution becomes small compared
with the distance between the peak and the origin as Ka/Kd increases (see also
figure 3.7).
3.3.5 Finite-volume release: time-dependent release gen-
eral solution
The solution φδ(z, t) = t−2/3yδ(η) is the response of the system described by the
effective advection–diffusion equation (3.15) to a finite volume released instanta-
neously at t = 0 and distributed spatially according to a Dirac delta function δ(z).
Due to the linearity of equation (3.15), we can construct from this ‘fundamental’
solution φδ an integral expression for the general solution φg for a finite volume B
being released at the origin z = 0 over a period of time such that φg(0, t) = f(t).
61
3 Model for the streamwise transport, dispersion and mixing
Ka/Kd
β B
β
Theory
Data
βB = 0.5
0 5 10 15 20 25
0
0.2
0.4
0.6
0.8
1
(a)
Ka/Kd
ξ B
Theory
Data
0 5 10 15 20 25
0
1
2
3
(b)
Figure 3.9: Finite-volume case for an instantaneous release: (a) plot of the theoret-
ically predicted variation of βB (defined in (3.81)), the portion of the total volume of
tracers released which travels ahead of the advective front ηa (defined in (3.40)), as a
function of Ka/Kd (plotted with a solid line), with the experimentally determined value
(obtained from the best fit of the constant-flux case shown in figure 4.7) marked with
a cross, and the asymptotic value βB = 0.5 is plotted with a dashed line; (b) plot of
the theoretically predicted variation of ξB (defined in (3.83)), the normalized distance
between the average location of the volume of tracers travelling ahead of the advective
front and the location of the advective front ηa, as a function of Ka/Kd (plotted with a
solid line), with the experimentally determined value (obtained from the best fit of the
constant-flux case shown in figure 4.7) marked with a cross.
Without loss of generality, we choose to normalize the source function f(t):
∫ ∞
−∞
f(t) dt = 1. (3.84)
Therefore, the general solution φg can be expressed as the following integral
φg(z, t) =
∫ t
0
(t− τ)−2/3 yδ(ητ )f(τ) dτ, with ητ =
z
(t− τ)2/3M01/3
. (3.85)
The case of a truly instantaneous release of a finite volume at (z, t) = (0, 0)
is physically impossible to realize in an experiment. It is also not ideal in the
modelling of real flows. A more realistic set of initial boundary conditions is
to have a finite volume released at a constant flux over a finite period of time
0 ≤ t ≤ T0. This problem can be defined in terms of the following conditions
φT0(z, t) → 0 as z →∞,
∫ ∞
0
φT0(z, t) dz =



Bt
T0
, 0 ≤ t ≤ T0
B, T0 < t
, (3.86a,b)
62
3.3 Mathematical model
with φT0 satisfying the general equation (3.15) for z > 0, t > 0. The solution to
this initial boundary value problem can be computed using equation (3.85) with
the source function
fT0(t) =
H(t)−H(t− T0)
T0
, (3.87)
where H is the Heaviside function (i.e. H(t) = 0 for all t < 0 and H(t) = 1 for all
t > 0). We find that the solution to the integral (3.85) with the source function
fT0 described by (3.87) is
φT0(z, t) =
2Bz1/2
3KdM01/2T0Γ
[
2
3
(
Ka
Kd
+ 1
)]

Γ
[
2
3
(Ka
Kd
− 1
2
)
, 4z
3/2
9KdM01/2t
]
−



0, 0 < t ≤ T0
Γ
[
2
3
(Ka
Kd
− 1
2
)
, 4z
3/2
9KdM01/2(t− T0)
]
, T0 < t

 . (3.88)
The upper incomplete Gamma function, Γ[g, ι] =
∫∞
ι hg−1e−h dh, requires g > 0,
hence this solution is well-defined only for Ka > Kd/2. As we mentioned previ-
ously, we will find later that for our experimental data Ka appears to be substan-
tially greater than Kd. Note that this solution cannot be written in similarity
form because of the dependence on the time constant T0.
We can prove (see Appendix A.1) that the solution φT0(z, t), described in (3.88),
satisfies
φT0(z, t) = φδ(z, t), for
t
T0
≫ 1. (3.89)
So, the general solution for a rectangular source function converges asymptotically
to the fundamental solution φδ(z, t) (defined by (3.73) and (3.66)) in the limit
t ≫ T0. It is interesting to study how fast φT0 converges towards φδ. We can
non-dimensionalize the distance z and the time t using the scalings for length and
time scales T01/3M01/3 and T0, respectively, such that
z = T01/3M01/3z˘, t = T0t˘, (3.90a,b)
where breves denote non-dimensional variables. The evolution in time of the
normalized absolute deviation of the general solution φT0 from the fundamental
63
3 Model for the streamwise transport, dispersion and mixing
t/T0
de
v
Ka = 1, Kd = 0.1
Ka = 1, Kd = 1
Ka = 10, Kd = 1
0 5 10 15 20 25 30
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 3.10: Variation with scaled time t˘ = t/T0 of the normalized absolute deviation
dev(t˘), defined in (3.91), of the general solution φT0(z, t), defined in (3.88), and the
fundamental solution φδ(z, t) (defined by (3.73) and (3.66)). The data are computed
numerically for different values of the advection and dispersion parameters.
solution φδ is
dev(t˘) =
∫∞
0 |φT0(z˘, t˘)− φδ(z˘, t˘)| dz˘∫∞
0 φδ(z˘, t˘) dz˘
, for tT0
≥ 1, (3.91)
a non-dimensional quantity which only depends on the advection and dispersion
parameter Ka and Kd, and in particular does not depend on the total injected
volume of tracers B, on the initial momentum M0 or on the period of injection
T0. We plot dev(t˘) in figure 3.10 for 1 ≤ t/T0 ≤ 30. We compute the deviation
numerically for three different sets of values of Ka and Kd. We can see that all
the curves decrease asymptotically towards 0 as t/T0 increases. The deviation is
smaller than 0.1 (which can be considered as a threshold value of near conver-
gence) for t/T0 > 11, t/T0 > 4.7 and t/T0 > 11 for the sets of advection and dis-
persion parameters (Ka = 1, Kd = 0.1), (Ka = 1, Kd = 1) and (Ka = 10, Kd = 1),
respectively. It appears that the deviation depends mainly on the ratio Ka/Kd
and only very weakly on Kd.
Furthermore, we can note that in equation (3.88), if we take the limit T0 →∞
64
3.3 Mathematical model
and define F = B/T0, then we find
φT0→∞(z, t) = t1/3yF (η), (3.92)
with η = z/
(
t2/3M01/3
)
consistently with (3.25). So, equation (3.37) is equivalent
to the asymptotic solution of the general solution φT0 if the period of release T0
extends to infinity.
We have developed in this chapter a theoretical model, for various source con-
ditions, describing the streamwise transport and dispersion in quasi-two-dimen-
sional jets. In the following chapter we test the predictions of this model through
comparison with a range of experimental measurements.
65
Chapter 4
Streamwise transport, dispersion and
mixing in quasi-two-dimensional jets:
experimental results
4.1 Experimental procedure
We conduct our experiments in a slight modification of the experimental ap-
paratus we presented in Chapter 2, as shown schematically in figure 4.1. We
conduct three distinct sets of experiments using two qualitatively different tech-
niques. Each set of experiments is designed to provide experimental data that
can be compared with the three theoretical predictions derived in Chapter 3
for: a constant-flux release; an instantaneous finite-volume release; and a non-
instantaneous finite-volume release. In the first set of experiments (whose results
are presented in § 4.2.2), we measure the distribution of the concentration of dye
67
4 Experimental results for the streamwise dispersion and mixing
x
z
d = 5mm
Electronic valves
Peristaltic
pump
Overflow
H
=
1
m
L = 1m
W = 0.01m
u
w
2b(z)
CCD camera
Blue back-light tape
PIV study area
Figure 4.1: Schematic diagram of the experimental apparatus.
as it is released at a constant flux at the source of quasi-two-dimensional steady
turbulent jets. The second set of experiments (whose results are presented in
§ 4.2.3) involves what we believe to be a new technique, which consists of track-
ing large quantities of virtual particles evolving as passive tracers in the velocity
field of quasi-two-dimensional steady turbulent jets. The velocity field is mea-
sured in experiments with real jets (as opposed to numerically computed jets) by
using particle image velocimetry. We designed this technique, which we desig-
nate as virtual particle tracking, to obtain data for an instantaneous release to
compare with our mathematical model (derived in Chapter 3). In the third set
of experiments (whose results are presented in § 4.2.4), we measure the distribu-
tion of the concentration of dye as it is released as finite volumes at the origin of
quasi-two-dimensional steady turbulent jets. For physical reasons, which will be
detailed below, we cannot release finite volumes of dye instantaneously in the jets,
and so such physical dye releases inevitably extended over a finite time interval.
4.1.1 Constant-flux releases of dye
We fill the 1m (L)×0.01m (W )×1m (H) tank displayed in figure 4.1 with fresh
tap water. A vertical jet of constant source volume flow rate is discharged into the
68
4.1 Experimental procedure
tank using a peristaltic pump (520DU/R2 Watson-Marlow variable speed pump)
fed by a constant-head tank.
The injection mechanism for the constant-flux releases of dye in steady turbu-
lent quasi-two-dimensional jets consists of a syringe-pump connected to a small
needle inserted into a single main tube. The needle is located 0.2 m upstream of
the nozzle. After the jet has reached a steady state in the tank, a mixture of red
food dye ‘Fiesta Red’ (Allura Red AC, E129) and tap water (with a dye concen-
tration of 1.8% per weight) is injected at a constant flow rate, 0.11 cm3 s−1 . We
study 19 constant-flux releases of dye in steady turbulent jets with jet Reynolds
number 2240 ≤ Rej = dws/ν ≤ 3870, where ws is the source velocity and ν is the
kinematic viscosity of water.
To measure the dye concentration, we perform the experiments in a dark room.
Following Dalziel et al. (2008) we attach a 0.54 m × 0.54 m electroluminescent
Light Tape (Electro-LuminX Lighting Corporation) to the external surface of
the rear side of the tank, centred on the jet axis and with the bottom of the
tape at the height of the nozzle. It provides a constant and uniform source of
near-monochromatic cyan light of approximately 400 cd m−2. This wave length is
close to the peak of the ‘Fiesta Red’ dye absorption spectrum. We measure the
transmitted light intensity with a high-speed 8 bit grey-scale camera (Fastcam
SA1.1 - Photron) mounted with an 85 mm focal-length lens (f-stop 5.6). The
camera is located 3 m away from the tank, which is sufficient to have negligible
parallax error. We also take care to reduce any light pollution from reflection or
other sources, in particular by installing a black frame around the study area.
The camera records 640× 848 pixel images covering the entire study area, which
spans −40 ≤ x/d ≤ 40 and 0 ≤ z/d ≤ 100 (where x is the coordinate in the
lateral, cross-jet direction, and z is the coordinate in the streamwise direction;
the origin is at the centre of the nozzle slot and d = 5 mm is the nozzle width),
and part of the black frame (in order to have a black intensity reference). For
each video we set the origin in time, t = 0, at the image preceding the first image
in which dye is seen by the camera. The frequency of image acquisition is set
at 60 frames per second. Following the calibration method and the algorithm
described by Coomaraswamy (2011) and based on Cenedese & Dalziel (1998), we
perform the calibration in situ. We record the intensity measured by the camera
for 23 known concentrations of dye, ranging from 0 to 2 % per weight. A fitting
69
4 Experimental results for the streamwise dispersion and mixing
curve using a third-order polynomial in the logarithm of the normalized intensity
gives us a continuous and monotonic relationship between the intensity and the
spanwise- (or y-) integrated concentration. All the images recorded by the camera,
either for the calibration process or for the experiments, are analysed using the
software code DigiFlow (Sveen & Dalziel, 2005). This procedure enables us to
obtain accurate measurements of the (spanwise-integrated) dye concentration in
time and space for each experiment.
4.1.2 Instantaneous finite-volume releases of clusters of vir-
tual particles
We track virtual particles in experimentally measured velocity fields of quasi-
two-dimensional steady turbulent jets. We use the velocity fields measured by
us previously as presented in Chapter 2 and obtained using a particle-image-
velocimetry technique (as described in Sveen & Dalziel, 2005). We measure the jet
velocity in a 0.4m×0.4m study area centred on the jet axis (as shown in figure 4.1)
and covering a height from z = 0.2–0.6 m. We use the camera described above
(mounted with a 62mm focal-length lens) at a frequency of image acquisition 250
frames per second and for a duration of 21.8 s. The 1024 × 1024 pixel images
provide us with spatially and temporally resolved velocity fields for six steady
turbulent jets at source volume flow rates 33.2, 37.0 and 40.3 cm3 s−1 . The jet
Reynolds number ranges from 3320 ≤ Rej ≤ 4030. We find that the divergence of
each velocity field is insignificant (typically mean(|∇·u|)/mean(|∇×u|) ≈ 5%,
wheremean(·) represents an average in time and space), so they can be considered
as incompressible. Using these computed velocity fields, we seed in each of them
201×51pixel clusters of (massless) virtual particles located in a rectangular evenly-
distributed cluster at −8.8 ≤ x/d ≤ 7 and 44.4 ≤ z/d ≤ 48.3 (i.e. within the
characteristic local width of the jet). The release can be considered instantaneous
as a cluster of virtual particles is injected in the flow field within a single time
step. The possibility of releasing instantaneously a large number of particles
constitutes the main reason for the use of this technique in this study. This
important advantage, compared with the non-instantaneous dye finite-volume
releases (discussed below), allows us to reproduce more easily the instantaneous
release constraint imposed in the mathematical model in (3.65a).
We release individual clusters every 0.4s in each experiment and study a total of
70
4.1 Experimental procedure
256 clusters representing 2,624,256 virtual particles. For each cluster the virtual
particles evolve in time and space as passive tracers transported by the flow. For
each simulation we set the origin in time, t = 0, at the first image in which the
particle cluster is seeded. The simulation of a cluster stops as soon as a virtual
particle reaches the top boundary of the velocity field. Finally, we record the
location in time and space of the tracers and analyse the results using DigiFlow.
By averaging 256 virtual-particle experiments we obtain a smooth distribution
of the particle concentration in time and space, which we compare with the dye
experiments and the theoretical prediction in § 4.2.
Different techniques involving particle tracking have been used to study disper-
sion, mixing and transport in jets or other types of flows. In previous studies,
the particles were either real and tracked by imaging analysis technique (see e.g.
Yang et al., 2000; Sveen & Dalziel, 2005), or purely numerical and evolving in nu-
merically resolved flows (see e.g. Dutkiewicz et al., 1993; Luo et al., 2006; Picano
et al., 2010). However, we have not been able to find any mention in the literature
of using virtual particles in the velocity field of real flows. This technique requires
a spatially and temporally resolved computation of the velocity field, which can
be done, for example, using a particle-image-velocimetry technique. We can then
seed some (massless) virtual particles in the velocity field and track their trajec-
tory as they are transported as passive tracers by the flow. The advantages of this
technique are numerous: the resolution is only limited by the resolution of the
acquisition of the velocity field; it is not restricted to the computation limitations
encountered in full numerical simulations, but can be used for any laboratory
experiments; a large quantity of virtual particles can be seeded instantaneously
in the jet (thus satisfying, in our case, the constraint imposed in the theoretical
model for an instantaneous finite-volume release); and their initial distribution
can be completely arbitrary.
4.1.3 Finite-volume releases of dye
The experimental procedure for the finite-volume releases of dye in steady turbu-
lent quasi-two-dimensional jets is very similar to the experimental procedure for
the constant-flux releases of dye (described in § 4.1.1). We fill the tank displayed
in figure 4.1 with fresh tap water. A vertical jet of constant source volume flow
rate is discharged into the tank using the same peristaltic pump described above
71
4 Experimental results for the streamwise dispersion and mixing
and fed by a constant-head tank.
For the injection mechanism of the finite-volume releases of dye, the main tube
divides into two approximately 80 cm before the nozzle (see figure 4.1). The two
tubes are recombined approximately 15 cm before the nozzle. Two valves located
just before the recombining junction control the flow for each pipe separately.
We monitor the valves to allow the flow to go through one section or the other
exclusively. We open and close the valves electronically so that a steady jet flow
is maintained in the tank before and after switching the valves. Although we
observe a small perturbation (a pressure wave) in the tank we believe it does
not perturb the experiment significantly. The purpose of this two-tube system
is to release a finite volume of dye in a steady turbulent jet. The procedure for
each experiment is as follows. We inject a 5 cm3 mixture of the same red food
dye described above and tap water (with a dye concentration of 2 % per weight)
into the closed tube approximately 5 mm upstream of the valve. Meanwhile,
water flows at a constant source volume flow rate through the other tube to
produce a turbulent jet in the tank. After the jet reaches a steady state, we
switch the valves to redirect the whole flow into the section containing the red
dye, thus releasing a finite volume of dye into the established jet. We conduct 26
finite-volume releases of dye in steady turbulent jets with jet Reynolds number
2170 ≤ Rej ≤ 4870. It is important to note that, although great care is taken
during the experiments and different protocols have been tested, instantaneous
finite-volume releases of dye cannot be achieved for practical reasons. We find
that the time of injection, although relatively short (of the order of 0.5 s), cannot
be considered as instantaneous, as we will discuss in § 4.2.4. We believe that
the main reason for this injection delay is due to some Taylor dispersion (Taylor,
1953) of the dye as it is transported in the short section of tube (approximately
0.2 m long) leading to the tank.
We perform the measurements of the dye concentration for the finite-volume
releases using exactly the same technique as described for the constant-flux re-
leases. From the transmitted light intensity recorded by the high-speed camera
described above, we can compute the dye concentration in the study area, span-
ning −40 ≤ x/d ≤ 40 and 0 ≤ z/d ≤ 100, at a frequency of 60 frames per second.
We obtain accurate measurements of the dye concentration in time and space for
each experiment.
72
4.2 Experimental results
4.2 Experimental results
Similarly to (2.7a,d), we find that the natural scalings for length and time in our
problem are d, the source width, and (d2/Q0), respectively. Therefore when con-
sidering our experimental data we will always scale quantities with these scalings,
i.e.
z = dz˜, t =
( d2
Q0
)
t˜, (4.1a,b)
where tildes denote non-dimensional variables. Although the initial momentum
fluxM0 is also a natural scaling parameter in the theoretical model (see equations
(2.5b), (3.15) and (3.20)), we do not use it as a scaling parameter in this section
because we could not measure it directly in the experiments. Instead ofM0, we use
the equivalent ratio Q02/d (in § 2.4, we found M0 ≈< M >= 0.55
(
Q02/d
)
, where
< M > is the space- and time-averaged momentum flux in quasi-two-dimensional
jets). In particular, the non-dimensional similarity variable η = z/
(
t2/3M01/3
)
,
defined in the model (see § 3.3.1), is replaced by ηexp = z/
(
t2/3
(
Q02/d
)1/3)
, so
that
ηexp
η =
(dM0
Q02
)1/3
≈ 0.82. (4.2)
This non-dimensionalization also affects slightly the advection and dispersion pa-
rameters Ka and Kd, defined in the model (see § 3.2). As a consequence, the
advection and dispersion parameters Ka,exp and Kd,exp, that we use in this sec-
tion, are related to Ka and Kd such that
Ka,exp
Ka
=
Kd,exp
Kd
=
(dM0
Q02
)1/2
≈ 0.74. (4.3)
We omit the subscript exp in ηexp, Ka,exp and Kd,exp hereafter in this section.
To test our turbulent model hypothesis developed in § 3.2 and which led to the
general effective advection–diffusion (3.15), we choose to compare the theoretical
predictions, developed in § 3.3, first with experiments realized in the constant-
flux case. The initial boundary and integral conditions (3.24a–c) imposed in the
constant-flux case are simpler to satisfy experimentally than the initial boundary
and integral conditions imposed in the finite-volume case (3.65a–c), which require
an instantaneous release of finite volumes of tracers. Instantaneous finite-volume
releases of virtual particles are then tested against the theoretical prediction,
73
4 Experimental results for the streamwise dispersion and mixing
before studying the more challenging case of a non-instantaneous finite-volume
release of dye. In each case, we are particularly interested in whether the natural
scaling of the model z ∝ t2/3 agrees with the experimental results and, if so, we
then estimate from the experimental data the two key parameters: the advection
parameter Ka and the dispersion parameter Kd. Since the experiments in the
constant-flux case are simpler to realize, we believe that the estimates of Ka and
Kd measured in this case are more accurate than in the other two cases. There-
fore, we consider the values of Ka and Kd measured in the constant-flux case as
reference values, while the values measured in the other two cases are used to
determine the confidence interval of Ka and Kd. Before presenting the quanti-
tative experimental results, we give below a qualitative assessment of our tur-
bulent model hypothesis and motivate the utility of the virtual-particle-tracking
technique (described in § 4.1.2) in understanding the transport, dispersion and
mixing properties of quasi-two-dimensional jets.
4.2.1 Qualitative assessment
The purpose of this qualitative assessment is two-fold. Firstly, we want to study
how the dynamical structure of steady turbulent quasi-two-dimensional jets affects
their transport and dispersion properties. We have developed our turbulent model
hypothesis, stated in § 3.2, from the qualitative understanding of these properties.
Secondly, we use in this study a new technique to analyse the transport and
dispersion properties of the jets, which we introduced in the previous section as
virtual particle tracking. We give a qualitative overview of this technique, as well
as some justifications and motivations for its use in a more systematic and rigorous
approach to obtain quantitative results (which will be presented in § 4.2.3).
As we discussed in Chapter 2, in the far-field of quasi-two-dimensional jets (i.e.
z ≥ 20 d for W = 2 d Dracos et al., 1992), the core forms a high-speed undulating
region, which grows on average in an expanding straight-sided triangular section.
Outside the core we observe large counter-rotating eddies, which develop on al-
ternate sides of the core and grow linearly with distance. Moreover, we showed
in Chapter 2 that the core–eddy structure is self-similar with distance z. The
characteristic sinuous core and the large growing eddies can be observed in fig-
ure 4.2(a), which is an instantaneous grey-scale picture of a constant-flux release
of dye in a steady-state quasi-two-dimensional jet with Rej = 3850 (shown five
74
4.2 Experimental results
0
20
40
60
80
100
120
140
160
z
(d
)
(a) (b)
0.0
200.0
400.0
600.0
800.0
1000.0
(c)
(d)
0.0
200.0
400.0
600.0
800.0
1000.0(e)
Figure 4.2: (a) Grey-scale picture of a dyed jet (Rej = 3850) rising in the tank. The
average dye edges are plotted with black lines (half-spreading angle, < θdye >= 12.4◦,
as measured in figure 2.5). (b) Passive tracers (Pliolite particles) shown as streaks in
a typical jet (Rej = 4080). (c) Trajectories of the passive tracers shown in (b) and
identified by imaging analysis (for a duration of 0.2 s). (d) Instantaneous velocity field
(arrows) of the jet shown in (b). (e) Trajectories of virtual particles (for approximately
0.3 s) seeded at the same initial locations as the particles identified in (c) and evolving
as passive tracers in the time-dependent velocity field shown in (d).
seconds after injection; the average dye edges are plotted with black lines, half-
spreading angle < θdye >= 12.4◦, as measured in figure 2.4). The instantaneous
core–eddy structure can also be seen in figure 4.2(b). In figure 4.2(b), a superpo-
sition of 50 images (i.e. for a duration of 0.2 s) of the filming of an experiment
(see § 2.2), where passive tracers (0.23 mm Pliolite VTAC particles) were mixed
with a quasi-two-dimensional jet (Rej = 4080), depicts the tracers as streaks,
thus revealing the Eulerian structures in the flow (see discussion in § 2.5).
We compute two different types of results from the experiment with passive
tracers shown in figure 4.2(b). We can consider the tracers as Lagrangian parti-
cles and track their trajectory in time using a particle tracking algorithm imple-
mented in DigiFlow (Dalziel, 1992; Sveen & Dalziel, 2005). Figure 4.2(c) shows
75
4 Experimental results for the streamwise dispersion and mixing
the trajectories identified by the algorithm, at the same time instant as the jet
displayed in figure 4.2(b). Particles have been tracked for 50 images (i.e. for a
duration of 0.2 s) and reveal very similar flow patterns to the streaks in figure
4.2(b). However, this technique has some limitations as the number of particles
tracked for a certain time period decreases quickly with increasing time period.
We have also very little control over the initial distribution of the particles (usu-
ally spatially homogeneous), and cannot, for example, reproduce an instantaneous
finite-volume release of these particles. To remedy these limitations, we have de-
veloped a virtual-particle-tracking technique, which we presented in § 4.1.2. We
seed in the velocity field (displayed in figure 4.2d) of the experimental jet shown
in figure 4.2(b) some virtual particles in order to track their trajectory as they
are advected as passive tracers by the flow. As a qualitative validation of this
technique, we have seeded the virtual particles so that their initial distribution is
identical to the initial distribution of the (real) particles identified in figure 4.2(c).
The resulting trajectories of the virtual particles are plotted in figure 4.2(e) for
a period of approximately 0.3 s. The trajectories of the virtual particles are very
similar to the trajectories of the particles in figures 4.2(b) and 4.2(c), and thus
reveal the same core–eddy structure. We believe that the virtual-particle-tracking
technique can provide meaningful information about the transport and dispersion
properties of quasi-two-dimensional jets.
The schematic diagram displayed in figure 4.3(a) summarizes the structure
of quasi-two-dimensional jets. The time-averaged mean picture of quasi-two-di-
mensional jets is associated with a triangular shape encapsulating all the flow
structures, while the time-dependent picture shows a sinuous core flanked by large
growing eddies. We believe that the interaction between the core and the eddies
results in large streamwise dispersion as the fluid experiences intense stretching at
the interface between the core and the eddies. The eddies also play a crucial role in
the entrainment and mixing of ambient fluid. From the observations of dyed jets
such as the jet illustrated in figure 4.2(a), we find that fluid can be entrained from
the ambient by the eddies and then either drawn within the eddies or incorporated
into the core. We also believe that fluid can be exchanged between the eddies and
the core. On the other hand, we have not observed any dyed fluid being detrained
completely from the jet to the ambient.
These processes can be revealed by applying the virtual-particle-tracking tech-
76
4.2 Experimental results
nique to the core and the eddies of a quasi-two-dimensional jet. In the velocity
field of the jet presented in figure 4.2(d) and reproduced in figure 4.3(b), we seed
three clusters of virtual particles. The first cluster, composed of 3721 virtual
particles, distributed in a square and initially seeded at the centre of an eddy is
shown in light grey in figure 4.3(b). The second cluster, composed of 7381 virtual
particles, distributed in a rectangle and initially seeded between the eddy and
the core is shown in grey in figure 4.3(b). The last cluster, composed of 3721
virtual particles, distributed in a square and initially seeded in the core of the jet
is shown in dark grey in figure 4.3(b). Figure 4.3(c) shows the typical trajectories
of one single particle from each cluster. The particle locations are plotted every
0.02 s and each colour corresponds to a time period of 0.2 s (see colour scale).
The particle starting in the eddy (plotted with pluses) moves slower than the
other two particles and its trajectory forms two loops characteristic of the fact
that it is transported within the eddy. The particle starting in the core (plotted
with crosses) is transported quickly and has a slightly sinuous trajectory, which is
characteristic of the transport within the core. On the other hand, the trajectory
of the particle chosen approximately at the interface between the eddy and the
core (see § 2.5 for a thorough discussion on the identification of the core and eddy
structures) is often more complex (plotted with squares) and can be transported
from the core to the eddy, or indeed from the eddy to the core. In the present case
the particle starts in the core and then is drawn into the neighbouring eddy as the
trajectory forms one loop. This is a simple illustration of the possible exchange
of fluid parcels between the different structures.
Figure 4.4 shows the simultaneous evolution in time of all the particles in the
three clusters as they are passively transported by the jet velocity field shown in
figure 4.3(b). Each colour corresponds to a particular time instant, starting from
black and finishing with white and with a time step of 0.2 s between each colour
(we use the same colour scale to that used in figure 4.3c). Again, we can clearly
see that the virtual particles are transported much faster in the core of the jet
(see figure 4.4c) than in the eddy (see figure 4.4a). On the other hand, mixing
is more intense in the eddy than in the core. The cluster initially seeded in the
eddy disintegrates very rapidly compared to the cluster initially seeded in the
core. The cluster initially seeded between the eddy and the core (see figure 4.4b)
experiences considerable stretching in the streamwise direction (its streamwise
77
4 Experimental results for the streamwise dispersion and mixing
(a) (b)
0 t (s)1 2 3
(c)
Figure 4.3: (a) Schematic diagram describing the structure of quasi-two-dimensional
jets. (b) Instantaneous velocity field displayed in figure 4.2(d) with three rectangular
clusters of virtual particles initially seeded: at the centre of an eddy (plotted in light
grey); between the eddy and the core (plotted in grey); and in the core of the jet
(plotted in dark grey). (c) Typical trajectories of three virtual particles evolving in
the time-dependent velocity field shown in (b) and initially seeded: in an eddy (cluster
outlined in light grey) (plotted with pluses); between the eddy and the core (cluster
outlined in grey) (plotted with squares); and in the core (cluster outlined in dark grey)
(plotted with crosses). The particle locations are plotted every 0.02 s and each colour
corresponds to a time period of 0.2 s (see colour scale).
maximum extent is ten times larger than its cross-stream maximum extent after
a few time steps), owing to the shear layer at the interface between the core and
the eddy. We can notice that some virtual particles are drawn into the eddy
while others remain in the core. This emphasizes the time-dependent exchange
of fluids between the core and the eddies pointed out above. We can also observe
the delaying effect (with the colour scheme) of the eddies, in which tracers have
a longer residency time than in the core. In Chapter 5, we investigate further the
turbulent relative dispersion of the particle clusters presented in figure 4.4.
When ensemble-averaged, we believe that the streamwise dispersive mecha-
nisms revealed by the virtual particles in figure 4.4 can be modelled as an en-
hanced dispersion coefficient, as stated in the turbulent hypothesis presented in
§ 3.2. The main assumption we make in equation (3.8), pertaining to the tur-
bulent eddy diffusive coefficient (Dzz ∝ b wm, where Dzz is the streamwise com-
ponent of the turbulent eddy diffusive tensor), can be physically justified from
the study of both the structures and the velocity profile of quasi-two-dimensional
jets (see figures 2.5, 2.7 and 2.13 for velocity measurements in quasi-two-dimen-
78
4.2 Experimental results
(a)
0 t (s)1 2 3
(b) (c)
Figure 4.4: Evolution in time of the virtual particles seeded in the velocity field
shown in figure 4.3(b) as they are transported by the flow (each colour corresponds to
a particular time instant): (a) cluster initially distributed at the centre of an eddy and
shown in light grey in figure 4.3(b); (b) cluster initially distributed between the eddy
and the core and shown in grey in figure 4.3(b); (c) cluster initially distributed in the
core of the jet and shown in dark grey in figure 4.3(b). Each colour corresponds to a
time period of 0.2 s, the colour scale shown at the bottom of (b) is the same to that
used in figure 4.3(c).
sional jets). The core–eddy structure is self-similar with height, thus the local
characteristic size of the jet, b(z), appears as a relevant length-scale. Moreover,
the local maximum time-averaged streamwise velocity is the second physically
meaningful variable in the problem of dispersion, because all mixing and disper-
sive mechanisms should scale like wm(z). In the rest of this section, we compare
ensemble-averaged experimental results with the theoretical predictions found in
§ 3.3 and based on our turbulent model hypothesis.
79
4 Experimental results for the streamwise dispersion and mixing
4.2.2 Constant-flux releases of dye
We present in figures 4.5(a–c) experimental results and theoretical predictions
of constant-flux releases of dye in quasi-two-dimensional steady turbulent jets.
The spatial distribution of the concentration C(x, z, t) is plotted using a colour
scale (see colour scale at the top of figures 4.5a–c) at different non-dimensional
times, 74 ≤ t˜ ≤ 374, to show the evolution of the dye concentration in the jet.
In figure 4.5(a), we plot the ensemble-averaged concentration of the 19 experi-
ments, which were conducted at different jet Reynolds number, 2240 ≤ Rej ≤
3870 (see § 4.1.1). We also plot the average dye edges (half-spreading angle,
< θdye >= 12.4◦) with thick white lines and the average boundaries of the core
(half-spreading angle, 7◦ starting from z = 20 d) with thin white lines. We can
observe some dispersion of the dye at the leading edge, which indicates the stream-
wise dispersion discussed above. It is also apparent that the dye is transported
first through the core (i.e. within the thin white lines) before mixing across the
full width of the jet (i.e. filling the triangle delimited by the average dye edges
shown with thick white lines). The characteristic sinuous instability of the core
(clearly visible in figure 4.2a) does not appear in figure 4.5(a) because of the
averaging process.
Our model is inherently one-dimensional, and so obviously cannot predict the
distribution of the concentration across the jet (i.e. in the x-direction). In order
to be able to solve the partial differential equation (3.4), we integrate the concen-
tration along the x-axis and study the evolution of φ(z, t) rather than C(x, z, t).
We present the laterally-integrated experimental concentration φF,exp(z, t) in fig-
ure 4.5(b) in normalized and re-distributed form using
C(x, z, t) =



φF,exp(z, t)
2l(z) , −l(z) ≤ x ≤ l(z)
0, otherwise
, (4.4)
where
l(z) = tan (< θdye >)(z − z0), for z ≥ 0 (4.5)
is the local lateral distance between the average dye edges (plotted with thick
white lines in figure 4.5a) and z0 is the space virtual origin defined below in
(4.7a). Alongside in figure 4.5(c), we show the equivalent theoretical prediction
80
4.2 Experimental results
t
~
 = 74
(a) (b) (c)
0 1.8C (% wt)
t
~
 = 56
(d) (e) (f )
0 2C (% wt)
t
~
 = 149
(a) (b) (c)
t
~
 = 108
(d) (e) (f )
t
~
 = 224
(a) (b) (c)
t
~
 = 222
(d) (e) (f )
t
~
 = 299
(a) (b) (c)
t
~
 = 332
(d) (e) (f )
t
~
 = 374
(a) (b) (c)
t
~
 = 533
(d) (e) (f )
Figure 4.5: Distribution in space and non-dimensional time t˜ = t/(d2/Q0) of the
concentration of dye (plotted using the two colour scales shown at the top for figures
a–c and d–f, respectively) in the case of constant-flux releases (a–c) and finite-volume
releases (d–f ) in quasi-two-dimensional jets for: (a) ensemble average of 19 experiments,
the average dye edges are plotted with thick white lines (half-spreading angle, < θdye >=
12.4◦, as measured in figure 2.4) and the average boundaries of the core are plotted
with thin white lines (half-spreading angle, 7◦ starting from z = 20 d, as measured
in figure 2.12); (b) spatial lateral average of the distribution shown in (a) (defined in
(4.4)); (c) theoretical prediction based on (3.37) and using Ka = 1.65 and Kd = 0.09;
(d) ensemble average of 26 experiments, similarly to (a) the average dye edges are
plotted with thick white lines and the average boundaries of the core are plotted with
thin white lines; (e) spatial lateral average of the distribution shown in (d) (defined
in (4.9)); (f ) theoretical prediction based on (3.88) using Ka = 1.65, Kd = 0.09 and
T0 = 183
(
d2/Q0
)
.
81
4 Experimental results for the streamwise dispersion and mixing
computed from equation (3.37) for yF (η), based on the assumption of a constant-
flux release at the origin of the jet. To compute the theoretical prediction yF , we
useKa = 1.65 andKd = 0.09 for the advection and dispersion parameters, respec-
tively. These parameters are optimized by obtaining the best least-squares fit be-
tween the experimental concentration yF,exp (i.e. the similarity form of φF,exp(z, t),
transformed using (3.25)), and the theoretical prediction yF . Before plotting the
theoretical prediction yF in figure 4.5(c), we transform yF into its physical form
φF (z, t) using (3.25), then normalize it (similarly to φF,exp(z, t)) with the local dis-
tance 2l(z) between the average dye edges, and finally re-distribute it uniformly,
assuming a top-hat spatially-averaged profile, within these boundaries, i.e.
C(x, z, t) =



