Turbulent convection from an area source in a confined space The density stratification produced by a turbulent lazy plume in a cylindrical container Francesco Ciriello UNIVERSITY OF CAMBRIDGE Department of Engineering This dissertation is submitted for the degree of Doctor of Philosophy Homerton College December 2018 Preface The research described in this thesis was carried out from October 2014 to December 2018 in the Engineering Department at the University of Cambridge. I hereby declare that except where specific reference is made to the work of others, the contents of this dissertation are original and have not been submitted in whole or in part for consideration for any other degree or qualification in this, or any other university. This dissertation is my own work and contains nothing which is the outcome of work done in collaboration with others, except as specified in the text. This dissertation contains fewer than 65,000 words including appendices, bibliography, footnotes, tables and equations and has fewer than 150 figures. I would like to thank my supervisor, Professor G. R. Hunt for his support and advice during these years. Words cannot express the depth of my gratitude towards him and the way he has supported me. I would also like to thank Samantha Lawton for additional and essential life support. I would like to thank our industrial sponsors from the Dyson Environmental Control Group: particularly Frederic Nicolas, Tim Jukes, Jimmy Lirvat and Peter Harley, for the many insightful discussions during our visits at Dyson HQ in Malmesbury and theirs at CUED. I would like to thank the members of my research group, past and present: Henry Burridge, Andy Acred, Recha Baumle, Jeff X. Wu, Ram Loganathan, Shahid Padhani, Antoine Debugne, James Richardson, Nicholas Wise and Megan Davies Wykes, for their help and support during these years. These thanks are also extended to the many friendly faces which have stopped for a chat about my work as they passed around the lab. I am especially grateful to the technical support staff of the Hopkinson Laboratory Workshop and would like to thank Roy Slater, Robert Leroy, John Harvey, Ken Griggs and Mark Garner for their suggestions and assistance in the design and construction of the experimental rig used during the PhD campaign. Financial support was provided under an EPSRC Industrial Case Award sponsored by Dyson Technology Ltd. I gratefully thank Homerton College for additional funding provided for the purchase of laboratory equipment. Francesco Ciriello December 14th, 2018 i Abstract Title: Turbulent convection from an area source in a confined space Author: Francesco Ciriello The study of turbulent convection is fundamental to our understanding of many flows that occur in the natural and built environment. This dissertation focusses on the density stratification produced by a turbulent lazy plume that is formed by the release of buoy- ant fluid from a circular source in a cylindrical container. This configuration allows us to explore a host of convection problems that uncover important features of the dynam- ics of turbulent plumes, Rayleigh-Taylor convection, gravity currents and stratification processes. Initially, the dynamics of lazy plumes are investigated, i.e. plumes formed by sources for which the Richardson number exceeds Γ0 > 1. Particular attention is dedicated to quantifying the near-source entrainment of very lazy plumes. For Γ0 & 103, newly- acquired experimental data presented herein reveals that an excess of dilution occurs in the proximity of the source relative to that predicted by classic integral models. Obser- vationally, the increased dilution coincides with the appearance of features resembling the growth and breakdown of Rayleigh-Taylor fingers in the near-source region. Sim- ple estimates to quantify the enhancement in entrainment are deduced based on classic quadratic laws for the growth of Rayleigh-Taylor convective layers. These estimates find good agreement with measurements of volume flux in the plume based on a separate suite of flow visualisation and particle image velocimetry experiments. A discussion on the shape of the mechanism is presented to discuss how the mechanism is expected to vary with increasing degree of plume ‘laziness’ (viz. via an increase Γ0). The study is subsequently extended to examine the flow that a lazy plume produces within the confines of a container, referred to herein as the filling-box problem. En route to describing the filling behaviour of a lazy plume, the dynamics of pure (Γ0 = 1) plumes whose source radius is much smaller than the height and width of the container are first examined. An experimental campaign was launched to examine the filling-box flow patterns that occur for a plume in cylindrical containers of radius-to-height aspect ratio 0.25 . φ := R/H . 2.0. New physical insights into the internal structure of these flows were acquired by simultaneously interrogating them with light-induced fluorescence and particle image velocimetry. A classification of filling-box regimes is reported in which five distinct flow patterns are identified: breakdown, rolling, slumping, blocking and dis- placement filling. Finally, this classification is extended to the stratification produced by a lazy plume. A theoretical model is developed to describe filling-box flows for a wide range of possible combinations of plume source conditions, characterised by the source Richardson number of the plume, 1 ≤ Γ0 . 106, the container aspect ratio, 0.7 . φ . 1.3 and relative size of the source compared to the height of the environment, 0.01 . β0 := b0/H . 0.4, b0 being the source radius. An experimental campaign of flow visualisations and particle image velocimetry is conducted to address key assumptions contained within the theoretical model. iii To my wife, Shivani v Nomenclature Roman Symbols At Atwood number [-]. a Dimensionless amplitude of Rayleigh-Taylor (RT) instability [-]. B Buoyancy flux [m4/s3]. B Cubical expansion coefficient of a gas [1/K]. b Plume radius [m]. c Circle of ‘confusion’, i.e. scale of focus on a camera sensor pixel [m]. C Coefficient [-]. D Diffusivity of a scalar [m2/s]. E Edge detection in image frame [binary]. e Unit vector [-]. FI Number of in-plane particles lost from an interrogation area in PIV. FO Number of out-of-plane particles lost from an interrogation area in PIV. f Dimensionless buoyancy flux [-]. FL Focal length [m]. F# F-stop of camera aperture [-]. f Spatial frequency of RT instability [1/m]. G Gaussian filter applied to image frame [px2]. g Gravity vector, g = (0, 0, g) [m/s2]. g Acceleration due to gravity [m/s2]. g′ Reduced gravity [m/s2]. H Height of container [m]. h Depth of current [m]. I Intensity matrix in image frame [8-bit]. i Constant √−1. j Discrete time-step counter in numerical schemes [-]. K Circulation [m2/s]. k Kernel size [px]. k Thermal conductivity [W/mK]. K Spreading rate of a pure plume [-]. L Length [m]. M Momentum flux [m4/s2]. m Dimensionless momentum flux [-]. vii N Number of fingers in RT instability [-]. n Refractive index [-]. N2 Vertical density gradient of the environment [1/s2]. NI Number of particles in an interrogation area in PIV [-]. n Number of steps in a numerical scheme [-]. P Pressure [kg/ms2]. Pe Peclét number [-]. Q Volume flux [m3/s]. q Dimensionless volume flux [-]. R Radius of container [m]. R Rotational component of the velocity gradient tensor [1/s]. R Confinement ratio R/b [-]. R Ideal gas constant [J/mol K]. Re Reynolds number [-]. Ri Richardson number [-]. r Radial coordinate [m]. S Interrogation area of vortex location scheme [m2]. S Strain component of the velocity gradient tensor [1/s]. S Source/sink of scalar [kg/m3s]. s Skewness [-]. T Temperature [K]. t Time [s]. U Mean vertical velocity of the environment surrounding plume [m/s]. u Velocity vector, u = (u, v,w) [m/s]. u Horizontal velocity in the current [m/s]. V Volume [m3]. v Azimuthal velocity [m/s]. W Width of (rectangular-based) container [m]. w Vertical velocity [m/s]. x Horizontal in-plane coordinate [m]. y Horizontal out-of-plane coordinate [m]. z Vertical coordinate [m]. Greek Symbols α Entrainment coefficient [-]. β Dimensionless radius of plume [-]. βg Dimensionless gross momentum flux coefficient [-]. γi Coefficient for momentum loss at impingement [-]. γ2 Weighted-average measure for location of vortex locus [-]. viii γg Dimensionless gross energy flux coefficient [-]. Γ Scaled Richardson number [-]. δg Dimensionless gross turbulent production coefficient [-]. ε Dimensionless entrainment function [-].  Coefficient [-]. ζ Dimensionless vertical coordinate [-]. η Displacement amplitude of instability [-]. θ Angle measured from the horizontal [rad]. θm Dimensionless buoyancy flux coefficient [-]. κ Thermal diffusivity [m2/s]. µ Dynamic viscosity [kg/ms]. ν Kinematic viscosity [m2/s]. ρ Density [kg/m3]. σ Standard deviation. τ Dimensionless time [-]. Φ Concentration of passive scalar [kg/m3]. ϕ Term in the series solutions to the plume conservation equations [-]. φ Radius-to-height aspect ratio of container [-]. χ Staircase function. ω Exponent [-]. Ω Vorticity [1/s]. Ω Vorticity vector [1/s]. Superscripts ∗ Effective. ( · ) Time and spatially-averaged. ( ·˜ ) Median. ( ·ˆ ) Dimensionless. Subscripts a Ambient. avs Asymptotic virtual origin correction. a0 Source acceleration. B Baroclinic. BT,H/3 Baines & Turner (1969) prediction for filling a container to one third of its depth. b Breakdown. C Contraction. c Outflow current. ix D Drag. δ Summation constant for virtual origin corrections (Hunt & Kaye, 2001). e Edge. F Filling. f First front / interface. G Gaussian. g Gross contribution in the coefficients of van Reeuwijk et al. (2016). gc Gravity current. HF Hyperfocal. I Inertial. i Impingement. J Pure jet. j Counter. L Layer. l Leading edge of current. m Mean contribution in the coefficients of van Reeuwijk et al. (2016). n Neck. p Pure plume. q Source radius scaling. Q Volume flux. RT Rayleigh-Taylor. r Inflow current. R Rolling. s Sinusoid. S Slumping. S t Stokes. T Turbulent. t Top. u Corner upflow. V Vortex. v Virtual origin correction. WD Working distance. w Vertical velocity. 0 Source value. x Acronyms AlO2 Aluminium dioxide (seeding particle). B&T69 Baines & Turner (1969). B91 Barnett (1991). C18 This dissertation. Col99 Colomer et al. (1999). DNS Direct numerical simulation. DOF Depth of field [m]. Eps01 Epstein & Burelbach (2001). E11 Ezzamel (2011). FOV Field of view matrix in image frame [pixels]. HCL01 Hunt et al. (2001). IA Interrogation area in a PIV window. K&H07 Kaye & Hunt (2007). K&H09 Kaye & Hunt (2009). LHS Left-hand side. LIF Light-induced fluorescence. LIF+PIV Simultaneous light-induced fluorescence and particle image velocime- try. MTT Morton et al. (1956). M17 Marjanovic et al. (2017). PIV Particle image velocimetry. P08 Plourde et al. (2008). PSP50 Polyamid seeding particle. RHS Right-hand side. RT Rayleigh-Taylor. TiO2 Titanium dioxide (seeding particle). vR16 van Reeuwijk et al. (2016). xi Table of Contents Preface i Abstract iii Nomenclature vii 1 Introduction 1 1.1 Aim, rationale and objectives . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Fundamentals 7 2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Turbulent entrainment in plumes . . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 Governing equations of motion . . . . . . . . . . . . . . . . . . . 11 2.2.2 Integral form of the plume equations . . . . . . . . . . . . . . . . 13 2.2.3 Pure, forced and lazy plumes . . . . . . . . . . . . . . . . . . . . 16 2.2.4 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.5 Entrainment in the adjustment region of a lazy plume . . . . . . . 19 2.2.6 Flows from uniformly distributed sources of buoyant fluid . . . . 23 2.2.7 Modelling shear exchanges . . . . . . . . . . . . . . . . . . . . . 24 2.3 The impingement of a plume with a horizontal boundary . . . . . . . . . 26 2.3.1 The impingement zone . . . . . . . . . . . . . . . . . . . . . . . 26 2.3.2 The plume outflow current . . . . . . . . . . . . . . . . . . . . . 27 2.3.3 Entrainment in horizontal density-driven currents . . . . . . . . . 30 2.4 The stratification produced by a plume in a container . . . . . . . . . . . 31 2.4.1 Descriptions of filling-box flow patterns . . . . . . . . . . . . . . 33 2.4.2 Modelling of filling-box flows . . . . . . . . . . . . . . . . . . . 35 2.5 Bounds of the current investigation . . . . . . . . . . . . . . . . . . . . . 39 3 Methodology 41 3.1 Configuration of a typical experiment . . . . . . . . . . . . . . . . . . . 41 3.1.1 Design of nozzle and supply system . . . . . . . . . . . . . . . . 42 3.1.2 Source flow rate and density measurements . . . . . . . . . . . . 