φF (z, t)
2l(z) , −l(z) ≤ x ≤ l(z)
0, otherwise
, (4.6)
where l(z) is defined in (4.5). As we noted in § 3.3.2, we can see that the cross-
stream distribution of the concentration spreads linearly with distance. Law
(2006) modelled mathematically the cross-stream distribution of the concentra-
tion of passive tracers in round and plane turbulent jets. He also found that the
cross-stream distribution spreads linearly with distance. The model predicts that,
in steady state, the laterally-integrated concentration φF increases like z1/2. How-
ever, due to the cross-stream dispersion, the concentration C should decrease like
z−1/2. We can actually see in figure 4.5(b,c) that the experimental and theoretical
concentrations, respectively, decrease with distance.
Comparing the data (figure 4.5b) with the theoretical prediction (figure 4.5c),
we can see that the propagation of the front as well as its dispersion appear to have
been correctly modelled (i.e. the scaling is correct), with only a small difference
near the source. This mismatch is probably due to the zone of flow establishment
of the jet (see e.g. Yannopoulos & Noutsopoulos, 1990). There is a necessary time
and distance of adjustment before the experimental data can match the theoretical
prediction, because the theoretical prediction is based around the assumption that
the jet characteristic properties are given by the similarity power laws (2.5a,b).
Giger et al. (1991) and Dracos et al. (1992) reported that the structure of qua-
si-two-dimensional jets was different near the source, where three-dimensional
effects were important. They found that the self-similar core and eddy structure
82
4.2 Experimental results
(which is key in the dispersion mechanisms of the jet) only developed beyond
approximately z ≥ 20 d (for the aspect ratio W/d = 2). Therefore, we might
expect our model to be appropriate for z ≥ 20 d.
We display in figure 4.6(a) the evolution in time of the non-dimensional inte-
grated concentration of dye released in the experiments shown in figure 4.5(a).
We can see that the experimental data (plotted with pluses) increase approxi-
mately linearly in time (a linear fit is plotted with a black line). Therefore, the
constant-flux integral condition (3.24c) assumed in the model is satisfied experi-
mentally.
We show in figure 4.6(b) the evolution in time of the distribution in similarity
space of the normalized experimental data yF,exp, plotted for nine successive time
periods in the range 2 ≤ t˜ ≤ 353. As we explained earlier, yF,exp is computed
from the ensemble-averaged laterally-integrated experimental concentration for
the constant-flux releases φF,exp using equation (3.25) at every instant in time t˜.
We also use the following virtual origins in space (see equation (2.6)) and time:
z0 = −
Q02
4
√
2αM0
, t0 =
z0d
Q0
. (4.7a,b)
The space virtual origin z0 is simply the virtual origin of quasi-two-dimensional
jets. The time virtual origin t0 represents the time needed to travel the distance
|z0|, from the jet virtual origin to the nozzle, at the average source jet velocity
Q0/d. We shift the origins in space and time from (z = 0, t = 0) (where z = 0
corresponds to the height of the nozzle and t = 0 corresponds to the time instant
when the dye first appears from the nozzle) to (z0, t0) by applying the following
transformation between the new and old coordinates
znew = zold − z0, tnew = told − t0. (4.8a,b)
For simplicity, we omit the subscripts new and old hereafter. In Chapter 2, we
found α ≈ 0.068 and M0 ≈< M >= 0.55
(
Q02/d
)
. So, the non-dimensional
virtual origins in space and time are z˜0 = t˜0 ≈ −4.7. Except for the data in the
time interval, 2 ≤ t˜ ≤ 118 (plotted with dashed curves), the data corresponding
to the time interval, 118 ≤ t˜ ≤ 353 (plotted with thin solid curves), seem to have
a similar distribution. The experimental concentration distribution converges
rapidly, in time, towards an asymptotic profile in similarity space (y, η). We
83
4 Experimental results for the streamwise dispersion and mixing
t/(d2/Q0)
∫ ∞ 0
φ
F
,e
x
p
(z
,t
)
dz
/d
2 Data
Linear fit
0 100 200 300
0
0.01
0.02
0.03
0.04
(a)
η
yF,exp/
(
F
(
d/Q02
)1/3)
Time-averaged
2 ≤ t˜ ≤ 118
118 ≤ t˜ ≤ 353
0 0.5 1
0
1
2
3
(b)
Figure 4.6: (a) Evolution in time of the non-dimensional integrated concentration of
dye: the experimental data are plotted with pluses, a linear fit is plotted with a black
line. (b) Evolution in time of the distribution of the normalized ensemble-averaged
laterally-integrated experimental concentration shown in similarity form yF,exp in the
case of constant-flux releases (plotted with dashed curves against the similarity variable
η = z/
(
t2/3
(
Q02/d
)1/3) for the time interval 2 ≤ t˜ ≤ 118 and with thin solid curves
for the time interval 118 ≤ t˜ ≤ 353). The time-averaged data yF,exp, for 118 ≤ t˜ ≤ 353,
are plotted with a thick solid curve.
approximate this asymptotic distribution by the time-averaged distribution yF,exp
for 118 ≤ t˜ ≤ 353 (plotted with a thick solid curve in figure 4.6b). The rapid
convergence of the data in similarity space is very important because it means
that the similarity scalings derived from the model, φF (z, t) = t1/3yF (η) (with
η ∝ z/t2/3), are the appropriate scalings for this phenomenon. We can notice
in figure 4.6(b) that near η = 0 the data are incomplete. Small values of η ∝
z/t2/3 are equivalent to small values of z compared with t2/3, or large values of
t2/3 compared with z. The incomplete data near η = 0 are simply due to a
lack of spatial resolution near the source and a finite time of observation in the
experiments.
We present the experimental data yF,exp in figure 4.7 (the ensemble average is
plotted with pluses and the standard deviation, std, with dotted curves). We
compute the best least-squares fit using the theoretical formula (3.37), where Ka
and Kd are optimized under the constant-flux constraint (3.27b). The best fit
(plotted with a solid curve) is found for Ka = 1.65 and Kd = 0.09. We can see
that the model captures the main characteristics of the data. The concentration
increases from zero at the origin (where the first derivative is infinite) to a peak
value and then decreases smoothly at the front. The front of the curve agrees with
84
4.2 Experimental results
η
yF /
(
F
(
d/Q02
)1/3)
Average data yF,exp
std data
Best fit yF : Ka = 1.65, Kd = 0.09
0 0.2 0.4 0.6 0.8 1
0
0.5
1
1.5
2
2.5
3
3.5
Figure 4.7: Constant-flux case, in similarity form: plots of the ensemble average
(pluses) and standard deviation (std) (thin dotted curves) of the normalized exper-
imental dye concentration yF,exp (pluses) and best least-squares fit using yF from
(3.37) and with Ka = 1.65 and Kd = 0.09 (solid curve) against the similarity vari-
able η = z/
(
t2/3
(
Q02/d
)1/3).
the theoretical fit, and so, the dispersion processes appears to have been correctly
modelled. The rear of the experimental data appears slightly more linear than
the theoretical prediction. This mismatch is probably due to the zone of flow
establishment discussed above.
The ratio between the advection parameter and the dispersion parameter is
approximately Ka/Kd = 18.3. Using the advection parameter, we can compute
theoretically the location of the advective front (considering ‘top-hat’ velocity
profiles in the jet), ηa = 1.83, based on (3.40). We find that the position of the
centroid relative to ηa is, for the experimental data, µF,exp = 0.65 (computed
using (3.42)), which is close to the theoretical prediction µF = 0.62 (shown with
a cross in figure 3.2a and computed using (3.43) with Ka/Kd = 18.3). The
standard deviation of yF,exp is σF,exp = 0.29 (computed using (3.44)), which is
almost identical to the theoretical prediction σF = 0.30 (shown with a cross
in figure 3.2b and computed using (3.45) with Ka/Kd = 18.3). We can also
85
4 Experimental results for the streamwise dispersion and mixing
measure from the experimental data the portion of the dye which travels ahead
of the advective front βF,exp = 0.12 (computed using (3.46)), which is close to
the theoretical prediction βF = 0.10 (shown with a cross in figure 3.3a) based
on the ratio Ka/Kd = 18.3 and using (3.47). Thus, at each instant in time
a non-negligible proportion of the total volume of tracers having been released
travels ahead of the advective front ηa. Finally, we can also determine from the
experimental data the normalized distance between the average location of the
volume of tracers travelling ahead of the advective front and the location of the
advective front ηa, ξF,exp = 0.16 (computed using (3.48)). This value is slightly
larger than the theoretical prediction based on the ratio Ka/Kd = 18.3 and using
(3.49), ξF = 0.13 (shown with a cross in figure 3.3b). ξF is a measure of the
spread of the front of the distribution compared with the distance of the peak
from the origin.
All these agreements between the data yF,exp and the best least-squares fit yF
suggest that our model can predict the shape of the concentration distribution of a
finite-volume release of tracers in quasi-two-dimensional jets. We believe that the
constant-flux experiments are the most straightforward experiments performed in
this chapter. Therefore, the values of the advection and the dispersion parameters
Ka = 1.65 and Kd = 0.09, respectively, found in this case will be used in the next
cases as reference values. Furthermore, these results clearly reveal the importance
of dispersion processes in the transport of passive tracers by quasi-two-dimensio-
nal jets. As is clear in figure 4.7, the front of the distribution of the concentration
in the similarity space (y, η) is not sharp but smooth due to dispersion. Were
the transport of passive tracers by quasi-two-dimensional jets purely governed by
advective processes alone, the distribution of the concentration in similarity space
would drop much more rapidly at the front, as shown by the distributions of yF,a
in figure 3.1 (plotted with a thin solid curve and a thin dashed curve). It is also
important to note that more than 10% of the total volume of tracers released, at
any time, propagates ahead of the advective front.
We plot the normalized ensemble-averaged experimental results for the concen-
tration flux of dye yM,exp/F in figure 4.8 with pluses, while the standard deviation
of the data (std) is plotted with thin dotted curves. The experimental concentra-
tion flux of dye Mφ,exp is computed using the expression (3.54) with Ka = 1.65
(as found above for the best fit of yF,exp in the constant-flux case, see figure 4.7)
86
4.2 Experimental results
and the virtual origins described in (4.7a) and (4.7b). Then, according to (3.55),
the similarity form is yM,exp = Mφ,exp. We compute the theoretical prediction yM
(plotted with a solid curve) using the theoretical formula (3.55) with Ka = 1.65
and Kd = 0.09 (the reference values obtained in the constant-flux case for yF , see
figure 4.7). We also compute the best least-squares fit yM,fit using the theoret-
ical formula (3.55), where Ka,fit and Kd,fit are optimized. The best fit (plotted
with a dashed curve) is found for Ka,fit = 1.55 and Kd,fit = 0.07. (The values
of the advection and dispersion parameters for the best fit and the theoretical
prediction are actually very similar.) The theoretical prediction matches with the
data at the front, with the dispersion processes appearing to have been correctly
modelled, but near the origin the data drop towards zero instead of remaining
constant. The absence of a plateau near the origin in the experimental results
is presumably due to the time and distance of adjustment before the experimen-
tal data can match the theoretical prediction, which we mentioned previously as
being associated with the zone of flow establishment.
We find that the position of the centroid relative to ηa = 1.83 (computed using
(3.40) with Ka = 1.65) is, for the experimental data, µM,exp = 0.58 (computed
using (3.57)), which is somewhat larger than the theoretical prediction µM = 0.50
(shown with a cross in figure 3.5a and computed using (3.58) with Ka/Kd =
18.3). The standard deviation of yM,exp is σM,exp = 0.31 (computed using (3.59)),
which is identical to the theoretical prediction σM = 0.31 (shown with a cross
in figure 3.5b and computed using (3.60) with Ka/Kd = 18.3). We can also
measure from the experimental data the proportion of the dye flux being ahead
of the advective front βM,exp = 0.09 (computed using (3.61)), which is close to
the theoretical prediction βM = 0.06 (shown with a cross in figure 3.6a) based
on the ratio Ka/Kd = 18.3 and using (3.62). We can also determine from the
experimental data the normalized distance between the average location of the
volume of tracers travelling ahead of the advective front and the location of the
advective front (considering ‘top-hat’ velocity profiles in the jet) ηa, ξM,exp = 0.19
(computed using (3.63)). This value is somewhat larger than the theoretical
prediction based on the ratio Ka/Kd = 18.3 and using (3.64), ξM = 0.12 (shown
with a cross in figure 3.6b).
The study of the flux of dye in the constant-flux case also demonstrates the
ability of the model to predict both advective and diffusive processes. It is clear
87
4 Experimental results for the streamwise dispersion and mixing
η
yM/F
Average data yM,exp
std data
Theoretical prediction yM : Ka = 1.65, Kd = 0.09
Best fit yM,fit: Ka,fit = 1.55, Kd,fit = 0.07
0 0.5 1 1.5
0
0.5
1
1.5
2
2.5
3
3.5
Figure 4.8: Constant-flux case, in similarity form: plots of the ensemble average
(pluses) and standard deviation (std) (thin dotted curves) of normalized experimen-
tal dye flux yM,exp, theoretical prediction yM using (3.55) and with Ka = 1.65
and Kd = 0.09 (solid curve), and best least-squares fit yM,fit using (3.55) and
with Ka,fit = 1.55 and Kd,fit = 0.07 (dashed curve) against the similarity variable
η = z/
(
t2/3
(
Q02/d
)1/3).
from the observation of the front of the profile in figure 4.8 that quasi-two-dimen-
sional jets diffuse tracers in a qualitatively different manner from the ‘top-hat’
purely advective case yM,a presented in figure 3.4 (plotted with a thin solid curve
and a thin dashed curve). Moreover, we measure that approximately 10 % of the
total concentration flux of tracers is located ahead of the advective front.
4.2.3 Instantaneous finite-volume releases of clusters of vir-
tual particles
We now compare our effective advection–diffusion model with experiments con-
ducted using finite-volume releases of tracers in quasi-two-dimensional jets. The
initial boundary and integral conditions imposed in the finite-volume case (3.65a–
c) are more difficult to reproduce experimentally because they require an instan-
taneous release. An instantaneous release is not physically possible in laboratory
88
4.2 Experimental results
experiments (as we discussed in § 4.1.3), but it can be achieved using virtual
particles. So, we first investigate the case of finite volumes of virtual particles
released in the velocity field of quasi-two-dimensional jets, before analysing the
more difficult problem of finite-volume releases of dye (presented in § 4.2.4).
Figure 4.9(a) shows the dimensionless streamwise profile, at different times,
of the laterally-integrated normalized concentration φv,exp
(
z˜, t˜
)
/φmax(t˜) (where
φmax(t˜) is the maximum value of φv,exp(z˜, t˜) in time, and z˜ = z/d and t˜ =
t/ (d2/Q0) as defined in (4.1a) and (4.1b), respectively) of the ensemble average
of 256 virtual-particle clusters released instantaneously, as finite volumes, in the
experimental velocity field of quasi-two-dimensional jets with source volume flow
rates 33.2, 37.0 and 40.3 cm3 s−1 (see § 4.1.2). At each time, we bin the data into
100 evenly-spaced intervals extending from the origin to the maximum stream-
wise extent of the ensemble-averaged cluster. The thick dashed curve shows the
location of the front zf of the ensemble-averaged cluster in time, which reaches
the top boundary of the velocity field at approximately t˜ = 290, after the release
time. The location of the front follows the expected power law zf ∝ t2/3, derived
from (2.5b). As we can see, the ensemble-averaged cluster rapidly changes from an
initial rectangular shape to a smoother rounded profile as it is advected by the jet.
At early times t˜ ≤ 150, the dispersion of the particles appears to differ slightly
between the front and the rear of the ensemble-averaged cluster. The front is
sharper and drops more rapidly, while the rear has a longer tail. This is probably
due to the fact that at the beginning most particles are advected quickly by the
core of the jet, while the rest are trapped in the lateral eddies where they move
more slowly (the time-averaged streamwise speed of an eddy is approximately
25% of the maximum speed of the core, as measured in § 2.5). However, at later
times the cluster seems to spread more symmetrically between the front and the
rear. We believe this is due to the continuous exchange of material between the
core and the eddies.
We apply the similarity transformation (3.66) to the ensemble-averaged exper-
imental concentration φv,exp to obtain the similarity form yv,exp, normalized by
the total volume of virtual particles Bv,exp = 2,624,256. We use the space vir-
tual origin z0 defined in (4.7a). The time virtual origin cannot be the same as
defined in the simple equation (4.7b) because the jet velocity is not constant be-
tween the jet virtual origin z0 and the location of release of the virtual particles
89
4 Experimental results for the streamwise dispersion and mixing
0 50 100 150 200 250 300 350
t/(d2/Q0)
50
60
70
80
90
100
110
120
z/
d
φv,exp(z˜, t˜)/φmax(t˜)
Front
(a)
η
yv,exp/
(
B
(
d/Q02
)1/3)
Time-averaged
48 ≤ t˜ ≤ 205
205 ≤ t˜ ≤ 401
0 0.5 1 1.5 2 2.5
0
1
2
3
4(b)
Figure 4.9: (a) Streamwise distribution of the normalized laterally-integrated con-
centration of virtual particles φv,exp(z, t)/φmax(t) (solid curves) at different non-
dimensional times. The results have been averaged for 256 releases of identical clusters
in the velocity fields of quasi-two-dimensional turbulent jets of source volume flow rates
33.2, 37.0 and 40.3 cm3 s−1 . The location of the front of the ensemble-averaged cluster
is plotted versus time with a thick dashed curve. (b) Evolution in time of the dis-
tribution in similarity form of the normalized ensemble-averaged laterally-integrated
experimental concentration of virtual particles yv,exp (dashed curves for the time in-
terval 48 ≤ t˜ ≤ 205 and with thin solid curves for the time intervals 205 ≤ t˜ ≤ 401)
against the similarity variable η = z/
(
t2/3
(
Q02/d
)1/3). The time-averaged data yv,exp,
for 205 ≤ t˜ ≤ 401, are plotted with a thick solid curve.
(i.e. 44.4 ≤ z/d ≤ 48.3). We determine the time virtual origin so that the loca-
tion of the front in time z˜f (t˜) (plotted with a thick dashed curve in figure 4.9a)
best fits (using a least-squares fit) a straight line in a log–log plot. We show in
figure 4.9(b) the evolution in time of the distribution in similarity space of the nor-
malized experimental data yv,exp, plotted for nine successive time periods in the
range 48 ≤ t˜ ≤ 401. We can see that yv,exp seems to converge towards an asymp-
totic distribution after 205 ≤ t˜ (the data for 48 ≤ t˜ ≤ 205 are plotted with dashed
curves, while the data for 205 ≤ t˜ ≤ 401 are plotted with thin solid curves). We
approximate the asymptotic distribution by the time-averaged distribution yv,exp
for 205 ≤ t˜ ≤ 401 (plotted with a thick solid curve in figure 4.9b). Similarly to the
constant-flux case, the convergence of these finite-volume data in similarity space
implies that the similarity scalings derived from the model, φδ(z, t) = t−2/3yδ(η)
(with η ∝ z/t2/3), are the appropriate scalings for this phenomenon.
In figure 4.10, we compare the time-averaged ensemble-averaged virtual par-
ticle data yv,exp (the ensemble average is plotted with crosses and the standard
90
4.2 Experimental results
deviation, std, is plotted with dotted curves) with the theoretical prediction of the
fundamental solution yδ (plotted with a solid curve), which assumes an instan-
taneous release. We compute the theoretical prediction yδ using equation (3.73)
with Ka = 1.65 and Kd = 0.09 (the reference values obtained in the constant-
flux case for yF , see figure 4.7). We also compute the best least-squares fit yδ,fit
(plotted with a dashed curve in figure 4.10) using (3.73), where Ka,fit and Kd,fit
are optimized under the finite-volume constraint (3.68b). The best least-squares
fit between yv,exp and yδ,fit is obtained for Ka,fit = 1.62 and Kd,fit = 0.09. Once
again, these best-fit values are quite similar to the reference values.
We can see that the model captures the main characteristics of the data. The
concentration increases from zero at the origin (where the first and second deriva-
tives also vanish) to a peak value and then decreases at the front, following the
theoretical prediction yδ. The location of the peak of yv,exp (which is also the
location of the advective front) is at ηa,exp = 1.83. Using the advection parameter
Ka = 1.65, we can compute theoretically a very similar value ηa = 1.83, based
on (3.40). We find that the position of the centroid relative to ηa is, for the ex-
perimental data, µB,exp = 0.99 (computed using (3.76)), which is slightly smaller
than the theoretical prediction µB = 1.03 (shown with a cross in figure 3.8a and
computed using (3.77) with Ka/Kd = 18.3). Thus, the centroid is very close
to the location of the concentration peak. The standard deviation of yv,exp is
σB,exp = 0.17 (computed using (3.78)), which is close to the theoretical prediction
σB = 0.19 (shown with a cross in figure 3.8b and computed using (3.79) with
Ka/Kd = 18.3). We can also measure from the experimental data the portion of
the virtual particles which travels ahead of the advective front βB,exp = 0.49 (com-
puted using (3.80)), which is very close to the theoretical prediction βB = 0.54
(shown with a cross in figure 3.9a) based on the ratio Ka/Kd = 18.3 and using
(3.81). (A value βB of 0.5 means that the virtual particles are symmetrically
distributed with respect to the concentration peak.) Finally, we can also deter-
mine from the experimental data the normalized distance between the average
location of the volume of tracers travelling ahead of the advective front and the
location of the advective front (considering ‘top-hat’ velocity profiles in the jet)
ηa, ξB,exp = 0.13 (computed using (3.82)). This value is somewhat smaller than
the theoretical prediction based on the ratio Ka/Kd = 18.3 and using (3.83),
ξB = 0.17 (shown with a cross in figure 3.9b). ξB is a measure of the spread of
91
4 Experimental results for the streamwise dispersion and mixing
the distribution compared with the distance of the peak from the origin.
η
y/
(
B
(
d/Q02
)1/3)
Average data yv,exp
std data
Theoretical prediction yδ: Ka = 1.65, Kd = 0.09
Best fit yδ,fit: Ka,fit = 1.62, Kd,fit = 0.09
0 0.5 1 1.5
0
0.5
1
1.5
2
2.5
3
3.5
Figure 4.10: Finite-volume case, instantaneous release, in similarity form: plots of
the variation with similarity variable η = z/
(
t2/3
(
Q02/d
)1/3) of the ensemble av-
erage (pluses) and standard deviation (std) (thin dotted curves) of the normalized
time-averaged experimental concentration of virtual particles yv,exp (pluses), theoreti-
cal prediction yδ defined by (3.73) with Ka = 1.65 and Kd = 0.09 (solid curve), and
best least-squares fit using yδ,fit defined by (3.73) with Ka,fit = 1.62 and Kd,fit = 0.09
(dashed curve).
All these agreements between the data yv,exp and the theoretical prediction yδ,
and between the advection and dispersion parameters of the constant-flux case
and the finite-volume case, suggest that our model can predict the shape of the
concentration distribution of an instantaneous finite-volume release of tracers in
quasi-two-dimensional jets. Furthermore, it clearly reveals the importance of dis-
persion processes in the transport of passive tracers by quasi-two-dimensional jets.
As is clear in figure 4.9(b), the distribution of the concentration in the similarity
space (y, η) converges in time towards a distribution with a finite width. Were
the transport of passive tracers by quasi-two-dimensional jets purely governed
by advective processes alone, the distribution of the concentration in similarity
space would rather shrink towards a distribution of negligible width (similar to
a Dirac delta function), even with a non-instantaneous release of tracers. It is
92
4.2 Experimental results
also important to note that approximately half of the total volume of tracers in
figure 4.10 travels ahead of the advective front, at a normalized averaged distance
ξB ≈ 0.17 (defined in (3.83) with Ka/Kd = 18.3).
4.2.4 Finite-volume releases of dye
We also present in figures 4.5(d–f ) experimental results and theoretical predictions
of finite-volume releases of dye in quasi-two-dimensional steady turbulent jets.
The spatial distribution of the concentration C(x, z, t) is plotted using a colour
scale (see colour scale at the top of figures 4.5d–f ) at different non-dimensional
times, 56 ≤ t˜ ≤ 533, to show the evolution of the patch of dye as it is ad-
vected, mixed and dispersed by the jet. In figure 4.5(d) we plot the ensemble
average of the 26 experiments, which were conducted at different jet Reynolds
number, 2170 ≤ Rej ≤ 4870 (see § 4.1.3). We also plot the average dye edges
(half-spreading angle, < θdye >= 12.4◦) with thick white lines and the average
boundaries of the ‘core’ (half-spreading angle, 7◦ starting from z = 20d) with thin
white lines. Similarly to the constant-flux results presented in figures 4.5(a–c),
we can observe that the interaction between the core and the eddies, as described
in § 2.5, results in large streamwise dispersion. As we explained earlier, we model
this streamwise dispersion using an enhanced turbulent eddy diffusive coefficient
Dzz ∝ bwm.
We present the laterally-integrated experimental concentration φB,exp(z, t) in
figure 4.5(e) in normalized and re-distributed form using
C(x, z, t) =



φB,exp(z, t)
2l(z) , −l(z) ≤ x ≤ l(z)
0, otherwise
, (4.9)
where l(z) = tan (< θdye >) (z−z0), as defined in (4.5), is the local lateral distance
between the average dye edges (plotted with thick white lines in figure 4.5d), and
z0 is the space virtual origin defined in (4.7a). Alongside in figure 4.5(f ), we show
the equivalent theoretical prediction φT0(z, t) computed from equation (3.88) and
based on the assumption of a finite volume being released at a constant-flux during
a finite period of time T0 = 183 (d2/Q0) (we discuss this value in more detail
below). To compute φT0 , we use Ka = 1.65 and Kd = 0.09 for the advection and
dispersion parameters, respectively (the reference values obtained in the constant-
93
4 Experimental results for the streamwise dispersion and mixing
flux case for yF , see figure 4.7). Before plotting the theoretical prediction φT0 in
figure 4.5(f ), we normalize it (similarly to φB,exp(z, t)) with the local distance 2l(z)
between the average dye edges, and finally re-distribute it uniformly, assuming a
top-hat spatially-averaged profile, within these boundaries, i.e.
C(x, z, t) =



φT0(z, t)
2l(z) , −l(z) ≤ x ≤ l(z)
0, otherwise
, (4.10)
where l(z) is defined in (4.5).
Although the comparison between the experimental data in figure 4.5(e) and
the theoretical prediction in figure 4.5(f ) is not perfect at early times and near
the origin (the theoretical concentration seems to travel slightly slower than the
experimental concentration for t˜ ≤ 222), it improves at later times as the jet
advects and diffuses the dye. As we mentioned above, this mismatch is probably
due to the zone of flow establishment of the jet. There is a necessary time and
distance of adjustment before the experimental data can match the theoretical
prediction, because the theoretical prediction is based around the assumption that
the jet characteristic properties are given by the similarity power laws (2.5a,b).
In these experiments, we naturally are not able to release finite volumes of
dye instantaneously. Aspects of the experimental dye release are revealed in
figure 4.11(a), where we show the evolution in time of the integral of the dye con-
centration over the whole domain
∫∞
0 φB,exp(z, t) dz (plotted with pluses). These
data represent the total volume of dye ‘seen’ by the imaging analysis in the win-
dow frame −40 ≤ x/d ≤ 40 and 0 ≤ z/d ≤ 100. The dashed line indicates
the time instant t˜90 = 183, when approximately 90 % of the total volume of dye
has entered the tank. We can see that the total volume of dye increases almost
steadily for t˜ ≤ t˜90. Then, the total volume of dye reaches a maximum at t˜ ≈ 290
before decreasing smoothly as the dye is transported outside the window frame.
These data clearly show that the release of dye occurs over a finite period of time
and not instantaneously.
The effect of the spreading in time of the release of dye can also be seen in the
evolution in time of the concentration distribution in the jets. In figure 4.11(b),
we show the non-dimensional experimental concentration in similarity form yB,exp
(computed from φB,exp using (3.66) at each instant in time). We normalize yB,exp
94
4.2 Experimental results
t/(d2/Q0)
∫ ∞ 0
φ
(z
,t
)
dz
Data
Theory based on φδ
Theory based on φT0
t˜90
0 200 400 600 800
0
0.02
0.04
0.06
0.08
(a)
800
η
yB,exp/
(
B
(
d/Q02
)1/3)
0 0.5 1
0
1
2
3
(b)
Figure 4.11: (a) Evolution in time of the integral of the dye concentration over the
whole domain for: the experimental data φB,exp (pluses); the theoretical prediction φδ
(solid line), defined by (3.73); and the theoretical prediction φT0 (dotted line), defined
by (3.88). The dashed line indicates the time instant t˜90 = 183 when approximately
90 % of the total integrated concentration of dye has entered the tank. (b) Plots of
the variation with similarity variable η = z/
(
t2/3
(
Q02/d
))
of the evolution in time
of the normalized ensemble-averaged laterally-integrated experimental concentration
plotted in similarity form, yB,exp (computed from φB,exp using (3.66)), in the case of
finite-volume releases of dye. The data are plotted at 12 different time instants for
0 ≤ t˜ ≤ 979, with time increasing as the amplitude of the data increases.
with the total injected volume B and plot it at 12 different instants in time for
0 ≤ t˜ ≤ 979, with time increasing as the amplitude of the data increases. The
space and time virtual origins described in (4.7a) and (4.7b) are used to compute
yB,exp. Ideally, if the dye were released instantaneously at the origin (as described
in the integral and initial boundary conditions (3.65a–c)) all the curves should be
identical and collapse on a single profile. Instead, we observe a gradual increase
of the area under the curves. The data do not appear to have yet reached an
asymptotic distribution. It can also be noticed that the curves at late times (for
290 ≤ t˜) are not plotted over the whole range 0 ≤ η ≤ 3.5, but stop at some
values η < 3.5. These curves are incomplete because for 290 ≤ t˜, the front of the
dye (located at the height zf ) has already moved outside the image frame, i.e.
zf/d > 100, and thus we cannot visualize the full distribution of the dye in space.
It is clear from both figures 4.11(a) and 4.11(b) that the release of the dye is not
instantaneous and that the data have not yet reached an asymptotic distribution
in similarity space. Thus, we cannot use the theoretical prediction yδ defined in
(3.73) to model these experiments (as we did in the case of finite-volume releases
95
4 Experimental results for the streamwise dispersion and mixing
of virtual particles presented above) because the fundamental solution yδ assumes
an instantaneous release of the finite volume of tracers (see the integral and initial
boundary conditions (3.65a–c)). Therefore, we compare the experimental data
φB,exp(z, t) with the general solution φg(z, t), described in (3.85) and based on the
convolution of the fundamental solution φδ with a source function f(t) = φg(0, t).
The source function can model the more general and realistic case of a time-
dependent release.
To compute the general solution φg(z, t), we need to define the source function
f(t), which represents the rate at which the overall integrated volume of tracers
changes with time. In figure 4.11(a), we observe that the total integrated con-
centration of dye
∫∞
0 φB,exp(z, t)dz increases almost linearly with time for t˜ ≤ t˜90.
Hence, we choose to model the source function as simply a non-zero constant for
0 ≤ t˜ ≤ t˜90 and zero for t˜90 ≤ t˜,
ft˜90(t˜) =
H(t˜)−H(t˜− t˜90)
t˜90
, (4.11)
where H is the Heaviside function. Using such a rectangular source function, the
general solution φg(z, t) corresponds to the particular solution φT0 (with T0 = t90),
described in (3.88). We plot the resulting theoretical integrated concentration∫∞
0 φT0(z, t) dz with a dotted curve in figure 4.11(a). We can see that the match
with the data (plotted with pluses) is, at least until the dye is advected beyond
the spatial range of the camera (for t˜ ≤ 290), better than for the model assuming
an instantaneous release φδ (plotted with a solid line).
We compute the theoretical prediction φT0 , based on the source function ft˜90
with T0 = t90 = 183(d2/Q0), using the virtual origins described in (4.7a,b). We
compare the distribution of the experimental data φB,exp (plotted with pluses)
and the theoretical prediction φT0 (plotted with solid curves) in figure 4.12 at
nine different times for 0 ≤ t˜ ≤ 418. We compute φT0 using the advection and
dispersion parametersKa = 1.65 andKd = 0.09, respectively (the reference values
obtained in the constant-flux case for yF , see figure 4.7). We also show the best
least-squares fit φT0,fit (plotted with dashed curves in figure 4.12), computed using
the theoretical formula (3.88) and the source function ft˜90 (see equation (4.11))
with T0 = t90 = 183(d2/Q0). The advection and dispersion parameters Ka,fit
and Kd,fit, respectively, are optimized under the finite-volume constraint (3.65c).
96
4.2 Experimental results
The best least-squares fit between φB,exp and φT0,fit is obtained for Ka,fit = 1.75
and Kd,fit = 0.09, still quite close to the reference values. Overall, we observe
a reasonable agreement between φT0 and φB,exp. At early time, for t˜ ≤ 100, the
match between the data and the model is not perfect because the experimental
concentration profile adjusts partially due to the lack of self-similarity in the jet
(this issue is related to the zone of flow establishment discussed previously). Then,
both the advection (location of the peak in time) and the dispersion (width of
the curve) seem to agree. There is a consistent mismatch at the rear where the
data seem to be more spread out. This is probably due to some residue of dye in
the tube still being injected in the jet at late time, and apparently stretching and
diffusing the experimental dye concentration.
According to equation (3.89), the solution φT0(z, t) converges in time towards
φδ(z, t). Hence, we also expect the data φB,exp(z, t) to converge in time towards
φδ(z, t). We demonstrate this convergence by plotting in figure 4.13 the similarity
form of the theoretical prediction φT0 at t˜ = 150 (plotted with a thin solid curve),
t˜ = 300 (plotted with a dotted curve) and t˜ = 450 (plotted with a dashed curve).
We also show the asymptotic solution yδ, defined by (3.73) and computed using
the (reference) advection and dispersion parameters Ka = 1.65 and Kd = 0.09.
We can measure the absolute deviation, based on equation (3.91), between yT0
at t˜ = 300 (when the integrated concentration of dye is approximately maxi-
mum, see figure 4.11a) and the asymptotic solution yδ. We find dev = 0.85,
computed for t˜/T˜0 = 300/183 ≈ 1.64 and using (3.91). If we consider that con-
vergence is ‘achieved’ if dev ≤ 0.1, then we find that our experimental data would
be expected to achieve convergence for t˜/T˜0 ≥ 13.6, or at t˜ ≥ 2488. We can
estimate that the distance at which we should observe the concentration distri-
bution of the finite volumes of dye converge towards an asymptotic distribution
is z ≥ ηa(13.6T˜0)2/3 d ≈ 2 m (based on the location of the concentration peak at
convergence). Finally, we can predict the key characteristics of yδ, the asymptotic
distribution of yB,exp (the similarity form of φB,exp computed using (3.66)), which
are actually identical to the characteristics of the theoretical prediction found
for the virtual particles because the advection and dispersion parameters are the
same. So, we can expect that the maximum concentration of the asymptotic
distribution of yB,exp is located at ηa = 1.83, based on (3.40) with Ka = 1.65.
The position of the centroid relative to ηa is µB = 1.03 (shown with a cross in
97
4 Experimental results for the streamwise dispersion and mixing
z/
d
1000φ/d
t/(d2/Q0) = 0
Data
Theoretical prediction
Best fit
0 0.5 1
0
20
40
60
80
100
z/
d
1000φ/d
t/(d2/Q0) = 51
0 0.5 1
0
20
40
60
80
100
z/
d
1000φ/d
t/(d2/Q0) = 103
0 0.5 1
0
20
40
60
80
100
z/
d
1000φ/d
t/(d2/Q0) = 155
0 0.5 1
0
20
40
60
80
100
z/
d
1000φ/d
t/(d2/Q0) = 207
0 0.5 1
0
20
40
60
80
100
z/
d
1000φ/d
t/(d2/Q0) = 259
0 0.5 1
0
20
40
60
80
100
z/
d
1000φ/d
t/(d2/Q0) = 311
0 0.5 1
0
20
40
60
80
100
z/
d
1000φ/d
t/(d2/Q0) = 363
0 0.5 1
0
20
40
60
80
100
z/
d
1000φ/d
t/(d2/Q0) = 418
0 0.5 1
0
20
40
60
80
100
Figure 4.12: Plots at various times of the streamwise distribution of the laterally-
integrated concentration of dye against the non-dimensional distance z/d in the case of
finite-volume releases for: ensemble-averaged experimental data φB,exp (pluses); theo-
retical prediction φT0 (solid curves), based on equation (3.88) using the reference advec-
tion and dispersion parameters Ka = 1.65 and Kd = 0.09, respectively, and the source
function ft˜90(t) as defined in (4.11); and best least-squares fit φT0,fit (dashed curves),
based on equation (3.88) using the advection and dispersion parameters Ka,fit = 1.75
and Kd,fit = 0.09, respectively, and the source function ft˜90(t) as defined in (4.11).
figure 3.8a and computed using (3.77) with Ka/Kd = 18.3). The theoretically
predicted standard deviation is σB = 0.19 (shown with a cross in figure 3.8b and
computed using (3.79) with Ka/Kd = 18.3). The portion of the virtual particles
which travels ahead of the advective front is βB = 0.54 (shown with a cross in
figure 3.9a), based on the ratio Ka/Kd = 18.3 and using (3.81). The average lo-
cation of the volume of tracers travelling ahead of the advective front is ξB = 0.17
(shown with a cross in figure 3.9b), based on the ratio Ka/Kd = 18.3 and using
(3.83).
98
4.3 Statistical significance of the experimental results
η
y/
(
B
(
d/Q02
)1/3)
Theoretical prediction yT0 : t˜ = 150
Theory prediction yT0 : t˜ = 300
Theory prediction yT0 : t˜ = 450
Theory prediction yT0 : t˜ → +∞
0 0.5 1 1.5
0
0.5
1
1.5
2
2.5
3
3.5
Figure 4.13: Finite-volume case, in similarity form: plots of the variation with simi-
larity variable η = z/
(
t2/3
(
Q02/d
))
of the non-dimensionalized theoretical prediction
yT0 , based on equation (3.88), computed at t˜ = 150 (thin solid curve), t˜ = 300 (dot-
ted curve), and t˜ = 450 (dashed curve), using the reference advection and dispersion
parameters Ka = 1.65 and Kd = 0.09, respectively, and the source function ft˜90(t) as
defined in (4.11). The asymptotic distribution of yT0 (thick solid curve) is equal to yδ
and can be computed using (3.73) with Ka = 1.65 and Kd = 0.09.
4.3 Statistical significance of the experimental re-
sults
We investigate the statistical significance of the experimental results, presented
in § 4.2 above, for the constant-flux releases of dye and the instantaneous finite-
volume releases of virtual particles in quasi-two-dimensional jets. We compute
the probability density function (p.d.f.) of all the measurements of the laterally-
integrated concentration y (in similarity form) at different values of the similarity
variable η ∝ z/t2/3.
We do not compute the p.d.f. of the experimental results found in the case of
finite-time finite-volume releases of dye. (We describe the experiments in § 4.1.3
and plot the ensemble-averaged concentration in figure 4.12 with pluses). As we
discuss in § 4.2.4, the distribution of the concentration has not yet reached an
99
4 Experimental results for the streamwise dispersion and mixing
asymptotic distribution in similarity form. Therefore, the study of the statistical
significance is not meaningful while the concentration profile is in a transient
time-dependent regime.
In the problem of river pollution, predicting and assessing the risk of encoun-
tering harmful concentration levels is crucial. We show in this section how we can
predict and assess this risk for the constant-flux and instantaneous finite-volume
cases. Moreover, we discuss how this risk varies in time and space, depending on
the p.d.f. of the measurements of the concentration.
4.3.1 Constant-flux release of dye
We compute numerically the p.d.f. fF of the laterally-integrated experimental
concentration yF,exp (in similarity form) for the Ne = 19 constant-flux release
experiments (presented in § 4.1.1) such that, for all 1 ≤ l ≤ Nη and 1 ≤ m ≤ Ny
(with l and m two integers)
fF (ym, ηl) =
Ne,Nz ,Nt∑
i,j,k
∆ei,zj ,tk(ym, ηl)
Ny∑
m
Ne,Nz ,Nt∑
i,j,k
∆ei,zj ,tk(ym, ηl) δy
, (4.12)
where δy = y2 − y1 is the concentration step and, for all 1 ≤ i ≤ Ne, 1 ≤ j ≤ Nz
and 1 ≤ k ≤ Nt (with i, j and k three integers),
∆ei,zj ,tk(ym, ηl) =



1 if ym−1 ≤
yF,exp(ei, zj , tk)
F
(
d/Q02(ei)
)1/3 < ym
and ηl−1 ≤ η =
zj
tk2/3
(
Q02(ei)/d
)1/3 < ηl,
0 otherwise
,
(4.13)
where zj (the discretized streamwise coordinate) are linearly distributed from
approximately 0 to 100 d (depending on the experiment and with Nz = 756), tk
(the kth frame of the experiment) are linearly distributed from approximately
0 to tNt (with Nt of the order of 300, depending on the experiment), ηl (the
discretized similarity variable) are linearly distributed from 0 to 3.5 (with Nη =
200, the number of bins) and ym (the discretized laterally-integrated concentration
100
4.3 Statistical significance of the experimental results
0 0.2 0.4 0.6 0.8 1.0
yF /
(
F
(
d/Q02
)1/3)
0
5
10
15
20
25
30
35
f F
η = 0.45
η = 0.89
η = 1.33
η = 1.76
η = 2.20
η = 2.64
η = 3.08
(a)
0 0.5 1.0 1.5 2.0 2.5
η
0
0.2
0.4
0.6
0.8
1.0
P
∗ F
t˜ = 1
t˜ = 2
t˜ = 3
t˜ = 4
t˜ = 5
t˜ = 6
t˜ = 7
t˜ = 8
(b)
Figure 4.14: (a) Probability density function of the experimental dye concentration
yF,exp/
(
F
(
d/Q02
)1/3), non-dimensionalized and in similarity form, at different values
of the similarity variable η in the case of constant-flux releases. We describe the exper-
iments in § 4.1.1 and plot the ensemble-averaged concentration yF,exp in figure 4.7 with
pluses. (b) Distribution against the similarity variable η at different non-dimensional
times (plotted with different colours) of the probability that the concentration of trac-
ers φF,exp is greater than a critical value φ∗ in the case of a constant-flux release in a
quasi-two-dimensional jet.
in similarity form) are linearly distributed from 0 to 1.07 (with Ny = 200, the
number of bins).
In figure 4.14(a), we show the p.d.f. fF , computed in (4.12), at seven different
values of the similarity variable η for 0.45 ≤ η ≤ 3.08 (plotted with different
colours).The distribution of the ensemble-averaged concentration yF,exp is plotted
in figure 4.7 with pluses. As we can see in figure 4.14(a), the p.d.f. fF decreases
with increasing concentration yF,exp. In figure 4.7, the maximum of yF,exp is found
at η = 1.17, which corresponds, in figure 4.14(a), to the rightmost and lowest
profile of fF (see light green curve at η = 1.33). The amplitude of fF is large for
either large values of η or small values: η > 2.20 and η < 0.45. In fact, we find
that the standard deviation of the p.d.f. of yF,exp grows approximately linearly
with its average value, although with a hysteresis between the values before the
concentration peak (i.e. η < 1.17) and the values after the concentration peak
(i.e. η > 1.17).
In the case of a constant-flux release of pollutants in a quasi-two-dimensional
turbulent jet, the probability P ∗F to find concentrations of pollutants larger than a
critical concentration level φ∗ (laterally-integrated concentration) at a given value
101
4 Experimental results for the streamwise dispersion and mixing
η is for t > 0
P ∗F (η, t˜) =
∫ ∞
t˜−1/3φ˜∗
fF (y˜F,exp, η) dy˜F,exp, (4.14)
where tildes denote non-dimensional values (see (4.1a,b)), and where we use equa-
tion (3.25), φ(z, t) = t1/3y(η). It is interesting to note that the distribution of
P ∗F increases in time, starting from 0 at t = 0. The increase in time of P ∗F is due
to the constant flux of tracer concentration at the source of the jet and to the
decrease (like z−1/2) of the velocity of the jet with distance. Thus, the laterally-
integrated tracer concentration tends to increase at a fixed value of η ∝ z/t2/3 as
time increases (i.e. for z increasing).
As an example, we have plotted P ∗F in figure 4.14(b) against η at eight different
non-dimensional times 1 ≤ t˜ ≤ 8 (plotted with different colours), for the critical
non-dimensional concentration φ˜∗ = 1. We can clearly see that the probability P ∗F
increases rapidly in time. As time increases, the maximum value of P ∗F appears to
move to the left, towards η = 0, and is found in the range 0.9 ≤ η ≤ 1.3 (the peak
of yF,exp, in figure 4.14a, is found at η = 1.17). We find that, in this example,
the peak of the probability P ∗F first becomes greater than 0.05 (i.e. statistically
significant) from t˜ ≥ 1.0 at the location η = 1.24, corresponding to z˜ = 1.2.
The peak of the probability P ∗F becomes greater than 0.95 from t˜ ≥ 12.0 at the
location η = 0.95, corresponding to z˜ = 5.0.
4.3.2 Instantaneous finite-volume release of virtual parti-
cles
Similarly to the constant-flux case presented above, we compute numerically the
p.d.f. fv of the laterally-integrated numerical concentration yv,exp (in similarity
form), for the Ne = 256 clusters, representing 2,624,256 virtual particles, released
instantaneously in the velocity field of quasi-two-dimensional jets (see § 4.1.2).
The p.d.f. fv is, for all 1 ≤ l ≤ Nη and 1 ≤ m ≤ Ny (with l and m two integers),
fv(ym, ηl) =
Ne,Nz ,Nt∑
i,j,k
∆ei,zj ,tk(ym, ηl)
Ny∑
m
Ne,Nz ,Nt∑
i,j,k
∆ei,zj ,tk(ym, ηl) δy
, (4.15)
102
4.3 Statistical significance of the experimental results
where δy = y2 − y1 is the concentration step and, for all 1 ≤ i ≤ Ne, 1 ≤ j ≤ Nz
and 1 ≤ k ≤ Nt (with i, j and k three integers),
∆ei,zj ,tk(ym, ηl) =



1 if ym−1 ≤
yv,exp(ei, zj , tk)
B
(
d/Q02(ei)
)1/3 < ym
and ηl−1 ≤ η =
zj
tk2/3
(
Q02(ei)/d
)1/3 < ηl,
0 otherwise
,
(4.16)
where zj (the discretized streamwise coordinate) are linearly distributed from
approximately 47 d to 128 d (depending on the experiment and with Nz = 1022),
tk (the kth frame of the experiment) are linearly distributed from approximately
0 to tNt (with Nt of the order of 300, depending on the experiment), ηl (the
discretized similarity variable) are linearly distributed from 0 to 5 (with Nη = 200,
the number of bins) and ym (the discretized laterally-integrated concentration in
similarity form) are linearly distributed from 0 to 5 (with Ny = 400, the number
of bins).
We present in figure 4.15(a) the p.d.f. fv, computed in (4.15), at eight different
values of the similarity variable η for 1.52 ≤ η ≤ 3.50 (plotted with different
colours). The distribution of the ensemble-averaged concentration yv,exp is plotted
in figure 4.10 with pluses. As we can see in figure 4.15(a), the p.d.f. fv decreases
even more rapidly than fF (shown in figure 4.14a) with increasing concentration
yv,exp. In figure 4.10, the maximum of yv,exp is found at η = 1.83, which is close
to the rightmost and lowest profile of fv in figure 4.15(a) (see curve at η = 2.07).
Similarly to fF , the amplitude of the fv is the largest for either large values of η
or small values: η > 2.4 and η < 1.8. Moreover, we also find that the standard
deviation of the p.d.f. of yv,exp grows approximately linearly with its average
value, although with more scatter than for the p.d.f. of yF,exp and with a stronger
hysteresis between the values before the concentration peak (i.e. η < 1.83) and
the values after the concentration peak (i.e. η > 1.83).
In the case of an instantaneous finite-volume release of pollutants in a quasi-two-
dimensional turbulent jet, the probability P ∗δ to find concentrations of pollutants
larger than a critical concentration level φ∗ at a given value η is for t > 0
P ∗δ (η, t˜) =
∫ ∞
t˜2/3φ˜∗
fv(y˜v,exp, η) dy˜v,exp, (4.17)
103
4 Experimental results for the streamwise dispersion and mixing
0 0.1 0.2 0.3 0.4 0.5
yv,exp/
(
B/
(
d/Q02
)1/3)
0
10
20
30
40
50
f v
η = 1.52
η = 1.80
η = 2.07
η = 2.37
η = 2.65
η = 2.92
η = 3.22
η = 3.50
(a)
0 0.5 1.0 1.5 2.0 2.5
η
0
0.2
0.4
0.6
0.8
1.0
P
∗ δ
t˜ = 1
t˜ = 2
t˜ = 3
t˜ = 4
t˜ = 5
t˜ = 6
t˜ = 7
t˜ = 8
(b)
Figure 4.15: (a) Probability density function of the concentration of virtual parti-
cles yv,exp/
(
B
(
d/Q02
)1/3), non-dimensionalized and in similarity form, at different
values of the similarity variable η in the case of instantaneous finite-volume releases.
We describe the experiments in § 4.1.2 and plot the ensemble-averaged concentration
yv,exp in figure 4.10 with pluses. (b) Distribution against the similarity variable η at
different non-dimensional times (plotted with different colours) of the probability that
the concentration of tracers φv,exp is greater than a critical value φ∗ in the case of an
instantaneous finite-volume release in a quasi-two-dimensional jet.
where we use equation (3.66), φ(z, t) = t−2/3y(η). It is interesting to note that,
contrary to P ∗F , the probability P ∗δ decreases in time. The probability P ∗F de-
creases in time because finite volumes of tracers become more dilute, due to the
streamwise dispersion, as they are transported by the jet.
As an example, we have plotted P ∗δ in figure 4.15(b) against η at eight different
non-dimensional times 1 ≤ t˜ ≤ 8 (plotted with different colours), for the critical
non-dimensional concentration φ˜∗ = 1. We can clearly see that the probability P ∗δ
decreases rapidly in time. The maximum value of P ∗δ is located at approximately
η = 0.7, a secondary, much smaller, local maximum is found in the range 1.1 ≤
η ≤ 1.5. We find that, in this example, the peak of the probability P ∗δ first
becomes less than 0.95 from t˜ ≥ 0.3 at the location η = 0.82, corresponding to
z˜ = 0.4. The peak of the probability P ∗δ becomes less than 0.05 (i.e. statistically
insignificant) from t˜ ≥ 9.1 at the location η = 0.77, corresponding to z˜ = 3.4.
4.3.3 Discussion
Owing to the large number of experiments conducted in the cases of constant-flux
releases of dye and instantaneous releases of finite volumes of virtual particles,
104
4.3 Statistical significance of the experimental results
we have been able to compute the statistical significance of the measurements of
the laterally-integrated concentration, in similarity form. We find that, in both
cases, the p.d.f. of the concentration tends to decrease and spread rapidly with
increasing concentration. This means that near the location of the concentration
peak, the difference between the concentration predicted by the model and the
experimental (or real) concentration is likely to be much larger than at the tail
or front of the distribution where the concentrations are smaller. Therefore, the
model is less accurate in the prediction of the values of the largest concentrations
than in the prediction of the values of the lowest concentrations.
From the experimental results of the p.d.f. in the constant-flux and instanta-
neous finite-volume cases, we can determine the probability of having a certain
range of tracer concentrations at a certain location in time and space. We discuss
the problem, relevant to pollution control, of how to calculate the probability to
find concentration levels greater than a critical value in a quasi-two-dimensional
jet. We find that in the constant-flux case, the probability increases rapidly in
time. On the other hand, the probability decreases for a finite-volume release. In
both cases, the location (in terms of the similarity variable η) of the maximum of
the probability seems to remain constant in time.
It is important to note that, in this section, we present and discuss the results
of the laterally-integrated concentration φ instead of the actual concentration C.
We observed previously in figure 4.5 that the concentration C tends to disperse
linearly with distance across the jet, for both the constant-flux and the finite-
volume cases. Therefore, the concentration of tracers is diluted not only because
of the streamwise dispersion but also because of the lateral spreading of the jet. As
a result, the distribution of the probability P ∗δ to encounter some concentrations
C greater than a critical value should decrease more rapidly in time in the case of
finite-volume releases of tracers. In the case of constant-flux releases, we believe
that instead of increasing in time (for φ) the distribution of the probability P ∗F
should actually decrease in time for the concentration C. This is related to the
fact that, in steady state, the concentration CF decreases like z−1/2 whereas the
laterally-integrated concentration φF increases like z1/2 (as discussed in § 3.3.2).
105
4 Experimental results for the streamwise dispersion and mixing
4.4 Conclusion
In Chapters 3 and 4, we have analysed the time-dependent transport and dis-
persion properties along the streamwise direction of quasi-two-dimensional jets.
We model the evolution in time and space of the concentration of passive tracers
using a one-dimensional time-dependent effective advection–diffusion equation.
We integrate the concentration across the jet in order to be able to solve the
effective advection–diffusion equation (3.4). From the analysis of experimental
results we find that this simplification appears to be appropriate, because the
tracer distribution remains confined within the quasi-two-dimensional jet between
two linearly-expanding straight-sided boundaries (see figure 2.4). Neglecting any
molecular diffusion, we assume a streamwise turbulent eddy diffusive coefficient
Dzz proportional to the product of the local half-width of the jet b(z) ∝ z and
the local time-averaged maximum streamwise velocity wm(z) ∝ z−1/2 (essentially
based on mixing length theory). The streamwise turbulent eddy diffusive diffusion
coefficient models physically the interaction between the core and eddy structures
of quasi-two-dimensional jets. (In § 2.5, we showed that the core–eddy structure
was self-similar with height, with characteristic local length-scale b(z), and with
characteristic local velocity scale wm(z).)
Using Dzz ∝ z1/2 we are able to transform the effective advection–diffusion
equation into a similarity form. We solve analytically the resulting ordinary dif-
ferential equation in the cases of a constant-flux release and an instantaneous
finite-volume release yielding a ‘fundamental solution’. The solutions depend on
two parameters, an advection parameterKa and a dispersion parameterKd, which
we determine using experimental measurements. We also provide an integral for-
mulation for the general problem of an arbitrary time-dependent release of tracers
governed by a source function. The integral formulation for this more realistic
case is the convolution between the fundamental solution found for the instan-
taneous finite-volume release and the source function. We present an analytical
solution for the general problem in the case of a rectangular source function (i.e.
the flux of tracers at the jet source is constant for a finite period of time, T0,
and zero otherwise, thus releasing a finite volume). At large time (t ≫ T0), this
solution converges towards the fundamental solution found for the instantaneous
finite-volume release. On the other hand, for T0 → ∞, this solution converges
towards the solution found for the constant-flux release.
106
4.4 Conclusion
Furthermore, we show theoretically that, owing to dispersion mechanisms, a
non-negligible portion of the total volume of tracers released travels ahead of
the advective front, in both the finite-volume and the constant-flux cases. The
advective front corresponds to the location of the volume of tracers (in the finite-
volume case) or the front of the tracer distribution (in the constant-flux case) if all
dispersion mechanisms are ignored and Kd = 0. We also find that the streamwise
dispersion increases in time as t2/3.
In this chapter, we compare the theoretical model developed in Chapter 3 with
experimental measurements obtained by tracking the concentration of dye or vir-
tual particles in time and space. We conduct both constant-flux and finite-volume
releases of dye in quasi-two-dimensional steady turbulent jets. We also release fi-
nite volumes of virtual particles (transported as passive tracers) instantaneously in
the fully resolved time-dependent velocity fields of quasi-two-dimensional steady
turbulent jets. We consider the experimental data for constant-flux releases of
dye more accurate because the initial, boundary and integral conditions imposed
in the theoretical model are more straightforward to satisfy experimentally. We
find that the experimental results agree well with the theoretical prediction, using
either the laterally-integrated concentration of dye φ or the streamwise concentra-
tion flux of dyeMφ as defined in (3.6) and (3.50), respectively. The similarity scal-
ing derived from the model η ∝ z/t2/3 is appropriate to study this phenomenon.
We find that what we refer to as our ‘reference’ values for the advection and dis-
persion parameters are Ka = 1.65 and Kd = 0.09, respectively, determined from
the study of the concentration in the constant-flux dye experiments.
We largely confirm these results by the experimental data obtained with finite-
volume releases of virtual particles. The data converge in similarity form towards
the fundamental theoretical solution assuming an instantaneous finite-volume re-
lease. The similarity scaling η ∝ z/t2/3 is also appropriate in this case. We find
that the best fits to the advection and dispersion parameters are Ka = 1.62 and
Kd = 0.09, respectively. In the case of finite-volume releases of dye, we find
that the experimental concentration distribution has not converged towards the
asymptotic fundamental solution assuming an instantaneous release. We believe
that this is principally due to the fact that the dye could not be released instan-
taneously in the experiment. The duration of the dye release introduces a new
time scale T0, which affects the concentration distribution. Until t ≫ T0, the
107
4 Experimental results for the streamwise dispersion and mixing
Case Theory Ka Kd Ka/Kd ηa µ σ β ξ
CF yF 1.65 0.09 18.3 1.83 0.65–0.62 0.29–0.30 0.12–0.10 0.16–0.13
CF yM 1.55 0.07 23.6 1.83 0.58–0.50 0.31–0.31 0.09–0.06 0.19–0.12
IFV yδ 1.62 0.09 18 1.83–1.83 0.99–1.03 0.17–0.19 0.49–0.54 0.13–0.17
FV φT0 1.75 0.09 19.4 1.83
a 1.03a 0.19a 0.54a 0.17a
aTheoretical value after φT0 converges to φδ.
Table 4.1: Summary of the key experimental results found in the constant-flux case
(CF) for dye releases, in the instantaneous finite-volume case (IFV) for virtual-particle
releases and the finite-volume case (FV) for dye releases. The values for the advection
and dispersion parameters Ka and Kd are obtained from the best least-squares fit of
the experimental data. On the other hand, ηa, µ, σ, β and ξ are computed theoretically
using the ‘reference’ parameters Ka = 1.65 and Kd = 0.09, found in the constant-flux
case; if two values are indicated: the first value is measured experimentally while the
second value is computed theoretically using Ka = 1.65 and Kd = 0.09.
concentration distribution is in a transition regime, which we model using the
general model φT0 defined in (3.88), assuming a rectangular source function. We
find that the best fits to the advection and dispersion parameters are Ka = 1.80
and Kd = 0.08, respectively. We also calculate that, in this case, the distribution
should ‘converge’ (i.e. the normalized absolute deviation between φT0 and φδ, de-
fined in (3.91), is smaller than 0.1) towards the fundamental solution φδ defined
in (3.73) after a duration equal to approximately 14 times the time of release of
the dye (i.e. t ≥ 14T0). In other words, the dye distribution should converge
towards an asymptotic distribution at z ≈ 2m (i.e. at a distance larger than four
times the maximum distance of our study area).
Our model appears to be robust to variations in the initial boundary conditions
of the experiments. In the experiments with finite-volume releases of virtual par-
ticles, even though the particles are released instantaneously but far away from
the source, the particle concentration distribution seems to converge rapidly in
time towards a stable asymptotic distribution predicted by the model. In the
experiments with finite-volume releases of dye, even though the dye is released
near the source but not instantaneously, we can prove that the dye concentra-
tion distribution will eventually converge in time towards a stable asymptotic
distribution predicted by the model. Moreover, we can estimate the time before
convergence and provide a model for the transition regime.
Overall, the model largely appears to agree with the data, especially at the
dispersive front of the distribution. In table 4.1 we collect all the various key
108
4.4 Conclusion
experimentally determined quantities. By comparing the various models with all
the experiments, we are able to give an estimated range for the advection and
dispersion parameters. We find that the advection and dispersion parameters are
Ka = 1.65 ± 0.10 and Kd = 0.09 ± 0.02 respectively, and the ratio between the
two is within the range 18 ≤ Ka/Kd ≤ 23.6. For both the constant-flux case
and the instantaneous finite-volume case, the location in similarity space of the
advective front (as defined in (3.40)) is found at ηa = 1.83. Then, in the case of
constant-flux releases of tracers, we find that the ratio between the centroid and
the advective front is approximately µF = 0.635±0.015 with a standard deviation
normalized with ηa σF = 0.295 ± 0.005. At each instant in time, approximately
βF = 11 % ±1 % (as defined in (3.47)) of the total volume of tracers having
already been released is transported ahead of the advective front, at an averaged
normalized distance in similarity space ξF = 0.145± 0.015 (as defined in (3.49)).
In the case of an instantaneous finite-volume release of tracers, the ratio between
the centroid and the advective front is approximately µB = 1.01 ± 0.02 with a
standard deviation normalized with ηa σB = 0.18± 0.01. At each instant in time,
approximately βB = 51.5 % ±2.5 % (as defined in (3.81)) of the total volume of
tracers released is transported ahead of the advective front ηa, at an averaged
normalized distance in similarity space ξB = 0.15± 0.02 (as defined in (3.83)).
The analysis of the statistical significance of the experimental measurements
of the laterally-integrated concentration reveals that experimental or real concen-
trations are more likely to differ from the concentrations predicted by the model
at large concentration levels than at low concentration levels. We find that the
distribution, against the similarity variable η, of the probability to encounter
laterally-integrated concentrations greater than a critical value increases in time
for the case of constant-flux releases of tracers. On the other hand, the distribu-
tion of the analogous probability decreases in time for the case of finite-volume
releases of tracers. However, if we study the actual (non-laterally-integrated) con-
centration, we believe that the probability distribution in the constant-flux case
should also decrease in time due to lateral dispersion across the jet with distance.
In § 3.1, we discussed the importance of modelling correctly the transport and
dispersion of tracers in quasi-two-dimensional jet flows. We believe that the model
developed in Chapter 3 provides not only a strong insight into these mechanisms
but also a quantitative basis to predict them. In this chapter, comparisons with
109
4 Experimental results for the streamwise dispersion and mixing
experimental data obtained using different techniques support the predictions of
the model. From this comparison, we can also measure accurately the strength
of the advection and the strength of the dispersion in quasi-two-dimensional jets,
using only an advection parameter Ka and a dispersion parameter Kd, respec-
tively. Finally, we have discovered that the streamwise dispersion increases in
time like t2/3. In other words, a significant amount of tracers released in quasi-
two-dimensional jets is transported faster than the speed predicted by a simple
advection model. Such predictions are crucial to many applications, particularly
in the event of environmental pollutions in rivers and lakes.
110
Chapter 5
Two-point statistics for turbulent
relative dispersion in
quasi-two-dimensional jets
5.1 Introduction
The dispersion and mixing mechanisms in the turbulent flow of quasi-two-di-
mensional jets are closely related to the dynamics of the large-scale structures
identified as core and eddies. The core and eddy structures display very different
flow properties. The velocity field of the core is very high in the streamwise
direction, and it appears to be subject to a sinuous instability. The velocity
field of the eddies is inherently vortical, with a time-averaged mean component
in the streamwise direction. At the interface between the eddies and the core,
the streamwise velocity has a large lateral (or cross-jet) gradient. Moreover, the
111
5 Two-point statistics for turbulent relative dispersion
flow is turbulent everywhere in a quasi-two-dimensional jet. We believe that such
distinctive Eulerian characteristics (of the flow in the core, in the eddies and at the
interface between the two) also have distinctive dispersive and mixing properties.
From the interaction between these structures, in time and space, results the
global, mean dispersion mechanism of quasi-two-dimensional jets, which we model
in Chapter 3 along the streamwise direction.
Conversely, in this chapter, we adopt a Lagrangian approach to investigate
the dispersion and mixing properties of the core and eddy structures of quasi-
two-dimensional jets. In figure 4.4 presented in the previous chapter, we showed
the evolution in time of clusters of virtual particles (or passive tracers) released
in different parts of a quasi-two-dimensional jet: in an eddy (see figure 4.4a),
between the eddy and the core (see figure 4.4b), and in the core (see figure 4.4c).
We qualitatively described how the clusters of particles disperse and mix, while
being transported by the jet. The virtual particles seeded in the eddy travel
significantly slower than the virtual particles seeded in the core. The virtual
particles seeded in the eddy appear to experience more vigorous stirring than the
virtual particles seeded in the core. We also noticed that the virtual particles
seeded in the core disperse laterally as they are advected by the flow. On the
other hand, the virtual-particle cluster seeded between the eddy and the core
display intense streamwise stretching.
The aim of this study is to quantify these observations about the dispersion
and mixing of the virtual particles in figure 4.4. We use statistical analysis to
understand the underlying physical mechanisms. We study the probability dis-
tribution of two-point properties, such as the lateral (or x-) distance between
two points, the streamwise (or z-) distance between two points, the distance be-
tween two points, and the ratio of the lateral distance to the streamwise distance
between two points. We apply these probabilistic tools to clusters of virtual parti-
cles released in quasi-two-dimensional jets (such as those in figure 4.4), where the
two-point measurements are made between pairs of virtual particles. We compute
the probability distribution of the two-point properties at each instant in time to
obtain meaningful quantitative insight about the temporal and spatial dispersive
dynamics of the jet structures.
The work of Richardson (1926) pioneers the use of two-point statistics to study
diffusion in turbulent flows (see e.g. Sawford, 2001; Salazar & Collins, 2009, for
112
5.1 Introduction
recent reviews). Observing considerable discrepancies (by more than ten orders
of magnitude) in the measurements of the atmospheric diffusivity, he argued that
two-point statistics are more appropriate to explain diffusion in the atmosphere
than single-point statistics (used previously to measure the diffusivity in the sense
described by Fick’s law). Two-point statistics (such as the time average of the
distance between two points) enable the study of the dispersion in the flow at each
spatial scale (defined, for example, by the eddy size), without being influenced by
the larger scales. From the probability density function of the distance between
particles, Richardson derived his famous 4/3 law of diffusion. Batchelor (1952)
developed a rigorous mathematical framework for the idea of Richardson (1926) to
use two-point statistics in order to study turbulent relative dispersion. He applied
two-point statistics to the diffusion of passive scalars in homogeneous isotropic
turbulence.
The concept of two-point statistics has then been used to study turbulent dis-
persion in the ocean and in the atmosphere (see e.g. Monin & Yaglom, 1975,
pp. 556–567, for a review). Salazar & Collins (2009) and Yeung (2002) give
a summary of experimental and numerical works investigating turbulent relative
dispersion. In experimental turbulent flows, two-point statistics can be calculated
by tracking Lagrangian particles. According to Toschi & Bodenschatz (2009), the
most successful current technique to perform Lagrangian particle tracking is called
particle tracking velocimetry. For example, Bourgoin et al. (2006) used particle
tracking velocimetry to measure the mean square distance between particles in
a turbulent flow (generated “between coaxial counter-rotating baffled disks in a
closed chamber”). They confirmed the theoretical prediction of Batchelor (1950)
that the temporal evolution of the distance between pairs of particles during the
superdiffusion stage (i.e. the regime when the mean square distance between par-
ticles increases in time like tα with α > 1, Bourgoin et al., 2006) is influenced
by the initial distance separation of the particles. Bourgoin et al. (2006) also
commented on the scarcity of direct experimental evidence for turbulent relative
dispersion. Toschi & Bodenschatz (2009) attributed the lack of experimental ev-
idence to the technical difficulties of the implementation of Lagrangian particle
tracking in fully turbulent flow.
We believe that applying two-point statistics to the turbulent flow of quasi-
two-dimensional jets can give new insight about turbulent relative dispersion in
113
5 Two-point statistics for turbulent relative dispersion
the case of a non-homogeneous and anisotropic turbulent flow. We use what
we believe to be a new method to calculate these two-point statistics, which we
call virtual particle tracking (see § 4.1). The virtual-particle-tracking technique
(which we use to produce the results shown in figure 4.4, mentioned above) con-
sists of seeding and tracking virtual passive tracers in velocity fields measured
using particle image velocimetry. The results presented in this study focus pri-
marily on the dispersion properties of quasi-two-dimensional jets, but not directly
on the transport or turbulent mixing properties. By definition, two-point statis-
tics do not depend on mean transport motion, and thus cannot investigate it.
(The transport properties of the jet have actually been studied extensively in
Chapters 2, 3 and 4.) On the other hand, we believe that mixing properties can-
not be directly examined from the results we present in this thesis for technical
reasons. The measurements of the velocity fields (performed using particle image
velocimetry), though well-resolved in time (the time resolution is one order of
magnitude smaller than the Kolmogorov time scale, τηK ≈ 40 ms), do not have
the spatial accuracy necessary to investigate the finest scales of turbulence in our
flow (the Kolmogorov length scale is of the order of ηK ≈ 0.2 mm, as discussed
in § 2.2.2). In Chapter 4, we quantify the mixing through the dilution of the dye
concentration. Likewise, in this chapter, we infer indirectly the turbulent mixing
processes from the dispersion, stretching and folding of our particle distributions.
In order to comprehend fully the temporal evolution of the probability distri-
butions of two-point properties applied to virtual-particle clusters seeded in the
different parts of the flow of quasi-two-dimensional jets, we compare our results
with other flow fields. As a preliminary study, we apply our statistical tools to
simple distributions of points (such as a circle, an ellipse and a square) evolving in
diverging velocity fields. The purpose of this preliminary study is to understand
how the probability distributions of two-point properties are related to a given
initial distribution of particles, and how they evolve in time. Then, we compare
the results for the time-dependent flow field of a quasi-two-dimensional jet with
results obtained using the time-averaged flow field of the same jet. This compari-
son allows us to identify some key dispersive mechanisms due to the core and eddy
structures and emphasizes the importance of their time-dependent interactions.
The rest of this chapter is organized as follows. In § 5.2, we describe mathe-
matically how to compute the probability distribution of the two-point properties
114
5.2 Mathematical definitions of two-point probability distributions
stated above, in both the continuous case and the discrete case. In § 5.3, we
present a preliminary study of three analytical and numerical test cases: an ax-
isymmetric expansion of a circular domain, a non-axisymmetric expansion of an
elliptical domain, and a diffusive expansion of a square domain. In § 5.4, we
present the results of the probability distributions for the three clusters of virtual
particles seeded in the quasi-two-dimensional jet shown in figure 4.4. We compare
these results with similar results obtained in the equivalent time-averaged velocity
field of the jet. Finally, we draw our conclusions in § 5.5.
5.2 Mathematical definitions of two-point proba-
bility distributions
5.2.1 Continuous formulation
The probability density function (p.d.f.) fY of a real-valued random variable
Y is the derivative of the cumulative distribution function (c.d.f.) F Y (see e.g.
Pope, 1985) such that, for any real number y ,
fY (y ) =
dF Y (y )
dy . (5.1)
The c.d.f. can be defined as the probability that the random variable Y takes on
a value less than or equal to y ,
F Y (y ) = P (Y ≤ y ). (5.2)
In the present study, we wish to compute the p.d.f. of four characteristic
properties between pairs of points (x1,x2) distributed in a domain A . The first
characteristic property is the lateral distance between two points:
H (x1,x2) = |x1 − x2|, with x1 = (x1, z1), x2 = (x2, z2) ∈ A . (5.3)
The second characteristic property is the streamwise distance between two points:
V (x1,x2) = |z1 − z2|, with x1 = (x1, z1), x2 = (x2, z2) ∈ A . (5.4)
115
5 Two-point statistics for turbulent relative dispersion
The third characteristic property is the (Euclidean) distance between two points:
D (x1,x2) = ‖x1 − x2‖, with x1 = (x1, z1), x2 = (x2, z2) ∈ A , (5.5)
where ‖x1 − x2‖ =
(
(x1 − x2)2 + (z1 − z2)2
)1/2
. The fourth characteristic prop-
erty is the ratio of the lateral distance to the streamwise distance between two
points:
M (x1,x2) =
|x1 − x2|
|z1 − z2|
, with x1 = (x1, z1), x2 = (x2, z2) ∈ A . (5.6)
The probability that the random variable Y (x1,x2) (= H (x1,x2), V (x1,x2),
D (x1,x2) or M (x1,x2)), with x1, x2 ∈ A , takes on a value less than or equal
to y (= h , v , d or m , respectively) is
PA (Y (x1,x2) ≤ y ) =
1∫
A
ς(x1)dτ1
∫
A
PA (Y (x1,x2) ≤ y |x1)ς(x1) dτ1, (5.7)
where dτ1 is an appropriate differential for the domain A with respect to the
first point x1, and ς(x) is the density of the probability distribution (i.e. it is a
measure of the local probability, which may not be uniform, at the point x ∈ A ).
PA (Y (x1,x2) ≤ y |x1) is the conditional probability that the random variable
Y (x1,x2) takes on a value less than or equal to y knowing x1 and is defined as
PA (Y (x1,x2) ≤ y |x1) =
1∫
A
ς(x2)dτ2
∫
A ∩B(x1,y )
ς(x2) dτ2, (5.8)
where the domain B(x1, y ) is defined such that x2 ∈ B(x1, y ) if Y (x1,x2) ≤ y
with x1 known, and dτ2 is an appropriate differential for the domain A with
respect to the second point x2. Finally, according to (5.1) and (5.2), the p.d.f. of
the random variable Y in the domain A is
fYA (y ) =
dPA (Y (x1,x2) ≤ y )
dy , (5.9)
where the probability PA (Y (x1,x2) ≤ y ) is defined by (5.7) and (5.8).
116
5.3 Test studies in diverging velocity fields
5.2.2 Discrete formulation
The probability density functions of the four characteristic properties of the dis-
tribution of particles defined in (5.3), (5.4), (5.5), and (5.6), for H , V , D and
M , respectively, can also be formulated in the discrete case. For n particles
xi = (xi, zi) of weight ωi (similarly to the concept of density ς used previously for
the continuous case) distributed in a domain A , the discrete p.d.f. of a random
variable Y (x1,x2) (= H (x1,x2), V (x1,x2), D (x1,x2) or M (x1,x2)) in this
domain is
fYA (yk) =
1
n∑
i>j
ωiωj δy
n∑
i>j
Yi,j(yk), k ∈ N, 1 ≤ k ≤ N, (5.10)
with δy = y1 − y0 and, for all 1 ≤ i ≤ n and 1 ≤ j ≤ n (where i and j are two
integers),
Yi,j(yk) =
{
ωiωj if yk−1 ≤ Y (xi,xj) < yk
0 otherwise
. (5.11)
Here, yk are distributed from y0 = 0 to yN , the maximum value taken by
Y (x1,x2) in the domain A , while N is the number of bins. The distribution
of yk is linear for Y = H , V and D , and logarithmic for Y = M .
5.3 Test studies in diverging velocity fields
We calculate analytically or numerically the time evolution of the p.d.f.s of the
two-point properties H , V , D , and M (as defined in (5.3), (5.4), (5.5) and
(5.6), respectively) in the case of continuous or discrete distributions of points in
simple two-dimensional domains. Firstly, we study a circular domain expanding
axisymmetrically in a diverging velocity field. Secondly, we investigate the non-
axisymmetric expansion of an elliptical domain, in order to understand the effect
on the p.d.f.s of a variation in the aspect ratio of the domain. Thirdly, we study
the effect of molecular diffusion on the p.d.f.s for an initially square distribution
of discrete points.
Following the example of Richardson (1926), who derived the p.d.f. of the dis-
tance between particles distributed on a straight line, we believe this preliminary
117
5 Two-point statistics for turbulent relative dispersion
analysis can help us understand the turbulent relative dispersion in quasi-two-di-
mensional jets, which is studied in § 5.4. In these test studies, the actual velocity
fields in which we set our domains (the disc, the ellipse and the square) are not
important because we do not intend to relate them directly to the velocity fields
of quasi-two-dimensional jets. The velocity fields of the test studies are merely a
means to change the shape of our three domains in time. In fact, this is rather
the effect of the time evolution of the shape of our domains on the p.d.f.s that
we intend to compare with the time evolution of the p.d.f.s for the three particle
clusters presented in § 5.4.
5.3.1 Circular domain in an axisymmetric diverging veloc-
ity field
We define in R2 a continuous uniform distribution Dt where the initial density
ςDt(x, 0) ≡ 1 for all x = (x, z) such that x2 + z2 ≤ R02 (where R0 is the initial
radius of the disc Dt) and ςDt(x, 0) ≡ 0 otherwise. The domain Dt evolves in time
due to a constant diverging radial velocity field (u, w) defined by
u(x) = x, w(z) = z. (5.12)
Hence, the radius of the disc increases uniformly in time at an exponential rate:
R(t) = etR0; (5.13)
and the density decreases in time such that
ςDt(x, t) =