43 3.2 Flow visualisation & measurement techniques . . . . . . . . . . . . . . . 46 3.2.1 Backlit dye visualisations . . . . . . . . . . . . . . . . . . . . . . 46 3.2.2 Light-induced fluorescence (LIF) visualisations . . . . . . . . . . 46 3.2.3 Robust edge detection routines for flow visualisation . . . . . . . 46 3.2.4 Planar particle image velocimetry (PIV) measurements . . . . . . 48 3.2.5 Simultaneous LIF & PIV . . . . . . . . . . . . . . . . . . . . . . 51 xiii 3.2.6 Analysing PIV data . . . . . . . . . . . . . . . . . . . . . . . . . 52 4 Rayleigh-Taylor lazy plumes 57 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.2.1 Basic definitions & existing solutions . . . . . . . . . . . . . . . 61 4.3 Theoretical modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3.1 A volume-flux-based correction . . . . . . . . . . . . . . . . . . 63 4.3.2 Simple models for entrainment in the contracting region . . . . . 66 4.3.3 A model for the morphology of the near-source region . . . . . . 67 4.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.4.1 Flow visualisation . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.4.2 Tracking of the first front . . . . . . . . . . . . . . . . . . . . . . 73 4.4.3 Particle image velocimetry (PIV) measurements . . . . . . . . . . 74 4.4.4 Flow control & measurement . . . . . . . . . . . . . . . . . . . . 74 4.5 Observations & measurements of the contracting region . . . . . . . . . . 75 4.5.1 Visualisations of the near-source region . . . . . . . . . . . . . . 78 4.6 Volume flux measurements . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.6.1 Volume flux measurements by filling rates . . . . . . . . . . . . . 80 4.6.2 Volume flux measurements by particle image velocimetry . . . . 80 4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.7.1 Comparison to other entrainment models . . . . . . . . . . . . . . 82 4.7.2 Matchings for βˆn and ζn solutions . . . . . . . . . . . . . . . . . . 83 4.7.3 A revised virtual origin correction . . . . . . . . . . . . . . . . . 85 4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5 A classification for filling-box flows 89 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.3 Filling mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.3.1 Breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.3.2 Rolling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.3.3 Slumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.3.4 Blocking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.4 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.4.1 Sidewall penetration . . . . . . . . . . . . . . . . . . . . . . . . 104 5.4.2 Vortex paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6 The filling of a container by a lazy plume 111 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.2 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.3 A theoretical model for slumping . . . . . . . . . . . . . . . . . . . . . . 116 6.3.1 Plume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.3.2 The plume impingement region . . . . . . . . . . . . . . . . . . . 120 6.3.3 Outflow current . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.3.4 Corner intrusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 xiv 6.3.5 Inflow current . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.3.6 Displacement filling . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.4.1 Flow visualisation . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.4.2 Velocimetry measurements in the near-impingement region . . . . 128 6.5 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.5.1 Stages of filling . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.5.2 Formation time and depth of initial layer . . . . . . . . . . . . . . 130 6.5.3 Filling rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.5.4 Impingement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.5.5 Entrainment in the plume outflow current . . . . . . . . . . . . . 138 6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.6.1 The evolution of the stratification . . . . . . . . . . . . . . . . . . 140 6.6.2 The internal structure of the slumping layer . . . . . . . . . . . . 141 6.6.3 Late stage filling rates . . . . . . . . . . . . . . . . . . . . . . . . 141 6.6.4 Expected regime bounds . . . . . . . . . . . . . . . . . . . . . . 147 6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.A A theoretical model for a non-idealised two-dimensional ‘line’ plume in a filling box during the slumping regime . . . . . . . . . . . . . . . . . . 152 7 Conclusions 155 7.1 Summary of research undertaken . . . . . . . . . . . . . . . . . . . . . . 155 7.2 The physics of lazy plumes and filling boxes . . . . . . . . . . . . . . . . 157 7.3 Choice of methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 7.4 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 7.5 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 7.6 Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 Bibliography 163 xv chapter 1 Introduction The plumes considered in this dissertation are formed from an isolated and continuous release of buoyant miscible fluid into an environment (Morton et al., 1956; Turner, 1973). Readers will be familiar, for example, with the smoke plumes that issue from chimney stacks, figure 1.1a. The turbulent billows that characterise the perimeter of the plume are indicative of the vigorous mixing that occurs as light fluid penetrates vertically through denser surroundings. Fundamental to the understanding of these flows is the mixing (viz. turbulent entrainment) of external fluid into the plume. Focus is on the dynamics of a turbulent plume that ensues from an isolated area source, first, in an unbounded environment, where the plume is free to develop, and then, within the confines of a container. The flow in the proximity of a plume source is known to display an array of different properties that depends on the source geometry, and also on both the relative density between source fluid and environment and the flow rate at which the plume is supplied. To illustrate this, an example visualisation of a highly buoyant plume, a so-called lazy plume (after Morton & Middleton 1973, see definition in Chapter 2, equation 2.38), is shown in figure 1.1b. Note that the near-source region of the plume is visibly distinct from that of the relatively forced plume shown in figure 1.1a. All plumes reach a state of local balance with height of the forces acting upon them. Lazy plumes are those with an excess of buoyancy supplied at source (Hunt & Kaye, 2001, 2005). As a result of this ‘imbalance’, the plume undergoes a pronounced gravitational acceleration which results in a rapid contraction of the plume to a neck. Note in figure 1.1b, how the lazy plume outline can be seen to contract to a neck before expanding. These con- tracting regions behave in peculiar ways compared to the well-understood behaviour of the expanding (self-similar) plume. The particular nature of how these imbalances affect turbulent mixing in the contracting region of a lazy plume constitutes a large portion of the research presented herein. Lazy plumes are a canonical class of flow in environmental fluid mechanics (Morton & Middleton, 1973; Caulfield & Woods, 1995; Hunt & Kaye, 2001; Fanneløp & Web- ber, 2003; Hunt & Kaye, 2005; Marjanovic et al., 2017). They are ubiquitous in nature and industry, and have been studied, for example, to examine the dispersal of smoke particulates in fires (Zukoski et al., 1981; Webber et al., 1992; Giannakopoulos et al., 2013), the heating and cooling of rooms (Hunt & Holford, 2000; Hunt & Kaye, 2010; Vauquelin et al., 2017) and episodic releases of contaminants in natural environments; e.g. blowouts of light fuel gas from underwater pipelines (Billeter & Fanneløp, 1989) and of effluents from waste storage tanks (Epstein & Burelbach, 2001). Despite this wide interest, there remain open questions regarding lazy plume behaviour that concern the near-field contracting region and entrainment, and this provides considerable motivation for the current study. Moreover, when confined with a container, the manner in which lazy plumes stratify the environment had not been considered prior to the investigations 1 1. Introduction (a) H un t& va n de n B re m er 20 11 (b) Figure 1.1: (a) An image of a smoke plume issuing from a chimney (Hunt & van den Bremer, 2011). Reprinted with permission. (b) An (vertically-inverted) backlit dye visualisation image of a lazy plume formed by the release of saline from a circular nozzle in a freshwater-filled tank. reported herein. Given their occurrence in rooms and buildings, e.g. due to a sun patch heating the floor (Hunt & Kaye, 2005) or to a thermally-driven flow through a vent (Hunt & Holford, 2000), the study of lazy plumes in a container could provide considerable practical benefit. A review of previous work (presented in chapters 2 and 4) highlights that there has been much debate over the characterisation of entrainment in the proximity of lazy plume sources (Colomer et al., 1999; Kaye & Hunt, 2009; van Reeuwijk et al., 2016; Carlotti & Hunt, 2017; Marjanovic et al., 2017). An excess of dilution has been observed relative to the reference case of a ‘self-similar’ (pure) plume (see discussions in Chapter 2). In other words, this would suggest that enhanced entrainment occurs in the near-source region. Theoretical models that have been developed to predict the dynamics of plumes, including lazy plumes, have typically assumed a height invariant entrainment behaviour (Morton & Middleton, 1973; Caulfield & Woods, 1995; Hunt & Kaye, 2001; Fanneløp & Webber, 2003; Hunt & Kaye, 2005). In the past, a detailed validation of the models for predicting the flow of lazy plumes has not been possible due to the relatively limited number of ex- perimental measurements conducted on them in the near field of interest (Colomer et al., 1999; Epstein & Burelbach, 2001; Hunt & Kaye, 2001; Kaye & Hunt, 2009). Herein, a detailed treatment on the dynamics of the near-source flow is presented. In doing so, the phenomenology of mixing in the near-source region is exposed and thereby explanations given for enhanced entrainment. 2 plume boundary outflow current Figure 1.2: Fluorescence visualisation of the impingement of a plume with a horizontal boundary and the ensuing radial outflow current that is formed along it. To develop an understanding of the dynamics of a lazy plume in a confined space, the impingement of a plume with a horizontal boundary is first examined. This impingement results in the formation of a current that propagates radially outwards along the boundary, figure 1.2. We will see that the dynamics of impingement and entrainment in these plume outflow currents are fundamental to understanding the development of stratification pro- cesses. Indeed, by varying the distance between the source and boundary a range of different dynamical plume behaviours on impact are anticipated. While radially spread- ing plume outflow currents have been previously examined by Kaye & Hunt (2007) for the case of a pure plume source, it was unclear at the outset how this current is modified for a lazy plume source. Additional confinement is then considered in the form of a cylindrical container that surrounds the plume. The source of the plume is centred at the bottom of the container as shown in the example visualisations presented in figures 1.3 and 1.4a-d. The container allows plume fluid to collect and progressively stratify the space. This class of flows is referred to as the filling box, after Turner (1973), in reference to the ‘filling’ of the con- tainer with buoyant fluid. The classic accepted pattern of filling was originally described by Baines & Turner (1969) as a thin and non-turbulent buoyant layer formed by the plume, which gradually deepens as it is vertically displaced by the subsequent incoming plume fluid. We will see that in practice an array of filling dynamics may occur depending on the container aspect ratio and plume source. The images that are shown in figure 1.3 (taken from Chapter 5), for example, are indicative of the multiple flow patterns that the outflow current formed by a plume of fixed source conditions creates in containers of different aspect ratio. Previous work on filling boxes has primarily focussed on the stratification produced by a plume whose source is much smaller in size than the height and width of the container (Baines & Turner, 1969; Hunt et al., 2001; Kaye & Hunt, 2007; Ezzamel, 2011); herein characterised by the ratio of the plume source radius b0 to the container height H and radius R (historically b0/R < 0.1 and b0/H < 0.1, see Chapter 2, table 2.1). While this work has exposed many facets of the stratification produced by plumes, herein we seek to build a comprehensive picture of filling-box flows. To this end, initial focus was on the stratification produced by nominally small-source pure plumes to develop a classification of flow patterns. We examine flows in containers of aspect ratio in the range 0.25 ≤ φ := R/H ≤ 2.0. Due to the presence of lateral boundaries, multiple patterns of overturning were observed as the plume outflow current impinged against the sidewalls, forcing the flow to reverse against the direction of buoyancy. These observations confirm 3 1. Introduction that there is not a single pattern of filling, viz. the Baines & Turner (1969) filling box, but strongly contrasting filling both in terms of density profiles and filling rates. This work was then extended to classify filling-box flows driven by lazy plume sources. Particular attention was dedicated to understanding how the flow patterns in the space vary as a function of the source conditions of the plume and the geometry of the container. 