R02
R2(t) = e
−2t ∀ x = (x, z), x2 + z2 ≤ R2(t)
0 otherwise
. (5.14)
Since the domain Dt is axisymmetric and expands radially at all times, the p.d.f.
of the lateral distance between two points is equal to the p.d.f. of the streamwise
distance between two points in the same domain, i.e. fHDt = fVDt . Moreover, the
p.d.f. is self-similar in time and depends only on the radius R(t). Using equations
(5.7) and (5.8) with Y = H (or V ), defined in (5.3) (and (5.4)), y = h (or v ),
118
5.3 Test studies in diverging velocity fields
the c.d.f. of H (x1,x2) (or V (x1,x2)) with x1, x2 in A = Dt is, for h ≥ 0,
FHDt (h ) =
4
πR2(t)
∫ R(t)
0
PDt(H (x1,x2) ≤ h |x1)
√
R2(t)− x12 dx1, (5.15)
where we use the fact that the c.d.f. is symmetric with respect to both the x-
axis and the z-axis, the conditional probability does not depend on z1 (as long
as x12 + z12 ≤ R2(t)) and the density is uniform over the whole domain Dt. For
0 ≤ h ≤ R(t), the conditional probability is
PDt(H (x1,x2) ≤ h |x1) = 1pi
(
arcsin
(
x1+h
R(t)
)
+(
x1+h )
R(t)
√
1−
(
x1+h
R(t)
)2
−arcsin
(
x1−h
R(t)
)
− (x1−h )R(t)
√
1−
(
x1−h
R(t)
)2
)
, 0≤x1≤R(t)−h , (5.16)
and
PDt(H (x1,x2) ≤ h |x1) = 1pi
(
pi
2−arcsin
(
x1−h
R(t)
)
− (x1−h )R(t)
√
1−
(
x1−h
R(t)
)2
)
, R(t)−h≤x1≤R(t).
(5.17)
For R(t) ≤ h ≤ 2R(t), the conditional probability is
PDt(H (x1,x2) ≤ h |x1) = 1, 0 ≤ x1 ≤ −R(t) + h , (5.18)
and
PDt(H (x1,x2) ≤ h |x1) = 1pi
(
pi
2−arcsin
(
x1−h
R(t)
)
− (x1−h )R(t)
√
1−
(
x1−h
R(t)
)2
)
, −R(t)+h≤x1≤R(t).
(5.19)
For 2R(t) ≤ h
PDt(H (x1,x2) ≤ h |x1) = 1, 0 ≤ x1 ≤ R(t). (5.20)
Details about the calculation of the conditional probability PDt(H (x1,x2) ≤
h |x1) can be found in Appendix B.1. We calculate the c.d.f. FHDt (as defined in
(5.15)) by computing the integral numerically. We then obtain the p.d.f. fHDt (or
fVDt) by differentiating this calculated c.d.f. numerically, i.e.
fHDt (h ) =
dFHDt (h )
dh
. (5.21)
119
5 Two-point statistics for turbulent relative dispersion
We plot the non-dimensional p.d.f. fHDtR(t) in figure 5.1(a) against the non-
dimensional variable h /R(t). We can see that fHDtR(t) decreases smoothly from
fHDt (0) = 32/ (3π2R(t)) to fHDt (2R(t)) = 0 (see details in Appendices B.2 and
B.3).
The p.d.f. fHSt of H in a square domain St defined by the density ςSt(x, 0) ≡ 1
for all x = (x, z) such that −R0 ≤ x ≤ R0 and −R0 ≤ z ≤ R0 and 0 otherwise
(with R(t) described by (5.13) and ςSt(x, t) = e−2t if −R(t) ≤ x ≤ R(t) and
−R(t) ≤ z ≤ R(t) and 0 otherwise) is
fHSt(h ) =
2R(t)− h
2R2(t) . (5.22)
The full derivation of (5.22) can be found in Appendix B.4. We can see in figure
5.1(a) that the ‘disc’ fHDt is somewhat similar to the ‘square’ fHSt (plotted with a
dashed line), which decreases linearly from fHSt(0) = 1/ (R(t)) to fHSt (2R(t)) = 0.
Similarly, we can compute the c.d.f. of the distance between two points using
equations (5.7) and (5.8) with Y = D (defined in (5.5)), y = d , with x1, x2 in
A = Dt. We have in polar coordinates, for d ≥ 0,
FDDt(d ) =
2
R2(t)
∫ R(t)
0
PDt(D (x1,x2) ≤ d |x1)r1dr1, (5.23)
where r1 =
√
x12 + z12, and where we use the fact that the conditional probability
is independent of the angle θ1 (as long as r1 ≤ R(t)) and the density is uniform
over the whole domain Dt. For 0 ≤ d ≤ R(t), the conditional probability is
PDt(D (x1,x2) ≤ d |x1) =
d 2
R2(t) , 0 ≤ r1 ≤ R(t)− d , (5.24)
and
PDt(D (x1,x2) ≤ d |x1) = d 2piR2(t)
(
pi−arccos(xI−r1d )+
(xI−r1)
d
√
1−(xI−r1d )
2
)
+
1
pi
(
arccos( xIR(t))−
xI
R(t)
√
1−( xIR(t))
2
)
, R(t)−d ≤r1≤R(t), (5.25)
where
xI =
R2(t) + r12 − d 2
2r1
, (5.26)
120
5.3 Test studies in diverging velocity fields
h /R(t)
f
H D
t
R
(t
)
Domain: disc
Domain: square
0 0.5 1.0 1.5 2.0
0
0.2
0.4
0.6
0.8
1.0
(a)
d /R(t)
f
D D
t
R
(t
)
0 0.5 1.0 1.5 2.0
0
0.2
0.4
0.6
0.8
1.0
(b)
square
m
f
M D
t
(t
)
10−5 1 105
10−10
10−5
1
(c)
Figure 5.1: Probability density functions in the case of a uniformly distributed disc
Dt in an axisymmetric diverging velocity field described in (5.12) for: (a) the lateral (or
streamwise) distance between two points fHDt (solid curve), defined in (5.21), the p.d.f.
of the lateral (or streamwise) distance between two points in a square fHSt (defined in
(5.22)) is plotted with a dashed line for comparison; (b) the distance between two points
fDDt , defined in (5.30); (c) the ratio of the lateral distance to the streamwise distance
between two points fMDt , defined in (5.34).
is the x-coordinate of the intersection between the perimeter of Dt and the circle
defined by x = (x, z) such that (x− r1)2 + z2 = d 2. For R(t) ≤ d ≤ 2R(t), the
conditional probability is
PDt(D (x1,x2) ≤ d |x1) = 1, 0 ≤ r1 ≤ −R(t) + d , (5.27)
121
5 Two-point statistics for turbulent relative dispersion
and
PDt(D (x1,x2) ≤ d |x1) = d 2piR2(t)
(
pi−arccos(xI−r1d )+
(xI−r1)
d
√
1−(xI−r1d )
2
)
+
1
pi
(
arccos( xIR(t))−
xI
R(t)
√
1−( xIR(t))
2
)
, −R(t)+d ≤r1≤R(t). (5.28)
For 2R(t) ≤ d ,
PDt(D (x1,x2) ≤ d |x1) = 1, 0 ≤ r1 ≤ R(t). (5.29)
Details about the calculation of the conditional probability PDt(D (x1,x2) ≤
d |x1) can be found in Appendix B.5. As before, we calculate the c.d.f. FDDt
(as defined in (5.23)) by computing the integral numerically. We then obtain the
p.d.f. fDDt by differentiating this calculated c.d.f. numerically, i.e.
fDDt(d ) =
dFDDt(d )
dd
. (5.30)
We plot the non-dimensional p.d.f. fDDtR(t) in figure 5.1(b) against the non-
dimensional variable d /R(t). It is interesting to note that the p.d.f. fDDt starts
from 0 at d = 0 (as can be proved from (5.23), (5.24) and (5.25) using a similar
technique to that used in Appendix B.2), increases to a maximum value (which
appears to occur for d /R(t) < 1) and then vanishes at d = 2R(t) (as can be
proved from (5.23), (5.27) and (5.28) using a similar technique to that used in
Appendix B.3).
We can also compute the c.d.f. of the ratio of the lateral distance to the
streamwise distance between two points using equations (5.7) and (5.8) with Y =
M , defined in (5.6), y = m , and with x1, x2 in A = Dt. We have in polar
coordinates, for m ≥ 0,
FMDt (m , t) =
4
πR2(t)
∫ R(t)
0
∫ pi
2
0
PDt(M (x1,x2) ≤ m |x1)r1 dr1dθ1, (5.31)
where x1 = r1 cos θ1 and z1 = r1 sin θ1, and where we use the fact that the
density is uniform over the whole domain Dt and that the conditional probability
is symmetric with respect to both the x-axis and the z-axis. The conditional
122
5.3 Test studies in diverging velocity fields
probability, which, in this case, depends on θ1, is
PDt(M (x1,x2) ≤ m |x1) = F (π − υ)−F (υ) + F (−υ)−F (υ − π), (5.32)
with υ = arctan (1/m ), and where
F (θ2) = 1piR2(t)

R2(t)
2 θ2+
r12
4 sin(2(θ2−θ1))+
R2(t)
2
(
arccos( r1R(t) sin(θ2−θ1))−
r1
R(t) sin(θ2−θ1)
√
1−( r1R(t) sin(θ2−θ1))
2
))
. (5.33)
Details about the calculation of the conditional probability PDt(M (x1,x2) ≤
m |x1) can be found in Appendix B.6. As before, we calculate the c.d.f. FMDt
(as defined in (5.31)) by computing the integral numerically. We then obtain the
p.d.f. fMDt by differentiating this calculated c.d.f. numerically, i.e.
fMDt (m ) =
dFMDt (m )
dm . (5.34)
We plot the dimensionless p.d.f. fMDt in figure 5.1(c) against the dimensionless
variable m using a logarithmic scale. We can see that the p.d.f. vanishes at r → 0
and r → ∞. Moreover, fMDt is symmetric with respect to m = 1, owing to the
axisymmetry of the domain Dt at all time. In other words, fMDt (m ) = fMDt (1/m ).
5.3.2 Elliptical domain in a non-axisymmetric diverging ve-
locity field
Now, we study the case of a non-axisymmetric diverging velocity field. We define
in R2 a continuous uniform distribution Lt, which is identical to the disc Dt
(described previously) at t = 0. The initial density of Lt is ςLt(x, 0) ≡ 1 for
all x = (x, z) such that x2 + z2 ≤ R02 (where R0 is the initial radius of Lt)
and ςDt(x, 0) ≡ 0 otherwise. For t > 0, the domain Lt evolves in time due to a
constant diverging non-axisymmetric velocity field (u, w) defined by
u(x) = x, w(z) = zc , (5.35)
123
5 Two-point statistics for turbulent relative dispersion
where c > 1 is a constant. The case c = 1 corresponds to the axisymmetric
expansion of the disc studied in the previous section. Also, we do not need
to study the case c < 1 owing to the symmetry between the x and z spatial
coordinates (or u and w components of the velocity field). The domain Lt has
an elliptical contour for t > 0, whose semi-major axis and semi-minor axis are
denoted a and b in the x- and z-directions, respectively. The semi-axes a and b
increase in time at different exponential rates:
a(t) = etR0, b(t) = et/cR0, (5.36a,b)
and the density decreases in time such that
ςLt(x, t) =



R02
a(t)b(t) = e
−t(c+1)/c ∀ x = (x, z),
(
x
a(t)
)2
+
(
z
b(t)
)2
≤ 1
0 otherwise
. (5.37)
The domain Lt expands radially at all time but not axisymmetrically. The
domain Lt remains symmetric with respect to both the x-axis and the z-axis.
The p.d.f. of the lateral distance between two points fHLt is not equal to the p.d.f.
of the streamwise distance between two points fVLt , but can be computed in a
similar manner by substituting a(t) and b(t). Using equations (5.7) and (5.8)
with Y = H , defined in (5.3), y = h , the c.d.f. of H (x1,x2) with x1, x2 in
A = Lt is, for h ≥ 0,
FHLt (h ) =
4
πa(t)b(t)
∫ a(t)
0
∫ b(t)
√
1−(x1/a(t))2
0
PLt(H (x1,x2) ≤ h |x1) dz1dx1, (5.38)
where we use the fact that the c.d.f. is symmetric with respect to both the x-axis
and the z-axis, and the density is uniform over the whole domain Lt. Since the
conditional probability does not depend on z1 (as long as (x1/a(t))2+(z1/b(t))2 ≤
1), we can integrate (5.38) with respect to z1, to obtain
FHLt (h ) =
4
πa2(t)
∫ a(t)
0
PLt(H (x1,x2) ≤ h |x1)
√
a2(t)− x12 dx1. (5.39)
We can notice that (5.39) is exactly the same as the c.d.f. of the lateral distance
between two points in the domain Dt FHDt (see equation (5.15)), but with R(t) =
124
5.3 Test studies in diverging velocity fields
(a) (b) (c)
Figure 5.2: Distribution of a cluster of virtual particles seeded in the non-axisymmetric
diverging velocity field described in (5.35) with c = 5 at successive non-dimensional
times: (a) t = 0; (b) t = 1; (c) t = 2.
a(t). Thus, the conditional probability in (5.39) is given by equations (5.16),
(5.17), (5.18), (5.19) and (5.20) with R(t) = a(t). Therefore, the p.d.f. fHLt is
equivalent to fHDt , but with R(t) = a(t). Similarly, we find by symmetry that the
p.d.f. of the streamwise distance between two points in the domain Lt (i.e. fVLt)
is equivalent to fVDt , but with R(t) = b(t). We have plotted the non-dimensional
p.d.f. fHLta(t) and fVLtb(t) in figures 5.3(a) and 5.3(b), respectively, with solid
curves. We can see that fHLta(t) and fVLtb(t) are similar when plotted against
h /a(t) and v /b(t), respectively. In comparison, we plot the non-dimensional
p.d.f. of the lateral (streamwise) distance between two points in a rectangular
domain (defined such that −a(t) ≤ x ≤ a(t) and −b(t) ≤ z ≤ b(t)) with a dashed
line in figure 5.3(a) (5.3b, respectively). The p.d.f. of the lateral (streamwise)
distance between two points in a rectangular domain is equivalent to the p.d.f. in
a square domain, described in (5.22), with R(t) = a(t) (R(t) = b(t), respectively).
The calculation of the p.d.f. of D (the distance between two points) and M
(the ratio of the lateral to the streamwise distances between two points) in the
elliptical domain Lt is apparently more difficult. Instead of computing fDLt and
fMLt analytically using the continuous formulation, we use the discrete formulation
described in § 5.2.2. We distribute 7845 virtual passive tracers (or particles) of
similar weight ω = 1 uniformly in a disc of initial radius R0 = 50 centred at the
origin of a two-dimensional (x, z) infinite domain. The particles are seeded in the
non-axisymmetric diverging velocity field described in (5.35), with c = 5. Using
a discrete time step δt = 1, we find that the position of a given particle at t is
xt = 2tx0, zt =
(c+ 1
c
)t
z0. (5.40)
We display in figures 5.2(a–c) the distribution of the particles at dimensionless
times t = 0, t = 1 and t = 2, respectively.
125
5 Two-point statistics for turbulent relative dispersion
From the location of all the particles at each instant in time, given by (5.40), we
can compute the p.d.f. of the distance between two particles fDLt using the discrete
formulation described in (5.10) and (5.11), with Y (x1,x2) = D (x1,x2), yk = dk
and N = 100. We plot the non-dimensional p.d.f. fDLta(t) in figure 5.3(c) against
d /a(t) for t = 0 (black), t = 1 (blue) and t = 2 (red). We can clearly see that the
p.d.f. is no longer self-similar in time. The peak of the curve increases and move
towards d = 0 (i.e. to the left) as time increases. The spurious fluctuations that
can be seen in the p.d.f. fDLt are due to discretization issues. These fluctuations,
particularly prominent at t = 0, are due to the fact that the discrete particle
distribution has not enough randomness. Thus, despite a large number of pairs
of particles (30,768,090) there cannot be a statistically good partition of all their
separation distances among the N = 100 bins of the discretized variable dk.
Again, from the location of all the particles at each instant in time, given by
(5.40), we can compute the p.d.f. of the ratio of the lateral distance to the stream-
wise distance between two particles fMLt using the discrete formulation described
in (5.10) and (5.11), with Y (x1,x2) = M (x1,x2), yk = m k and N = 100. We
plot the dimensionless p.d.f. fMLt in figure 5.3(d) against m for t = 0 (black),
t = 1 (blue) and t = 2 (red). We can clearly see that for t > 0 the p.d.f. is not
symmetric with respect to m = 1, as it is at t = 0 when the domain is circular.
The p.d.f. fMLt displaces to the right as the aspect ratio a(t)/b(t) of the domain
increases in time. Therefore, the evolution in time of fMLt can reveal a change in
the aspect ratio of the domain studied. The spurious fluctuations that can be seen
in the p.d.f. fMLt are also due to the discretization issue mentioned previously.
5.3.3 Square cluster of virtual particles in a diffusive veloc-
ity field
Now, we study the case of a diffusive velocity field. We distribute 3721 virtual
passive tracers (or particles) of similar weight ω = 1 uniformly in a square of unit
size centred at the origin of a two-dimensional (x, z) infinite domain. At each
time step, the particles move following a two-dimensional random walk of length
500(t+1). We designate by Kt the diffusing distribution of particles. We display
in figures 5.4(a–c) the distribution of the particles at dimensionless times t = 0,
t = 1 and t = 2, respectively.
From the location of all the particles at each instant in time, we can compute
126
5.3 Test studies in diverging velocity fields
h /a(t)
f
H L
t
a(
t)
Domain: ellipse
Domain: rectangle
0 0.5 1.0 1.5 2.0
0
0.2
0.4
0.6
0.8
1.0
(a)
v /b(t)
f
V L
t
b(
t)
Domain: ellipse
Domain: rectangle
0 0.5 1.0 1.5 2.0
0
0.2
0.4
0.6
0.8
1.0
(b)
0 0.5 1.0 1.5 2.0
d /a(t)
0
0.2
0.4
0.6
0.8
1.0
1.2
f
D L
t
a(
t)
t = 0
t = 1
t = 2
(c)
1.0
0.01 0.1 1 10 100
m
0.01
0.1
1
f
M L
t
t = 0
t = 1
t = 2
(d)
Figure 5.3: Evolution in time of the probability density functions in the case of a
uniformly distributed elliptical domain Lt in a non-axisymmetric diverging velocity
field described in (5.35) for: (a) the lateral distance between two points fHLt (solid
curve) computed in the continuous case using (5.15) with R(t) = a(t), the p.d.f. of the
lateral distance between two points in a rectangle (defined in (5.22) with R(t) = a(t))
is plotted with a dashed line for comparison; (b) the streamwise distance between two
points fVLt (solid curve) computed in the continuous case using (5.15) with R(t) = b(t),
the p.d.f. of the streamwise distance between two points in a rectangle (defined in
(5.22) with R(t) = b(t)) is plotted with a dashed line for comparison; (c) the distance
between pairs of particles fDLt computed in the discrete case using (5.10) and (5.11)
with n = 7845 and N = 100 at times t = 0 (black), t = 1 (blue) and t = 2 (red);
and (d) the ratio of the lateral distance to the streamwise distance between pairs of
particles fMLt computed in the discrete case using (5.10) and (5.11) with n = 7845 and
N = 100 at times t = 0 (black), t = 1 (blue) and t = 2 (red), note that, in this case,
the distribution is in a log–log plot.
the p.d.f. of the lateral distance between two particles fHKt using the discrete
formulation described in (5.10) and (5.11), with Y (x1,x2) = H (x1,x2), yk = hk
and N = 100. We plot the p.d.f. fHKt in figure 5.5(a) against h for t = 0
127
5 Two-point statistics for turbulent relative dispersion
(a) (b) (c)
Figure 5.4: Time evolution of an initially square distribution of particles following
random walks, at successive non-dimensional times: (a) t = 0; (b) t = 1; (c) t = 2.
(black), t = 1 (blue) and t = 2 (red). We can see that starting from the expected
distribution for a square domain, the p.d.f. rapidly drops and becomes smoother
(similarly to the p.d.f. for the circular domain fHDt displayed in figure 5.1a).
Similarly, we can compute the p.d.f. of the streamwise distance between two
particles fVKt using the discrete formulation described in (5.10) and (5.11), with
Y (x1,x2) = V (x1,x2), yk = vk and N = 100. We plot the p.d.f. fVKt in figure
5.5(b) against v for t = 0 (black), t = 1 (blue) and t = 2 (red). The p.d.f. of the
streamwise distance is very similar to the p.d.f. of the lateral distance shown in
figure 5.5(a), owing to the axisymmetry of the diffusion process.
Similarly, we can compute the p.d.f. of the distance between two particles fDKt
using the discrete formulation described in (5.10) and (5.11), with Y (x1,x2) =
D (x1,x2), yk = dk and N = 100. We plot the p.d.f. fDKt in figure 5.5(c) against
d for t = 0 (black), t = 1 (blue) and t = 2 (red). The p.d.f. gradually drops
and spreads in time. For t > 0, the p.d.f. is similar to the p.d.f. for the circular
domain fDDt displayed in figure 5.1(c).
Similarly, we can compute the p.d.f. of the ratio between the lateral distance to
the streamwise distance between two particles fMKt using the discrete formulation
described in (5.10) and (5.11), with Y (x1,x2) = M (x1,x2), yk = m k and N =
100. We plot the p.d.f. fMKt in figure 5.5(d) against m for t = 0 (black), t =
1 (blue) and t = 2 (red). It is clear that the distribution of particles remain
symmetric as the p.d.f. is centred around m = 1 at all time.
Again, the spurious fluctuations that can be observed in the p.d.f.s fHKt , fVKt ,
fDKt , and fMKt are due to the discretization issue mentioned previously.
128
5.3 Test studies in diverging velocity fields
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
h
0
0.5
1.0
1.5
2.0
f
H K
t
t = 0
t = 1
t = 2
(a)
2
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
v
0
0.5
1.0
1.5
2.0
f
V K
t
t = 0
t = 1
t = 2
(b)
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
d
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
f
D K
t
t = 0
t = 1
t = 2
(c)
2
10−2 0.1 1 10 102
m
10−2
0.1
1
f
M K
t
t = 0
t = 1
t = 2
(d)
Figure 5.5: Evolution in time of the probability density functions in the case of a
diffusing domain Kt of virtual particles for: (a) the lateral distance between pairs of
particles fHKt computed in the discrete case using (5.10) and (5.11) with n = 3721 and
N = 100 at times t = 0 (black), t = 1 (blue) and t = 2 (red); (b) the streamwise
distance between pairs of particles fVKt computed in the discrete case using (5.10) and
(5.11) with n = 3721 and N = 100 at times t = 0 (black), t = 1 (blue) and t = 2
(red); (c) the distance between pairs of particles fDKt computed in the discrete case
using (5.10) and (5.11) with n = 3721 and N = 100 at times t = 0 (black), t = 1 (blue)
and t = 2 (red); and (d) the ratio of the lateral distance to the streamwise distance
between pairs of particles fMKt computed in the discrete case using (5.10) and (5.11)
with n = 3721 and N = 100 at times t = 0 (black), t = 1 (blue) and t = 2 (red), note
that, in this case, the distribution is in a log–log plot.
5.3.4 Conclusion of the test studies
We have analysed the probability distributions of two-point properties for: a circu-
lar domain in axisymmetric expansion, an elliptical domain in non-axisymmetric
expansion and a square domain expanding due to a diffusion-like process. From
these studies, we have learnt that:
129
5 Two-point statistics for turbulent relative dispersion
• the p.d.f. of the lateral and streamwise distance between two points, H and
V respectively, characterizes the extent of the domain along each specific
direction. The p.d.f. of H or V do not seem to depend on the actual shape
of the distribution. In normalized form, the results of fH or fV were iden-
tical between the disc and the ellipse (see solid curves in figures 5.1a,b and
figures 5.3a,b), and were very similar to the results for the square domain
and the diffusing domain (see black lines in figures 5.5a,b, and blue and red
curves in figures 5.5a,b).
• Regardless of the orientation, the p.d.f. of D characterizes the average
distance between particles, and thus the shape of the domain. The p.d.f. fD
of an axisymmetric domain (i.e. a disc in two dimensions) has its maximum
value the furthest away from d = 0 (see figure 5.1c). The more elongated
the distribution (e.g. an ellipse with large aspect ratio), the closer the peak
of fD is to 0 (see figure 5.3c).
• The ratio between the lateral and the streamwise distance between two
points, M characterizes the symmetry between the x-direction and the z-
direction. fM characterizes the aspect ratio of the extension of the domain
along these two directions.
Furthermore, through the evolution in time of these probability distributions we
can note that
• stretching or shrinking of the domain along the specific x- and z-directions
can be quantified with fH and fV . This is also characterized by the shifting
of the maximum of fM away from m = 1 (see figure 5.3d).
• If fD remains self-similar in time, then the transformation seems to preserve
the form (see the evolution of fD for the circular domain in figure 5.1c
compared with the elliptical domain in figure 5.3c).
• In a diffusion process (see figure 5.4), the distribution of particles tends to
become more axisymmetric. In figures 5.5(a,b) the p.d.f. fHKt and fVKt are
linear at t = 0 and then become smoother at each time step, similarly to
fHLt and fVLt .
130
5.4 Analysis of the virtual particles in the jet structures
5.4 Analysis of the virtual particles in the jet
structures
5.4.1 Virtual particles: time-dependent versus time-ave-
raged velocity fields
In the light of the preliminary study presented above, we analyse the statistical
properties of particles seeded in the core and eddy structures of quasi-two-dimen-
sional jets. We use the time-dependent velocity field of the jet shown previously
in figure 4.3(b), where we seed a square cluster of n = 3721 virtual particles in an
eddy (shown in light grey), a rectangular cluster of n = 7381 virtual particles and
aspect ratio 2 at the interface between the core and the eddy (shown in grey), and
a square cluster of n = 3721 virtual particles in the middle of the core (shown
in dark grey). We have reproduced the time evolution of the particle clusters
seeded in this time-dependent velocity field (previously shown in figures 4.4a–c)
in figures 5.6(a–c).
We now repeat the same process in the time-averaged velocity field (with an
average duration time of 21.8 s, as explained in § 2.2.2) of the jet used in fig-
ures 5.6(a–c). We show in figures 5.6(d–f ) the time evolution (the colour-scale
used is the same to that used in figures 5.6a–c) of three clusters of virtual particles
seeded in this time-averaged velocity field. The clusters in figures 5.6(d–f ) have
the same size and are initially located at the same position in the velocity field
as the clusters in figures 5.6(a–c), respectively.
The comparison between the evolution of the particle clusters in the time-
dependent and the time-averaged velocity fields reveals crucial information about
the dynamics of the core and eddy structures. The cluster seeded at the location
of an eddy in the time-averaged velocity field (see figure 5.6d) stretches in the
streamwise direction and rotates counter-clockwise. The evolution of the corre-
sponding cluster in the time-dependent velocity field (see figure 5.6a) does not
display the same streamwise stretching, but rather expands isotropically. Strong
stirring and turbulent mixing at the location of the eddy also seem to be fea-
tures of the time-dependent velocity field only. At the interface between the eddy
and the core, the time-averaged velocity field (see figure 5.6e) merely stretches
the particle cluster in the streamwise direction. On the other hand, the cluster
131
5 Two-point statistics for turbulent relative dispersion
seeded in the time-dependent velocity field (see figure 5.6b) not only experiences
streamwise dispersion (clear at early times, shown with dark and red colours),
but also divides into two as some particles are drawn into the neighbouring eddy.
The particles drawn into the eddy experience the same isotropic dispersion and
strong turbulent stirring as observed for the cluster in figure 5.6(a), whereas the
particles that remain in the core are rapidly transported away. The particle clus-
ter seeded in the core of the time-dependent velocity field (see figure 5.6c) has a
similar evolution as the corresponding particle cluster seeded in the time-averaged
velocity field (see figure 5.6f ). We can observe slightly more streamwise disper-
sion in the time-averaged velocity field, whereas the time-dependent velocity field
rather seems to stretch the particles in the cross-jet direction.
5.4.2 Two-point statistics: time-dependent versus time-a-
veraged velocity fields
We now study the time evolution of the two-point statistics of the three particle
clusters evolving in the time-dependent velocity field (presented in figures 5.6a–
c), as well as the three particle clusters evolving in the time-averaged velocity
field of the same jet (presented in figures 5.6d–f ). For every cluster of both
velocity fields, we compute the p.d.f., using the discrete formulation (5.10) and
(5.11) (with N = 100, the number of bins), for the lateral distance H (except
at t˜ = t/(d2/Q0) = 0 where we use the theoretical prediction defined in (5.22)
for a rectangular domain), the streamwise distance V (except at t˜ = 0 where
we use the theoretical prediction defined in (5.22) for a rectangular domain), the
(Euclidean) distance D , and the ratio of the lateral distance to the streamwise
distance between pairs of particles M , as defined in (5.3), (5.4), (5.5) and (5.6),
respectively. We present the distributions of the p.d.f.s at three or four different
times, linearly distributed from t˜ = 0 (the time we initially seed the particle
cluster in the velocity field) to the time instant a particle of the cluster reaches
the top boundary of the velocity field (this time varies between each cluster).
In the eddy
In figures 5.7(a–d), we present the non-dimensional p.d.f. of the lateral dis-
tance fHEt d (where d = 0.5 cm is the nozzle width of the experimental appara-
132
5.4 Analysis of the virtual particles in the jet structures
(a)
0 t (s)1 2 3
(b) (c)
(d) (e) (f )
Figure 5.6: Comparison of the evolution in time of the virtual particles seeded in the
velocity field shown in figure 4.3(b): (a–c) correspond to the time-dependent velocity
field, while (d–f ) correspond to the time-averaged velocity field. (a,d) Particle cluster
initially distributed at the centre of an eddy and shown in light grey in figure 4.3(b).
(b,e) Particle cluster initially distributed between the eddy and the core and shown in
grey in figure 4.3(b). (c,f ) Particle cluster initially distributed in the core of the jet
and shown in dark grey in figure 4.3(b). Each colour corresponds to a time period of
∆t = 0.2 s (or ∆t˜ = 33 in dimensionless time), the colour scale shown at the bottom of
(b) is the same to that used in figure 4.3(b).
133
5 Two-point statistics for turbulent relative dispersion
tus presented in figure 4.1), the streamwise distance fVEt d, the ratio between the
lateral and streamwise distances between pairs of virtual particles fMEt and the
distance fDEt d for the cluster initially seeded at the location of an eddy in the
time-dependent velocity field (see figure 5.6a). We plot the p.d.f. against the
non-dimensional variables h /d, v /d, m and d /d in figures 5.7(a–d), respectively.
The p.d.f.s are plotted at four different times, from t˜ = 0 to 392, using different
colours. The colour scale used here is the same as shown in figure 5.6(b). Sim-
ilarly, we show in figures 5.7(e–h) the evolution in time of the non-dimensional
p.d.f.s fHEt d, fVEt d, fDEt d and fMEt for the cluster initially seeded at the location of
an eddy in the time-averaged velocity field (see figure 5.6d).
As we can see in figure 5.7(a), the range of fHEt d increases slightly in time from
approximately 0 ≤ h /d ≤ 5 at t˜ = 0 to 0 ≤ h /d ≤ 7 at t˜ = 392. In the streamwise
direction, the distribution of particles stretches more than in the lateral direction,
as the range of fVEt d (shown in figure 5.7b) increases from approximately 0 ≤
v /d ≤ 5 at t˜ = 0 to 0 ≤ v /d ≤ 13 at t˜ = 392. The small change in aspect ratio
can also be observed in figure 5.7(c), where fMEt is no longer exactly symmetric
with m = 1 for t˜ > 0. Moreover, the smooth profile of fMEt at t˜ = 0 seems to be
disturbed near m = 1 for t˜ > 0. This disturbance could suggest changes in the
distribution of the particles with time. The evolution in time of the p.d.f. of the
distance between two points fDEt , shown in figure 5.7(d), also reveals important
changes in the distribution of the particles. The increase in the range of fDEt (from
approximately 0 ≤ d /d ≤ 6 at t˜ = 0 to 0 ≤ d /d ≤ 13 at t˜ = 392) means that
the particles have been spread over a larger domain. Moreover, the characteristic
profile of fDEt at t˜ = 0 (which corresponds to a square domain) quickly vanishes,
thus suggesting a radical change in the shape of the domain. Finally, for t˜ > 0,
fDEt displays large fluctuations and peaks (different from the small fluctuations at
t˜ = 0 which are due to the resolution problem mentioned previously), which vary
in amplitude and location with time.
In the light of the observations made in § 5.4.1, we can notice some major
differences between the p.d.f.s for the time-dependent flow field (shown in figures
5.7a–d) and the p.d.f.s for the time-averaged flow field (shown in figures 5.7e–
h). Firstly, the aspect ratio of the distribution in the time-averaged flow field
deviates considerably in time from the aspect ratio of the distribution in the
time-dependent flow field. In figure 5.7(e), the range of fHEt d decreases slightly in
134
5.4 Analysis of the virtual particles in the jet structures
time from approximately 0 ≤ h /d ≤ 5 at t˜ = 0 to 0 ≤ h /d ≤ 3 at t˜ = 196. In
figure 5.7(f ), the range of fVEt d increases considerably in time from approximately
0 ≤ h /d ≤ 5 at t˜ = 0 to 0 ≤ v /d ≤ 35 at t˜ = 196. The aspect ratio (between
the lateral and streamwise extent) drops from 1 to less than 0.1, as clearly shown
by fMEt plotted in figure 5.7(g). Secondly, the distribution of the p.d.f. of the
distance fDEt for the time-averaged velocity field (plotted in figure 5.7h) is much
smoother than for the time-dependent velocity field (plotted in figure 5.7d). We
believe these continuous and rapid variations in time of the profile of fDEt for
the time-dependent flow field can be related to the intense stirring effect of the
turbulent eddy. The chaotic dynamics of the turbulent flow in the eddy perturbs
the distribution of particles. This manifests itself in the rapid displacement of the
peaks in the distribution of fDEt , for the time-dependent flow field.
At the interface between the core and the eddy
In figures 5.8(a–d), we present the non-dimensional p.d.f. of the lateral distance
fHItd, the streamwise distance fVItd, the ratio between the lateral and streamwise
distances between pairs of virtual particles fMIt and the distance fDItd for the clus-
ter initially seeded at the interface between the eddy and the core (see figure 5.6b).
We plot the p.d.f. against the non-dimensional variables h /d, v /d, m and d /d
in figures 5.8(a–d), respectively. The p.d.f.s are plotted at four different times,
from t˜ = 0 to 98, using different colours. The colour scale used here is the same
as shown in figure 5.6(b). Similarly, we show in figures 5.8(e–h) the evolution in
time of the non-dimensional p.d.f.s fHItd, fVItd, fDItd and fMIt for the cluster ini-
tially seeded at the interface between the eddy and the core in the time-averaged
velocity field (see figure 5.6e).
In figure 5.8(a), the range of fHItd first decreases in time from approximately
0 ≤ h /d ≤ 10 at t˜ = 0 to 0 ≤ h /d ≤ 5 at t˜ = 33 before increasing to 0 ≤ h /d ≤ 13
at t˜ = 98. On the other hand, the distribution of particles steadily stretches in
the streamwise direction, as the range of fVItd (shown in figure 5.8b) increases
from approximately 0 ≤ v /d ≤ 5 at t˜ = 0 to 0 ≤ v /d ≤ 55 at t˜ = 98. This
considerable change in the aspect ratio of the distribution, from 2 to less than 1/4
is also clearly revealed in figure 5.8(c), where the peak of fMIt rapidly moves from
the right-hand side of m = 1 to the left-hand side. The evolution in time of the
p.d.f. of the distance between two points fDIt , shown in figure 5.8(d), is different
135
5 Two-point statistics for turbulent relative dispersion
Time-dependent flow field
0 2 4 6 8 10
h /d
0
0.1
0.2
0.3
0.4
0.5
0.6
f
H E
t
d
t˜ = 0
t˜ = 131
t˜ = 262
t˜ = 392
(a)
Time-averaged mean flow field
0 1 2 3 4 5
h /d
0
0.2
0.4
0.6
0.8
1.0
1.2
f
H E
t
d
t˜ = 0
t˜ = 66
t˜ = 131
t˜ = 196
(e)
Time-dependent flow field
0 2 4 6 8 10 12 14
v /d
0
0.1
0.2
0.3
0.4
f
V E
t
d
t˜ = 0
t˜ = 131
t˜ = 262
t˜ = 392
(b)
Time-averaged mean flow field
0 5 10 15 20 25 30 35
v /d
0
0.1
0.2
0.3
0.4
f
V E
t
d
t˜ = 0
t˜ = 66
t˜ = 131
t˜ = 196
(f )
Time-dependent flow field
10−4 10−2 1 102 104
m
10−4
10−3
10−2
0.1
1
f
M E
t
t˜ = 0 t˜ = 131
t˜ = 262
t˜ = 392
(c)
Time-averaged mean flow field
10−4 10−2 1 102 104
m
10−4
10−3
10−2
0.1
1.0
f
M E
t
t˜ = 0
t˜ = 66
t˜ = 131
t˜ = 196
(g)
Time-dependent flow field
0 2 4 6 8 10 12 14
d /d
0
0.1
0.2
0.3
f
D E
t
d
t˜ = 0
t˜ = 131
t˜ = 262
t˜ = 392
(d)
Time-averaged mean flow field
0 5 10 15 20 25 30 35
d /d
0
0.1
0.2
0.3
f
D E
t
d
t˜ = 0
t˜ = 66
t˜ = 131
t˜ = 196
(h)
Figure 5.7: Comparison between the time evolutions (the colour-scale used is the
same to that used in figure 5.6) of the p.d.f.s in the case of the cluster of virtual
particles initially seeded in the eddy of the time-dependent (a–d) (see figure 5.6a) and
the time-averaged (e–h) (see figure 5.6d) velocity fields, for: (a,e) the dimensionless
lateral distance between pairs of particles fHEt d computed using (5.10) and (5.11) with
n = 3721 and N = 100; (b,f ) the dimensionless streamwise distance between pairs of
particles fVEt d computed using (5.10) and (5.11) with n = 3721 and N = 100; (c,g) the
ratio of the lateral distance to the streamwise distance between pairs of particles fMEt
computed using (5.10) and (5.11) with n = 3721 and N = 100 (in a log–log plot); and
(d,h) the dimensionless distance between pairs of particles fDEt d computed using (5.10)
and (5.11) with n = 3721 and N = 100.
136
5.4 Analysis of the virtual particles in the jet structures
from the evolution of fDEt for the cluster seeded in the eddy (shown in figure 5.7d).
The increase in the range of fDIt (from approximately 0 ≤ d /d ≤ 10 at t˜ = 0
to 0 ≤ d /d ≤ 55 at t˜ = 98) means that the particles have been spread over a
larger domain very rapidly. Contrary to fDEt , the characteristic profile of fDIt at
t˜ = 0 (which corresponds to a square domain) does not completely change. The
main peak recedes towards d → 0 as time increases, thus suggesting a significant
thinning of the distribution, probably owing to the intense lateral shear at the
interface between the core and the eddy. Note that, in this case, the statistical
study stops at t˜ ≈ 98, which corresponds to the time when the first particle
reaches the top of the visualization window (see figure 5.6b).
We can notice one major difference between the p.d.f.s for the time-dependent
flow field (shown in figures 5.8a–d) and the p.d.f.s for the time-averaged flow field
(shown in figures 5.8e–h). The distribution of the p.d.f.s fHIt , fVIt and fDIt are
smoother for the time-averaged velocity field (plotted in figures 5.8e,f,h, respec-
tively) than for the time-dependent flow field (plotted in figures 5.8a,b,c, respec-
tively). We believe that the jaggedness observed for the time-dependent flow field
is, similarly to the case of the eddy, related to the unstable and turbulent flow of
the shear layer at the interface between the eddy and the core.
In the core
In figures 5.9(a–d), we present the non-dimensional p.d.f. of the lateral distance
fHCt d, the streamwise distance fVCt d, the ratio between the lateral and streamwise
distances between pairs of virtual particles fMCt and the distance fDCtd for the
cluster initially seeded in the core (see figure 5.6c). We plot the p.d.f. against the
non-dimensional variables h /d, v /d, m and d /d in figures 5.9(a–d), respectively.
The p.d.f.s are plotted at four different times, from t˜ = 0 to 98, using different
colours. The colour scale used here is the same as shown in figure 5.6(b). Similarly,
we show in figures 5.9(e–h) the evolution in time of the non-dimensional p.d.f.s
fHCt d, fVCt d, fDCtd and fMCt for the cluster initially seeded at the location of the core
in the time-averaged velocity field (see figure 5.6f ).
The p.d.f. of the lateral distance fHCt , presented in figure 5.9(a), seems to
remain linear until approximately t˜ = 66, and then becomes bimodal at t˜ = 98
with a peak close to h = 0 and the other one near h = 13. The p.d.f. of the
streamwise distance fVCt , shown in figure 5.9(b), decreases approximately linearly
137
5 Two-point statistics for turbulent relative dispersion
Time-dependent flow field
0 2 4 6 8 10 12 14
h /d
0
0.1
0.2
0.3
0.4
f
H I
t
d
t˜ = 0
t˜ = 33
t˜ = 66
t˜ = 98
(a)
Time-averaged mean flow field
0 2 4 6 8 10
h /d
0
0.05
0.10
0.15
0.20
f
H I
t
d
t˜ = 0
t˜ = 33
t˜ = 66
(e)
Time-dependent flow field
0 10 20 30 40 50
v /d
0
0.1
0.2
0.3
0.4
f
V I
t
d
t˜ = 0
t˜ = 33
t˜ = 66
t˜ = 98
(b)
Time-averaged mean flow field
0 5 10 15 20 25 30
v /d
0
0.1
0.2
0.3
0.4
f
V I
t
d
t˜ = 0
t˜ = 33
t˜ = 66
(f )
Time-dependent flow field
10−4 10−2 1 102 104
m
10−4
10−3
10−2
0.1
1
f
M I
t
t˜ = 0 t˜ = 33
t˜ = 66
t˜ = 98
(c)
Time-averaged mean flow field
10−4 10−2 1 10−2 10−4
m
10−4
10−3
10−2
0.1
1
f
M I
t
t˜ = 0 t˜ = 33
t˜ = 66
(g)
Time-dependent flow field
0 10 20 30 40 50
d /d
0
0.05
0.10
0.15
0.20
f
D I
t
d
t˜ = 0
t˜ = 33
t˜ = 66
t˜ = 98
(d)
Time-averaged mean flow field
0 5 10 15 20 25 30
d /d
0
0.05
0.10
0.15
0.20
f
D I
t
d
t˜ = 0
t˜ = 33
t˜ = 66
(h)
Figure 5.8: Comparison between the time evolutions (the colour-scale used is the
same to that used in figure 5.6) of the p.d.f.s in the case of the cluster of virtual
particles initially seeded between an eddy and the core of the time-dependent (a–d)
(see figure 5.6b) and the time-averaged (e–h) (see figure 5.6e) velocity fields, for: (a,e)
the dimensionless lateral distance between pairs of particles fHItd computed using (5.10)
and (5.11) with n = 7381 and N = 100; (b,f ) the dimensionless streamwise distance
between pairs of particles fVItd computed using (5.10) and (5.11) with n = 7381 and
N = 100; (c,g) the ratio of the lateral distance to the streamwise distance between
pairs of particles fMIt computed using (5.10) and (5.11) with n = 7381 and N = 100
(in a log–log plot); and (d,h) the dimensionless distance between pairs of particles fDItd
computed using (5.10) and (5.11) with n = 7381 and N = 100.
138
5.5 Discussion and Conclusion
at all time, with its range increasing only slightly (from 0 ≤ v /d ≤ 5 at t˜ = 0 to
approximately 0 ≤ v /d ≤ 8 at t˜ = 98). The p.d.f. of the ratio between the lateral
and streamwise distances fMCt , shown in figure 5.9(c), moves away to the right of
the axis of symmetry m = 1. Therefore, the distribution stretches in the cross-jet
or lateral direction, conversely to the cluster of particles seeded between the core
and the eddy discussed above. This lateral stretching is probably due to the linear
time-averaged lateral spreading of the jet velocity field with z. Finally, we can
notice that, similarly to fHCt , the p.d.f. of the distance fDCt , shown in figure 5.9(d),
also becomes more and more bimodal with time. The bimodality can be related
to the gradual splitting of the cluster of virtual particles, as it becomes thinner
along the centreline of the jet (see figure 5.6c). This effect must originate from
the divergence of the lateral mean flow along the jet axis.
We point out two minor differences between the p.d.f.s for the time-dependent
flow field (shown in figures 5.9a–d) and the p.d.f.s for the time-averaged flow field
(shown in figures 5.9e–h). Firstly, the range of the p.d.f. fHCt for the time-averaged
velocity field (plotted in figure 5.9e) increases less (from 0 ≤ h /d ≤ 5 at t˜ = 0
to approximately 0 ≤ h /d ≤ 7 at t˜ = 66) than for the time-dependent flow field
(plotted in figure 5.9a). The spurious fluctuations, which can be noticed in the
distribution of fHCt in figure 5.9(e) at t˜ = 33 and 66 (and also, to some extent, in
figures 5.9d,h at t˜ = 0), are due to the discretization issue mentioned previously
(the problem does not occur at t˜ = 0 where we plot the theoretical prediction
defined in (5.22)). Secondly, the bimodality of the distribution of the p.d.f. fDCt
for the time-dependent flow field (shown in figure 5.9d) is not clear for the time-
averaged flow field (shown in figure 5.9h), though it may develop at later time
due to the time-averaged mean diverging lateral velocity near the centreline of
the jet.
5.5 Discussion and Conclusion
In this chapter, we investigate turbulent relative dispersion in the flow field of
quasi-two-dimensional jets using two-point statistics. To obtain the data neces-
sary to compute these two-point statistics, we have developed what we believe
to be a new method which allows us to perform effectively Lagrangian particle
tracking in the turbulent flow of the jets. We use virtual particle tracking, which
139
5 Two-point statistics for turbulent relative dispersion
Time-dependent flow field
0 2 4 6 8 10 12 14 16 18
h /d
0
0.1
0.2
0.3
0.4
f
H C
t
d
t˜ = 0
t˜ = 33
t˜ = 66
t˜ = 98
(a)
Time-averaged mean flow field
0 2 4 6 8 10
h /d
0
0.1
0.2
0.3
0.4
f
H C
t
d
t˜ = 0
t˜ = 33
t˜ = 66
(e)
Time-dependent flow field
0 1 2 3 4 5 6 7 8 9
v /d
0
0.1
0.2
0.3
0.4
f
V C
t
d
t˜ = 0
t˜ = 33
t˜ = 66
t˜ = 98
(b)
Time-averaged mean flow field
0 2 4 6 8 10
v /d
0
0.1
0.2
0.3
0.4
f
V C
t
d
t˜ = 0
t˜ = 33
t˜ = 66
(f )
Time-dependent flow field
10−4 10−2 1 102 104
m
10−4
10−3
10−2
0.1
1
f
M C
t
t˜ = 0 t˜ = 33
t˜ = 66
t˜ = 98
(c)
Time-averaged mean flow field
10−4 10−2 1 102 104
m
10−4
10−3
10−2
0.1
1
f
M C
t
t˜ = 0 t˜ = 33
t˜ = 66
(g)
Time-dependent flow field
0 2 4 6 8 10 12 14 16 18
d /d
0
0.1
0.2
0.3
f
D C
t
d
t˜ = 0
t˜ = 33
t˜ = 66
t˜ = 98
(d)
Time-averaged mean flow field
0 2 4 6 8 10 12
d /d
0
0.1
0.2
0.3
f
D C
t
d
t˜ = 0
t˜ = 33
t˜ = 66
(h)
Figure 5.9: Comparison between the time evolutions (the colour-scale used is the
same to that used in figure 5.6) of the p.d.f.s in the case of the cluster of virtual
particles initially seeded in the core of the time-dependent (a–d) (see figure 5.6c) and
the time-averaged (e–h) (see figure 5.6f ) velocity fields, for: (a,e) the dimensionless
lateral distance between pairs of particles fHCt d computed using (5.10) and (5.11) with
n = 3721 and N = 100; (b,f ) the dimensionless streamwise distance between pairs of
particles fVCt d computed using (5.10) and (5.11) with n = 3721 and N = 100; (c,g) the
ratio of the lateral distance to the streamwise distance between pairs of particles fMCt
computed using (5.10) and (5.11) with n = 3721 and N = 100 (in a log–log plot); and
(d,h) the dimensionless distance between pairs of particles fDCtd computed using (5.10)
and (5.11) with n = 3721 and N = 100.
140
5.5 Discussion and Conclusion
consists of seeding and tracking clusters of virtual particles (or passive tracers) in
experimentally-measured velocity fields. As we discussed in the previous chapter
(see § 4.1.2), there are numerous advantages to using this technique. The spatial
and temporal resolutions are only limited by the resolution of the acquisition tech-
nique used to measure the velocity field.1 Virtual particle tracking can potentially
be applied to any laboratory flows, with a possible range of Reynolds numbers,
Schmidt numbers or Prandtl numbers far exceeding the capabilities of numerical
simulations. A large quantity of virtual particles can be seeded instantaneously
in the flow field, with any arbitrary initial distribution, and then tracked over a
spatial range only limited by the size of the measured velocity field. One could ar-
gue that virtual particle tracking is not adapted to the study of three-dimensional
flow fields. With only a two-dimensional velocity field of a three-dimensional flow
field, it is true that virtual particle tracking cannot give meaningful information,
because the trajectories of real Lagrangian particles are also three-dimensional.
We believe that the recent development of volumetric particle image velocimetry
to measure the three components of the velocity in three-dimensional domains
(see e.g. Kitzhofer et al., 2011; Cierpka & Kaehler, 2012, for recent reviews) can
address this shortcoming.
The flow in quasi-two-dimensional jet is appropriate for the application of par-
ticle tracking velocimetry because the three-dimensionality of the flow can be
considered insignificant in the first order. In § 4.1.2 we report that the mean di-
vergence of the flow is small compared with the mean vorticity. Moreover, Dracos
et al. (1992) found that the flow of quasi-two-dimensional jets is primarily gov-
erned by a two-dimensional inverse cascade of turbulence, except at scales of the
order of (or less than) the gap width of the tankW . Therefore, we believe that par-
ticle tracking velocimetry can give physically meaningful information about the
dispersion in quasi-two-dimensional jets. However, the three-dimensional small-
scale turbulence, typically of the order of W = 1cm or less, cannot be adequately
resolved in this study, with only a two-dimensional velocity field.
Bearing in mind the limited spatial resolution of our data, we have probed the
large-scale dispersion of the (large-scale) eddy and core structures of the flow.
1It can be noted that particle image velocimetry, a common technique to measure veloc-
ity fields, is considered technically less demanding than experimental Lagrangian particles
tracking techniques, such as particle tracking velocimetry or other optical particle tracking
techniques (Kitzhofer, Nonn & Bru¨cker, 2011)
141
5 Two-point statistics for turbulent relative dispersion
The time evolution of the probability distributions of key two-point properties
(such as the lateral distance, the streamwise distance, the Euclidean distance and
the ratio of the lateral distance to the streamwise distance between two points) in
the main structures of quasi-two-dimensional jets has shown different behaviours
for the different parts of the flow. We compare the results of the two-point statis-
tics obtained in the time-dependent velocity field with results obtained in the
time-averaged velocity field of the same jet and results obtained with simple ge-
ometrical distributions of points (a circle, an ellipse and a square). From the
study of these simple geometrical distributions, we have been able to understand
how the variation in time of general shape characteristics of the distribution af-
fects the p.d.f.s of the two-point properties. In particular, we have been able to
measure that, in the eddy, the distribution of particles disperses slowly and in
a rather axisymmetric manner. At the interface between the core and the eddy,
the distribution of particles stretches considerably in the streamwise direction at
a high rate. This is accompanied by thinning of the particle cluster. In the core
of the jet, the particle distribution disperses slowly in the cross-jet direction and
splits along the jet axis. Finally, we believe that the comparison between the
p.d.f.s for the time-averaged flow field and the p.d.f.s for the time-dependent flow
field demonstrates the intense stirring (and potentially the resulting vigorous tur-
bulent mixing) occurring within the eddy and, to some extent, at the interface
between the eddy and the core. This aspect is revealed by the rapid displacement
through time of the peaks in the distribution of fDEt (t) (the time evolution of the
p.d.f. of the distance between two particles initially seeded in the eddy) for the
time-dependent velocity field of the eddy. The chaotic dynamics of the turbulent
flow in the eddy strongly perturbs the distribution of the virtual particles, which
manifests itself in the time evolution of the p.d.f. for the separation distance
between particles.
Future research about the turbulent relative dispersion of the flow of quasi-two-
dimensional jets could investigate the ideas of Richardson (1926) and Batchelor
(1952) to describe the relative dispersion in the jet by a differential equation based
on the p.d.f.s of two-point properties. In Chapter 3, we propose a model for the
transport and streamwise dispersion in the jet, based on the Eulerian description
of the flow. Forming the connection between the Eulerian and the Lagrangian
descriptions of the turbulent dispersion could provide invaluable insight in the
142
5.5 Discussion and Conclusion
physics of anisotropic turbulent processes. One particular question of interest
is to relate the streamwise turbulent eddy diffusivity KdM1/20 z1/2 in the general
effective advection–diffusion equation (3.15) (obtained using a mixing length hy-
pothesis) to the p.d.f. of the streamwise distance between two points obtained
directly from virtual particle tracking, in an effort to identify and parameterize the
cumulative quantitative effect of the complex time-dependent flow on streamwise
dispersion.
Another possible avenue of research would be to improve the spatial resolution
of the velocity field, and perhaps to measure a truly three-dimensional velocity
field of the flow. With a fully resolved velocity field in time (i.e. resolving Kol-
mogorov time scale τηK ≈ 40 ms) and in space (i.e. resolving Kolmogorov length
scale ηK ≈ 0.2 mm), we could explore, for instance, the two-point dispersion
model of Batchelor (1950). As Bourgoin et al. (2006) pointed out, there is a
need for more experimental evidence. A comparison between the results of two-
point statistics for the flow field of quasi-two-dimensional jets with the results for
three-dimensional turbulent flows could shed new light on the physics of turbulent
relative dispersion.
143
Chapter 6
Flow induced by a
quasi-two-dimensional jet in a
confined rectangular domain
6.1 Introduction
A turbulent momentum jet induces a flow towards the jet in the surrounding
ambient fluid. The entrainment of the ambient fluid, which is the result of a
complex turbulent dynamics at the boundary of the jet, was modelled by Morton
et al. (1956). Using dimensional analysis, they related the lateral velocity of
the entrained fluid at the boundary of the jet as simply being proportional to
the time-averaged maximum axial (or streamwise) velocity in the jet. Contrary
to round axisymmetric jets, the velocity of the fluid entrained by a plane jet
does not decay with distance away from the jet axis. The flow induced by plane
145
6 Flow induced by a jet in a confined domain
jets is important in industrial applications such as chemical reactors and mixing
chambers (Jirka & Harleman, 1979). As we discuss in the introduction of Chapter
2 (see § 2.1) rivers flowing into lakes or oceans can be modelled as quasi-two-di-
mensional turbulent jets (Giger et al., 1991; Dracos et al., 1992; Rowland et al.,
2009). Studying the flow induced by rivers emerging into lakes or oceans, Joshi
& Taylor (1983) revealed the impact on the local sediment transport and the
possible coastal erosion.
Taylor (1958) calculated the stream function of the flow in the ambient of
plane jets and axisymmetric jets, when emerging from either a plane wall into a
semi-infinite domain or directly into unbounded space, and for both buoyant and
non-buoyant jets. Assuming an inviscid incompressible potential flow in the am-
bient, using a slip boundary condition at the wall (if present) and modelling the
jet as a distribution of sinks, he solved Laplace’s equation to obtain the stream
function. However, Schneider (1981) demonstrated that the hypothesis of an in-
viscid fluid and the use of a slip boundary condition at the wall gave an incorrect
result for the streamlines in the ambient of axisymmetric jets. Comparing Tay-
lor (1958)’s analytical solution and Schneider (1981)’s numerical solution in the
case of a laminar axisymmetric jet, Zauner (1985) confirmed experimentally the
importance of viscosity in the ambient flow and the need to satisfy the condition
of zero tangential velocity at the wall. Nevertheless, inviscid potential theory
using slip boundary conditions at the walls is still valid in the case of the flow
induced by plane turbulent jets, because the Reynolds number in the ambient
flow is comparable to the jet Reynolds number (Schneider, 1981).
The flow induced by a turbulent jet can also influence the axial momentum
flux of the jet (Kotsovinos, 1978; Schneider, 1985; Kotsovinos & Angelidis, 1991).
According to Kotsovinos & Angelidis (1991), this influence occurs through two
factors. The momentum flux of the induced flow can contribute positively or neg-
atively to the jet momentum flux depending on the angle between the streamlines
of the induced flow and the direction of the jet flow at the jet boundary. The pres-
sure field at the boundary of the jet contributes negatively to the jet momentum
flux. Therefore, predicting the streamlines of the induced flow is important to
determine the rate of change of the jet momentum flux. In the case of a plane jet
emerging from a wall into a semi-infinite domain, the streamlines of the induced
potential flow form two sets of confocal parabolas with axes perpendicular to the
146
6.1 Introduction
jet axis (Taylor, 1958). Thus, the streamlines are opposed to the jet flow and the
jet momentum flux slowly decays with distance. However, the more realistic case
of a plane jet emerging from a wall into a confined domain does not seem to have
been solved in the literature.
The case of a plane jet emerging from a wall into a domain confined in the
axial, lateral and spanwise directions is a common problem, because, in practice,
semi-infinite domains or fully unbounded domains (as assumed by Taylor, 1958;
Schneider, 1981) do not exist. In his model, Schneider (1981) analysed how the
angle between the jet axis and the wall (from which the jet emerged) influences the
streamlines of the induced flow. Revuelta, Sa´nchez & Lin˜a´n (2002) investigated
numerically the case of an axisymmetric laminar jet confined in an axisymmet-
ric domain. They predicted the size and the induced pressure drop of a long
recirculating region surrounding the jet, before the jet expands across the whole
domain. Jirka & Harleman (1979) studied experimentally and theoretically the
stability and mixing of plane jets confined in the axial direction, but unconfined
in the lateral (or cross-stream) direction. For non-buoyant jets, they observed
on both sides of the jet the formation of alternating recirculation cells. The cell
closer to the jet is driven by two distinct mechanisms. As the vertical upward
jet impinges on the free surface at the top, the flow spreads laterally outwards
along the free surface. Along the bottom boundary, the flow is driven inwards
by the jet entrainment process. The size and the total mass flow of the cell are
controlled by the growth characteristics of the jet and the associated entrainment
mechanism. Moreover, Jirka & Harleman (1979) noted that if passive tracers are
injected at the source of the jet, their concentration in the jet increases due to
the recirculation in the cell.
In this chapter, we are interested in the flow induced by a quasi-two-dimen-
sional turbulent jet emerging from a plane wall into a fully confined domain (see
experimental apparatus presented in figure 2.1). In this domain, the distance
between the source and the lateral or axial boundaries is much larger than the
nozzle width, d = 0.5 cm. As we discuss in Chapter 2, the flow in the jet does
not seem to be affected by the streamwise confinement for 0 ≤ z/d ≤ hi/d ≈ 120,
where hi is the height at which the impingement region starts (see § 2.4). For
hi/d ≤ z/d ≤ hf/d = 183 (where hf is the height of the free surface), the flow
experiences a transition as it impinges on the free surface. The vertical upward
147
6 Flow induced by a jet in a confined domain
flow from the jet spreads laterally outwards, symmetrically with respect to the jet
axis. Two overflows, located close to the lateral walls, maintain the free surface
at a constant hf = 91.5 cm. A large portion of the flow spreading along the
free surface recirculates inside the tank as it reaches the lateral boundaries and
produces the counterflow mentioned in § 2.4 (with volume flux Qr). Somewhat
similarly to the recirculation cells observed by Jirka & Harleman (1979), the flow
in our experimental apparatus also displays a recirculation cell on either side of
the jet, but in our case, the recirculation cells are confined laterally by rigid walls.
In this study, we do not model the flow in the impingement region located
directly above the jet nor the recirculation flow at the lateral boundaries near the
free surface (i.e. the region ranging hi ≤ z ≤ hf in the streamwise direction and
spanning the entire domain in the lateral (or x-) direction and the spanwise (or y-)
direction). We only model the flow on the left-hand side of the jet axis (the flow
on the right-hand side can be obtained by symmetry), before the transition from a
jet flow to an impingement flow. We assume that the jet is a distribution of sinks.
The domain of study, which we designate as Ds, ranges 0 ≤ x/d ≤ xj/d = 90
in the lateral or cross jet direction (where xj represents the lateral coordinate of
the jet nozzle, considering the origin of the domain (x = 0, y = 0, z = 0) at the
bottom left-hand-side corner of the tank), 0 ≤ z ≤ hi in the streamwise direction,
and spans the entire domain in the spanwise (or y-) direction, W/2 ≤ y ≤ W/2
(where W = 1 cm is the gap width). We distinguish two aspect ratios in this
study: the aspect ratio of the inner dimensions of the experimental apparatus,
(2xj)/hf ≈ 1; and the aspect ratio of the domain Ds, ζ = xj/hi = 3/4, or the jet
aspect ratio.
In § 6.2, we develop a model of the ambient flow field in the domain Ds using
two-dimensional potential theory. We present the results for the potential field,
the stream function, and the velocity field. In § 6.3, we compare the theoretical
results with results from dyed quasi-two-dimensional turbulent jets and particle
image velocimetry experiments for the stream function, the velocity field, the
volume flux and the momentum flux of the induced flow. We draw our conclusions
in § 6.4.
148
6.2 Potential flow model
6.2 Potential flow model
6.2.1 Description of the entrainment problem
We consider a similar experimental apparatus to that which is depicted in figure
2.1. We model the ambient flow field at the left-hand side of a quasi-two-dimen-
sional turbulent jet using potential theory. We assume a steady laminar plane
flow in the rectangular domain Ds. In Cartesian coordinates (x, z) with the origin
at the bottom left-hand corner of the inside of the experimental apparatus,1, the
domain is bounded at the bottom (z = 0m) and on the left-hand side (x = 0m) by
rigid walls. For the top boundary, we do not wish to consider the impingement
region observed by Jirka & Harleman (1979) near the free surface (located at
z = hf = 0.915 m). So, instead of choosing the free surface as the top boundary,
we choose the height of transition between the jet region and the impingement
region, which is at z = hi ≈ 0.6 m (see § 2.4 and figure 2.4). On the right-hand
side, the domain is delimited by the jet boundary, which we assume to be along
the jet axis (Taylor, 1958) at x = xj ≈ 0.45 m. Although the distance between
the jet boundary and the axis increases with z, we believe that this assumption
is valid because the jet velocity spread rate b(z) (which is of the same order of
magnitude as the jet boundary) is much smaller than the lateral dimension of the
domain at all height: b/xj ≤ 0.2 for 0 ≤ z ≤ hi according to (2.5a) using an
entrainment coefficient α = 0.068. Since we assume a two-dimensional plane flow
in the domain (similarly to the model in § 2.4), we do not consider the boundaries
in the spanwise (or y-) direction in this model.
The domain Ds is delimited by: 0 ≤ x ≤ xj and 0 ≤ z ≤ hi. We use the
following boundary conditions for the velocity field u = (u, w). The normal
velocity vanishes at the walls, u(x = 0, z) = 0 and w(x, z = 0) = 0, and the
slip condition applies for the tangential velocity at the walls (Taylor, 1958). At
z = hi, we assume a uniform constant line source, the flux per unit length is
w(x, z = hi) = −ℓ, (6.1)
where ℓ ≥ 0 is a constant, which is determined below. The boundary condition
(6.1) represents the recirculation of the flow in the experimental apparatus. The
1Note that in the model developed in § 2.4 the origin of the domain is in the middle of the
bottom wall of the experimental apparatus.
149
6 Flow induced by a jet in a confined domain
last boundary condition corresponds to the jet at x = xj. Similarly to Taylor
(1958), we assume that, in the ambient, the influence of the jet can be considered
as a line sink along the z-axis of strength j(z), varying with height,
j(z) = u(x = xj, z) = αwm, (6.2)
where we use the entrainment assumption of Morton et al. (1956), and where
wm is defined by (2.5b) (which assumes a constant momentum flux). Hence, the
strength of the line sink is
j(z) = Kj
√
d√z − z0
, with Kj =
( αM0
2
√
2d
)1/2
, (6.3)
where M0 is the initial momentum of the jet at z = 0, and z0 the space virtual
origin defined in (2.6). By continuity, the volume flux of the line source must
equal the volume flux of the line sink
∫ xj
0
ℓ dx =
∫ hi
0
j(z) dz. (6.4)
Finally, we assume that the flow is irrotational and incompressible in the do-
main. Therefore, the velocity field u = (u, w) derives from a potential ϕ, such
that u = ∇ϕ (where ∇ is the gradient operator), which must satisfy Laplace’s
equation in the domain:
∇2ϕ = 0 for 0 ≤ x ≤ xj , 0 ≤ z ≤ hi. (6.5)
We scale all spatial variables with the height of the impingement region hi, such
that x˜ = x/hi and z˜ = z/hi (where wide tildes denote non-dimensional variables).
We define ζ = xj/hi = 3/4, the jet aspect ratio of the domain Ds corresponding
to our particular experimental problem. (As mentioned previously, we distinguish
the jet aspect ratio ζ from the aspect ratio of the apparatus 2xj/hf ≈ 1.) We
scale velocities with Kj (which is proportional to the streamwise velocity at the
nozzle) defined in (6.3), such that u˜ = u/Kj and w˜ = w/Kj.
We summarize the entrainment problem in figure 6.1. The non-dimensional
150
6.2 Potential flow model
potential ϕ˜ is the solution to Laplace’s equation in the domain Ds:
∇˜
2ϕ˜ = 0 for 0 ≤ x˜ ≤ ζ, 0 ≤ z˜ ≤ 1, (6.6)
subject to the Neumann boundary conditions