1.1 Aim, rationale and objectives The principal aim is to understand the dynamics of the flow produced by a lazy turbulent plume in a container; in other words, to understand the filling-box driven by a lazy plume. The configuration considered is shown in figure 1.5. The container is cylindrical, there is no buoyancy transfer to the boundaries and the source is located centrally at the bottom of the container. In pursuit of the principal aim, the following objectives were identified and tackled in the chapters cited: (i) to review existing models for the dynamics of lazy plumes and filling boxes (Chapter 2); (ii) to devise and conduct an experimental campaign both for the near-field lazy plume and the filling box (Chapter 3); (iii) to address gaps in our understanding of lazy plumes that were made evident by the review and current experimental campaign (Chapter 4); (iv) to develop a theoretical model to predict the dynamics of lazy plumes guided by the findings of the experimental campaign (Chapter 4); (v) to observe and describe the fundamental dynamics of filling-box flows (Chap- ter 5); (vi) to classify the dynamics of filling-box flows (Chapter 5-6); (vii) to extend the classification of filling-box flows to lazy plumes (Chapter 6); (viii) to develop a theoretical model for the dynamics of filling-box flows for lazy plumes (Chapter 6). The structure of the dissertation follows naturally from these objectives. Coupled ex- perimental and simplified theoretical approaches have proved highly successful in earlier plume and filling-box studies (e.g. Morton et al. 1956; Baines & Turner 1969), and this provided the overriding reasoning for adopting this approach herein. 1.2 Orientation The layout of this document continues as follows. Chapter 2 outlines the fundamen- tals of existing plume and filling-box theories, describing their historical context and the bounds of the current investigation. The foundations of the modelling approach adopted are overviewed and the rationale followed explained. Chapter 3 covers the experimental aspects of the study which address the research questions exposed in Chapter 2. 4 Figure 1.3: Fluorescence visualisation images of different regimes of filling-box behaviour for a ‘small-source’ plume. The flows produced by a plume in a container display an array of patterns which are discussed in Chapter 5. (a) (b) (c) (d) Figure 1.4: Backlit dye visualisation images of a flow produced by a lazy plume formed from the release of buoyant fluid from a circular area source centred at the base of a cylindrical container. The images that are shown are frames that progress in time from (a) to (d). 1. Introduction source containerlayering plume outflow currents corner flows Figure 1.5: Schematic of the flow configuration which is considered for the investigation. A turbulent plume is formed by the release of buoyant fluid from a circular source which is centrally located at the bottom of a cylindrical container. The source conditions of the plume and the aspect ratio of the container are varied to explore an array of flow patterns. Chapters 4, 5 and 6 each deal with different aspects of the behaviour of the plume. The relevant literature is reviewed as appropriate at the start of each chapter. Chapter 4 examines entrainment in the near-source region of a lazy plume. We will see that near- field plume behaviour is crucial to understanding the behaviour of the entire flow field as the region conditions the dynamics of the ensuing flows downstream. Chapter 5 exam- ines filling-box dynamics for a small-source plume with a focus on classifying these into distinct regimes. Chapter 6 presents a theoretical model to predict the dynamics of filling- box flows. The chapter focuses on the modelling of a single regime of filling behaviour (viz. ‘slumping’, see Chapters 5 and 6) to exemplify how filling flows can be modelled. Indications are provided to show how the modelling framework presented in Chapter 6 can be extended to other patterns of filling-box flows. Chapter 7 summarises the research, discusses the broad implications of the work and explores future avenues. 6 chapter 2 Fundamentals When a turbulent plume forms in an enclosed space, it tends to stratify the internal envi- ronment. The motions that occur during stratification create distributions of velocity and density within the space that are continuously evolving in time. These patterns of flow are complex and vary as a function of the source conditions of the plume and the geometry of the container. These complexities have in a way shaped the approach that has traditionally been adopted to describe these flows. Descriptions typically rely on simplified theoretical models which require inputs from experimental measurements to quantify the extent the flow mixes within the environment. Historical descriptions of lazy plumes and filling-box flows are overviewed in this chapter to discuss how similar mixing processes have been idealised in the past. Examining this information exposed gaps in our understanding of mixing in both plume and filling-box flows. Before reviewing the literature, the basic configuration and conventions used hence- forth are outlined in §2.1. In the following section (§2.2), quantitative descriptions of a turbulent lazy plume are discussed. Several assumptions are introduced to demonstrate, following standard derivation procedures (e.g. Turner 1973), how the Navier-Stokes equa- tions can be simplified to the shear layer equations for a buoyant flow in a cylindrical coordinate system. The shear layer equations are then integrated to derive the governing equations for the plume. These integral equations are referred to as the plume conser- vation equations and rely on the conservation of mass, momentum and buoyancy fluxes to describe its dynamics. As a system of equations, they are under-determined (four un- knowns, three equations) and can be closed, at first order, by introducing assumptions quantifying the effects of turbulent entrainment. Relying on these derivations, we discuss the suitability of certain assumptions concerning entrainment, their relationship to the concept of self-similarity and the extent to which they are applicable to the modelling of lazy plumes. The shear layer equations can also be employed to describe the outflow current formed by the impingement of a plume with a horizontal boundary. As we shall see in Chapter 6, these currents play a pivotal role in stratification. A variety of corner flow behaviours can occur depending on the dynamics of the current when it impinges against the side- wall. The governing equations for the current are derived by integrating the shear layer equations and introducing assumptions quantifying entrainment. These are discussed in §2.3. As a means to review theoretical models for plumes and currents, solutions to the governing equations are presented to discuss the fundamental dynamics of these flows. Finally, previous work on filling-box flows is reviewed (§2.4). We compare descrip- tions of flow patterns and overview existing theoretical models. These reviews help delimit the bounds of our study around research areas that require further investigation (§2.5). 7 2. Fundamentals b0 R H ρa > ρ0 w0, ρ0 z r Figure 2.1: Schematic of the basic configuration for the flow investigated. A turbulent plume is supplied with buoyant fluid from a circular source of radius b0 at a uniform source velocity w0 and source density ρ0 into an initially uniform (density ρa) cylindrical environment of radius R and height H. A coordinate system is introduced with vertical coordinate z measured from the level of the source and radial coordinate r measured from the axis of the plume. 2.1 Basic definitions An idealised configuration of the flow considered is drawn in figure 2.1. The configuration allows to delimit the investigation and introduce the basic notation and conventions used henceforth. A turbulent plume is examined which is steadily released from a circular source of buoyant fluid into a cylindrical container. Motions are discussed with buoyancy taken as positive upwards, i.e. as if heated air were rising through the environment. The direction of motion is immaterial to the dynamics as the investigated density variations are prescribed to be ‘small’. Laboratory experiments, for convenience, use plumes of salt water descending through a tank of fresh water. The source of the plume is located at the centre of the base of the container and is of radius b0, uniform mean velocity w0 and density ρ0. The subscript (.)0 refers to the source value and the overbar (.) to the fact that the quantity is time, horizontally and azimuthally-averaged. The environment is initially uniform in density, with ambient density ρa that is denser than the source fluid density, ρa > ρ0, and quiescent before the activation of the plume. The difference in density between the plume source fluid and the environment is prescribed to be sufficiently small, |ρ0 − ρa|/ρa  1, so as to neglect the inertial contributions provided by the density differences in the momentum flux equations (cf. the Boussinesq assumption presented in §2.2). The internal radius of the container is denoted by R and the height as H. The bound- aries are impermeable and the effects of buoyancy transfer between the container and the flow are neglected. The coordinate system is defined relative to the source of the plume. The vertical coordinate z is measured in the streamwise direction from the level of the source and the radial coordinate r is measured outwards from the axis of the plume (fig- ure 2.1). The lower boundary of the container (b0 < r ≤ R at z = 0) is open to allow for a finite volume flux to enter the container via the plume source (in contrast with the classic filling-box study of Baines & Turner 1969 in which this boundary is closed and 8 2.2. Turbulent entrainment in plumes the source volume flux is zero). 