∂ϕ˜
∂x˜ = 0 for x˜ = 0, 0 ≤ z˜ ≤ 1,
∂ϕ˜
∂x˜ = j˜ for x˜ = ζ, 0 ≤ z˜ ≤ 1,
∂ϕ˜
∂z˜ = 0 for 0 ≤ x˜ ≤ ζ, z˜ = 0,
∂ϕ˜
∂z˜ = −ℓ˜ for 0 ≤ x˜ ≤ ζ, z˜ = 1
, (6.7)
and with the continuity condition (from (6.4))
∫ 1
0
j˜(z˜) dz˜ = ζℓ˜, (6.8)
where
j˜(z˜) =
(
d˜
z˜ − z˜0
)1/2
, (6.9)
according to (6.3). Therefore, the strength of the line source is, in non-dimensional
form,
ℓ˜ = 2
√
d˜
ζ
(√
1− z˜0 −
√
−z˜0
)
. (6.10)
6.2.2 Decomposition of the problem
To simplify our problem and to eventually improve the convergence of the numeri-
cal calculation of our analytical solution, we split the non-dimensional potential ϕ˜,
by virtue of the superposition principle for linear problems, into two components:
ϕ˜ = ϕ˜u + ϕ˜p, (6.11)
151
6 Flow induced by a jet in a confined domain
x
z
1
∂ϕ˜
∂x˜ = 0
0 ∂ϕ˜
∂z˜ = 0
ζ
∂ϕ˜
∂z˜ = −ℓ˜
∂ϕ˜
∂x˜ = j˜∇˜
2ϕ˜ = 0
Figure 6.1: Description of the entrainment problem of which ϕ˜ is solution.
where ϕ˜u is the solution of a ‘uniform problem’ in the domain Ds, with a non-
dimensional uniform line source of strength ℓ˜ at z˜ = 1 and a non-dimensional
uniform line sink of strength ζℓ˜ at x˜ = ζ, as described in figure 6.2(a), and
where ϕ˜p is defined as a perturbation to this uniform problem. The ‘perturbation
problem’ is represented in figure 6.2(b). In the perturbation problem, we have
no-flux boundary conditions at x˜ = 0, z˜ = 0 and z˜ = 1, and a varying flux at
x˜ = ζ such that
∂ϕ˜p
∂x˜ (x˜ = ζ, z˜) = j˜ (z˜)− ζℓ˜. (6.12)
6.2.3 Solution to the uniform problem ϕ˜u
Solving Laplace’s equation in the domain described in figure 6.2(a), the solution
to the uniform problem, with a uniform line source at z˜ = 1 and a uniform line
sink at x˜ = ζ, is
ϕ˜u =
1
2
(
x˜2 − z˜2
)
for 0 ≤ x˜ ≤ ζ, 0 ≤ z˜ ≤ 1. (6.13)
As a solution of Laplace’s equation under Neumann boundary conditions, ϕ˜u is
the unique solution, to within a constant, to the uniform problem.
In a two-dimensional inviscid and incompressible flow, we can also define a
152
6.2 Potential flow model
(a)
x
z
1
∂ϕ˜u
∂x˜ = 0
0 ∂ϕ˜u
∂z˜ = 0
ζ
∂ϕ˜u
∂z˜ = −ℓ˜
∂ϕ˜u
∂x˜ = ζℓ˜∇˜
2ϕ˜u = 0
(b)
x
z
1
∂ϕ˜p
∂x˜ = 0
0 ∂ϕ˜p
∂z˜ = 0
ζ
∂ϕ˜p
∂z˜ = 0
∂ϕ˜p
∂x˜ = j˜ − ζℓ˜∇˜
2ϕ˜p = 0
Figure 6.2: We decompose the entrainment problem into two problems (see equation
(6.11)): (a) a uniform problem of which ϕ˜u is solution; (b) a perturbation problem of
which ϕ˜p is solution.
stream function ψ such that
∇ψ ·∇ϕ = 0, (6.14)
i.e. the streamlines are orthogonal to the equipotential lines in the domain. The
corresponding non-dimensional stream function ψ˜u for the uniform problem de-
scribed in figure 6.2(a) is
ψ˜u = x˜z˜ for 0 ≤ x˜ ≤ ζ, 0 ≤ z˜ ≤ 1. (6.15)
In figures 6.3(a,b), we show the non-dimensional potential ϕ˜u and the non-
dimensional stream function ψ˜u, respectively, for the uniform problem described
in figure 6.2(a). For the aspect ratio, we use ζ = xj/hi = 3/4, the jet aspect ratio
of our particular case. The flow field in the right-hand-side half of the tank can
be found by symmetry with respect to the jet axis. As we can see in figure 6.3(b),
the streamlines are hyperbolas.
By definition we have u = ∇ϕ, so the velocity field of the uniform problem
u˜u = (u˜u, w˜u) can be derived from the potential ϕ˜u described in (6.13). We find
u˜u = x˜, w˜u = −z˜ for 0 ≤ x˜ ≤ ζ, 0 ≤ z˜ ≤ 1. (6.16a,b)
153
6 Flow induced by a jet in a confined domain
x˜
z˜
0 0.2 0.4 0.6
0
0.2
0.4
0.6
0.8
1.0
(a)
x˜
z˜
0 0.2 0.4 0.6
0
0.2
0.4
0.6
0.8
1.0
(b)
Figure 6.3: (a) Non-dimensional potential ϕ˜u (defined by (6.13)), and (b) non-
dimensional stream function ψ˜u (defined by (6.15)) for the uniform problem described
in figure 6.2(a), using ζ = xj/hi = 3/4.
The velocity components are linear in their coordinate direction and constant in
the orthogonal direction. The non-dimensional velocity field (u˜u, w˜u) is presented
in figures 6.4(a,b), respectively.
x˜
z˜
0 0.2 0.4 0.6
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
0.2
0.4
0.6
0.8
1.0(a)
x˜
z˜
0 0.2 0.4 0.6
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.2
0.4
0.6
0.8
1.0
(b)
Figure 6.4: (a) Non-dimensional lateral velocity u˜u (defined by (6.16a)), and (b)
non-dimensional streamwise velocity w˜u (defined by (6.16b)) for the uniform problem
described in figure 6.2(a), using ζ = xj/hi = 3/4.
154
6.2 Potential flow model
6.2.4 Solution to the perturbation problem ϕ˜p
A solution to the perturbation problem can be found by the method of separation
of variables. The solution ϕ˜p consists of an infinite linear combination of the
product of hyperbolic cosines and cosines, i.e.
ϕ˜p =
∞∑
n=1
An cosh (nπx˜) cos (nπz˜) for 0 ≤ x˜ ≤ ζ, 0 ≤ z˜ ≤ 1, (6.17)
where An are coefficients which can be determined using the non-homogeneous
boundary condition at x˜ = ζ,
∂ϕ˜p
∂x˜
∣∣∣∣
x˜=ζ
=
∞∑
n=1
Annπ sinh (nπζ) cos (nπz˜) = j˜(z˜)− ζℓ˜ for 0 ≤ z˜ ≤ 1, (6.18)
according to (6.12). We define the coefficients Bn such that
Bn = Annπ sinh (nπζ) for n ≥ 1. (6.19)
Re-writing equation (6.18),
∞∑
n=1
Bn cos (nπz˜) = j˜(z˜)− ζℓ˜ for 0 ≤ z˜ ≤ 1, (6.20)
we can clearly see that the coefficients Bn are the Fourier coefficients of an even
function E˜ defined as
E˜(z˜) = j˜(|z˜|)− ζℓ˜ for − 1 ≤ z˜ ≤ 1. (6.21)
Therefore, we can calculate the coefficients Bn as follows
Bn =
∫ 1
−1
E˜(z˜) cos (nπz˜) dz˜ for n ≥ 1, (6.22)
which simplifies to
Bn = 2
∫ 1
0
j˜(z˜) cos (nπz˜) dz˜, (6.23)
because j˜(|z˜|) and cos (nπz˜) are even functions for −1 ≤ z˜ ≤ 1 and ζℓ˜ cos (nπz˜)
integrates to zero in the interval −1 ≤ z˜ ≤ 1 for n ≥ 1. Using equation (6.9), we
155
6 Flow induced by a jet in a confined domain
have
Bn = 2
√
d˜
∫ 1
0
cos (nπz˜)√
z˜ − z˜0
dz˜. (6.24)
Then, applying the transformation q = nπ(z˜ − z˜0) we find
Bn =
2
√
d˜√nπ
(
cos (nπz˜0)
∫ npi(1−z˜0)
−npiz˜0
cos q√q dq − sin (nπz˜0)
∫ npi(1−z˜0)
−npiz˜0
sin q√q dq
)
.
(6.25)
Finally, we can apply another transformation s =
√
2q/π, which gives
Bn =
2
√
2d˜√n
(
cos (nπz˜0)
[
C(y)
]√2n(1−z˜0)
√
−2nz˜0
− sin (nπz˜0)
[
S(y)
]√2n(1−z˜0)
√
−2nz˜0
)
(6.26)
for n ≥ 1, where we have introduced the Fresnel C and S integrals (see e.g.
Abramowitz & Stegun, 1972) defined as
C(y) =
∫ y
0
cos
(π
2
s2
)
ds and S(y) =
∫ y
0
sin
(π
2
s2
)
ds. (6.27)
It can be noted that the coefficient B0 equals zero from the condition of continuity
stated in equation (6.8). Therefore, we have found a unique solution ϕ˜p (defined
by equations (6.17), (6.19) and (6.26)) to the perturbation problem described
in figure 6.2(b). According to (6.14), the corresponding non-dimensional stream
function ψ˜p for the perturbation problem described in figure 6.2(b) is
ψ˜p =
∞∑
n=1
An sinh (nπx˜) sin (nπz˜) for 0 ≤ x˜ ≤ ζ, 0 ≤ z˜ ≤ 1, (6.28)
where the coefficients An are given by (6.19) and (6.26).
We present in figures 6.5(a,b) the non-dimensional potential ϕ˜p and the non-
dimensional stream function ψ˜p, respectively, for the perturbation problem de-
scribed in figure 6.2(b). We compute the series ϕ˜p (defined by equations (6.17),
(6.19) and (6.26)) and ψ˜p (defined by equations (6.28), (6.19) and (6.26)) numer-
ically for nmax = 100, the (finite) number of terms of both series. We use the jet
aspect ratio ζ = xj/hi = 3/4 of our particular case, and the space virtual origin
z0 = −4.7 d (computed from (2.6) using α = 0.068, < M > /
(
Q02/d
)
= 0.55).
Similarly to u˜u, the velocity field of the perturbation problem u˜p = (u˜p, w˜p) can
156
6.2 Potential flow model
x˜
z˜
0 0.2 0.4 0.6
0
0.2
0.4
0.6
0.8
1.0
(a)
x˜
z˜
0 0.2 0.4 0.6
0
0.2
0.4
0.6
0.8
1.0
(b)
Figure 6.5: (a) Non-dimensional potential ϕ˜p (defined by equations (6.17), (6.19)
and (6.26) with nmax = 100 the number of terms of the series, and z0 = −4.7 d the
space virtual origin), and (b) non-dimensional stream function ψ˜p (defined by (6.28),
(6.19) and (6.26) with nmax = 100, the number of terms of the series, and z0 = −4.7 d
the space virtual origin) for the perturbation problem described in figure 6.2(b), using
ζ = xj/hi = 3/4.
be derived from the potential ϕ˜p defined by equations (6.17), (6.19) and (6.26).
We find for 0 ≤ x˜ ≤ ζ, 0 ≤ z˜ ≤ 1,
u˜p =
∞∑
n=1
nπAn sinh (nπx˜) cos (nπz˜), (6.29)
w˜p = −
∞∑
n=1
nπAn cosh (nπx˜) sin (nπz˜), (6.30)
where the coefficients An are given by (6.19) and (6.26). We present the non-
dimensional velocities u˜p and w˜p for the perturbation problem in figures 6.6(a,b),
respectively. Similarly to ϕ˜p and ψ˜p, we compute the series u˜p (defined by equa-
tions (6.29), (6.19) and (6.26)) and w˜p (defined by equations (6.30), (6.19) and
(6.26)) numerically for nmax = 100, the number of terms of both series. We use
the aspect ratio ζ = xj/hi = 3/4, and the space virtual origin z0 = −4.7 d (com-
puted from (2.6) using α = 0.068, < M > /
(
Q02/d
)
= 0.55). We note that both
the lateral and the streamwise velocities are strongly affected by the origin of the
157
6 Flow induced by a jet in a confined domain
x˜
z˜
0 0.2 0.4 0.6
-0.05
0
0.05
0.1
0.15
0.2
0.25
0
0.2
0.4
0.6
0.8
1.0
(a)
x˜
z˜
0 0.2 0.4 0.6
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.2
0.4
0.6
0.8
1.0
(b)
Figure 6.6: (a) Non-dimensional lateral velocity u˜p (defined by (6.29), (6.19) and
(6.26) with nmax = 100 the number of terms of the series, and z0 = −4.7 d the space
virtual origin), and (b) non-dimensional streamwise velocity w˜p (defined by (6.30),
(6.19) and (6.26) with nmax = 100 the number of terms of the series, and z0 = −4.7 d
the space virtual origin) for the perturbation problem described in figure 6.2(b), using
ζ = xj/hi = 3/4.
line sink at (x˜ = ζ, z˜ = 0). This is due to the singularity at the virtual origin
z˜ = z˜0 < 0. We can see in figure 6.6(a) that the lateral velocity u˜p is maximum
at the bottom right-hand corner, (x˜ = ζ, z˜ = 0). In figure 6.6(b), we can see that,
along the right-hand side boundary x˜ = ζ, the streamwise velocity w˜p sharply
decreases from the bottom boundary z˜ = 0 to approximately z˜ ≈ 0.1. Then,
w˜p slowly increases again for z˜ > 0.1, to eventually vanish at the top boundary
z˜ = 1.
Since there is no flux inwards or outwards at the boundaries x˜ = 0, z˜ = 0 and
z˜ = 1, by continuity, the total integrated flux along the right-hand side boundary
at x˜ = ζ must also be zero,
∫ 1
0
∂ϕ˜p
∂x˜ (x˜ = ζ, z˜) dz˜ = 0, (6.31)
according to equation (6.8) and (6.12), thus
∂ϕ˜p
∂x˜ (x˜ = ζ, z˜) =
√
d˜
(
1√
z˜ − z˜0
− 2
(√
1− z˜0 −
√
−z˜0
))
, (6.32)
according to (6.12), with j˜(z˜) defined by (6.9) and ℓ˜ defined by (6.10). There is
158
6.2 Potential flow model
a local non-uniform flux along the right-hand side boundary at x˜ = ζ oriented
inwards for z ≥ z˜c and outwards for z ≤ z˜c, where z˜c is defined by
∂ϕ˜p
∂x˜ (x˜ = ζ, z˜c) = 0. (6.33)
Hence, the non-dimensional height z˜c is
z˜c =
1
4
(√
1− z˜0 −
√
−z˜0
)2 + z˜0. (6.34)
We show in figure 6.7 the distribution of the non-dimensional perturbation flux
∂ϕ˜p/∂x˜(x˜ = ζ, z˜) = j˜(z˜)− ζℓ˜ along the right-hand side boundary. The analytical
formula (6.32), for the flux along the right-hand side boundary, is plotted with
a solid curve. We can see the relative importance of the line sink and the line
source. The varying line sink j˜(z˜) is stronger than the line source ζℓ˜ and oriented
outwards (i.e. in the positive direction) for z˜ < z˜c (where z˜c ≈ 0.33, defined
in (6.34), is marked with dashed lines). The flux ∂ϕ˜p/∂x˜(x˜ = ζ, z˜) increases
steeply approaching the origin, i.e. as z˜ → 0, as expected near the singularity at
z˜ = z˜0 < 0. For z˜ > z˜c, the uniform line source dominates and the local flux is
oriented inwards (i.e. in the negative direction). The exchange flow at the right-
hand-side boundary is also depicted clearly by the distribution of the streamlines
shown in figure 6.5(b).
We compute the potential ϕ˜p (defined in (6.17)), the stream function ψ˜p (defined
in (6.28)), and the velocity field u˜p = (u˜p, w˜p) (defined in (6.29) and (6.30),
respectively) numerically and we truncate their infinite series to a finite number
of terms nmax. In order to test the accuracy of the numerical computation of these
truncated series, we compare the numerical computation of the truncated series of
the lateral velocity along the right-hand side boundary, designated by u˜nmaxp (x˜ =
ζ, z˜) for the first nmax terms of the series (6.29), with the analytical formula (6.32)
of the flux imposed at the same boundary, the perturbation flux ∂ϕ˜p/∂x˜(x˜ = ζ, z˜).
We measure the mismatch (introduced by the truncation) between the truncated
series and the analytical formula by calculating their standard deviation, as a
function of the number of terms of the series nmax. The standard deviation is
159
6 Flow induced by a jet in a confined domain
z˜
ϕ˜p
∂x˜ (z˜)
Analytical formula (6.32)
Numerical truncated series (6.36)
z˜c = 0.33
-0.1 0 0.1 0.2 0.3
0
0.2
0.4
0.6
0.8
1.0
Figure 6.7: Distribution of the non-dimensional flux along the right-hand side bound-
ary. The analytical formula of the imposed boundary condition, ∂ϕ˜p/∂x˜(x˜ = ζ, z˜)
defined by (6.32), is plotted with a solid curve. We plot the numerical truncated se-
ries of the flux u˜nmaxp (defined in (6.36), with nmax = 100 the number of terms of the
truncated series, and z0 = −4.7 d the space virtual origin) with pluses. The location z˜c
(defined by (6.34)), where the flux vanishes and changes sign, is marked with dashed
lines.
defined, in discrete form, by
σp(nmax) =
(
1
Nz
Nz∑
i=1
(
u˜nmaxp (ζ, z˜i)−
∂ϕ˜p
∂x˜ (ζ, z˜i)
)2)1/2
, (6.35)
where z˜i are linearly distributed from z˜1 = 0 to z˜Nz = 1, with Nz = 1001 the
discretization number, and
u˜nmaxp (ζ, z˜i) =
nmax∑
n=1
Bn cos (nπz˜i), for all 1 ≤ i ≤ Nz, (6.36)
according to (6.29) and (6.19), where the coefficients Bn are described by (6.26)
for n ≥ 1.
We plot in figure 6.8(a) the standard deviation σp (defined in (6.35)) against
160
6.2 Potential flow model
nmax. We can see that σp decreases rapidly and monotonically as nmax increases.
Thus, the numerical truncated series u˜nmaxp (ζ, z˜i) converges rapidly towards the
analytical formula for ∂ϕ˜p/∂x˜(ζ, z˜i). We find that for nmax = 100 the standard
deviation is very small, σp ≤ 0.1 %. As a comparison with the analytical formula
(6.32) for the perturbation flux ∂ϕ˜p/∂x˜(x˜ = ζ, z˜), we plot in figure 6.7 the numer-
ical truncated series u˜nmaxp (ζ, z˜) (defined in (6.36), for nmax = 100) with pluses.
As expected, the match is excellent.
To study the smoothness of the (non-truncated) Fourier series u˜∞p (ζ, z˜), we
present in a log–log plot in figure 6.8(b) the coefficients of the series Bn (described
by (6.26) and shown with pluses), for 1 ≤ n ≤ 200. As we can observe, the
coefficients Bn appear to fall off like O(1/n2) (the function 1/n2 is plotted with
a red line) rather than O(1/n) (the function 1/n is plotted with a black line). A
decrease of O(1/n2) means that the Fourier series u˜∞p (ζ, z˜) is continuous while its
first derivative (with respect to z˜) is discontinuous over the interval 0 ≤ z˜ ≤ 1.
The Fourier series u˜∞p (ζ, z˜) (defined in (6.36), with nmax = ∞) converges precisely
to the even continuous function E(z˜) = j˜(|z˜|) − ζℓ˜ (except perhaps on a set of
measure zero, see e.g. Ko¨rner, 1988) defined in the periodic interval −1 ≤ z˜ ≤ 1,
according to (6.20) and (6.21). The function E is continuous over this interval,
but its first derivative is discontinuous at z˜ = 0 mod Tp = 2 (where Tp is the
period of the function E) owing to the absolute value, and at z˜ = 1 mod Tp by
construction of the periodic function E. Since the Fourier series u˜∞p (ζ, z˜) also
satisfies Dirichlet’s conditions (see e.g. Kahane & Lemarie´-Rieusset, 1998), then
it converges to the analytical formula (6.32) for the right-hand side boundary
condition ∂ϕ˜p/∂x˜(x˜ = ζ, z˜), for all point 0 ≤ z˜ ≤ 1, with a smoothness of order
2.
We believe that the numerical computations of the truncated series for the
potential ϕ˜p (defined in (6.17)), the stream function ψ˜p (defined in (6.28)), and
the velocity field u˜p = (u˜p, w˜p) (defined in (6.29) and (6.30), respectively) should
all be sufficiently accurate with nmax = 100.
6.2.5 Solution to the entrainment problem
According to the superposition principle (6.11), we can combine the potential for
the uniform solution ϕ˜u (defined by (6.13)) with the potential for the perturbation
solution ϕ˜p (defined by (6.17), (6.19) and (6.26)). We find a unique analytical
161
6 Flow induced by a jet in a confined domain
nmax
σ p
0 50 100 150 200
0
0.01
0.02
0.03
0.04
0.05
(a)
n
B
n
Coefficients Bn
1/n2
1/n
1 10 100
10−4
10−2
1
(b)
Figure 6.8: (a) Standard deviation σp (defined in (6.35)) between the truncated se-
ries u˜nmaxp (ζ, z˜) (defined in (6.36)) and the analytical formula ∂ϕ˜p/∂x˜(ζ, z˜i) (defined
in (6.32)). (b) Log-log plot of the coefficients Bn (defined in (6.26) and plotted with
pluses) of the series u˜nmaxp (ζ, z˜) (defined in (6.36)) versus n. The function 1/n is plotted
with a black curve and the function 1/n2 is plotted with a red curve.
solution, to within a constant, for the potential of the entrainment problem
ϕ˜ = 1
2
(
x˜2 − z˜2
)
+
∞∑
n=1
An cosh (nπx˜) cos (nπz˜) for 0 ≤ x˜ ≤ ζ, 0 ≤ z˜ ≤ 1.
(6.37)
where An are given by (6.19) and (6.26). Again, applying the superposition
principle, the corresponding stream function is
ψ˜ = x˜z˜ +
∞∑
n=1
An sinh (nπx˜) sin (nπz˜) for 0 ≤ x˜ ≤ ζ, 0 ≤ z˜ ≤ 1, (6.38)
where the coefficients An are given by (6.19) and (6.26).
We show in figures 6.9(a,b) the non-dimensional potential ϕ˜ (defined by (6.37),
(6.19) and (6.26)) and the non-dimensional stream function ψ˜ (defined by (6.38),
(6.19) and (6.26)) for the entrainment problem described in figure 6.1. We com-
pute the series ϕ˜ and ψ˜ numerically for nmax = 100, the number of terms of both
series. Again, we use the jet aspect ratio ζ = xj/hi = 3/4 of our particular case,
and the space virtual origin z0 = −4.7 d (computed from (2.6) using α = 0.068,
< M > /
(
Q02/d
)
= 0.55). The streamlines are very similar to those found in the
uniform problem in figure 6.3(b). The difference is that they are slightly steeper
along the right-hand-side boundary, probably due to the singularity at the virtual
origin z˜0 of the line sink. In figure 6.9(b), we also plot with red curves the Taylor’s
162
6.2 Potential flow model
solution for the streamlines ψ˜T (Taylor, 1958)
ψ˜T =
(√
x˜2 + z˜2 − z˜
)1/2
for 0 ≤ x˜ ≤ ζ, 0 ≤ z˜ ≤ 1. (6.39)
The stream function ψ˜T corresponds to the two-dimensional incompressible and
irrotational flow induced by a plane jet emerging from a plane wall (at x˜ = ζ) into
a semi-infinite domain. As we can observe in figure 6.9(b), the streamlines pre-
dicted by Taylor (1958) are qualitatively different from our solution. The stream
function ψ˜T produces concave streamlines, whereas our solution ψ˜ produces con-
vex streamlines. The discrepancy between Taylor’s stream function ψ˜T and our
stream function ψ˜ is due to the fact that we consider a fully confined domain,
which induces a recirculation flow on either side of the jet, whereas Taylor (1958)
considered fully unbounded domains or the case of a jet emerging from a wall
into a semi-infinite domain, thus ignoring the possibility of recirculation in the
ambient flow. Nevertheless, the streamlines of both stream functions are pointing
downwards, i.e. opposite to the jet direction.
The velocity field for the entrainment problem is, for 0 ≤ x˜ ≤ ζ, 0 ≤ z˜ ≤ 1,
u˜ = x˜+
∞∑
n=1
nπAn sinh (nπx˜) cos (nπz˜), (6.40)
w˜ = −z˜ −
∞∑
n=1
nπAn cosh (nπx˜) sin (nπz˜), (6.41)
where the coefficients An are given by (6.19) and (6.26).
We show in figures 6.10(a,b) the non-dimensional lateral velocity u˜ (defined by
(6.40), (6.19) and (6.26)) and the non-dimensional streamwise velocity w˜ (defined
by (6.41), (6.19) and (6.26)) for the entrainment problem described in figure 6.1.
We compute the series u˜ and w˜ numerically for nmax = 100, the number of terms of
both series. We use the aspect ratio ζ = xj/hi = 3/4, and the space virtual origin
z0 = −4.7 d (computed from (2.6) using α = 0.068, < M > /
(
Q02/d
)
= 0.55). As
we can see, the velocity field u˜ is very similar to the velocity field of the uniform
problem u˜u presented in figure 6.4(a,b). The influence of the varying line sink
mainly appears in its vicinity (i.e. along the right-hand-side boundary at x˜ = ζ).
The perturbation of the velocity field is stronger near the source of the jet due to
the singularity at the virtual origin (x˜ = ζ, z˜ = z˜0).
163
6 Flow induced by a jet in a confined domain
x˜
z˜
0 0.2 0.4 0.6
0
0.2
0.4
0.6
0.8
1.0
(a)
x˜
z˜
0 0.2 0.4 0.6
0
0.2
0.4
0.6
0.8
1.0
(b)
Figure 6.9: (a) Non-dimensional potential ϕ˜ (defined by equations (6.37), (6.19) and
(6.26) with nmax = 100 the number of terms of the series, and z0 = −4.7 d the space
virtual origin), and (b) non-dimensional stream function ψ˜ (defined by (6.38), (6.19)
and (6.26) with nmax = 100, the number of terms of the series, and z0 = −4.7 d
the space virtual origin) for the entrainment problem described in figure 6.1, using
ζ = xj/hi = 3/4. We also plot with red curves in (b) the Taylor’s solution (defined by
(6.39)) for the streamlines of a flow induced by a plane jet emerging from a plane wall
(at x˜ = ζ) into a semi-infinite domain (Taylor, 1958).
The aspect ratio ζ and the virtual origin z0 influence the flow field of the
full problem only through the perturbation part of the problem. The aspect
ratio ζ and the virtual origin z0 appear only in the coefficients An of the series,
in (6.19) and in (6.26) respectively. We find that, qualitatively, increasing the
aspect ratio (i.e. stretching the domain Ds in the x-direction) tends to ‘stretch’
the streamlines in the lateral direction throughout the domain and decrease the
angle (with respect to the x-axis) made by the streamlines at the right-hand side
boundary (i.e. the ambient fluid enters the jet more perpendicular to the jet axis).
Decreasing the aspect ratio produces the opposite effects. Changing the virtual
origin has only a local impact along the right-hand side boundary. An increase
in |z0| (i.e. the virtual origin is further away below the right-hand side bottom
corner) diminishes the influence of the singularity on the flow field, decreases the
strength of the sink line, and thus decreases the angle (with respect to the x-axis)
made by the streamlines at the right-hand side boundary. Decreasing |z0| (i.e.
164
6.3 Experimental results
x˜
z˜
0 0.2 0.4 0.6
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
0.2
0.4
0.6
0.8
1.0(a)
x˜
z˜
0 0.2 0.4 0.6
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.2
0.4
0.6
0.8
1.0
(b)
Figure 6.10: (a) Non-dimensional lateral velocity u˜ (defined by (6.40), (6.19) and
(6.26) with nmax = 100 the number of terms of the series, and z0 = −4.7 d the space
virtual origin), and (b) non-dimensional streamwise velocity w˜ (defined by (6.41), (6.19)
and (6.26) with nmax = 100 the number of terms of the series, and z0 = −4.7 d the
space virtual origin) for the entrainment problem described in figure 6.1, using ζ =
xj/hi = 3/4.
the virtual origin is closer to the right-hand side bottom corner), has the opposite
effect.
Our model, and in particular the assumption of a uniform flux at the top
boundary, cannot apply for all ranges of aspect ratios. In the case of a very large
aspect ratio, Jirka & Harleman (1979) observed alternating recirculation cells on
either side of the jet. Thus, the assumption of a uniform flux at the top boundary
seems to be incorrect for approximately ζ > 3, from the measurements of the
size of the primary recirculation cells made by Jirka & Harleman (1979). In the
case of a small aspect ratio, we believe that the width of the jet (whose boundary
expands at a rate of approximately 0.22 from the z-axis) could also affect the flux
at the top boundary for ζ < 2/3.
6.3 Experimental results
6.3.1 Experimental procedure
The experimental procedure is almost identical to the experimental procedure
described in § 2.2. The main difference concerns the location of the PIV study
areas. To investigate the flow induced by the jet, we choose two PIV study areas
165
6 Flow induced by a jet in a confined domain
x
z
d = 5 mm
Constant-head tank
Overflow
H
=
1
m
L = 1 m
W = 0.01 m
u
w
2b(z)
CCD camera
Red filter
(dyed jets)
Study area
4 (PIV)
Study area
5 (PIV)
Figure 6.11: Schematic diagram of the experimental apparatus. The two PIV study
areas are shown with dashed lines along the left-hand-side inner wall.
located on the left-hand-side of the jet axis along the left-hand side inner wall,
as shown in figure 6.11. Study area 4 covers a height from z = 0.425 to 0.85 m,
while study area 5 covers a height from z = 0 to 0.425m. Both study areas cover
a width from x = 0 to 0.425 m. We analyse three steady turbulent jets in each
study area. In study area 4, the jet flow rates are 32.2, 36.8 and 40.3 cm3 s−1 .
The corresponding jet Reynolds number (based on the jet source characteristics,
such that Rej = dws/ν) are in the range 3220 ≤ Rej ≤ 4030. In study area
5, the jet flow rates are 33.5, 37.5 and 40.3 cm3 s−1 . The corresponding jet
Reynolds number are in the range 3350 ≤ Rej ≤ 4030. The frequency of image
acquisition is set at 60 frames per second for both study areas. The duration of
every experiment is approximately 91 s.
6.3.2 Qualitative observations
In figure 6.12(a), a superposition of 20 images (i.e. a duration of 0.33 s) of
the filming of two experiments, where passive tracers (0.23 mm Pliolite VTAC
particles) were mixed with a quasi-two-dimensional jet (Rej = 4030), depicts the
tracers as streaks. One experiment is filmed in the PIV study area 5 (see figure
166
6.3 Experimental results
6.11) located at 0 ≤ x/d ≤ 85, 0 ≤ z/d ≤ 85. The other experiment is filmed in
the PIV study area 4 (see figure 6.11) located at 0 ≤ x/d ≤ 85, 85 ≤ z/d ≤ 170.
In figure 6.12(a), we can visualize the recirculation of the flow on the left-hand
side of the jet. Along the right-hand side border of the picture (x/d ≈ 80), we
can see three characteristic eddies (located at approximately z/d = 45, z/d = 75
and z/d = 110) of the flow in a quasi-two-dimensional jet (the axis of the jet,
not visible in the experiment, is plotted with a dot-dashed line at x = xj.). The
top eddy (z/d = 110) is close to the boundary between the jet region and the
impingement region, shown with a red dashed line at z = hi = 120 d. As it
approaches the free surface (not visible in this experiment, but identified with a
black dashed line at z = hf ), the flow spreads laterally. The flow is eventually
redirected downwards along the wall at x = 0, before being re-entrained by the
jet. The location of the transition height hi = 120 d, between the jet region
and the impingement region, was chosen in § 2.4 because we found that the jet
was no longer self-similar beyond this height. In figure 6.12(a), we can see that
hi also corresponds approximately to the location of the last eddy of the quasi-
two-dimensional jet. We actually have not seen eddies beyond this height. The
size of the eddies approaching the height hi is around 30 d. This size is still
small compared with the distance between the left-hand-side wall and the jet
axis, xj/d = 90, but the eddy may start becoming affected by the left-hand-side
wall. Jirka & Harleman (1979) found a transition height between the jet region
and the impingement region at approximately 85 % of the total depth hf . In
our experiment, we find that the ratio between the transition height and the free
surface is hi/hf ≈ 65 %. We believe that this discrepancy is due to the lateral
confinement, which was insignificant in the experiments of Jirka & Harleman
(1979).
In figure 6.12(b), we show a picture of a quasi-two-dimensional turbulent dyed
jet (Rej ≈ 4000) in the region delimited by −40 ≤ (x − xj)/d ≤ 40 and 0 ≤
z/d ≤ 100. The dye was injected at a constant rate for approximately 3 s in
a steady jet, following the experimental procedure detailed in § 2.2.1. We took
the picture approximately 4.2 s after the injection of the dye, hence the brighter
intensity of the non-dyed fluid in the jet compared with the ambient flow. The
dye streaks outside the jet reveal the re-entrainment process of the flow owing
to the recirculation in the domain. As Jirka & Harleman (1979) noted, the re-
167
6 Flow induced by a jet in a confined domain
entrainment process leads to an increase in dye concentration in the jet. The
boundary between the jet and the ambient fluid can be identified by the sharp
contrast between the light intensity in the induced ambient flow and the light
intensity in the turbulent jet flow. We find in figure 2.4 that the location of
the average dye edge of quasi-two-dimensional jets is a linear function of height
with a slope equal to 0.22 from the z-axis (calculated for 20 ≤ z/d ≤ 120), this
corresponds to an average angle of approximately 12◦ from the z-axis. Kotsovinos
(1978) also defined the jet boundary as the separation between the turbulent jet
flow and the ambient flow field. He too used the sharp contrast between dyed
jets and the ambient fluid to determine the boundaries of plane turbulent jets.
In figure 2.14, we can observe that the location of the jet boundary is almost
at the location x0(z), where the time-averaged streamwise velocity vanishes and
changes sign. It means that, in average, the flow velocity is purely lateral at
the jet boundary. Therefore, we choose x0(z) ≈ 0.22z as the location of the jet
boundary in this study. We can also see that, at the jet boundary, the streamlines
of the induced ambient flow are oriented in the opposite direction to the jet flow,
thus producing a counterflow which can affect the momentum flux of the jet
(Kotsovinos & Angelidis, 1991).
6.3.3 Quantitative results
In figure 6.13, we compare the streamlines predicted theoretically in equations
(6.38), (6.19) and (6.26) (plotted with solid curves) with the streamlines of an
ensemble-averaged experimental flow field (plotted with dotted curves). The ex-
perimental flow field is the ensemble average of the time-averaged flow fields of
the three jets studied (see § 6.3.1). Both the experimental and theoretical stream-
lines start at the same locations along the top boundary of the PIV study area
5, i.e. at z/d = 85. As we can see, the theory and the data agree in the far field
away from the jet axis. The streamlines of the experimental data, which point
downwards in the ambient fluid, change direction at the boundary of the jet to
point upwards.
In figure 6.14(a), we compare the time-averaged lateral velocity predicted the-
oretically in equations (6.40), (6.19) and (6.26) (plotted with solid curves) with
the time-averaged lateral velocity (plotted with dotted curves) of the ensemble-
averaged experimental flow field shown in figure 6.13. We show the experimen-
168
6.3 Experimental results
0
20
40
60
80
100
120
140
160
hf
d
z/
d
0 20 40 60 80 xj
dx/d
(a) (b)
Figure 6.12: (a) Passive tracers (Pliolite particles) shown as streaks in a typical jet
(Rej = 4030). The axis of the jet is plotted with a dot-dashed line at x = xj . The
free surface is plotted with a black dashed line at z = hf . The transition between the
jet region and the impingement region is shown with a red dashed line at z = hi. (b)
Grey-scale picture of a turbulent quasi-two-dimensional dyed jet (Rej ≈ 4000) and the
induced flow in the region delimited by −40 ≤ (x− xj)/d ≤ 40 and 0 ≤ z/d ≤ 100.
tal and theoretical lateral distributions of the lateral velocity at four different
heights (indicated by dashed lines) in the PIV study area 5 (0 ≤ x/d ≤ 85,
0 ≤ z/d ≤ 85). We normalize the time-averaged lateral velocity u with the maxi-
mum time-averaged streamwise velocity wmax measured in the domain (at height
z/d ≈ 75, see figure 6.14b). The normalized velocity u/wmax is then scaled so
that the maximum amplitude (i.e. wmax/wmax = 1) matches the z-separation
between two heights of measurement (shown with dashed lines in figures 6.14a).
Similarly to the streamlines, the theory and the data agree in the far field away
from the jet boundary, except for the lowest curve at z/d = 10. We believe that
the difference at z/d = 10 is due to the fact that the jet is very close to the source,
and thus the flow of the quasi-two-dimensional jet (as well as its induced flow) is
not yet established (see § 2.4).
169
6 Flow induced by a jet in a confined domain
0 10 20 30 40 50 60 70 80
x/d
0
20
40
60
80
100
z/
d
Data
Theory
Figure 6.13: Experimental (dotted curves) and theoretical (solid curves) distributions
of the time-averaged streamlines of the flow induced by quasi-two-dimensional jets in
the PIV study area 5, 0 ≤ x/d ≤ 85, 0 ≤ z/d ≤ 85 (see equations (6.38), (6.19) and
(6.26) for the theoretical curve).
In figure 6.14(b), we compare the time-averaged streamwise velocity predicted
theoretically in equations (6.41), (6.19) and (6.26) (plotted with solid curves)
with the time-averaged streamwise velocity (plotted with dotted curves) of the
ensemble-averaged experimental flow field shown in figure 6.13. We show the ex-
perimental and theoretical lateral distributions of the lateral velocity at four dif-
ferent heights (indicated by dashed lines) in the PIV study area 5 (0 ≤ x/d ≤ 85,
0 ≤ z/d ≤ 85). Similarly to the time-averaged lateral velocity, we normal-
ize the time-averaged streamwise velocity w with the maximum time-averaged
streamwise velocity wmax measured in the domain (at height z/d ≈ 75). The
normalized velocity w/wmax is then scaled so that the maximum amplitude (i.e.
wmax/wmax = 1) matches the z-separation between two heights of measurement
(shown with dashed lines in figures 6.14b). The theory seems to agree with the
data in the far field away from the jet boundary at least to leading order. How-
ever, we can see that the experimental time-averaged streamwise velocity is not
exactly uniform across the study area. In particular, the experimental streamwise
velocity vanishes along the right-hand-side boundary at x = 0, contrary to the
theoretical streamwise velocity which assumes a slip-boundary condition.
In figure 6.15(a), we plot the non-dimensional time-averaged volume fluxQr/Q0
170
6.3 Experimental results
0 1 20 30 40 50 60 70 80
x/d
0
20
40
60
80
100
z/
d
Data
Theory
(a)
0 10 20 30 40 50 60 70 80
x/d
0
20
40
60
80
100
z/
d
Data
Theory
(b)
Figure 6.14: Experimental (dotted curves) and theoretical (solid curves) lateral distri-
butions of the time-averaged velocity field of the flow induced by quasi-two-dimensional
jets in the PIV study area 5 (0 ≤ x/d ≤ 85, 0 ≤ z/d ≤ 85) at four different heights (plot-
ted with dashed lines) for: (a) the normalized time-averaged lateral velocity u/wmax
(see equations (6.40), (6.19) and (6.26) for the theoretical curve), with wmax the max-
imum time-averaged streamwise velocity measured in the domain (at height z/d ≈ 75,
see b); (b) the normalized time-averaged streamwise velocity w/wmax (see equations
(6.41), (6.19) and (6.26) for the theoretical curve).
of the return flow (combining the two sides of the jet) versus non-dimensional
height z/d. The experimental data, plotted with a dotted curve, are computed
from the streamwise velocity of the ensemble-averaged experimental flow field
shown in figure 6.14(b), such that
Qr,exp(z) = −2
∫ xj−x0(z)
0
w(x, z) dx, (6.42)
where x0(z) ≈ 0.22z defines the location where w = 0, as discussed in § 6.3.2.
The first theoretical prediction, plotted with a solid curve, is computed using the
streamwise velocity predicted by potential theory and defined in equations (6.41),
(6.19) and (6.26), such that
Qr,pot(z) = −2
∫ xj
0
w(x, z) dx. (6.43)
Note the difference between the top boundaries of the integral (6.42), where we
choose the boundary of the jet, and the integral (6.43), where we choose the jet
axis (because the jet is modelled as a line sink). Using equation (6.41), we find
that the volume flux Qr,pot increases linearly with distance z. However, equation
171
6 Flow induced by a jet in a confined domain
(6.43) is valid only in the region where the jet is self-similar and the momentum
flux of the jet is conserved, i.e. for z ≤ hi. The second theoretical prediction,
plotted with a dashed curve, is computed using conservation of volume flux at
every height. By continuity, the downward volume flux of the return flow on both
sides of the jet Qr,cont is equal to the upward volume flux of the jet Q (defined in
(2.4b)) minus the source volume flux Q0, at every height: Qr,cont(z) = Q(z)−Q0.
We find
Qr,cont(z) = Q0
(
4
√
2αM0zQ02
+ 1
)1/2
−Q0. (6.44)
The volume flux Qr,cont increases like z1/2 with distance. However, similarly to
Qr,pot, Qr,cont can only model the return flow for z ≤ hi. The theoretical predic-
tions Qr,pot and Qr,cont have different growth rates because of the difference in the
boundary conditions. Qr,pot is computed with fixed boundary conditions (the jet
is modelled as a line sink located at x = xj), whereas the boundary conditions
for Qr,cont change linearly with distance (the jet boundary is a function of height,
x0(z) ∝ z).
We can see in figure 6.15(a) that the experimental return-flow volume flux
is increasing between the two theoretical curves. The first prediction (6.43),
based on potential theory, underestimates the volume flux, whereas the second
prediction (6.44), based on continuity, overestimates the volume flux. We can
also note that the volume flux of the return flow becomes rapidly larger than the
initial volume flux, Qr/Q0 ≈ 3 at mid-height in the tank z = hf/2. Therefore,
the volume flux of the return flow is of the order of magnitude of the jet volume
flux away from the source, i.e. Qr ≈ Q for z ≥ hf/2. As predicted, Qr,pot
increases linearly while Qr,cont increases like z1/2 with distance. We believe that
the assumption we make to model the jet as a fixed line sink is valid only if
the distance between the jet axis and the lateral wall (at x = 0) is large. The
trend of the experimental data Qr,exp is not accurate enough in figure 6.15(a) to
distinguish a linear growth rate, as predicted by Qr,pot, or a growth rate of z1/2,
as predicted by Qr,cont.
Similarly to the volume flux, we plot in figure 6.15(b) the non-dimensional
time-averaged momentum flux M r/
(
Q02/d
)
of the return flow (combining the
two sides of the jet) versus non-dimensional height z/d. The experimental data,
plotted with a dotted curve, are computed from the time-averaged streamwise
172
6.3 Experimental results
velocity of the ensemble-averaged experimental flow field shown in figure 6.14(b),
such that
M r,exp(z) = 2
∫ xj−x0(z)
0
(w)2(x, z) dx. (6.45)
The first theoretical prediction, plotted with a solid curve, is computed using the
streamwise velocity predicted by potential theory and defined in equations (6.41),
(6.19) and (6.26), such that
M r,pot(z) = 2
∫ xj
0
(w)2(x, z) dx. (6.46)
Using equation (6.41), we find that the momentum flux M r,pot increases like z2
with distance. However, as we mentioned for the volume flux, equation (6.47) is
valid only in the region where the jet is self-similar and the momentum flux of
the jet is conserved, i.e. for z ≤ hi. The second theoretical prediction, plotted
with a dashed curve, is computed using the volume flux Qr,cont defined in (6.44)
and assuming a uniform velocity outside the jet (with slip boundary condition at
the walls). We find
M r,cont(z) =
(
Qr,cont
)2
(z)
2 (xj − x0(z))
. (6.47)
The momentum flux M r,cont increases like z2 with distance for x0(z)/xj ≪ 1.
But, unlike M r,pot, we find that in the limit x0(z)/xj → 1, M r,cont increases like
1/(1 − x0(z)/xj) with distance z. Nevertheless, and similarly to M r,pot, M r,cont
can only model the return flow for z ≤ hi, so that x0(z)/xj < 1 and M r,cont only
increases like z2. This discrepancy between the asymptotic behaviours of the two
theoretical predictions M r,pot and M r,cont is, again, due to the difference between
the boundary conditions.
Similarly to the volume flux, we can see in figure 6.15(b) that the experimental
return-flow momentum flux is increasing between the two theoretical curves. The
first prediction (6.46), based on potential theory, underestimates the momentum
flux, whereas the second prediction (6.47), based on continuity, overestimates
the momentum flux. As predicted, we can observe that both M r,pot and M r,cont
increases like z2, which seems to be also the case for the experimental momentum
flux M r,exp. From consideration of figure 2.6, we find that the time-averaged
momentum flux of the jet is approximately constant with height at an average
173
6 Flow induced by a jet in a confined domain
0 10 20 30 40 50 60 70 80
z/d
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Q
r/
Q
0
Data: Qr,exp
Theory: Qr,pot
Theory: Qr,cont
(a)
0 10 20 30 40 50 60 70 80
z/d
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
M
r/
( Q
0
2
/d
) Data: Mr,expTheory: Mr,pot
Theory: Mr,cont
(b)
Figure 6.15: Experimental and theoretical distributions against non-dimensional
height z/d of: (a) the experimental normalized time-averaged volume flux Qr,exp/Q0
(dotted curve) computed using (6.42), the theoretical prediction based on potential the-
ory Qr,pot/Q0 (solid curve) and computed using (6.43), the theoretical prediction based
on continuity Qr,pot/Q0 (dashed curve) and computed using (6.44); (b) the experi-
mental normalized time-averaged momentum flux M r,exp/
(
Q02/d
)
(dotted curve) com-
puted using (6.45), the theoretical prediction based on potential theoryM r,pot/
(
Q02/d
)
(solid curve) and computed using (6.46), the theoretical prediction based on continuity
M r,cont/
(
Q02/d
)
(dashed curve) and computed using (6.47).
value of < M > /
(
Q02/d
)
= 0.55. The non-dimensional momentum flux of the
return flow increases from M r/
(
Q02/d
)
= 0 at z/d = 0 to approximately 0.06
at z/d = 80. Therefore, in our domain, the momentum flux of the return flow
is rather insignificant compared with the jet momentum flux. This finding is
completely different from the results for the volume flux of the return flow, which
is not insignificant because it has to balance the volume flux of the flow. This
crucial difference, which enables us to neglect the influence of the momentum flux
of the return flow on the jet flow, is related to the distance between the jet and
the lateral boundaries (i.e. to the lateral confinement of the jet) and justifies the
assumptions we make in Chapter 2 that the return flow has a weak effect on the
dynamics of the evolving jet.
6.4 Conclusion
In this chapter, we study the flow induced by a steady quasi-two-dimensional
turbulent jet in a confined rectangular domain. Using two-dimensional potential
theory, we determine the induced flow in a representative domain Ds of aspect
174
6.4 Conclusion
ratio ζ = xj/hi = 3/4 (corresponding to our particular case) between the stream-
wise dimension and the cross-stream dimension. The domain is delimited by the
jet axis at the right-hand-side boundary (at x = xj), the walls of the experimental
apparatus at the left-hand-side and bottom boundaries and the transition height
hi between the jet flow region and the impingement region, at the top boundary.
The jet is modelled as a line sink (located on the jet axis) with a lateral flux per
unit length varying with height in a similar way to the entrainment velocity due
to a quasi-two-dimensional jet. The transition height hi is modelled as a uniform
line source, whose total inwards flux matches the total outwards flux of the line
sink.
To solve Laplace’s equation in the domain Ds, we decompose the problem
into a uniform problem with a uniform line source and a uniform line sink, and
a perturbation problem accounting for the varying line sink condition at the
jet boundary. We find an analytical solution for the potential field, the stream
function and the velocity field in the domain Ds. It appears that in the far field,
away from the jet, the results are dominated by the uniform problem with uniform
boundary conditions. The influence of the varying line sink (i.e. the entrainment
process of the jet) is strong near the source of the jet, because of the singularity
at the virtual origin of the jet (located outside the domain below the bottom
boundary).
We observe qualitative discrepancies between our analytical solution for the
streamlines of the induced flow compared with the solutions of Taylor (1958)
or Schneider (1981). The second derivative of the streamlines with respect to
the lateral or cross-jet coordinate (x) have a different sign. Our streamlines
are convex, whereas the streamlines of Taylor (1958) or Schneider (1981) are
concave. This difference is due to the fact that we consider a fully confined
domain, which induces a recirculation flow on either side of the jet, whereas
Taylor (1958) or Schneider (1981) considered fully unbounded domains or the
case of a jet emerging from a wall into a semi-infinite domain, thus ignoring the
possibility of recirculation in the ambient flow.
We compare our theoretical flow field with experimental data from quasi-two-
dimensional turbulent jets in a confined experimental apparatus of aspect ratio 1
(the ratio between the inner dimensions of the tank). The theoretical streamlines
agree with the data in the far-field, away from the boundary of the jet. We find
175
6 Flow induced by a jet in a confined domain
that the boundary of the jet, defined as the boundary between the turbulent jet
flow and the ambient flow (Kotsovinos, 1978), also corresponds to the location
x0(z) ≈ 0.22z where the flow is, in average, purely lateral because the time-
averaged streamwise velocity vanishes and changes sign at x0(z). In our model,
we assume that the jet boundary coincides exactly with the jet axis, instead of
being at an angle of approximately 12◦. We find that this assumption is valid in
the far-field away from the jet boundary and for z ≤ hi.
We find that, to the leading order, the experimental velocity field agrees with
the model. Differences are seen near the rigid boundaries, where the experimental
time-averaged tangential velocity vanishes at the walls, contrary to the theoretical
tangential velocity which is assumed to satisfy a slip boundary condition. Also,
near the jet source, the experimental data differ from the model because the flow
of the jet is not yet fully established. Finally, the experimental measurements
for the volume flux and the momentum flux of the return flow agree to leading
order with the model based on potential theory and a model based on volume
conservation. In particular, we find that the time-averaged momentum flux of the
return flow increases like z2 to approximately 10 % of the jet momentum flux at
mid-height in the experimental apparatus.
We believe that a jet emerging from a wall into a fully confined domain is a
more realistic case than the case of a jet in an unbounded or semi-infinite domain.
The streamlines of the induced flow are strongly modified by the recirculation
cells observed on either side of the jet. This phenomenon is important in mixing
problems because the re-entrainment process tends to increase the concentration
in the jet of passive tracers injected in the fluid. The momentum flux of the jet can
also become negatively affected by the counter-flow after a certain distance. The
core and eddy structures also become affected by the confinement at a distance
hi approximately equal to 65 % of the depth of the flow, for an experimental
apparatus of aspect ratio 1 (i.e. the ratio between the inner dimensions of the
tank) or a jet aspect ratio ζ = xj/hi = 3/4 (i.e. the ratio of the distance between
the jet and the lateral boundary to the transition height of the impingement
region). We believe that our model, and in particular the assumption of a uniform
flux at the top boundary and the assumption of a jet boundary parallel to the
z-axis on the right-hand side of the domain, is valid for a range of jet aspect ratios
2/3 < ζ = xj/hi < 3. At higher aspect ratios, secondary recirculation cells could
176
6.4 Conclusion
form on either side of the jet (Jirka & Harleman, 1979), thus affecting the flux
at the top boundary. On the other hand, at lower aspect ratios, the expansion of
the jet boundary becomes significant compared with the size of the domain, and
thus can influence the flux at the top boundary.
177
Chapter 7
Dynamics of particle-laden jets in
quasi-two-dimensional environments
7.1 Introduction
Two-phase flows involving mixtures of solid particles and liquids are common in
industrial applications. One example can be found in the coking process of the
residue, or heavy-tar, from the refinement of crude oil, which serves to produce
graphite electrodes for smelting applications (see e.g. Lee et al., 1997). During
the coking, chemical reactions normally provoke a gradual phase transition of the
heavy-tar into solid sponge coke; however, this process can also occasionally lead
to the formation of a less valuable product called shot coke (Eser et al., 1986).
Eser et al. (1986) reproduced the chemical reactions that can cause the production
of shot coke, but much less is known about the dynamics of the flow and its mixing
properties, when the heavy-tar is injected into the reactor. Another important
179
7 Dynamics of particle-laden jets in quasi-two-dimensional environments
industrial application of particle-laden jets is for fluidized beds, which are used