2.2 Turbulent entrainment in plumes Before delving into the basics of plume theory, required to address the limitations of the modelling, the reasoning behind the choice of the approach is outlined. Quantitative descriptions of turbulent plumes can be broadly delineated in three groups: integral, nu- merical and empirical. As we shall see, the distinction between these is not clear cut as they tend to rely on each other, but is a useful simplification for those familiar with the subject. The merits of these approaches are discussed to explain why integral techniques were the natural choice for the investigation and to indicate how information from numer- ical and experimental approaches was used in support of these integral models. Integral techniques for the modelling of a turbulent plume stem from the description of thin-shear flows (figure 2.2). Shear flows occur when localised regions of relatively high velocity exist within a flow field. A comprehensively studied example of a thin- shear flow is the purely-momentum driven turbulent jet (see, for example, the reviews of Fischer et al. 1979 and List 1982 for introductions to the subject). A jet forms from an isolated and continuous release of fluid into a uniform environment of equal density (ρ0/ρa = 1). Provided the jet is fully turbulent in that locally, inertial forces far exceed viscous forces, then at sufficient streamwise distances from the release, turbulence within the jet reaches a state of local equilibrium (Tennekes & Lumley, 1970). This equilibrium results in an invariant (self-similar) shape for the cross-stream profiles of velocity and scalar concentration, and their associated statistics (Hussein et al., 1994). As a result, similarity solutions to the shear layer equations for the jet can be deduced, for example, as those derived by Tollmien (1926), in which turbulent transfer between the jet and the environment was modelled using Prandtl’s (1926) mixing-length hypothesis. Solutions for a turbulent plume can be derived using analogous approaches. Simi- larity solutions employing the mixing-length assumption were derived for the turbulent plume, for instance, by Corrsin (1943). In practice, however, solutions for a plume that employ a mixing-length or higher order closures are often cumbersome in that they overly complicate the solution procedure and limit its applicability to unconfined and unstratified environments. For these reasons, as pointed out in Morton et al. (1956), Taylor (1946) introduced a simpler assumption about turbulent transport in a plume, which bypasses the complexity associated with modelling how turbulent eddies redistribute mass, momentum and density. Taylor’s (1946) approach relies on a turbulence closure which assumes that the velocity induced into the plume by turbulent entrainment is proportional to a charac- teristic local streamwise velocity in the plume. This assumption allowed Taylor (1946) to derive analytical solutions for the idealised case of a turbulent plume ensuing from a point source of buoyancy flux. It was popularised in a lecture by Batchelor (1954) and is now typically referred to as the entrainment hypothesis, or constant-α approach of Morton et al. (1956), n.b. after the constant of proportionality, herein αp. Self-similar solutions, such as those of Corrsin (1943) and Morton et al. (1956), re- quire that turbulence within the plume has reached a state of local equilibrium such that the mechanism inducing the turbulent transfer is invariant along the streamwise extent of 9 2. Fundamentals the flow (Turner, 1986). A state of self-similarity however cannot be achieved in many situations where a plume occurs (Morton et al., 1956), for example, when a plume is re- leased in a stratified environment or in the rapidly adjusting region in the proximity of the source. Following this reasoning, it can be deduced that the turbulence closure of Morton et al. (1956) may not be theoretically suitable for the description of a lazy plume in a container, which is indeed the case we would like to examine. In spite of this, the entrainment hypothesis of Morton et al. (1956) has proven to be adaptable to situations where the condition of self-similarity is not satisfied (Morton et al., 1956; Morton, 1959; van Reeuwijk & Craske, 2015). This hypothesis (or adaptations of it) has successfully yielded satisfactory predictions of the dynamics of a plume in which the turbulent mechanism that induces the mixing does not vary rapidly with streamwise dis- tance (see, for example, the predictions for ‘weakly’ lazy plumes of Hunt & Kaye 2005). We will show in the upcoming sections that the validity of the self-similarity assumption can be breached in several instances without compromising the predictive capability of the models presented. We will also show that this is not always the case and that one needs to be aware of the circumstances in which breaching this assumption yields grossly incorrect predictions for the dynamics of the flow. The extent to which the entrainment hypothesis suitably predicts plume flows is discussed in §2.2.3 -2.2.5 and in further detail in Chapter 4. Not only can the integral modelling approach of Morton et al. (1956) yield satisfac- tory predictions of the dynamics of the plume, but it is also well suited as a foundation for more elaborate theoretical modelling. A substantial merit, which will become evident in the upcoming sections, is indeed the possibility to incorporate additional effects within the governing equations of motion, e.g. modified near-source entrainment, stratification, co- or counter-flows, detrainment, drag or heat transfer. The first inclusion that will be discussed (§2.2.5) are techniques to model entrainment in regions where self-similarity is not satisfied (so-called entrainment functions). The second major inclusion involves the characterisation of turbulent exchanges between co- and counter-flows. It stems from the work of Morton (1961) and is discussed in §2.2.7. The last inclusion is the effect of a time-varying stratification which is discussed in §2.4. We will point out how other effects can be incorporated into this framework at appropriate locations throughout the text. Alternatives to the integral modelling approach also played an important role in our investigation. Indeed, fundamental to the formulation of Morton et al. (1956) is the value of a constant of proportionality, αp, within the entrainment hypothesis. The value of this constant has been deduced both by the means of experimental measurements (e.g. Baines 1983) and numerical simulations (e.g. van Reeuwijk & Craske 2015). Previous experimental (Colomer et al., 1999; Epstein & Burelbach, 2001; Kaye & Hunt, 2009) and numerical (Plourde et al., 2008; van Reeuwijk et al., 2016; Marjanovic et al., 2017) work that has been conducted on the lazy plume is limited to few studies and to the authors knowledge, has never examined the effect of the stratification produced by the plume. These works are further reviewed at the beginning of Chapter 4. The foundations of integral plume modelling are outlined in §2.2.1-2.2.7. In sum- mary, the reasons behind the choice of this approach are threefold: (i) it is practically more viable compared to its alternatives, i.e. analytical and numerical solutions involving other turbulent closures, (ii) it has proven to effectively predict the dynamics of similar 10 2.2. Turbulent entrainment in plumes flows and (iii) it is adaptable to a range of scenarios, crucially, to that of a lazy plume stratifying a container. Addressing the limitations of predictions based on integral mod- els requires a detailed inspection of the underlying assumptions of plume theory and of existing models for stratification which we proceed to analyse in this chapter. This is of particular relevance to the work presented in Chapter 6, where we will see that, as we consider the behaviour of plumes formed from a source which is not small compared to the size of the container, the non-self similar behaviour of the plume has a profound influence on the pattern of stratification. 2.2.1 Governing equations of motion Integral models for turbulent plumes are based on the shear layer equations. In this sec- tion, we show how the shear layer equations for an axisymmetric flow in a cylindrical coordinate system are deduced from the Navier-Stokes equations to highlight the main assumptions involved in these descriptions. For the sake of the common application of these models in engineering and related practice, the descriptions presented will encom- pass the behaviour of both aqueous-saline and thermal air plumes. In a generalised form and Eulerian frame of reference, the equations of motion for fluid flow, assuming there are no external body forces acting on the fluid except for grav- ity, can be expressed in terms of the conservation of mass and momentum fluxes: Dρ Dt = −∇. (ρ.u) , (2.1) Dρu Dt = −∇P + ρg + µ∇2u, (2.2) (Batchelor, 1967), where u is the velocity vector, ρ is the density, P is the pressure field, g = (0, 0,−g) is the gravity vector, and µ is the dynamic viscosity, taken to be constant for a Newtonian fluid such as air or water. Additionally, an advection-diffusion equation can describe a scalar quantity Φ, e.g. saline in water, that is transported by the flow: ∂Φ ∂t + u.∇Φ = D∇2Φ + S Φ, (2.3) where D is the molecular diffusivity of the scalar taken to be constant (i.e. time and space- invariant) and S Φ is a source or sink of the scalar (Crank, 1975). If the scalar considered is temperature T , such as in the case of thermal air plumes, then equation (2.3) can be expressed as: ∂T ∂t + u.∇T = κ∇2T + S T , (2.4) where thermal diffusivity κ = k/ρCp (≈ 19 mm2/s for air, defined by the ratio of thermal conductivity, k, to the specific heat capacity, Cp, of the fluid) and S T , being a source or sink of temperature. For applications in air, pressure, density and temperature are related by the ideal gas law: P = ρRT, (2.5) where R is the gas constant (≈ 287 J/kg K for air at atmospheric pressure and tempera- ture). 11 M or to n et al .( 19 56 ) V an D yk e (1 97 9) Sc ha nz et al .( 20 16 ) V an D yk e (1 97 9) 19 79 Sc ha nz et al .( 20 16 ) (a) (b) (c) (d) (e) (f) Figure 2.2: A collection of images of turbulent shear flows (plumes and jets) that shows the flows using different visualisation techniques. (a) Backlit image of a saline plume released in freshwater stained with dye, (b) front-lit image of a saline plume released in freshwater stained with fluorescent dye, (c) image of the central cross-section of a freshwater jet visualised with fluorescent dye illuminated by a planar light sheet, (d) shadowgraph of a carbon dioxide jet in air, (e)-(f) velocity vectors and iso-surfaces of vortical structures estimated using the Q-criterion (cf. §3.2.6) measured using high-density tomographic particle tracking velocimetry of a thermal plume in air. Images re-printed with permission. 2.2. Turbulent entrainment in plumes The simplified governing equations for a turbulent plume can be deduced from equa- tions (2.1)-(2.5) by means of several assumptions. The first assumption that will be in- voked is that the flow is nominally incompressible, such that Dρ/Dt = 0, as typically, in convective flows, local velocity magnitudes are significantly lower than the speed of sound in the medium. Additional simplifications can be employed which rely on the axisymmetry of the configuration. The plume develops from a circular source of buoy- ant fluid which is released vertically with no swirl. The environment is either laterally unbounded or cylindrical. It follows that it can be assumed that the flow is statistically axisymmetric and contains no swirl. Azimuthal velocities and their gradients are then negligible, such that v = 0 and ∂/∂θ = 0. Finally, assuming that the flow is nominally quasi-steady in that the plume source conditions considered are set to be steady, ∂/∂t = 0, the conservation of mass (2.1) is given by: 1 r ∂ruρ ∂r + ∂wρ ∂z = 0. (2.6) Analogous assumptions, i.e. v = 0, and ∂/∂θ = 0, ∂/∂t = 0, can be applied to the conser- vation of momentum (2.2). A further assumption is invoked to neglect the viscous shear terms (µ∇2u = 0) relying on the fact that we intend to describe fully turbulent flows, i.e. of sufficiently high Reynolds number, Re := ρUL/µ  1 where U and L are characteristic velocity and length scales. The conservation of radial and streamwise momentum fluxes, following from simplifying (2.2) by means of these assumptions, correspond to 1 r ∂ru2ρ ∂r + ∂uwρ ∂z = −∂P ∂r , (2.7) 1 r ∂rwuρ ∂r + ∂w2ρ ∂z = −∂P ∂z + g. (2.8) Finally, assuming that the pressure field varies hydrostatically, i.e. P = ρa gz, and that radial pressure gradients are negligible, ∂P/∂r = 0, the conservation of radial (2.7) and vertical (2.8) momentum flux are given by: 1 r ∂ru2ρ ∂r + ∂uwρ ∂z = 0, (2.9) 1 r ∂rwuρ ∂r + ∂w2ρ ∂z = g(ρ − ρa). (2.10) Equations (2.6), (2.9) and (2.10) are referred to as the shear layer equations for a buoyant incompressible axisymmetric flow in a cylindrical coordinate system. These equation are applied to describe the dynamics of the plume in §2.2.2 and of the current in §2.3.2. 2.2.2 Integral form of the plume equations Equations (2.6) and (2.10) can be integrated with respect to radial coordinate, r, from the axis of symmetry, r = 0, to the edge of the plume, r = b, to obtain the integral form of the conservation equations for mass and vertical momentum flux: d dz ∫ b 0 ρwr dr = ∫ b 0 ρu dr = ρabue, (2.11) 13 2. Fundamentals d dz ∫ b 0 ρw2r dr = ∫ b 0 (ρ − ρa)gr dr. (2.12) The entrainment hypothesis of Morton et al. (1956) has been introduced in (2.11), whereby an edge ‘entrainment’ velocity, ue, is defined and assumed to be proportional to the local vertical velocity, ue := αw. The entrainment hypothesis serves as a turbulence closure to the system of equations. It quantifies the entrainment of fluid from the environment into the plume. Assuming that the density differences are small, ∆ρ/ρa  1, and only considered in the generation of vertical momentum flux (viz. the so-called Boussinesq assumption, Turner 1973), the integral equations (2.11) and (2.12) can be expressed as: d dz ∫ b 0 wr dr = αbw, (2.13) d dz ∫ b 0 w2r dr = g′b2, (2.14) where the reduced gravity of the fluid is defined by: g′ := g ( ρ − ρa ρa ) . (2.15) Within the Boussinesq assumption it is implicit, as inertial contributions are neglected and buoyant contributions retained, that equation (2.12) is independent of whether gravity is acting in the same or in the opposing direction to that of the z-coordinate, i.e. dense fluid falling in lighter surroundings is dynamically equivalent to light fluid ascending in denser surroundings. The flows of a saline plume descending freshwater and of a thermal plume rising in air can be described using the same equations provided this assumption is valid. We restrict the investigation to Boussinesq plumes in order to reduce the complexity in the investigation of the near-field entrainment of a lazy plume (Chapter 4), but we acknowledge that the modelling approach could be readily extended to non-Boussinesq plumes following the work of Rooney & Linden (1996), Carlotti & Hunt (2005) and van den Bremer & Hunt (2010). For descriptions of thermal air plumes, the reduced gravity can be defined relative to the temperature of the gas. By defining a reference absolute temperature Ta relative to the density ρa of the environment, a cubical expansion coefficient B can be defined as: B(T − Ta) + O(P) = ρ − ρa ρa , (2.16) (Turner, 1973), where O(P) signifies higher-order pressure terms. Assuming the fluid obeys the ideal gas law (2.5) in an isobaric environment, i.e. dP = 0, to first order, additional pressure effects are negligible, i.e. O(P)  1, and as a result, the coefficient of expansion is inversely proportional to temperature: B := 1 V ( dV dT ) dP=0 = R PV = 1 T . (2.17) 14 2.2. Turbulent entrainment in plumes The reduced gravity can then be expressed as function of temperature differences, i.e. g′ = g(1 − T/Ta). A transport equation for the reduced gravity, g′, which is analogous to (2.3) and (2.4), is given by: ∂g′ ∂t + u.∇g′ = κ∇2g′ + S g′ , (2.18) with S g′ being a source or sink of reduced gravity. For a stratified environment, which is characterised a vertical reduced gravity gradient given by: N2 := ∂g′e ∂z , (2.19) with g′e being the horizontally-averaged reduced gravity of the environment, the relative density of the local stratification to that of the plume locally acts as a sink of buoyancy, denoted S g′ := −N2w. By assuming that the flow is quasi-steady, such that ∂g′/∂t = 0, and that the flow is of high Peclét number, i.e. PeD = UL/D  1 and Peκ = UL/κ  1, such that the transport by molecular and thermal diffusion is negligible, i.e. κ∇2g′ = 0, the transport of reduced gravity is reduced to: 1 r ∂rug′ ∂r + ∂wg′ ∂z = −N2w. (2.20) Equation (2.20) can be integrated with respect to r to obtain: d dz ∫ b 0 g′wr dr = − ∫ b 0 N2wr dr. (2.21) An assumption on the shape of the velocity and reduced gravity profiles is invoked to integrate the conservation equations. This assumption coincides with the assumption of self-similarity of the cross-stream profiles discussed at the onset of §2.2. Uniform (‘top- hat’) profiles of velocity w and reduced gravity g′ are assumed henceforth. The shape of profiles is chosen for consistency with the models presented in Chapter 6 for other regions in the flow. By substituting for the integrals in equations (2.13), (2.14) and (2.21) with the definition of integral quantities of volume, momentum and buoyancy fluxes given by: Q := 2 ∫ b 0 wr dr = wb2, (2.22) M := 2 ∫ b 0 w2r dr = w2b2, (2.23) B := 2 ∫ b 0 g′wr dr = g′wb2, (2.24) the plume conservation equation are obtained. Note that these are scaled by a factor of pi compared to the physical fluxes of volume, specific momentum and buoyancy. On integration, the plume conservation equations are deduced: dwb2 dz = 2αwb, (2.25) 15 2. Fundamentals dw2b2 dz = g′b2, (2.26) dg′wb2 dz = −N2wb2. (2.27) Henceforth, the overbars that denote time- and horizontally-averaged quantities in (2.22)- (2.24) are dropped to simplify notation. 2.2.3 Pure, forced and lazy plumes An alternative form of the plume conservation equations (2.25)- (2.27) is in terms of the fluxes defined in (2.22)-(2.24): dQ dz = 2αM1/2, dM dz = BQ M , dB dz = −N2Q. (2.28) Equation (2.28) can be obtained by substituting (2.22)-(2.24) into (2.25)- (2.27) and is equivalent to the form reported in Morton et al. (1956). Taylor (1946) showed that an analytical solution to (2.28) can be obtained for a plume that develops from a point-source of buoyancy flux B0, with zero source volume flux, Q0 = 0, and zero source momentum flux, M0 = 0, with the coefficient α taken to be constant (α = αp) and the environment unstratified, N2 = 0. Based on dimensional grounds, a solution to (2.28) can be sought in the form: Q = CpQB 1/3 0 z 5/3, M = CpMB 2/3 0 z 4/3, B = B0. (2.29) The constants of proportionality, CpQ and CpM, were then be solved for by substituting these forms (2.29) into plume equations (2.28), to give: CpQ = 6αp 5 ( 9αp 10 )1/3 , CpM = ( 9αp 10 )2/3 . (2.30) Taylor’s (1946) solutions reveal that a self-similar axisymmetric point-source plume adopts the following behaviour in terms of its local plume radius, velocity and reduced gravity: b = Q1/2 M = 6αp 5 z, (2.31) w = M Q = 5 6αp ( 9αp 10 )1/3 B1/30 z1/3 , (2.32) g′ = B0 Q = 5 6αp ( 10 9αp )1/3 B2/30 z5/3 . (2.33) In other words, the plume radius grows linearly with vertical distance b ∼ z and the veloc- ities and reduced gravities decay as power laws of w ∼ z−1/3 and g′ ∼ z−5/3 respectively. A dimensionless quantity can be formed by a ratio composed of (2.31)-(2.33). This quantity is referred to as the local Richardson number of the plume and is defined by: Ri := g′b w2 , (2.34) 16 2.2. Turbulent entrainment in plumes (Turner, 1973). The Richardson number characterises the balance of buoyancy to inertia in the plume. As a result, it is closely linked with the state of self-similarity in a plume developing from an area source. It can be readily estimated for the point-source plume, by substituting (2.31)-(2.33) into (2.34), which yields: Rip = 8αp 5 . (2.35) The scaling (2.35) suggests that in a self-similar state, the local Richardson number Rip of a plume is constant and independent of the magnitude of the buoyancy flux B0 and the streamwise distance z. The idealisation of a point source can be relaxed so that solutions can be extended to any combination of source fluxes Q0, M0 and B0. Morton (1959) showed that by doing so, different regimes of plume dynamics can be identified. By non-dimensionalising the system of equations (2.28) by the length scale Lq := 5b0/6αp and by the source fluxes: ζq := z Lq , q := Q Q0 , m := M M0 , f := B B0 , (2.36) the following dimensionless form of the conservation equations can be obtained: dq dζq = 5m1/2 3 , dm dζq = 4Γ0 3 q m , d f dζq = 0, (2.37) (Hunt & Kaye, 2005). A scaled source Richardson number, Γ0, results from the analysis. This is defined as the ratio between the Richardson number at the source of the plume compared to that of a self-similar (pure) plume: Γ0 := Ri0 Rip = 5 8αp B0Q20 M5/20 = 5 8αp g′0b0 w20 . (2.38) The parameter Γ0 segregates area-source plume behaviour into three regimes: forced 0 < Γ0 < 1, pure Γ0 = 1 and lazy Γ0 > 1 plumes (Morton, 1959; Morton & Middleton, 1973; Hunt & Kaye, 2005). The scaled source Richardson number Γ0 is a measure of the imbalance created at the source between gravitational and inertial forces relative to the balance observed in a self-similar plume. This balance is required for the plume from an area source to reach a state of self-similarity (by means of a streamwise invariance in the turbulent production mechanism, Turner 1986). It indeed naturally occurs in the plume with increasing dis- tance from the source as locally Γ(z) adjusts from a source value Γ0 to a value of Γ ≈ 1. If the plume is pure at source, Γ0 = 1, buoyancy and inertia are in balance and the flow conditions (viz. the local value of Γ = 1), following an effective ‘core’, are nominally invariant with streamwise distance. When the source is out of balance (Γ0 , 1), the flow adjusts to balance the excess (in forced plumes) or deficit (in lazy plumes) of inertia rel- ative to the local buoyancy. In doing so, locally the value of Richardson number tends to unity, Γ = 5BQ2 8αpM5/2 → 1. (2.39) 17 2. Fundamentals As a result, Γ also implicitly characterises the local deviation of the plume from the self-similar state. It is thus reasonable to assume that if Γ ≈ 1, then the plume has reached an approximately self-similar state. In this state, the solutions based on the assumption of a constant value of the entrainment parameter α are suitable in that locally turbulent en- trainment in the plume is expected to be statistically invariant. Conversely, the greater the imbalance in Γ0, the less applicable is the assumption of a constant-α in the adjustment region. The extents of this applicability is examined in the upcoming sections. In in- stances where this applicability does not hold, suitable descriptions of an ‘effective’ value of α in this region can be nevertheless determined from an extended analysis of the shear layer equations (as shown by van Reeuwijk & Craske 2015), which we proceed to discuss in §2.2.5. The work presented herein focusses primarily on the lazy plumes due to the relatively limited experimental data available concerning their near-source entrainment (discussed in Chapter 4). 2.2.4 Solutions The system of equations (2.37) can be solved, subject to the source conditions Q = Q0, M = M0 and B = B0 at z = 0, in different ways. Solution procedures fall into three cat- egories: approximate (typically via a virtual origin correction), analytical and numerical. Approximate and analytical solutions of (2.37) were deduced by Hunt & Kaye (2001, 2005) for a plume in an unstratified environment and rely on the assumption of a constant α. Their solution procedures are further discussed in Chapter 4. We present numerical solutions for (2.37) as these are relatively simple to obtain. A solution procedure can be implemented in an integration scheme (e.g. Euler, Crank-Nicolson, Runge-Kutta) in the form a first-order initial value problem; the source fluxes being the initial values. Numerical solutions for (2.37) are shown in figure 2.3 for a range of scaled source Richardson number Γ0, including the pure jet Γ0 = 0, the forced plume 0 < Γ0 < 1, the pure plume Γ0 = 1 and the lazy plume Γ0 > 1. These solutions are presented to review the fundamental dynamics of jets and plumes. We here limit ourselves to highlighting the most relevant features within these plots. The solutions indicate that, provided Γ0 , 0, the local Richardson number tends to pure plume-like behaviour, Γ→ 1, with increasing distance from the source. In the far-field, the relationships for Q, M, g′, b and w revert to the power laws of the point-source plume description (2.29) with vertical origin offsets (Hunt & Kaye, 2001). The spreading rates of lazy plumes are of particular interest to the results presented in Chapter 4, as the spreading behaviour of the plume is employed as a diagnostic to assess near-source entrainment. To demonstrate the understanding concerning entrainment that can be acquired from observing spreading rates, let us take, for example, the solutions for b/b0 of a pure jet (Γ0 = 0) and pure plume (Γ0 = 1) shown in figure 2.3b-c. For solutions that assume α is constant, a pure jet spreads at a rate that is a factor 5/3 greater than that of the plume. In practice, measurements reveal that jets and plume spread at nominally equal rates (Fischer et al., 1979). This suggests that in a pure jet the magnitude of α is lower than that in plumes. This was confirmed in experimental measurements, for example, as reviewed by List (1982), α j/αp ≈ 0.64. Following this reasoning, the magnitude of α is closely related to spreading rate. In the case of a comparison between a pure jet and a pure plume, a decrease in the relative spreading rate is indicative of a decrease in α/αp. 18 2.2. Turbulent entrainment in plumes This interpretation of entrainment by means of the plume spreading rate will help us understand the behaviour of a lazy plume in the near-source region. Hunt & Kaye (2001) showed that provided Γ0 > 2.5, a lazy plume contracts to a neck before expanding in the far-field. Theoretically, the plume contracts to a thinner neck, which occurs closer to the source, as Γ0 increases (figure 2.3c). We will see that the radius and location of the neck does not agree well with experimental observations and we proceed to discuss the implications of necking behaviour on entrainment in Chapter 4. 