in chemical reactors or in the transport of granular material (see e.g. Zoueshtiagh
& Merlen, 2007). The study of particle-laden jets is also relevant to geophysical
applications such as volcanic eruptions (see e.g. Sparks, 1986; Ernst et al., 1996;
Veitch & Woods, 2000; Walters et al., 2006), and the transport and resuspension
of sediments by jets (see e.g. Neves & Fernando, 1995; Colomer & Fernando, 1996;
Colomer, Casamitjana & Fernando, 1998; Cardoso & Zarrebini, 2002; Jiang, Law
& Cheng, 2005).
We have conducted different experiments in which a vertical water jet is dis-
charged below a flat bed of particles immersed in water. Similarly to the ex-
periments described in § 2.2 and § 4.1, these experiments are performed in a
quasi-two-dimensional environment. The jet and the bed are constrained be-
tween two close walls in the spanwise (y-) direction. Typically, the dimension
of the gap is two orders of magnitude smaller than the other dimensions. This
particular geometry allows us to visualize and study the evolution of the system
inside the bed of particles. Rich dynamical behaviours, characteristically differ-
ent from the three-dimensional case, appear in two dimensions. For instance,
the quasi-two-dimensional particle-laden jet (Q2DPL jet) presents an unreported
instability occurring at intermediate flow rates.
The objective is to understand and analyze the succession of regimes shown by
the evolution of the system while the jet flow rate is changed. We are interested
in the interaction between the jet and the bed of particles. The entrainment and
recirculation of the particles in the jet reveal an interesting coupling with the ge-
ometry of the eroded bed. The maximum height reached by the particles is a key
parameter to understand the dynamics of the whole system. As a model of the
Q2DPL jet, we draw a comparison with the non-buoyant quasi-two-dimensional
momentum jet studied in the previous chapters. We discuss the assumptions and
conditions under which the model holds on the basis of experimental and theo-
retical results. Future work will be to compare Q2DPL jets with heavy fountains
in a quasi-two-dimensional environment, in order to model the regimes where the
density difference between the Q2DPL jet and the ambient fluid is important.
The rest of this chapter is organized as follows. In § 7.2, we describe the
experimental procedure. In § 7.3, we describe the different phenomenological
regimes observed in the experiment, as the jet flow rate increases. In § 7.4, we
180
7.2 Experimental procedure
discuss a model to predict the maximum height reached by the particles in the
final dilute regime, based on the model for the time-averaged mean momentum
jet (presented in § 2.4). We draw our conclusions in § 7.5 and suggest new avenues
of research.
7.2 Experimental procedure
The 0.5 m (L) × 0.01 m (W ) × 0.5 m (H) quasi-two-dimensional experimental
apparatus is presented in figure 7.1.1 The (heavy) particles we use are 0.5 mm
glass beads (density: ρp = 2.5 g cm−3 ). The initial thickness of the bed ranges
from h0 = 1.75 to 8 cm. At the beginning of an experiment, we lay the bed flat at
the bottom of the tank, which is filled with water. The water jet, injected through
a circular nozzle of diameter d = 6 mm located at the middle of the bottom of
the tank, has a source flow rate ranging from Q0 = 0 to 33 cm3 s−1 . We generate
the flow either by gravity or using a peristaltic pump (the pulsing of the pump
had no influence on the system for Q0 > 4 cm3 s−1 ). We increase the flow rate in
a stepwise manner, with typical step 1 cm3 s−1 . After each increase in the flow
rate, we allow the system to reach steady state (characterized by a fixed shape of
the bed and by an approximately constant amount of particles in circulation in
the jet, i.e. particles no longer sediment in the far field). This typically takes 5 to
30 minutes. At steady state, we take (with a ruler) some characteristic geometric
measurements of the shape of the bed: the size of the cone formed by the erosion
of the bed, the thickness of the bed above the nozzle (hsource) and the angle of the
slopes of the cone (with accuracy of approximately ±5 mm for the lengths and
±5◦ for the angle). Moreover, we measure the maximum height reached by the
particles in the jet hmax (accuracy of ±5 to ±20mm) and the oscillation frequency
of the Q2DPL jet (accuracy of ±15 %).
1Note that the length L and the height H of the experimental apparatus used in this chapter
are half the length and half the height of the experimental apparatus used in Chapters 2 to
6.
181
7 Dynamics of particle-laden jets in quasi-two-dimensional environments
x
z
d = 0.6 cm
Liquid jet
overflow
H = 50 cm
L = 50 cm
W = 1 cm
Bed of particles h0
Liquid medium
Figure 7.1: Sketch of the experimental apparatus. The evolution of the system is
analyzed when a water jet of variable flow rate is injected at the bottom of the bed of
particles (0.5 mm glass beads).
7.3 Phenomenological description
7.3.1 Regime diagram
In figure 7.2, we present a schematic diagram of the successive regimes displayed
by the system (bed of particles and jet) as the jet flow rate Q0 increases, and for
various initial bed thicknesses h0.
We start with Q0 = 0 and a flat bed of particles. At very low flow rates, the
bed remains motionless because the pressure of the flow is insignificant compared
with the weight of the bed of particles per unit area. In this pre-regime (not
represented in figure 7.2), we have a porous medium flow, which can be modelled
using Darcy’s law (Zoueshtiagh & Merlen, 2007)
W = −κµ∇P, (7.1)
where W is the superficial velocity, κ is the permeability of the bed, µ is the
dynamic viscosity of the liquid and P the pressure. From this model, Zoueshtiagh
182
7.3 Phenomenological description
Source volume flux Q0
h0
In
it
ia
l
be
d
th
ic
kn
es
s
I Fluidized
bed
II Oscillatory flow
III Core and eddy flow
Figure 7.2: Schematic diagram showing the boundaries between the three different
phenomenological regimes observed as the source flow rate is increased from Q0 = 0 to
33 cm3 s−1 .
& Merlen (2007) deduced the flow velocity at the surface of the bed. Their
experimental measurements of the flow velocity at the bed surface agreed with
the theoretical prediction as long as the configuration could be considered as a
point source in an infinite domain (i.e. d/h0 ≤ 0.2).
As we increase the source flow rate (for a given bed thickness), the system
displays very different regimes, which we describe in detail below. At low flow
rates, we observe a fluidization regime after the Darcy flow regime (see regime
I in figure 7.2). Then, the jet starts eroding the bed if we increase Q0 further.
In this “oscillatory flow” regime (see regime II in figure 7.2), the Q2DPL jet is
unstable and oscillates in the eroded bed with respect to the vertical axis. Finally,
at large Q0 the bed is fully eroded and the jet can lift particles higher in the water
above the bed. The motion of the particles in the jet flow shows the same core
and eddy structures as we observe in particle-free quasi-two-dimensional jets. So,
in reference to the flow in particle-free quasi-two-dimensional jets, we name this
regime the “core and eddy flow” regime (see regime III in figure 7.2).
183
7 Dynamics of particle-laden jets in quasi-two-dimensional environments
7.3.2 Regime I: fluidized bed
At low flow rates, the jet fluidizes the bed, which maintains a flat surface (see
figure 7.3). We observe a strong recirculation of the particles inside a chimney,
or narrow cone, located above the nozzle (see the dashed blue lines in regime I,
fluidized bed, in figure 7.2). In fluidization models, the basic hypothesis states
that the pressure gradient of the flow inside the chimney is balanced by the weight
of the particles in the chimney (see e.g. Zoueshtiagh & Merlen, 2007), i.e.
∆p = φb∆ρgh0, (7.2)
where ∆p = (p(0) − p(h0)) is the pressure difference between the bottom of the
bed (at z = 0) and the top surface of the bed (at z = h0), φb is the volume fraction
of the bed,2 ∆ρ = ρp−ρ is the difference between the density of the particles and
the density of the liquid, and g is the constant of gravity. In the rest of the bed,
the flow follows Darcy’s law (described in (7.1)). Therefore, the total flow rate at
the nozzle exit is equal to the sum of the flow rate in the fluidized chimney and
the flow rate in the unfluidized part of the bed.
Zoueshtiagh & Merlen (2007) conducted a similar experiment in three dimen-
sions. It is interesting to note that our observations about the fluidization of
the bed in the quasi-two-dimensional case agree qualitatively with their report.
They also proposed a model for the fluidization process and the formation of the
chimney. However, they could not obtain any experimental evidence to validate
or invalidate the model because their experimental apparatus did not allow them
to visualize and measure the interior of the bed where the fluidization process
occurred. We believe that our quasi-two-dimensional experiment, which gives a
clear picture of the dynamics inside the bed, could provide quantitative data to
verify the fluidization model of Zoueshtiagh & Merlen (2007).
7.3.3 Regime II: oscillatory flow
As we increase the flow rate, the opening angle of the cone increases by an erosion
process, as depicted in figure 7.4. At an intermediate range of flow rates the
2Ojha, Menon & Durian (2000) determined experimentally the volume fraction of a slowly
defluidized bed of particles. They found that the volume fraction φb is independent of the
size or shape of the particles and is approximately equal to φb = 0.59.
184
7.3 Phenomenological description
Figure 7.3: Illustration of regime I, Q0 = 2.5 cm3 s−1 and the initial height h0 = 4cm.
At very low flow rates the jet coming through the bed at the bottom center of the
picture fluidizes the particles above it. The surface of the bed remains almost flat (only
a small hump is observed), showing almost no activity.
Q2DPL jet builds a mound of particles on each side of the cone. For every
increase in the flow rate, the mounds grow and move away from the cone until
they reach a steady state (i.e. a fixed position). In addition, we can observe a
surprising behaviour in this regime: the Q2DPL jet does not maintain a vertical
axis, but oscillates in the (x, z) plane about the z-axis (the origin of the domain
(0, 0, 0) is at the centre of the nozzle). As we can see in figure 7.5, a large vortical
structure develops alternately on each side of the z-axis at each semi-oscillation.
As an explanation of this oscillating phenomenon, we believe that the Q2DPL jet
becomes unstable as it emerges out of the bed and is no longer bounded by the
steep walls of the cone. An oscillation can start when some disturbance breaks
the symmetry of the sedimentation process. This leads to an asymmetry of the
avalanching process occurring on the slopes of the cone. The stronger avalanche
deflects the jet to the opposite side of the cone. This has the effect of reversing
the asymmetric avalanching process, which can then deflect the jet back to the
initial side of the oscillation, thus completing a whole cycle. The oscillation of
the jet is sustained by the kinetic energy of the jet. The oscillation frequency
of the jet appears to be steady for a given source flow rate and to decrease with
increasing source flow rate.
185
7 Dynamics of particle-laden jets in quasi-two-dimensional environments
Figure 7.4: Illustration of regime II, the jet flow rate is Q0 = 15 cm3 s−1 and the
frequency of image acquisition is 500 frames per second. At medium-high flow rates,
a mound of particles forms on each side of the cone. The opening angle of the cone
increases rapidly with the flow rate. The jet oscillates from side to side in the crater.
A steady state is reached when the jet cannot eject any particles above the mounds.
Figure 7.5: Illustration of the vortical structure in regime II, Q0 = 18.5 cm3 s−1 and
h0 = 4 cm. The Q2DPL jet displays a large vortex as it oscillates alternately on each
side of the z-axis (several images are superimposed on this picture to show particles as
streaks).
186
7.4 Core and eddy flow model
7.3.4 Regime III: core and eddy flow
As we can see in figure 7.6(a), the final regime of the system is characterized by a
fixed final shape of the bed. The bed forms a crater, whose slopes are at the angle
of repose (approximately 26◦ for our particles). Moreover, it is flanked by two
pyramidal mounds, whose added volumes (computed above the initial height of
the bed, i.e. for z > h0) account for the volume of the crater. In this regime, the
jet still entrains some particles, but the volume fraction is much lower. Unlike
in regime II, the flow in the jet does not seem to be very strongly affected by
the particles in this regime. We observe a sharp decrease in the frequency and
amplitude of the oscillations of the Q2DPL jet, and also an increase in the rate of
change of the particle maximum height hmax (plotted with triangles in figure 7.7)
with the source flow rate Q0.
The trajectory of the particles in the jet flow (shown as streaks in figure 7.6a,
where the velocity is large) reveals three large eddies (identified with yellow cir-
cular arrows) and a high speed core (identified with a yellow arrowed curve). The
resemblance to the core and eddy structures in a quasi-two-dimensional turbulent
jet (depicted with yellow circular arrows and a yellow arrowed curve, respectively,
in the picture of a dyed quasi-two-dimensional jet in figure 7.6b) is strong. From
our qualitative observations, we can report that the size of the eddies in the
Q2DPL jet also grows with distance z. Therefore, it appears that in the case of a
dilute concentration of particles, the momentum of the jet is not strongly affected
by the negative bulk density of the two-phase flow.
7.4 Core and eddy flow model
The experiment described above reveals the complexity of the interaction between
the jet and the bed of particles. In this section, we are interested in the final
regime, or core and eddy flow regime, which has strong similarities with the flow
in a quasi-two-dimensional jet, described in the previous chapters. Assuming that
the density of the particles does not affect the flow of the jet we propose a model
for the maximum height reached by the particles transported by the jet.
From the study of non-buoyant quasi-two-dimensional momentum jets pre-
sented in Chapter 2, we know the velocity field of a quasi-two-dimensional jet
in our apparatus. If we assume that, in the final regime (i.e. regime III), the con-
187
7 Dynamics of particle-laden jets in quasi-two-dimensional environments
(a) (b)
Figure 7.6: (a) Illustration of the core and eddy structures in regime III, Q0 =
23 cm3 s−1 and h0 = 4 cm. At very high flow rates, the slopes of the cone are at
the angle of repose and the bed has a fixed shape. The Q2DPL jet displays three large
growing eddies (identified with yellow circular arrows), which are advected upwards
by the flow, as well as a high speed core (identified with a yellow arrowed curve). In
this picture, the maximum height reached by the particles is approximately 20 cm. (b)
Grey-scale picture of a dyed steady quasi-two-dimensional turbulent jet rising in the
tank over a height of approximately 40 cm (this dyed jet was produced in the appara-
tus depicted in figure 7.1, following the experimental procedure described in § 2.2.1).
Similarly to (a), the eddies are identified with yellow circular arrows and the core is
identified with a yellow arrowed curve.
centration of the particles in the Q2DPL jets does not strongly affect the liquid
phase of the jet, the particles are passively advected by the flow. Therefore, we
can theoretically compute the maximum height reached by the particles htmax. We
make the simplifying assumption that the theoretical particle maximum height
htmax is equal to the height at which the maximum vertical velocity of the pure
momentum jet matches the particle settling velocity. For our particles, the set-
tling velocity is vs ≈ 7.2±0.4 cm s−1 in the tank at rest. We further assume that
the maximum vertical velocity of the jet is approximately equal to its maximum
time-averaged vertical velocity wm, described by (2.5b). Solving
wm(z = htmax) = vs, (7.3)
188
7.4 Core and eddy flow model
we find
htmax =
Q02
4
√
2α
(
2
( M0
vsQ0
)2
− 1
)
, (7.4)
where α = 0.068 is the entrainment coefficient (Morton et al., 1956) measured in
§ 2.4, and M0 is the source momentum flux. As discussed in § 2.4, we have the
relationship M0/
(
Q02/d
)
≈ 0.55.
In figure 7.7, we present the experimental results for the evolution of the non-
dimensional particle maximum height hmax/d against the non-dimensional source
flow rate Q0/(vsd). The experimental data are obtained for an initial bed thick-
ness h0 = 1.85 cm. The data plotted with triangles for hmax are obtained with
increasing Q0 (in the stepwise manner described in § 7.2). The data plotted with
pluses for hmax are obtained with decreasing Q0. We conduct this second phase
directly after the Q0-increasing phase, after having reached the maximum source
flow rate (which can lift the particles to the free surface) and the experiment has
reached a steady state (as described in § 7.2).
We also plot in figure 7.7 the thickness of the bed of particles above the nozzle
hsource (in the Q0-increasing phase of the experiment) with blue squares (multi-
plied by a factor 15, for clarity, and non-dimensionalized by d). The evolution
of hsource with Q0 indicates the transitions between the three regimes I, II and
III described in § 7.3. From Q0/(vsd) ≈ 0 to 0.8, hsource remains constant and
the bed is fluidized by the jet (regime I, on the left-hand side of the first dotted
line plotted in figure 7.7). In regime II (within the two dotted lines), or from
Q0/(vsd) ≈ 0.8 to 3.6, hsource decreases because of the erosion process above the
jet nozzle. In regime II, we can also notice that the Q0-increasing data for hmax
are increasing slowly, i.e. the particles rise only slightly higher than the initial
bed thickness. For Q0 > 3.6, or regime III (on the right-hand side of the second
dotted line in figure 7.7), the erosion process is finished. In regime III, the Q0-
increasing data for hmax are in a new regime: the rate of increase of hmax with
Q0 is larger than in regime II. Moreover, we show in figure 7.7 the impingement
transition height hi = 3/4H (plotted with a black dashed line), which we discuss
in Chapter 6. The transport of the particles by the jet is perturbed beyond this
height because the flow changes from a jet flow to an impingement flow. We
can see that, for z > hi, hmax no longer shows the same increasing trend, but
approximately plateaus.
189
7 Dynamics of particle-laden jets in quasi-two-dimensional environments
We plot in figure 7.7 the range of the non-dimensional theoretical prediction
htmax/d ± 12 % (computed using (7.4)) between two red dashed curves. We al-
low a variation of ±12 % in the calculation of htmax to account for the typical
0.4/7.2 = 5.5 % standard deviation in the measurements of the particle settling
velocity vs. As we can see, the theoretical curves lie above the Q0-increasing
data for hmax (plotted with triangles) and slightly below the Q0-decreasing data
for hmax (plotted with pluses). Moreover, we can observe a strong hysteresis be-
tween the two data sets: the particles rise lower in the Q0-increasing phase than
in the Q0-decreasing phase of the experiment. We believe that the main reason
for this hysteresis is because during the Q0-increasing phase the bulk density of
the Q2DPL jet is larger than during the Q0-decreasing phase. During the Q0-
increasing phase, the Q2DPL jet loses particles because, at each increase of the
source flow rate, particles in the jet can settle outside the cone of recirculation
until the steady state is reached. On the other hand, once the system has reached
a steady state at the maximum flow rate, no more particles can settle out of the
recirculation cone as the source flow rate is reduced. Therefore, the assumption
that the particle concentration does not affect the jet flow appears to be incor-
rect in the Q0-increasing phase of the experiment. In the Q0-decreasing phase,
the ‘dilute’ assumption seems to be valid because the theoretical prediction un-
derestimates the experimental data only slightly. This small mismatch could be
related to the (second) assumption that the time-dependent vertical velocity is
approximately equal to the time-averaged vertical velocity. Indeed, we find in
Chapter 2 that the vertical velocity in the high-speed core of the jet is different
from the Gaussian profile of the time-averaged vertical velocity, and we are un-
able to investigate whether the maximum height hmax reached by the particles is
attained exclusively with flow in the high-speed core.
7.5 Conclusion
7.5.1 Summary
We have studied the dynamics of quasi-two-dimensional particle-laden jets in
the case of a vertical jet injected below a bed of particles confined in a quasi-
two-dimensional environment. We have observed several regimes as we increase
the source flow rate and vary the initial bed thickness. Initially, we find the well-
190
7.5 Conclusion
0 1 2 3 4 5 6 7
Q0/(vsd)
0
10
20
30
40
50
60
70
80
h
/d
Increasing Q0: hmax
Decreasing Q0: hmax
Data: 15hsource
Theory: htmax ± 12%
hi
I II III
Regime separation lines
Figure 7.7: Evolution against the non-dimensional source flow rate Q0/(vsd) of: the
non-dimensional experimental particle maximum height hmax/d for increasing source
flow rate (black triangles), the non-dimensional experimental particle maximum height
for decreasing source flow rate (black pluses), the non-dimensional theoretical particle
maximum height plus or minus 12 percent htmax/d± 12% (red dashed curves), and the
non-dimensional experimental bed thickness (measured above the nozzle) 15hsource/d
(plotted with blue squares). We plot the non-dimensional impingement transition height
hi/d with a black dashed line. The different regimes (I, II and III), presented in the
regime diagram shown in figure 7.2 and discussed in § 7.3, are delimited with dotted
lines.
known Darcy flow regime and fluidization regime for low flow rates (or large initial
bed thicknesses). Then, the bed of particles evolves towards a triangular shape
because the jet erodes the bed of particles gradually. The jet entrains particles
above the bed, which settle and avalanche on the slope of the triangular eroded
bed. In this regime, we observe an instability characterized by the oscillation of
the Q2DPL jet. Finally, at large flow rates the particles are transported higher
by the jet and their bulk concentration in the jet decreases.
We propose a model for the final regime, in which the flow of the Q2DPL jet
displays the same characteristic core and eddy structure as the flow of the non-
buoyant quasi-two-dimensional momentum jet. Assuming that the concentration
191
7 Dynamics of particle-laden jets in quasi-two-dimensional environments
of the particles does not affect the flow of the jet and that the time-dependent
vertical velocity is approximately equal to the time-averaged vertical velocity, we
calculate the maximum height reached by the particles by equating their set-
tling velocity with the maximum time-averaged vertical velocity of non-buoyant
quasi-two-dimensional jets (as described in Chapter 2). The comparison with ex-
perimental results shows that the order of magnitude and the trend (with source
flow rate) of the particle maximum height are predicted by the model. However,
in the phase where the source flow rate increases, the assumption of a dilute
suspension of particles in the jet appears to be incorrect because the theoreti-
cal prediction overestimates the maximum height reached by the particles. On
the other hand, in the (hysteretic) phase where the source flow rate decreases,
the dilute assumption appears to be correct because the theoretical prediction
only slightly underestimates the maximum height reached by the particles. The
small mismatch is thought to be due to the difference between the time-dependent
vertical velocity in the high-speed core of the jet and the time-averaged vertical
velocity, which we assume in the simplified model presented here.
7.5.2 Future work
Study of heavy fountains
The next step in the understanding of the different regimes is to model the Q2DPL
jet as a heavy fountain in a quasi-two-dimensional environment. We believe that a
heavy fountain can account for the non-dilute regimes of the flow: the oscillatory
flow regime, or regime II, and the Q0-increasing phase of regime III.
A heavy fountain is a vertical upward jet with negative buoyancy (see e.g.
Baines, Turner & Campbell, 1990, for an introduction to the theory of heavy
fountains). From dimensional analysis, its maximum height is related to its initial
momentum flux M0 and its initial buoyancy flux B0
zmax = A
M0
|B0|2/3
, (7.5)
with
M0 = 2b0w02, B0 = 2b0
(ρf − ρh)
ρf
gw0, (7.6)
and where A is a constant of proportionality which can be determined experimen-
192
7.5 Conclusion
tally (Bower et al., 2008), ρf is the density of the ambient fluid, ρh is the density
of the heavy fountain, b0 is the initial half-width of the fountain, and w0 is the
initial time-averaged vertical velocity of the fountain. Then, we can re-write the
density of the fountain in terms of the densities of the fluid and the particles, and
the initial volume fraction of the particles φ0 in the fountain:
ρh = φ0 ρp + (1− φ0)ρf . (7.7)
Therefore, the particle maximum height is
zmax = A
(
φ0
(ρf − ρp)
ρf
g
)−2/3
(2b0)1/3w04/3. (7.8)
A future aim could be to apply this model to regime II and the Q0-increasing
phase of regime III of the Q2DPL jet and to verify it experimentally. We could
inject a homogenous heavy buoyant fluid in the quasi-two-dimensional tank in
order to investigate the maximum height reached by the particles as a function
of M0 and B0, and to measure the experimental constant A. However, relating
the heavy-fountain model to the Q2DPL jet is particularly challenging because
it requires an estimation of the initial buoyancy flux of the Q2DPL jet, and in
particular the initial volume fraction φ0. The rate of entrainment of the particles
at the source is one of the most critical and intriguing points: the recirculation
of the particles denotes the coupling between the solid phase and the flow of the
jet. Both the shape of the crater and the deposition of the particles determine
the avalanching flux feeding the jet. The jet entrains these particles, which in
turn affect the momentum of the jet by changing its bulk density. Thereafter,
the particles rise to the height where their settling velocity matches the vertical
velocity of the fluid. In order to close the recirculation problem, we must relate
the sedimentation rate of the particles to both the settling time of the particles
and their rising time inside the jet. In conclusion, a robust model accounting for
the recirculation of the particles is needed to understand the full dynamics of the
system.
193
7 Dynamics of particle-laden jets in quasi-two-dimensional environments
Changing the viscosity of the medium
To improve our model of the motivating industrial application (the problem
of shot-coke formation in a late-stage oil-refining process), the viscosity of the
medium in which the bed of particles is prepared could be different from the
viscosity of the jet. This experiment is expected to have very rich dynamics be-
cause instabilities such as finger-like structures can occur when viscous forces and
gravity forces play a key role (Sto¨hr & Khalili, 2006).
Study of the oscillation of the Q2DPL jet
Another puzzling and interesting issue in this study is our discovery of an insta-
bility at intermediate flow rates. The Q2DPL jet oscillates steadily at a fixed
flow rate and about a vertical axis. It also produces a large vortex as it tilts
sideways at every half oscillation. From these observations, we can wonder what
sets the frequency of the oscillation and why the frequency decreases with Q0.
There are different time scales that can influence the frequency: the rising time
of the particles in the jet, the settling time of the particles and the avalanching
time of the particles. Even the vortex recirculation time could be considered part
of the problem; however, the vorticity tends to increase with the flow rate, which
seems to be in contradiction with the fact that the oscillation frequency actually
decreases with it. Finally, we also noticed that the frequency rapidly drops at the
transition between regimes II and III, thus suggesting a different model for the
evolution of the frequency in these two regimes.
Injecting water and particles through the nozzle
As a further step in the understanding of the experiments, we could change the
experimental procedure by injecting both liquid and particles through the nozzle.
The experimental results should be closer to real applications, such as industrial
two-phase flows and volcanic eruptions. The complexity of this problem is likely
to increase: for example, there cannot be a steady state at fixed flow rate because
of the continuous injection of particles in the tank. We might find that the shape
of the bed evolves in the reverse order from that which occurs in the present ex-
periment as we increase the flow rate, i.e. passing through regime III, then regime
II and finally regime I as shown on figure 7.2. From preliminary experiments, we
194
7.5 Conclusion
observe the formation of an open and flatter crater from the sedimentation of the
particles (Jiang et al., 2005; Neves & Fernando, 1995). Moreover, we find that the
crater grows in size and its slopes become steeper, thus blocking the rise of the
Q2DPL jet. The final stages could also show a fluidization regime and eventu-
ally a porous medium flow. An interesting issue is to determine the mechanisms
accounting for the transition from one regime to the next.
Study of the three-dimensional case
Three-dimensional particle-laden jets have been studied by many scientists (see
e.g. Cardoso & Zarrebini, 2002; Colomer et al., 1998; Colomer & Fernando, 1996;
Ernst et al., 1996; Jiang et al., 2005; Neves & Fernando, 1995; Walters et al., 2006;
Zoueshtiagh & Merlen, 2007). However, our understanding of the quasi-two-di-
mensional experiment gives us the opportunity to analyze the three-dimensional
case from a different perspective. When comparing the two cases, it is possible to
consider a wide variety of interesting issues: the dynamics of the particle-laden
jet, the interaction between the particle-laden jet and the bed, the mechanisms of
recirculation of the particles, their mixing properties, and the three-dimensional
manifestation of the periodic oscillation observed in the quasi-two-dimensional
environment.
195
Chapter 8
Conclusion and future work
8.1 Review
In this thesis, we have studied experimentally and theoretically the dynamics of
steady quasi-two-dimensional turbulent jets. In Chapter 1, we present a brief
summary of past studies on quasi-two-dimensional jets, as well as some motiva-
tions for this study. Giger et al. (1991) and Dracos et al. (1992) gave the first
clear description of this particular type of jets, which occurs in the far field of
a plane turbulent jet confined between two close boundaries separated by a gap
widthW (i.e. for z > 10W , where z is the streamwise coordinate). They observed
that, in the far field, the unstable flow develops into a meandering core with large
counter-rotating eddies developing on alternate sides of the core. They found an
inverse cascade of quasi-two-dimensional turbulence, which affects not only the
structure of the flow but also the transport, dispersion and mixing properties.
197
8 Conclusion and future work
One particular application relevant to this study concerns the flow of rivers dis-
charging into lakes or oceans. Various phenomena are related to this type of flow:
sediment transport, coastal erosion, and the transport and dispersion of passive
tracers such as pollutants. Understanding the physics of the flow is crucial to the
prediction and assessment of the environmental impact.
In Chapter 2, we describe the phenomenology of the core and eddy structure of
the jet using detailed experimental measurements of the velocity field, obtained
with particle image velocimetry. We observe an inverse cascade typical of quasi-
two-dimensional turbulence where both the core and the eddies grow linearly with
z and travel at an average speed proportional to z−1/2. We find that quasi-two-
dimensional jets are self-similar and their mean properties are consistent with
both experimental results and theoretical models of the time-averaged properties
of fully unconfined planar two-dimensional jets. The experimental results for the
spatial statistical distribution of the core and eddy structure led us to believe that
the dynamics of the interacting core and large eddies accounts for the Gaussian
profile of the mean streamwise velocity. The lateral excursions (caused by the
propagating eddies) of the high-speed central core produce a Gaussian distribution
for the time-averaged streamwise velocity. In addition, we find that approximately
75% of the total momentum flux of the jet is contained within the core. The eddies
travel substantially slower (at approximately 25 % of the maximum speed of the
core) at each height and their growth is primarily attributed to entrainment of
ambient fluid. The frequency of occurrence of the eddies decreases in a stepwise
manner due to merging, with a well-defined minimum value of the corresponding
Strouhal number St = fb/wm ≥ 0.07 (where f is the eddy frequency, b is the
velocity spread rate of the jet and wm is the maximum time-averaged streamwise
velocity in the jet).
In Chapter 3, we investigate theoretically the streamwise transport and dis-
persion properties of quasi-two-dimensional jets. We model the evolution in time
and space of the concentration of passive tracers released in these jets using a
one-dimensional time-dependent effective advection–diffusion equation. Based on
the study of the flow field presented in Chapter 2, we make a mixing length hy-
pothesis to model the streamwise turbulent eddy diffusivity Dzz ∝ bwm, where
b is the jet velocity spread rate, wm is the maximum time-averaged streamwise
velocity, and Dzz is the streamwise component of the turbulent eddy diffusive
198
8.1 Review
tensor. Under these assumptions, the effective advection–diffusion equation for
φ(z, t), the cross-stream integral of the ensemble-averaged concentration, is of the
form:
∂tφ+KaM1/20 ∂z
(
φ/z1/2
)
= KdM1/20 ∂z
(
z1/2∂zφ
)
, (8.1)
where t is time, Ka (the advection parameter) and Kd (the dispersion parameter)
are empirical dimensionless parameters which quantify the importance of advec-
tion and dispersion, respectively, and M0 is the source momentum flux. We find
analytical solutions to this equation for φ in the cases of a constant-flux release
and an instantaneous finite-volume release. We also give an integral formulation
for the more general case of a time-dependent release, which we solve analytically
when tracers are released at a constant flux over a finite period of time.
In Chapter 4, we compare the theoretical predictions of the streamwise ad-
vection and dispersion model, derived in Chapter 3, with experimental evidence.
From our experimental results, whose concentration distributions agree with the
model, we find that Ka = 1.65±0.10 and Kd = 0.09±0.02, for both finite-volume
releases and constant-flux releases using either dye or virtual passive tracers. The
experiments also show that streamwise dispersion increases in time as t2/3. As a
result, in the case of finite-volume releases, more than 50% of the total volume of
tracers is transported ahead of the purely advective front (i.e. the front location
of the tracer distribution if all dispersion mechanisms are ignored, corresponding
formally to the assumption of ‘top-hat’ velocity profiles in the jet); and in the
case of constant-flux releases, at each instant in time, approximately 10 % of the
total volume of tracers is transported ahead of the advective front. Finally, we
assess the statistical significance of our results. We find that experimental or real
concentrations are more likely to differ from the concentrations predicted by the
model at large concentration levels than at low concentration levels. These find-
ings are important in problems of pollution control in rivers because they show
that pollutants can travel faster than expected and their concentration may be
higher than predicted.
In Chapter 5, we investigate turbulent relative dispersion in quasi-two-dimen-
sional turbulent jets. Following the seminal paper of Richardson (1926), we use
two-point statistics to describe the dispersion properties of the core and eddy
structure of the jet. The experimental data are obtained using what we believe
to be a novel Lagrangian-particle-tracking technique, which we refer to as vir-
199
8 Conclusion and future work
tual particle tracking. Virtual particle tracking, first introduced in Chapter 4,
consists of tracking (numerically) virtual passive tracers seeded in the experimen-
tally measured velocity field of a flow. We demonstrate that this technique can
yield valuable experimental data to compare with turbulent relative dispersion
models. We calculate the time evolution of the probability distributions of key
two-point properties (such as the lateral distance, the streamwise distance, the
Euclidean distance and the ratio of the lateral distance to the streamwise distance
between two points) in three different parts of the flow of quasi-two-dimensional
jets. We find that in the eddy, the distribution of particles disperses slowly and in
a rather axisymmetric manner. At the interface between the core and the eddy,
the distribution of particles stretches considerably in the streamwise direction at
a high rate. In the core of the jet, the particle distribution disperses slowly in the
cross-jet direction and splits along the jet axis. Finally, we believe that the rapid
change in time of the jagged distribution of the p.d.f. for the distance between
two points in the eddy reveals the intense stirring (and potentially the resulting
vigorous turbulent mixing) occurring within the eddy.
In Chapter 6, we use potential theory to describe the ambient flow induced
by a quasi-two-dimensional jet discharged vertically upwards in a fully confined
rectangular domain. In our experimental apparatus (of aspect ratio 1), we can
observe that at a height hi ≈ 0.65hf (where hf is the distance of the free surface
from the source) the jet flow becomes an impingement flow which spreads laterally
along the free surface, recirculates downwards along the lateral boundaries of the
apparatus, and is eventually re-entrained by the jet. In the domain, spanning
from the lateral rigid boundary to the jet axis in the x-direction and from the
bottom rigid boundary to the impingement transition height hi in the streamwise
direction, we solve Laplace’s equation. We assume slip boundary conditions at
the rigid boundaries, a sink link with varying strength proportional to (z−z0)−1/2
(where z0 is the space virtual origin of the jet) at the jet axis, and a uniform source
line (whose integrated volume flux matches the integrated volume flux of the
sink line) at the impingement transition height. The analytical stream function
and velocity field agree with our experimental measurements, except near the
boundary of the jet. We also find that (contrary to the volume flux) the time-
averaged momentum flux of the induced return flow is insignificant compared with
the time-averaged momentum flux of the jet, typically less than 10 %. We believe
200
8.2 Future work
that this means that the induced return flow in our experimental apparatus has
little impact on the flow structures of the quasi-two-dimensional jet studied in
the previous chapters.
In Chapter 7, we study the case of a momentum jet discharged below a bed of
particles in a quasi-two-dimensional environment. As the jet flow rate increases,
the interaction between the jet and the bed of particles evolves through three
main different regimes. At low flow rates or large initial bed thicknesses, the
jet fluidizes the bed. At intermediate flow rates, the jet erodes the bed and
form a pyramidal mound on either side of the jet axis. The particle-laden jet is
also unstable and oscillates about a vertical axis. At large flow-rates, the bed is
fully eroded and the flow of the particle-laden jet shows the same core and eddy
structure as the particle-free quasi-two-dimensional jet observed in Chapter 2. We
propose a model to predict the maximum height of rise reached by the particles
in the jet based on the time-averaged vertical velocity of a particle-free quasi-
two-dimensional jet. We find that the model agrees with experimental data for a
dilute suspension of particles in the jet.
8.2 Future work
This study has raised questions for future research. We highlight below the vari-
ous possible directions already discussed throughout the thesis. For instance, the
streamwise advection and dispersion model (developed in Chapter 3) could be ex-
tended to include advection and dispersion in the cross-jet direction of quasi-two-
dimensional jets. With a two-dimensional time-dependent model, the distribution
of the concentration of passive tracers in quasi-two-dimensional jets would be fully
resolved. Such a model would provide more accurate predictions for dispersion
and transport in river flows.
A relationship between the two-point statistics in the jet and our streamwise
advection and dispersion model (i.e. connecting Chapters 3 and 4 with Chap-
ter 5) could improve our understanding of relative dispersion in turbulent flows.
Furthermore, the spatial resolution of the results obtained with the technique of
virtual particle tracking (described in § 4.1.2 and used in Chapters 4 and 5) could
be enhanced to resolve the finest scale of turbulence. This would provide crucial
experimental data for comparison with the vast number of turbulent dispersion
201
8 Conclusion and future work
models.
We also believe that the technique of virtual particle tracking, developed in
this study, can be successfully applied to other flow problems. Virtual particle
tracking can resolve Lagrangian particle tracking (as shown in Chapter 5), as
well as identify Eulerian features in the flow (as performed in § 2.5 to study
quantitatively the core and eddy structures of quasi-two-dimensional jets).
The study of particle-laden jets in quasi-two-dimensional environments has also
opened many avenues of research. An analogy with heavy fountains could give
a basis to model the flow regimes with large concentrations of particles, and
potentially explain the physics of the oscillatory instability displayed by quasi-
two-dimensional particle-laden jets. Changing the viscosity of the liquid phase, or
injecting solid particles with the liquid phase at the source would show very rich
dynamics relevant to many industrial applications, such as coking, and geophysical
applications, such as volcanic eruptions.
The influence of quasi-two-dimensional confinement on buoyant jets or plumes
could also be studied. This problem is relevant to the study of natural ventilation
in buildings with line sources of heat. The question of the stability of the flow and
the conditions of emergence of the core and eddy structure can be raised when
buoyancy forces play an important role. Moreover, one might investigate whether
the entrainment, transport, dispersion and mixing mechanisms in quasi-two-di-
mensional buoyant jets or quasi-two-dimensional plumes are analogous to those
in the non-buoyant case studied in this thesis.
Finally, the fundamental modelling of the turbulence in the flow of quasi-two-di-
mensional jets could be investigated. Dracos et al. (1992) found an inverse cascade
of turbulence at scales larger than the gap width W and a three-dimensional
cascade of turbulence at smaller scales. Thus, the turbulence in quasi-two-dimen-
sional jets is neither purely two-dimensional nor exactly three-dimensional. The
study of the transfer of energy in this quasi-two-dimensional cascade of turbulence
could provide some insight about the general problem of turbulence.
202
Appendix A
Advection–diffusion model for
quasi-two-dimensional jets
A.1 Proof of equation (3.89)
If t > T0, we have, according to (3.88),
φT0(z, t) =
KT0
T0
z1/2
(∫ +∞
s(t)
ha−1e−h dh−
∫ +∞
s(t−T0)
ha−1e−h dh
)
, (A.1)
with
KT0 =
2B
3KdM01/2Γ [a+ 1]
, a = 2
3
(Ka
Kd
− 1
2
)
and s(t) = 4z
3/2
9KdM01/2t
.
(A.2a–c)
203
A Advection–diffusion model for quasi-two-dimensional jets
Combining the two integrals, in the limit t≫ T0, (A.1) becomes
φT0(z, t) ∼
KT0
T0
z1/2 (s(t))a−1 e−s(t) (s(t− T0)− s(t)) (A.3)
∼ KT0
z1/2
t (s(t))
a e−s(t) (A.4)
Using η = z/
(
t2/3M01/3
)
, (A.2b) and (A.2c), we obtain
φT0(z, t) ∼ t−2/3KT0M01/6
(
4
9Kd
)a
ηKa/Kd exp
[
− 4
9Kd
η3/2
]
. (A.5)
Finally, using (A.2a), we find
φT0(z, t) ∼t−2/3
B
(
3
2
)2a+1
(Kd)a+1 Γ [a+ 1]M01/3
ηKa/Kd exp
[
− 4
9Kd
η3/2
]
(A.6)
=t−2/3yδ(η), (A.7)
where yδ is defined in (3.73), and hence (3.89) follows.
204
Appendix B
Two-point statistics in circular
distributions
B.1 Conditional probability for the x-distance be-
tween two points in a disc
In equation (5.15), the conditional probability for the x- or lateral distance be-
tween two points H (x1,x2) (with 0 ≤ x1 ≤ R(t) fixed and x12 + z12 ≤ R2(t)) is,
using Cartesian coordinates,
PDt(H (x1,x2) ≤ h |x1) =
1
πR2(t)
∫∫
Dt∩H (x1,h )
dz2dx2, (B.1)
where the domain H (x1, h ) is defined such that x2 ∈ H (x1, h ) if |x1 − x2| ≤
h . Since H (x1, h ) does not depend on z2 (as long as x22 + z22 ≤ R2(t)) and
205
B Two-point statistics in circular distributions
Dt ∩ H (x1, h ) is symmetric with respect to the x-axis, (B.1) becomes, for 0 ≤
h ≤ R(t),
PDt(H (x1,x2) ≤ h |x1) =
2
πR2(t)
∫ x1+h
x1−h
√
R2(t)− x22 dx2, 0 ≤ x1 ≤ R(t)− h
(B.2)
and
PDt(H (x1,x2) ≤ h |x1) =
2
πR2(t)
∫ R(t)
x1−h
√
R2(t)− x22dx2, R(t)−h ≤ x1 ≤ R(t);
(B.3)
for R(t) ≤ h ≤ 2R(t),
PDt(H (x1,x2) ≤ h |x1) =
2
πR2(t)
∫ R(t)
−R(t)
√
R2(t)− x22 dx2, 0 ≤ x1 ≤ −R(t) + h ,
(B.4)
and
PDt(H (x1,x2) ≤ h |x1) =
2
πR2(t)
×
∫ R(t)
x1−h
√
R2(t)− x22 dx2, −R(t) + h ≤ x1 ≤ R(t);
(B.5)
and for 2R(t) ≤ h ,
PDt(H (x1,x2) ≤ h |x1) =
2
πR2(t)
∫ R(t)
−R(t)
√
R2(t)− x22 dx2, 0 ≤ x1 ≤ R(t).
(B.6)
Solving the integrals in (B.2), (B.3), (B.4), (B.5) and (B.6), we obtain the results
described in (5.16), (5.17), (5.18), (5.19) and (5.20), respectively.
206
B Two-point statistics in circular distributions
B.2 Value at the origin for the p.d.f. of the lateral
distance between two points in a disc
The value at the origin for the p.d.f. of the lateral distance between two points
in a disc fHDt is defined as
fHDt (0) = lim
δh→0
fHDt (δh )
δh
(B.7)
= lim
δh→0
4
πR2(t)
∫ R(t)
0
PDt(H (x1,x2) ≤ δh |x1)
δh
√
R2(t)− x12 dx1 (B.8)
≈ 4πR2(t)
∫ R(t)
0
lim
δh→0
PDt(H (x1,x2) ≤ δh |x1)
δh
√
R2(t)− x12 dx1, (B.9)
where we use (5.15). According to (5.16), the conditional probability is, for
δh → 0,
lim
δh→0
PDt(H (x1,x2) ≤ δh |x1)
δh
= limδh→0 1piδh
(
arcsin
(
x1+δh
R(t)
)
+(
x1+δh )
R(t)
√
1−
(
x1+δh
R(t)
)2
− arcsin
(
x1−δh
R(t)
)
− (x1−δh )R(t)
√
1−
(
x1−δh
R(t)
)2
)
(B.10)
= 2piR(t) dd( x1R(t))
[
arcsin( x1R(t))+(
x1
R(t))
√
1−( x1R(t))
2
]
(B.11)
=
4
πR(t)
√
1−
( x1
R(t)
)2
. (B.12)
We can replace (B.12) in (B.9), and integrate to find
fHDt (0) =
32
3π2R(t) . (B.13)
207
B Two-point statistics in circular distributions
B.3 Value at 2R(t) for the p.d.f. of the lateral
distance between two points in a disc
The value at h = 2R(t) for the p.d.f. of the lateral distance between two points
in a disc, fHDt is defined as
fHDt (2R(t)) = lim
δh→0
fHDt (2R(t))− fHDt
(
2R(t)− δh
)
δh
(B.14)
= lim
δh→0
1− fHDt
(
2R(t)− δh
)
δh
, (B.15)
where we use (5.20), then using to (5.18) and (5.19) we can show that for δh ≪ 1
fHDt
(
2R(t)− δh
)
= 1 +O
[
δh
]3/2
. (B.16)
Therefore, replacing (B.16) in (B.15), we obtain
fHDt (2R(t)) = 0. (B.17)
B.4 P.d.f of the x-distance between two points in
a square domain
The c.d.f. of the x− or lateral distance between two points in a square domain
St (described in § 5.3.1) is, using Cartesian coordinates,
FHSt (h ) = PSt(H (x1,x2) ≤ h ) =
1
R(t)
∫ R(t)
0
PSt(H (x1,x2) ≤ h |x1) dx1,
(B.18)
based on the general definition (5.7), and where we use the fact that the domain
St is symmetric with respect to the z-axis, the density is uniform over the whole
domain Dt and the conditional probability does not depend on z1 (as long as
−R(t) ≤ z1 ≤ R(t)). Similarly to the conditional probability for the disc described
208
B Two-point statistics in circular distributions
in § B.1, the conditional probability for the domain St is
PSt(H (x1,x2) ≤ h |x1) =
1
4R2(t)
∫
St∩H (x1,h )
dz2dx2, (B.19)
where the domain H (x1, h ) is defined such that x2 ∈ H (x1, h ) if |x1− x2| ≤ h .
We can integrate (B.19) with respect to both x1 and z1. We find, for 0 ≤ h ≤ R(t),
PSt(H (x1,x2) ≤ h |x1) =