2.2.5 Entrainment in the adjustment region of a lazy plume The constant-α formulation of Morton et al. (1956) has been adopted in a number of theoretical studies describing the dynamics of a lazy plume (Morton & Middleton, 1973; Caulfield & Woods, 1995; Hunt & Kaye, 2001; Fanneløp & Webber, 2003; Hunt & Kaye, 2005; Carlotti & Hunt, 2005; Michaux & Vauquelin, 2008; van den Bremer & Hunt, 2010). It is a very attractive option for the description of these plumes for two reasons. First, it enables the derivation of analytical solutions to the plume conservation equations. Secondly, it is effective in the prediction of the dynamics of both a plume which is not excessively imbalanced at source from a pure plume-like state, Γ0 = 1 (e.g. Hunt & Kaye 2005), and of a plume at further distance from the source, where Γ ≈ 1 (e.g. Baines & Turner 1969, Baines 1983 and Wang & Law 2002). Nevertheless, as argued in §2.2, constant-αmodels cannot predict near-source dynam- ics in all cases (viz. Γ0  1 and Γ  1). To compensate for regions of excessive imbal- ance of inertia to buoyancy, entrainment in these regions can be parameterised according to the local conditions in the plume, typically characterised by the local Richardson num- ber. Extensive reviews on the suitability of entrainment parameterisations are presented in Kaminski et al. (2005), Ezzamel et al. (2015) and van Reeuwijk & Craske (2015). In this section, our overview of entrainment parameterisations is restricted to their deploy- ment in the context of a lazy plume. Entrainment functions can be readily implemented by modifying the plume volume flux equation (2.37) into the following form: dq dζq = 5 3 εm1/2, (2.40) where the entrainment function ε is introduced and defined relative to the entrainment measured in a self-similar pure plume: ε := α αp = f (Γ). (2.41) An early example of an entrainment function for a forced plume is that deduced by Priestly & Ball (1955). The authors assumed a linear variation in α with Γ between values measured in pure jets (α j) and pure plumes (αp): ε := α j αp + ( 1 − α j αp ) Γ for 0 ≤ Γ0 ≤ 1. (2.42) Entrainment parameterisations for the lazy plume have also been proposed. Carlotti & Hunt (2017), to describe the increased entrainment in the adjustment region of lazy 19 10−7 10−3 101 105 10−6 10−4 10−2 100 (a) Γ ζq 0 5 10 15 0 2 4 6 8 10 0 0.5 1 0 0.2 0.4 (b) b/b0 (c) 100 101 102 103 104 10−3 10−2 10−1 100 101 (d) Q/Q0 ζq 1 1 5 3 10−2 10−1 100 101 102 103 10−2 10−1 100 101 (e) w/w0 1 -1 -1 3 101 103 105 10−3 10−2 10−1 100 101 (f) M/M0 ζq 4 3 10−4 10−3 10−2 10−1 100 10−2 10−1 100 101 (g) g′/g′0 3 -5 -1 1 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 Γ0 = 0 log10 Γ0 Γ0 = 1 Figure 2.3: Constant-α solutions for a plume for varying Γ0 (lines show steps of a single of order of magnitude) in terms of (a) local Richardson number Γ, (b) plume radius b/b0, with (c) being a close up of the near-source region, (d) volume flux Q/Q0, (e) vertical velocities w/w0, (f) mo- mentum flux M/M0 and reduced gravity g′/g′0. Colour indicates the value of Γ0 to be compared to the colourbar at the bottom of the figures. 10−7 10−3 101 105 10−6 10−4 10−2 100 (a) Γ ζq 0 5 10 15 0 2 4 6 8 10 0 0.5 1 0 0.2 0.4 (b) b/b0 (c) 100 101 102 103 104 10−3 10−2 10−1 100 101 (d) Q/Q0 ζq 10−2 10−1 100 101 102 103 10−2 10−1 100 101 (e) w/w0 101 103 105 10−3 10−2 10−1 100 101 (f) M/M0 ζq 10−4 10−3 10−2 10−1 100 10−2 10−1 100 101 (g) g′/g′0 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 Γ0 = 0 log10 Γ0 Γ0 = 1 Figure 2.4: Solutions for a plume for varying Γ0 (lines show steps of a single of order of mag- nitude), based on the α = f (Γ) derivation of van Reeuwijk & Craske (2015), where higher-order terms are modelled using the coefficients from the DNS measurements of lazy plumes provided in van Reeuwijk et al. (2016). 2. Fundamentals plumes, suggested a power law form where: ε := Γω for Γ0 ≥ 1. (2.43) Analytical solutions were possible for the special case of ω = 1/5. The suitability of this entrainment function is discussed in Chapter 4, Appendix A. Van Reeuwijk & Craske (2015) generalised a definition of α by examining how the plume conservation equations are derived from the shear layer equations and ensuring that the solutions are also consistent with the conservation of mean kinetic energy in the plume. They deduced an entrainment function that is a function of local Richardson number and can be computed by estimating coefficients for the modelling of higher-order terms in the integral form of the plume equations. The resulting entrainment function, assuming steadiness, takes the form: α = − δg 2γg + ( 1 βg − θm γg ) Ri, (2.44) where coefficients βg, γg, θm and δg are respectively associated to the dimensionless mo- mentum flux, energy flux, buoyancy flux and turbulent production in the plume, and sub- scripts (.)m denote a contribution from the mean terms and (.)g a gross contribution from the sum of mean, turbulent and pressure terms. The coefficients δg, γg, βg and θm could then be estimated from their DNS data. Following from the discussion of van Reeuwijk et al. (2016) in which they extend their analysis to lazy plumes, we select, for the fol- lowing discussions (including the ones presented in Chapter 4 and 6), values for these coefficients of: δg = −0.184, γg = 1.391 , βg = 1.076, and θm = 1.011. To the author’s knowledge, analytical solutions to the plume equations in an axisym- metric geometry that use an entrainment function in the form, ε = C1 + C2Γ; C1 ≈ 0.067 and C2 ≈ 0.203 being unspecified constants, have not been deduced yet. On the other hand, numerical solutions can be implemented straightforwardly. These solutions are shown in figure 2.4 based on the entrainment model proposed by van Reeuwijk et al. (2016) and prescribed by (2.44). Focus is primarily on discussing changes in the be- haviour of a lazy plume as a result of modified near-source entrainment effects by com- paring figures 2.3 and 2.4. A first result, which becomes discernible upon inspection of figure 2.4c, is the change in the necking dynamics of a lazy plume. On increasing the degree of ‘laziness’ of the source (via Γ0), the plume contracts to a lesser extent than what predicted by constant- α models. This is indicative, as discussed for the spreading rates of jets and plumes in §2.2.4, of an increase in effective α in this region. The same apparent increase in entrainment can be observed in the near-source region of the plume in terms of volume fluxes in figure 2.4d. A more detailed inspection of how α/αp might be expected to change in the near- source region is shown in figure 2.5, where the predictions based on (2.44) are shown in conjunction with values estimated from the DNS data of lazy plumes from Plourde et al. (2008) and Marjanovic et al. (2017). Effective magnitudes of entrainment α/αp increase by several order of magnitude with increasing source laziness Γ0. The largest increase in entrainment occurs closest to the source and over a region which is considerably smaller 22 2.2. Turbulent entrainment in plumes 10−1 100 101 102 103 10−4 10−3 10−2 10−1 100 101 ζq α/αp Γ0 ≈ 1 (M17) Γ0 ≈ 10 (M17) Γ0 ≈ 100 (M17) Γ0 ≈ ∞ (M17) Γ0 ≈ ∞ (P08) −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 Γ0 = 0 log10 Γ0 Γ0 = 1 Figure 2.5: Effective α/αp parameter in the near-source region of a plume from the theory of van Reeuwijk & Craske (2015) (shown as solid lines) and the data from the DNS studies of Plourde et al. (2008), denoted P08, and Marjanovic et al. (2017), denoted M17, (shown as markers, see legend in figure). Colour is indicative of the scaled source Richardson number Γ0 (see colour bar). N.b. the measurements with Γ0 = ∞ are effectively simulations of discs which induce a buoyancy transfer with the environment e.g. a heated disc. compared to the radius of the source (and thus hard to measure both experimentally and numerically). No detailed experimental validation of these models and numerical studies has been conducted prior to this investigation. Moreover, the reasons behind this increase in entrainment are of particular interest in this work. We will see in Chapters 4 and 6 that this near-source entrainment substantially changes the dynamics of the plume and of the ensuing stratification. 2.2.6 Flows from uniformly distributed sources of buoyant fluid A question that is raised in the discussion presented in §2.2.2-2.2.5 is that of the origin of the apparent increase in near-source entrainment observed in a lazy plume. Analysed under the perspective of the entrainment function (2.44), the major contribution to en- trainment in this region is attributed to the production of kinetic energy in the proximity of the source (van Reeuwijk et al., 2016) as most of the work is performed by the ex- cess of buoyancy that characterises their near field. Some general observations of flows produced from distributed sources of buoyancy are presented in preparation for the dis- cussions presented in Chapter 4. Visualisations of the near-source region of a lazy plume ensuing from the injection of brine into a freshwater environment are shown in figures 2.6a-c. Mixing in these plumes resembles in morphology that observed in thin-shear flows, as seen in the examples shown in figure 2.2. Observations of saline plumes presented in Chapter 4 will reveal that this particular mixing mechanism (i.e. shear driven) can change with respect to what is seen in figure 2.2. 23 2. Fundamentals Plumes ensuing from heated discs are shown in figures 2.6d-f. Theoretically, heated disc plumes correspond to the limit of an infinite source Richardson number in that the source velocity w0 ≡ 0 (Plourde et al., 2008; Marjanovic et al., 2017). Correspondingly, their source Reynolds number also tends to zero, Re0 = w0b0/ν, and as a result, they transition from a laminar close to the source into a turbulent regime further away from it. Plourde et al. (2008) present a discussion on the morphology of vortical structures in the near-source region of heated disc plumes. They observe, as shown in 2.6e-f, unsteady ‘helical’ structures that develop around the contracting portion of the plume. Shown in figures 2.6g-i are visualisations of buoyancy-driven flows formed by dis- tributed sources in classic two-layer Rayleigh-Taylor convection studies. Theoretically, these flows are typically considered as laterally unbounded given their wide horizontal extent. If an analogy to lazy plumes were to be drawn, these flows could be considered to be representative of the near-field region of an infinitely lazy plume in the limit of an infinitely large source, i.e. b0 ≡ ∞. Mixing in these flows is distinguished by finger-like intrusions that break down into turbulence as they penetrate deeper into each layer. We will see in Chapter 4 that the dynamics of injected plumes share similarities with both heated disc and Rayleigh-Taylor flows; and that the appearance of a finger-like mixing mechanism lays the reason behind the apparent increase in time-averaged entrainment. 2.2.7 Modelling shear exchanges In sections §2.2.5-2.2.6, the modelling of the near-source entrainment dynamics of a lazy plume was reviewed. In this section, a further manipulation to the plume conservation equations is discussed which becomes useful in the description of stratification. Morton (1961) devised a theoretical approach to idealise the turbulent exchange between a plume and a turbulent (co-/counter-)flowing environment. This approach is extended to the flow of a lazy plume in a filling box. In figure 2.7, we idealise the scenario of a flow interacting with a coflow of uniform velocity which surrounds the plume, denoted U and defined as positive in the streamwise direction. A control volume analysis can be applied to an infinitesimally thin slice of thickness δz of the plume, to derive expressions for the conservation of volume, vertical momentum and buoyancy fluxes:[ wb2 ] z+δz − [ wb2 ] z︸ ︷︷ ︸ ≡δQ = 2ueδz − 2Ueδz, (2.45) [ w2b2 ] z+δz − [ w2b2 ] z︸ ︷︷ ︸ ≡δM +2ueUδz = −g′b2δz, (2.46) [ g′wb2 ] z+δz − [ g′wb2 ] z︸ ︷︷ ︸ ≡δB = −2g′eUeδz. (2.47) In (2.45) and (2.47), the entrainment hypothesis, Ue := αU, characterises the horizontal velocity into the environment. The volume flux entrained into the plume is expected to depend on the shear between the two flows and is thus proportional to the difference in velocities between the plume and its environment, the conservation of volume flux being 24 R ay le ig h- Ta yl or he at ed di sc s in je ct ed pl um es Colomer et al. (1990) Hunt & Bremer (2014) Chapter 1 Pottenbaum & Gharib (2004) Plourde et al. (2008) Dalziel et al. (1999) Davies Wykes & Dalziel (2014) Boffetta & Mazzino (2017) (a) (b) (c) (d) (e) (f) (g) (h) (i) Figure 2.6: Visualisations of flows from distributed sources of buoyant fluid. (a)-(c) Saline so- lutions injected from microporous nozzles shown as (a) a fluorescence visualisation of the near- source region, (b) a shadowgraph visualisation of a plume at Γ0 ≈ 1500 and (c) a backlit (false- colour) dye visualisation of a plume at Γ0 ≈ 200. (d) A plume ensuing from a heated disc visu- alised using thermochromatic crystals. (e)-(f) A DNS study on a plume from a heated disc show- ing the temperature field (e) and isosurfaces of vorticity (f). (g)-(h) Fluorescence visualisations of Rayleigh-Taylor convection from an initially two-layer stratification of aqueous solutions. (i) A DNS study of Rayleigh-Taylor convection showing the instantaneous temperature field. Images re-printed with permission. 2. Fundamentals (a) [ piwb2 ] z+δz [ piwb2 ] z U 2pibUeδz 2pibueδz volume (b) [ piw2b2 ] z+δz [ piw2b2 ] z U 2pibuUeδz pig′b2 momentum (c) [ pig′wb2 ] z+δz [ pig′wb2 ] z U 2pibg′eUeδz buoyancy Figure 2.7: A schematic of the plume which is used for a control volume analysis, whereby the plume conservation equations are derived in a form that models the interaction between the plume and a uniform coflowing environment of velocity U, following the approach of Morton (1961). given by: dwb2 dz = 2αb(w − U). (2.48) It is evident that if the plume and environment velocities are equal, then there is no ve- locity gradient between the plume and the co-flow, and as a result no exchange of fluid between the regions. The conservation of momentum and buoyancy fluxes in the plume are given by: dw2b2 dz = g′b2 − 2αbwU, (2.49) dg′wb2 dz = −2αg′ebαU. (2.50) These idealisations are extended in Chapter 6 to the modelling of the exchanges between a plume and the return flow it induces in the environment of the container and of the exchanges that occur between currents in filling-box flows. 2.3 The impingement of a plume with a horizontal boundary In axisymmetric filling-box flows, a radial outflow current forms from the impingement of the plume against the horizontal boundary of the container (figure 2.1). The current plays a pivotal role in stratification. In this section, a theoretical model for the outflow current is discussed. The ‘source’ of the current is modelled following the approach proposed by Kaye & Hunt (2007). Their approach assumes that the source is cylindrical and that the source fluxes scale on the fluxes of the plume as it impinges against the boundary (§2.3.1). This modelling approach is extended by deriving governing equations for the current in §2.3.2, which rely on an entrainment assumption analogous to that proposed by Morton et al. (1956) for the plume. 2.3.1 The impingement zone A schematic of the impingement model, which we extend from Kaye & Hunt (2007), is shown in figure 2.8. The source of the current is modelled as a cylindrical source, which 26 2.3. The impingement of a plume with a horizontal boundary (a) current nose / leading vortex ring current stem plume ‘cylindrical’ impingement regionz′ r (b) wi hc0 uc0 r0 r rl Figure 2.8: (a) An instantaneous image of the impingement of a plume with a horizontal bound- ary, visualised via light-induced fluorescence. The coordinate system, (r, z′), shown in figure, is centred along the plume axis and measured from the level of the floor. (b) A schematic of the cylindrical impingement region that serves as the idealised source of the outflow current. issues buoyant fluid of volume, specific momentum and buoyancy fluxes respectively: Qc0 := uc0 r0 hc0, Mc0 := u 2 c0 r0 hc0, Bc0 := g ′ c0 uc r0 hc0, (2.51) where uc0 and g ′ c0 = g(ρc−ρa)/ρa are the time- and vertically-averaged source velocity and reduced gravity. The source fluxes of volume and buoyancy are estimated by outlining a control volume over the impingement region whose width is dictated by the radius of the plume as it impinges against the boundary bi = b(z = H), figure 2.8. Volume and buoyancy fluxes are conserved through the control volume and are estimated via the solutions of the plume conservation equations (2.37) at a distance of z = H − bi from the level of the source: Qc0 = Q(z = H − bi), Bc0 = B0. (2.52) While momentum flux is not conserved during the impingement of the plume with the boundary, it is expected that the source radial momentum flux of the current scales on the momentum flux of the plume as it enters the region: Mc0 = γiM(z = H − bi). (2.53) The proportionality constant γi is introduced to account for the kinetic energy flux losses experienced by the flow as it turns 90 degrees. The parameter γi, as shown by Kaye & Hunt (2007), is important for the modelling of filling-box flows as it affects the predicted dynamics of the current. An experimental campaign is conducted to measure the value of γi and is further discussed in Chapter 6. 2.3.2 The plume outflow current Conservation equations for the quasi-steady behaviour of the current can be derived by employing an analogous procedure to that described for the plume in §2.2.2. The equation 27 2. Fundamentals for the conservation of mass flux in the shear layer equations (2.6) can be integrated with respect to a vertical coordinate z′ = H − z (see figure 2.8). By integrating from the level of the floor z′ = 0, up to the depth of the current z′ = hc, equation (2.6) gives: d dr ∫ hc 0 ru dz′ = r ∫ hc 0 w dz′ = rwe. (2.54) In (2.54), the velocity at the edge of the current, we, is defined at a level hc from the floor to introduce an entrainment hypothesis, we := αc|u|, to close the system of equations, where αc is the entrainment coefficient for the current. The radial momentum flux (2.7) can be integrated accordingly. By assuming that pressure varies hydrostatically, i.e. P = ρa gz′, equation (2.10) becomes: 1 r d dr ∫ hc 0 ru2 dz′ + ∫ hc 0 uw dz′︸ ︷︷ ︸ = 0 = − d dr ∫ hc 0 (ρ − ρa)gz ρa dz. (2.55) The second integral term in (2.55) is equal to zero as, by definition, w = 0, for 0 ≤ z′ ≤ hc, and u = 0, for z′ ≥ hc, such that, uw = 0, for all z′. Finally, by integrating the advec- tion equation for the reduced gravity (2.8), it may be shown that radial buoyancy flux is conserved, i.e. 1 r d dr ∫ hc 0 g′ur dz′ + ∫ hc 0 wg′ dz′︸ ︷︷ ︸ = 0 = 0, (2.56) in the absence of sinks and sources. Following from the approach adopted for the plume conservation equations (2.22)-(2.24), top-hat profiles are assumed for the time- and vertically- averaged velocity, uc, and reduced gravity of the current, g ′ c := g ( ρc − ρa ) /ρa, to integrate equations (2.54)-(2.56); ρc being the mean density of the current. The radial volume, mo- mentum and buoyancy fluxes of the current are defined as: Qc := ∫ hc 0 ur dz′ = ucrhc, (2.57) Mc := ∫ hc 0 u2r dz′ = u2crhc, (2.58) Bc := ∫ hc 0 g′ur dz′ = g′cucrhc. (2.59) An expression for the conservation of these fluxes is obtained on integration: 1 r d dr (uc r hc) = αc|uc|, (2.60) 1 r d dr ( u2c r hc ) = − d dr ( g′c h 2 c 2 ) , (2.61) 1 r d dr ( g′c uc r hc ) = 0. (2.62) 28 2.3. The impingement of a plume with a horizontal boundary Alternatively, the conservation equations for the current, (2.60)-(2.62), can be expressed in terms of volume, momentum and buoyancy fluxes: 1 r dQc dr = αc ( Mc Qc ) , (2.63) 1 r dMc dr = − d dr ( BcQ3c 2M2c r2 ) , (2.64) 1 r dBc dr = 0. (2.65) To obtain further insight into the dynamics of the current, solutions to equations (2.63)- (2.65) are discussed. Initially, we assume an invariant entrainment behaviour with dis- tance from the plume axis, i.e. αc = cst. We first consider solutions for a source of the current which is (i) a point radial source of pure momentum flux Mc0 with no source volume and buoyancy flux Qc0 = 0 and Bc0 = 0, and then for (ii) a point radial source of buoyancy flux Bc0 with no source volume and momentum flux Qc0 = 0 and Mc0 = 0. For the first case, a solution to (2.63)-(2.65) is sought on dimensional grounds in the form: Qc = CcQM 1/2 c0 r, Mc = Mc0, Bc = 0. (2.66) By substituting (2.66) into (2.63)-(2.65), the value of the proportionality constant is evalu- ated: CcQ = α 1/2 c . As a result, for the idealised case of a current formed by a point-source of momentum flux, the depth grows linearly with radial distance and the velocities decay as a power law of M1/2c0 r −1: hc = Q2c Mcr = αcr, uc = Mc Qc = ( Mc0 αc )1/2 1 r . (2.67) For the case of a current formed by a point-source of buoyancy flux, a solution to (2.63)- (2.65) is sought on dimensional grounds in terms of: Qc = CcQB 1/3 0 r 5/3, Mc = CcMB 2/3 0 r 4/3, B = B0. (2.68) The constants of proportionality are evaluated to be equal to: CcQ = 3αc 52/32 , CcM = 3αc 51/322 . (2.69) It follows that, for this case, the current depth, velocity and reduced gravity vary as: hc = 3αc 5 r, uc = 51/3 2 (Bc0 r )1/3 , g′c = Bc0 Q = 52/32 3αc B2/3c0 r5/3 . (2.70) Comparing the solutions for a current formed by a point source of momentum flux to those for a current formed by a point source of buoyancy flux can provide insights into the expected behaviour of inertially- and buoyancy-driven currents. For example, the buoyant current spreads linearly with radial distance r at a rate which is 3/5 that of the inertial current. The velocities and reduced gravities in the buoyant current decay as power laws of uc ∼ B1/3c0 r−1/3 and g′c ∼ B2/3c0 r−5/3 as opposed to the inertial current, whereby uc ∼ M1/2c0 r−1 and g′c ∼ M−1/2c0 r−1. 29 2. Fundamentals 2.3.3 Entrainment in horizontal density-driven currents In practice, entrainment in inertial-buoyant currents cannot be parameterised by a simple constant (Ellison & Turner, 1959; Simpson, 1999; Ezzamel, 2011). The outflow current is effectively stable in configuration in that buoyancy is acting towards (and perpendicular to) the horizontal boundary. The extent to which the current mixes with the overlying environment depends on this stability relative to the shear experienced by the current, and may be characterised by a local Richardson number: Ric := g′chc u2c . (2.71) In a forced (inertially-dominated) configuration where Ric  1, the current is expected to entrain in similar fashion to a wall jet (Glauert, 1956) and its dynamics are expected to follow the power laws prescribed by the point-source momentum flux solutions (2.66). In a buoyancy-dominated configuration Ric  1, it is expected that the current ceases to entrain due to its stability (Ellison & Turner, 1959). It is of use then to consider solutions based on negligible entrainment, i.e. αc = 0. Point-source solutions can no longer be obtained from (2.63)-(2.65) as these are trivial (i.e. equal to zero). Solutions are instead considered for initial conditions given by finite source fluxes: Qc = Qc0, Mc = Mc0 and B = Bc0. By substituting: rˆ := α1/2c r h0 , q := Q Q0 , m := M M0 , f := B B0 = 1, (2.72) into equations (2.63)-(2.65), these are expressed in dimensionless form. The conservation of volume flux (2.63) and of momentum flux (2.64) are expressed as: 1 rˆ dqc drˆ = εc φc0 mc qc , (2.73) 1 rˆ dmc drˆ = −Γc0 ddrˆ ( q3c m2c rˆ2 ) . (2.74) An entrainment coefficient ratio is introduced in (2.73): εc := αc αc,re f , (2.75) which is defined relative to a reference case of a isodensity radial wall jet whose entrain- ment can be described by a constant entrainment coefficient αc = αc,re f . Two dimension- less groups result from equations (2.73) and (2.74) which correspond to the source aspect ratio: φc0 := r0 hc0 , (2.76) and the source Richardson number of the current, Ric, which is scaled in (2.74) as: Γc0 := α1/2c 2 Bc0Q3c0 2M3c0hc0 = α1/2c 2 g′c0hc0 u2c0 . (2.77) 30 2.4. The stratification produced by a plume in a container For the inertial current (Γc0 → 0), negligible entrainment (εc = 0) implies that the depth and mean velocity of the current vary as: hc hc0 = 1 r , uc uc0 = 1. (2.78) while for the buoyant current, in the absence of mixing, these parameters do not change from their initial values: hc hc0 = 1, uc uc0 = 1, g′c g′c0 = 1. (2.79) Kaye & Hunt (2007) showed that the plume outflow current is forced at source and tran- sitions into a buoyancy-dominated regime. The entrainment behaviour of the current is therefore expected to transition from a state of jet-like entrainment into a stable non- entraining gravity current. To compensate for this effect, a semi-empirical entrainment function is adopted following Ellison & Turner (1959), which assumes a variation be- tween these two extremes in the form: εc = max ( 1 − 1.25Ric 1 + 5Ric , 0 ) . (2.80) The entrainment ratio (2.80) is based on a reference value of αc,re f = 0.08 that corresponds to that of a purely-momentum driven wall jet (Ellison & Turner, 1959; Witze & Dwyer, 1976; Tanaka & Tanaka, 1977). Numerical solutions for (2.73) and (2.74) using constant- αc and the entrainment ratio (2.80) are shown in figure 2.9. Solutions that assume a constant αc generally follow the behaviours of the idealised point-source solutions (2.67), (2.70), (2.78) and (2.79). Notably, however, for the forced case, constant αc solutions become singular at certain distances from the source (marked by the crosses in figures 2.9a,c, and e). The occurrence