h
R(t) 0 ≤ x1 ≤ R(t)− h
R(t)− (x1 − h )
2R(t) R(t)− h ≤ x1 ≤ R(t)
; (B.20)
for R(t) ≤ h ≤ 2R(t),
PSt(H (x1,x2) ≤ h |x1) =



1
2R(t) 0 ≤ x1 ≤ −R(t) + h
R(t)− (x1 − h )
2R(t) −R(t) + h ≤ x1 ≤ R(t)
;
(B.21)
and for 2R(t) ≤ h ,
PSt(H (x1,x2) ≤ h |x1) =
1
2R(t) . (B.22)
Using (B.20), (B.21) and (B.22) we can integrate (B.18) to find
FHSt (h ) =
4R(t)h − h 2
4R2(t) . (B.23)
Finally, we can differentiate (B.23) with respect to h to obtain the p.d.f. fHSt
described in (5.22).
B.5 Conditional probability for the Euclidean di-
stance between two points in a disc
In equation (5.23), the conditional probability for the distance between two points
D (x1,x2) (with 0 ≤ x1 ≤ R(t) fixed and x12 + z12 ≤ R2(t)) is, using Cartesian
209
B Two-point statistics in circular distributions
coordinates,
PDt(D (x1,x2) ≤ d |x1) =
1
πR2(t)
∫∫
Dt∩J (x1,d )
dz2dx2, (B.24)
where the domain J (x1, h ) is defined such that x2 ∈ J (x1, d ) if (x1 − x2)2 +
z22 ≤ d 2 (we can fix z1 = 0 by axisymmetry). (B.24) becomes, for 0 ≤ d ≤ R(t),
PDt(D (x1,x2) ≤ d |x1) =
1
πR2(t)
∫ d
0
∫ 2pi
0
r2dr2dθ2, 0 ≤ r1 ≤ R(t)−d , (B.25)
with r1 =
√
x12 + z12 and r2 =
√
x22 + z22, and
PDt(D (x1,x2) ≤ d |x1) =
2
πR2(t)
(∫ xI
r1−d
√
d 2 − (x2 − r1)2 dx2
+
∫ R(t)
xI
√
R2(t)− x22 dx2
)
, R(t)− d ≤ r1 ≤ R(t),
(B.26)
where xI is defined in (5.26); for R(t) ≤ d ≤ 2R(t),
PDt(D (x1,x2) ≤ d |x1) =
1
πR2(t)
∫ R(t)
0
∫ 2pi
0
r2 dr2dθ2, 0 ≤ r1 ≤ −R(t) + d ,
(B.27)
and
PDt(D (x1,x2) ≤ d |x1) =
2
πR2(t)
(∫ xI
r1−d
√
d 2 − (x2 − r1)2 dx2
+
∫ R(t)
xI
√
R2(t)− x22 dx2
)
, −R(t) + d ≤ r1 ≤ R(t);
(B.28)
and for 2R(t) ≤ d ,
PDt(D (x1,x2) ≤ d |x1) =
1
πR2(t)
∫ R(t)
0
∫ 2pi
0
r2 dr2dθ2, 0 ≤ r1 ≤ R(t). (B.29)
210
B Two-point statistics in circular distributions
Solving the integrals in (B.25), (B.26), (B.27), (B.28) and (B.29), we obtain the
results described in (5.24), (5.25), (5.27), (5.28) and (5.29), respectively.
B.6 Conditional probability for the ratio of the
lateral distance to the streamwise distance
between two points in a disc
In equation (5.31), the conditional probability for the ratio of the lateral distance
to the streamwise distance between two points M (x1,x2) (with 0 ≤ r1 ≤ R(t)
and 0 ≤ θ1 ≤ π/2 fixed) is, using polar coordinates,
PDt(M (x1,x2) ≤ m |x1) =
1
πR2(t)
∫∫
Dt∩G (x1,m )
r2 dr2dθ2, (B.30)
where the domain G (x1,m ) is defined such that x2 ∈ G (x1, d ) if |x1 − x2|/|z1 −
z2| ≤ m . (B.30) becomes, for 0 ≤ m ,
PDt(M (x1,x2) ≤ m |x1) =
1
πR2(t)
(∫ pi−υ
υ
∫ l
0
r2 dr2dθ2 +
∫ −υ
−pi+υ
∫ l
0
r2 dr2dθ2
)
,
(B.31)
with υ = arctan (1/m ), and where
l = r1 cos (θ2 − θ1 − π) +
√
R2(t)− r12 sin2 (θ2 − υ − π) (B.32)
is the equation of the perimeter of Dt in polar coordinates with the origin at x1.
We can then integrate (B.31) with respect to r2. We obtain
PDt(M (x1,x2) ≤ m |x1) =
1
πR2(t)
(∫ pi−υ
υ
l2
2
dθ2 +
∫ −υ
−pi+υ
l2
2
dθ2
)
. (B.33)
Finally, replacing (B.32) into (B.33) and integrating (B.33) with respect to θ2, we
find equations (5.32) and (5.33).
211
Bibliography
Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables, 10th Edition. Dover
Publications.
Albertson, M. L., Dai, Y. B., Jensen, R. A. & Rouse, H. 1950 Diffusion
of submerged jets. Transactions of the American Society of Civil Engineers
115, 639–664.
Antonia, R. A., Satyaprakash, B. R. & Hussain, A. K. M. F. 1980
Measurements of dissipation rate and some other characteristics of turbulent
plane and circular jets. Physics of Fluids 23, 695–700.
Baines, W. D., Turner, J. S. & Campbell, I. H. 1990 Turbulent fountains
in an open chamber. Journal of Fluid Mechanics 212, 557–592.
Batchelor, G. K. 1950 The application of the similarity theory of turbulence
to atmospheric diffusion. Quarterly Journal of the Royal Meteorological Society
76, 133–146.
Batchelor, G. K. 1952 Diffusion in a field of homogeneous turbulence. 2 The
relative motion of particles. Proceedings of the Cambridge Philosophical Society
48, 345–362.
213
Bibliography
Bird, R. B., Stewart, W. E. & Lightfoot, E. N. 2007 Transport Phenom-
ena, 2nd Edition. John Wiley & Sons.
Bourgoin, M., Ouellette, N. T., Xu, H. T., Berg, J. & Bodenschatz,
E. 2006 The role of pair dispersion in turbulent flow. Science 311, 835–838.
Bower, D. J., Caulfield, C. P., Fitzgerald, S. D. & Woods, A. W.
2008 Transient ventilation dynamics following a change in strength of a point
source of heat. Journal of Fluid Mechanics 614, 15–37.
Cardoso, S. S. S. & Zarrebini, M. 2002 Sedimentation from surface currents
generated by particle-laden jets. Chemical Engineering Science 57, 1425–1437.
Cenedese, C. & Dalziel, S. B. 1998 Concentration and depth fields deter-
mined by the light transmitted through a dyed solution. In Proceedings of the
8th International Symposium on Flow Visualization.
Chen, D. & Jirka, G. H. 1998 Linear stability analysis of turbulent mixing
layers and jets in shallow water layers. Journal of Hydraulic Research 36, 815–
830.
Chen, D. & Jirka, G. H. 1999 LIF study of plane jet bounded in shallow water
layer. Journal of Hydraulic Engineering – ASCE 125, 817–826.
Cierpka, C. & Kaehler, C. J. 2012 Particle imaging techniques for volumetric
three-component (3D3C) velocity measurements in microfluidics. Journal of
Visualization 15, 1–31.
Colomer, J., Casamitjana, X. & Fernando, H. J. S. 1998 Resuspension
and sedimentation of particles from a sediment bed by turbulent jets. Applied
Scientific Research 59, 229–242.
Colomer, J. & Fernando, H. J. S. 1996 Resuspension of particle bed by
round vertical jet. Journal of Environmental Engineering – ASCE 122, 864–
869.
Coomaraswamy, I. A. 2011 Natural ventilation of buildings: time-dependent
phenomena. PhD thesis, University of Cambridge, Cambridge, UK.
214
Bibliography
Dalziel, S. B. 1992 Decay of rotating turbulence - Some particle tracking ex-
periments. Applied Scientific Research 49, 217–244.
Dalziel, S. B., Patterson, M. D., Caulfield, C. P. & Coomaraswamy,
I. A. 2008 Mixing efficiency in high-aspect-ratio Rayleigh–Taylor experiments.
Physics of Fluids 20 (065106).
De Young, D. S. 1997 Growth of large scale structures in two-dimensional
mixing layers. Physics of Fluids 9, 2168–2170.
Dimotakis, P. E., Miake-Lye, R. C. & Papantoniou, D. A. 1983 Structure
and dynamics of round turbulent jets. Physics of Fluids 26, 3185–3192.
Dracos, T., Giger, M. & Jirka, G. H. 1992 Plane turbulent jets in a
bounded fluid layer. Journal of Fluid Mechanics 241, 587–614.
Drayton, M. J. 1993 Eulerian and Lagrangian studies of inhomogeneous tur-
bulence generated by an oscillating grid. PhD thesis, University of Cambridge,
Cambridge, UK.
Dutkiewicz, S., Griffa, A. & Olson, D. B. 1993 Particle diffusion in a
meandering jet. Journal of Geophysical Research–Oceans 98, 16,487–16,500.
Ernst, G. G. J., Sparks, R. S. J., Carey, S. N. & Bursik, M. I. 1996
Sedimentation from turbulent jets and plumes. Journal of Geophysical Research
101, 5575–5589.
Eser, S., Jenkins, R. G., Derbyshire, F. J. & Malladi, M. 1986 Car-
bonization of coker feedstocks and their fractions. Carbon 24, 77–82.
Foss, J. F. & Jones, J. B. 1968 Secondary flow effects in a bounded rectangular
jet. Transactions of the ASME: Journal of Basic Engineering 90, 241–248.
Fung, J. C. H. 1990 Kinematic simulation of turbulent flow and particle motion.
PhD thesis, University of Cambridge, Cambridge, UK.
Giger, M., Dracos, T. & Jirka, G. H. 1991 Entrainment and mixing in plane
turbulent jets in shallow water. Journal of Hydraulic Research 29, 615–642.
215
Bibliography
Cervantes de Gortari, J. & Goldschmidt, V. W. 1981 The apparent flap-
ping motion of a turbulent plane jet – further experimental results. Transactions
of the ASME: Journal of Fluids Engineering 103, 119–126.
Gradshteyn, I. S. & Ryzhik, I. M. 2007 Table of Integrals, Series, and
Products, 7th Edition. Elsevier Academic Press.
Holdeman, J. D. & Foss, J. F. 1975 Initiation, development, and decay of
secondary flow in a bounded jet. Transactions of the ASME: Journal of Fluids
Engineering 97, 342–352.
Hussein, H. J., Capp, S. P. & George, W. K. 1994 Velocity measurements
in a high-Reynolds number, momentum-conserving, axisymmetric, turbulent
jet. Journal of Fluid Mechanics 258, 31–75.
Itoˆ, S. 1992 Diffusion Equations . American Mathematical Society.
Jiang, J. S., Law, A. W. K. & Cheng, N. S. 2005 Two-phase analysis of
vertical sediment-laden jets. Journal of Engineering Mechanics – ASCE 131,
308–318.
Jirka, G. H. 2001 Large scale flow structures and mixing processes in shallow
flows. Journal of Hydraulic Research 39, 567–573.
Jirka, G. H. 2006 Integral model for turbulent buoyant jets in unbounded strat-
ified flows. Part 2. Plane jet dynamics resulting from multiport diffuser jets.
Environmental Fluid Mechanics 6, 43–100.
Jirka, G. H. & Harleman, D. R. F. 1979 Stability and mixing of a vertical
plane buoyant jet in confined depth. Journal of Fluid Mechanics 94, 275–304.
Jirka, G. H. & Uijttewaal, W. S. J. 2004 Shallow flows: a definition. In
Shallow Flows (ed. G. H. Jirka & W. S. J. Uijttewaal), pp. 3–11. Taylor &
Francis.
Joshi, P. B. & Taylor, R. B. 1983 Circulation induced by tidal jets. Journal
of Waterway Port Coastal and Ocean Engineering – ASCE 109, 445–464.
Kahane, J.-P. & Lemarie´-Rieusset, P.-G. 1998 Se´ries de Fourier et On-
delettes . Cassini.
216
Bibliography
Kitzhofer, J., Nonn, T. & Bru¨cker, C. 2011 Generation and visualization
of volumetric PIV data fields. Experiments in Fluids 51, 1471–1492.
Ko¨rner, T. W. 1988 Fourier Analysis . Cambridge University Press.
Kotsovinos, N. E. 1975 A study of the entrainment and turbulence in a plane
buoyant jet. PhD thesis, California Institute of Technology, Pasadena, USA.
Kotsovinos, N. E. 1976 A note on the spreading rate and virtual origin of a
plane turbulent jet. Journal of Fluid Mechanics 77, 305–311.
Kotsovinos, N. E. 1978 A note on the conservation of the axial momentum of
a turbulent jet. Journal of Fluid Mechanics 87, 55–63.
Kotsovinos, N. E. & Angelidis, P. B. 1991 The momentum flux in turbulent
submerged jets. Journal of Fluid Mechanics 229, 453–470.
Kotsovinos, N. E. & List, E. J. 1977 Plane turbulent buoyant jets. Part 1.
Integral properties. Journal of Fluid Mechanics 81, 25–44.
Kuang, J., Hsu, C.-T. & Qiu, H. 2001 Experiments on vertical turbulent
plane jets in water of finite depth. Journal of Engineering Mechanics – ASCE
127, 18–26.
Landel, J. R., Caulfield, C. P. & Woods, A. W. 2012a Meandering due to
large eddies and the statistically self-similar dynamics of quasi-two-dimensional
jets. Journal of Fluid Mechanics 692, 347–368.
Landel, J. R., Caulfield, C. P. & Woods, A. W. 2012b, sub judice
Streamwise dispersion and mixing in quasi-two-dimensional steady turbulent
jets. Journal of Fluid Mechanics. .
Law, A. W. K. 2006 Velocity and concentration distributions of round and plane
turbulent jets. Journal of Engineering Mathematics 56, 69–78.
Lee, J. M., Baker, J. J., Murray, D., Llerena, R. & Rolle, J. G. 1997
Quality analysis of petroleum cokes and coals for export specifications required
in use of specialty products and utility fuels. In Preprints of Symposia, 214th
National Meeting, American Chemical Society 42, pp. 844–853.
217
Bibliography
List, E. J. 1982 Turbulent jets and plumes. Annual Review of Fluid Mechanics
14, 189–212.
Luo, K., Klein, M., Fan, J.-R. & Cen, K.-F. 2006 Effects on particle dis-
persion by turbulent transition in a jet. Physics Letters A 357, 345–350.
Mathieu, J. & Scott, J. 2000 An Introduction to Turbulent Flow . Cambridge
University Press.
Monin, A. S. & Yaglom, A. M. 1975 Statistical Fluid Mechanics, Volume 2.
MIT Press.
Morton, B. R., Taylor, G. I. & Turner, J. S. 1956 Turbulent gravitational
convection from maintained and instantaneous sources. Proceedings of the Royal
Society of London. Series A 234, 1–23.
National Institute of Standards and Technology 2011-08-29 Digital
Library of Mathematical Functions. http://dlmf.nist.gov/.
Neves, M. J. & Fernando, H. J. S. 1995 Sedimentation of particles from
jets discharged by ocean outfalls: a theoretical and laboratory study. Water
Science and Technology 32, 133–139.
Ojha, R., Menon, N. & Durian, D. J. 2000 Hysteresis and packing in gas-
fluidized beds. Physical Review E 62, 4442–4445.
Paranthoe¨n, P., Fouari, A., Dupont, A. & Lecordier, J. C. 1988 Dis-
persion measurements in turbulent flows (boundary layer and plane jet). Inter-
national Journal of Heat and Mass Transfer 31, 153–165.
Picano, F., Sardina, G., Gualtieri, P. & Casciola, C. M. 2010 Anoma-
lous memory effects on transport of inertial particles in turbulent jets. Physics
of Fluids 22 (051705).
Pope, S. B. 1985 Pdf methods for turbulent reactive flows. Progress in Energy
and Combustion Science 11, 119–192.
Pope, S. B. 2000 Turbulent Flows . Cambridge University Press.
218
Bibliography
Prandtl, L. 1925 A report on testing for built-up turbulence. Zeitschrift fu¨r
Angewandte Mathematik und Mechanik 5, 136–139.
Ramaprian, B. R. & Chandrasekhara, M. S. 1985 LDA measurements in
plane turbulent jets. Transactions of the ASME: Journal of Fluids Engineering
107, 264–271.
Revuelta, A., Sa´nchez, A. L. & Lin˜a´n, A. 2002 Confined axisymmetric
laminar jets with large expansion ratios. Journal of Fluid Mechanics 456, 319–
352.
Richardson, L. F. 1926 Atmospheric diffusion shown on a distance-neighbour
graph. Proceedings of the Royal Society of London. Series A 110, 709–737.
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Annual
Review of Fluid Mechanics 23, 601–639.
Rowland, J. C., Stacey, M. T. & Dietrich, W. E. 2009 Turbulent char-
acteristics of a shallow wall-bounded plane jet: experimental implications for
river mouth hydrodynamics. Journal of Fluid Mechanics 627, 423–449.
Salazar, J. P. L. C. & Collins, L. R. 2009 Two-particle dispersion in
isotropic turbulent flows. Annual Review of Fluid Mechanics 41, 405–432.
Sawford, B. 2001 Turbulent relative dispersion. Annual Review of Fluid Me-
chanics 33, 289–317.
Schefer, R. W., Kerstein, A. R., Namazian, M. & Kelly, J. 1994 Role
of large-scale structure in a nonreacting turbulent CH4 jet. Physics of Fluids
6, 652–661.
Schneider, W. 1981 Flow induced by jets and plumes. Journal of Fluid Me-
chanics 108, 55–65.
Schneider, W. 1985 Decay of momentum flux in submerged jets. Journal of
Fluid Mechanics 154, 91–110.
Shinneeb, A.-M., Bugg, J. D. & Balachandar, R. 2011 Coherent struc-
tures in shallow water jets. Transactions of the ASME: Journal of Fluids En-
gineering 133, 011203.
219
Bibliography
Sparks, R. S. J. 1986 The dimensions and dynamics of volcanic eruption
columns. Bulletin of Volcanology 48, 3–15.
Stanley, S. A., Sarkar, S. & Mellado, J. P. 2002 A study of the flow-field
evolution and mixing in a planar turbulent jet using direct numerical simulation.
Journal of Fluid Mechanics 450, 377–407.
Sto¨hr, M. & Khalili, A. 2006 Dynamic regimes of buoyancy-affected two-
phase flow in unconsolidated porous media. Physical Review E 73, 036301.
Sveen, J. K. & Dalziel, S. B. 2005 A dynamic masking technique for com-
bined measurements of PIV and synthetic schlieren applied to internal gravity
waves. Measurement Science and Technology 16, 1954–1960.
Taylor, G. I. 1953 Dispersion of soluble matter in solvent flowing slowly through
a tube. Proceedings of the Royal Society of London. Series A 219, 186–203.
Taylor, G. I. 1958 Flow induced by jets. Journal of the Aero/Space Sciences
25, 464–465.
Thomas, F. O. & Brehob, E. G. 1986 An investigation of large-scale structure
in the similarity region of a two-dimensional turbulent jet. Physics of Fluids
29, 1788–1795.
Toschi, F. & Bodenschatz, E. 2009 Lagrangian properties of particles in
turbulence. Annual Review of Fluid Mechanics 41, 375–404.
Turner, J. S. 1986 Turbulent entrainment: the development of the entrain-
ment assumption, and its application to geophysical flows. Journal of Fluid
Mechanics. 173, 431–471.
Uberoi, M. S. & Singh, P. I. 1975 Turbulent mixing in a two-dimensional jet.
Physics of Fluids 18, 764–769.
Veitch, G. & Woods, A. W. 2000 Particle recycling and oscillations of vol-
canic eruption columns. Journal of Geophysical Research 105, 2829–2842.
Walters, A. L., Phillips, J. C., Brown, R. J., Field, M., Gernon,
T., Stripp, G. & Sparks, R. S. J. 2006 The role of fluidisation in the
220
Bibliography
formation of volcaniclastic kimberlite: grain size observations and experimental
investigation. Journal of Volcanology and Geothermal Research 155, 119–137.
Wang, H. W. & Law, A. W.-K. 2002 Second-order integral model for a round
turbulent buoyant jet. Journal of Fluid Mechanics 459, 397–428.
Woods, A. W. & Caulfield, C. P. 1992 A laboratory study of explosive
volcanic eruptions. Journal of Geophysical Research 97, 6699–6712.
Xu, H. & Bodenschatz, E. 2008 Motion of inertial particles with size larger
than Kolmogorov scale in turbulent flows. Physica D 237, 2095–2100.
Yang, Y., Crowe, C. T., Chung, J. N. & Troutt, T. R. 2000 Experiments
on particle dispersion in a plane wake. International Journal of Multiphase Flow
26, 1583–1607.
Yannopoulos, P. & Noutsopoulos, G. 1990 The plane vertical turbulent
buoyant jet. Journal of Hydraulic Research 28, 565–580.
Yannopoulos, P. C. 2006 An improved integral model for plane and round
turbulent buoyant jets. Journal of Fluid Mechanics 547, 267–296.
Yeung, P. K. 2002 Lagrangian investigations of turbulence. Annual Review of
Fluid Mechanics 34, 115–142.
Zauner, E. 1985 Visualization of the viscous flow induced by a round jet. Journal
of Fluid Mechanics 154, 111–119.
Zoueshtiagh, F. & Merlen, A. 2007 Effect of a vertically flowing water jet
underneath a granular bed. Physical Review E 75, 056313–1.
221