of the singularity coincides with the approach of the current to a local Richardson number of Γc → 1. An analogy can be drawn with the stability of open-channel flows to understand the reasons behind the oc- currence of this singularity. The solution transitions suddenly from what may be regarded as a ‘supercritical’ (Ric < 1) to a ‘subcritical’ state (Ric > 1). There has been much debate whether internal jumps occur in gravity currents (Ungarish, 2009; Slim & Huppert, 2011). Solutions employing the entrainment function of Ellison & Turner (1959) however do not become singular. Forced flows transition, following an adjustment region in which the current contracts (and the solutions are affected by the aspect ratio of the source), from a linearly spreading (jet-like) current to a (gravity current-like) one travelling at relatively constant depth. The effect of the reduction of entrainment due to transition can be ob- served as an effective reduction in spreading rate and in a marginal increase in velocities and densities relative to the highly-forced case. 2.4 The stratification produced by a plume in a container The process by which a turbulent plume stratifies a container is referred to as the fill- ing box. The name stems from Turner’s (1973) description of the theoretical model for the stratification formed by a point-source plume presented by Baines & Turner (1969). The model has naturally evolved over the years (Germeles, 1975; Worster & Huppert, 31 0 5 10 15 0 1 2 3 h c /h c0 εc = 1 (–) & εc = 0 (- -) (a) 0 5 10 15 0 1 2 3 εc = f (Γc) , E&T (1959) (b) 0 5 10 15 0 0.5 1 1.5 u c /u c0 (c) 0 5 10 15 0 0.5 1 1.5 (d) 0 5 10 15 0 0.5 1 r/r0 g′ c /g 0 (e) 0 5 10 15 0 0.5 1 r/r0 (f) −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 Γc0 = 0 log10 Γc0 Γc0 = 1 Figure 2.9: Solutions to the current conservation equations (2.73)-(2.74) in terms of (a)-(b) depth, (c)-(d) time- and vertically-averaged velocity and (e)-(f) time- and vertically-averaged reduced gravity; taking a source aspect ratio of φc0 = 1.6, which is chosen to be most representative of the plume outflow current as a result of the discussions on impingement losses (i.e. on γi) presented later in Chapter 6. The left column shows solutions taking εc = 1 (solid lines) and εc = 0 (dashed lines). The right column shows solutions based on a description of entrainment that is given by the function εc = f (Γc) of Ellison & Turner (1959). 2.4. The stratification produced by a plume in a container 1983) and has been applied to an array of different scenarios (Baines et al., 1990; Linden et al., 1990). In this section, focus is on reviewing the work conducted in nominally- axisymmetric configurations (Barnett, 1991; Hunt et al., 2001; Kaye & Hunt, 2007; Ez- zamel, 2011). For further reference, table 2.1 summarises the parameter ranges explored in previous filling-box studies. For convenience, filling-box flow patterns are described in the laboratory reference frame of dense plumes released vertically downwards in lighter surroundings (but the descriptions are dynamically equivalent when flipped vertically due to the assumption of small-density differences). 2.4.1 Descriptions of filling-box flow patterns The original descriptions of Baines & Turner (1969) laid the foundations to our under- standing of the stratification produced by a plume. In their investigation, an aqueous- saline plume was activated at the start of each experiment and continuously released at a steady rate into a rectangular-based visualisation tank. The plume fluid was stained with dye at regular time intervals to assist visualisation (figure 2.10a). The first fluid that en- tered the tank sank to the floor and spread horizontally to form a heavy layer which was divided from the original environment by a sharp discontinuity in density (referred to as a front or interface). An example of the formation of this initial layer can be observed in the backlit dye visualisations of Hunt et al. (2001) in figure 2.10b. Turbulence within the layer dissipated as the layer established. The subsequent plume fluid slid underneath the layer and progressively displaced the original layer upwards (a behaviour which we here refer to as displacement filling). The layer gradually deepened as it continued to be penetrated by the plume with the rate of ascent of the front progressively reducing as the front approached the level of the plume source. Baines & Turner (1969) developed a theoretical model for the evolution of the stratification based on this description. The au- thors matched solutions to the plume equations to an unsteady one-dimensional advection equation to describe how the vertical density profiles of the environment evolved (§2.4.2). As a result, the model relied on the assumption of horizontal homogeneity in density, and of laminarity in the motions, in the developing layer. Baines & Turner (1969) note that this pattern of stratification is not unique and in fact its occurrence is restricted to containers which are short and wide, of radius-to-height aspect ratio exceeding φ := R/H & 1. They suggested that an instability can develop at the base of the container as the plume outflow current impinges against the sidewall and that the vertical penetration of the instability can radically change the dynamics via which the initial heavy layer is formed. To this end, Baines & Turner (1969) conducted a series of scoping experiments to investigate these motions, which they refer to as the overturning of the plume. They showed, by balancing the inertia supplied by the plume to the layer, I := pib2i w 2 i /2, to an estimate of the weight of the layer itself, B := pib 3 i g ′ i , that a Richardson number can be deduced to characterise this instability that is solely a function of the aspect ratio of the container: Riov := B I = 10 9αp φ2. (2.81) Their description of overturning is limited in length as these motions laid beyond the scope of their original paper. They simply suggest that a pattern of overturning develops 33 2. Fundamentals on decreasing the aspect ratio φ below one; and that for φ . 0.67, this overturning consists of a non-uniform ring vortex motion that fills more than half of the depth of the container. Hunt et al. (2001) examined the filling of a container by a forced plume. They noted that the penetration depth of an overturning intrusion can be expected to scale on the relative contribution of momentum to buoyancy of the current flow impinging against the corner: zL ∝ M3/4ci B1/20 . (2.82) In other words, they suggest that the penetration depth of the intrusion scales on a ‘jet- length’ (Fischer et al., 1979) constructed for the current at impingement with the corner, i.e. a classic measure of the physical length a forced plume or a fountain for which the flow is predominantly source-momentum driven (Morton, 1959). Kaye & Hunt (2007) extended this analysis when they considered the early transients of stratification in a dedi- cated treatment on overturning. They deduced dimensional scalings for the plume outflow current to estimate the penetration depth (zL) of the overturning intrusion which they ide- alised as a planar fountain. They suggest that two regimes of overturning exist, rolling and slumping (figure 2.10c-d), due to the transition of the plume outflow current from an inertially to a buoyancy-dominated regime (cf. §2.3). The relative vertical penetration depth of the overturning was measured to be a function of the aspect ratio of the container: zL H =  0.38φ−1/3 for 0.25 . φ . 0.66 (rolling),0.33 for 0.66 . φ . 1 (slumping). (2.83) In rolling, the current is inertially-driven as it impinges against the sidewall and thus rises higher than in the slumping case, where the current is buoyancy-driven. Kaye & Hunt (2007) describe that, after the initial overturning transients, the layer settles to a stable horizontal state. The layer gradually deepens as it is fed by the plume in similar manner to the displacement filling description of Baines & Turner (1969). Little effort was placed in describing the transition from these early transients to the gradual filling stages of stratification until the study of Ezzamel (2011). Ezzamel (2011) fur- ther inspected the early transients of filling in a set of separate experiments observing aqueous-saline plumes by fluorescence visualisations and thermal air plumes by particle image velocimetry. His descriptions of rolling and slumping are extended to include the behaviour succeeding the initial rise of the corner intrusion. Rolling was distinguished by the distinctive vortex motion, aforementioned by Baines & Turner (1969), that fills the bottom of the container (figure 2.10f). Slumping was instead identified by the unrolling of the vortex into a current that converges towards the centre of the container (figure 2.10e). Ezzamel (2011) identified a further regime in his thermal air experiments that occurs for containers of aspect ratio 1.25 . φ . 2.0 which he refers to as blocking. During blocking, fluid from the plume outflow current accumulates in the corners of the container. The author does not present fluorescence visualisations of blocking, and as result, these are omitted in figure 2.10. Further descriptions are postponed to Chapter 5 with aid of the flow visualisations presented therein. The bounds on φ for each regime were deduced experimentally and there is exact agreement between authors on their values (given in figure 2.11) over a wide range of conditions (Table 2.1). Overturning is not the only effect that can radically change the way a container strat- ifies. Barnett (1991) investigated the stratification produced in containers which instead 34 2.4. The stratification produced by a plume in a container Authors b0 g′0 w0 R H β0 Γ0 φ mm mm/s2 mm/s mm mm % B&T69 5 130 - 160 53 - 140 213* 200 - 400 1.2 - 2.5 0.2 - 2 0.5 - 1.1 B91 0.25 0.03 - 0.3 50 - 100 45, 71* 400-1310 < 0.06 ≈ 1 0.03 - 0.2 HCL01 5 40 - 210 130 - 680 228* 153 3.3 0.001 - 0.4 1.5 K&H07 2 n/a n/a 225 70 - 510 0.4 - 2.9 ≈ 1 0.3 - 4.5 E11 6.5 210 1 - 130 120, 225 110 - 410 1.6 - 5.9 0.1 - 5 0.5 - 2.0 C18 11 - 45 50 - 700 1 - 440 110, 300 200 - 450 3.6 - 40 6 - 106 0.2 - 1.5 Table 2.1: Table of parameters explored in axisymmetric filling-box studies: B&T69 (Baines & Turner, 1969), B91 Barnett (1991), HCL01 (Hunt et al., 2001), K&H07 (Kaye & Hunt, 2007), E11 (Ezzamel, 2011) and C18 (this study). The experiments of Baines & Turner (1969), Barnett (1991) and Hunt et al. (2001) were conducted in rectangular-based tanks of dimensions 577 by 427 mm2, 126 by 126 mm2 and 583 by 279 mm2 respectively. As a consequence, ‘effective’ measurements of radius, R∗ = (WL/pi)1/2, W and L being the width and length of the tank, are presented and distinguished by an asterisk ()*. were tall and narrow. He identified a new regime for a container of aspect ratio φ . 0.25. He noted that the return flow induced by the plume in the environment increases in mag- nitude to such an extent, for containers of φ . 0.25, that the shear between the two flows prevents the plume from reaching the base of the container. The plume instead breaks down into a turbulent mixing region at a level of about z/R ≈ 4. If the container aspect ratio is further reduced φ . 0.16, the turbulent mixing region forms a plug in the con- tainer that extends for a region of about 4 . z/R . 6 below which the flow mixes as a one-dimensional convective flow (similar to Rayleigh-Taylor convection displayed in figures 2.6g-i). Figure 2.11 summarises schematically the descriptions provided by these authors. 2.4.2 Modelling of filling-box flows The theoretical model of Baines & Turner (1969) was based on the assumption that the plume ensues from a point-source of buoyancy flux and that thus its dynamics can be described by the analytical solutions to the plume conservation equations: (2.29)-(2.33). This allowed to estimate the filling rate of the container, i.e. the rate of ascent of the first front, denoted z f , and how the mean vertical reduced gravity profile of the environment g′e(z, t) changed over time. Baines & Turner (1969) showed that the mean velocity of the flow surrounding the plume, denoted U, could be estimated by a simple continuity argument: U(R2 − b2) = −wb2. (2.84) As a result, the rate of ascent of the first front could be estimated via the volume flux of the plume penetrating the interface (Q f = CpQB 1/3 0 z 5/3 f ): dz f dt = −Q f R2 − b2 . (2.85) 35 B ai ne s & Tu rn er (1 96 9) H un te ta l. (2 00 1) K ay e & H un t( 20 07 ) E zz am el (2 01 1) B ar ne tt (1 99 0) displacement filling first front slumping rolling slumping rolling breakdown (a) (b) (c) (d) (e) (f) (g) (h) Figure 2.10: Visualisations of filling-box flows in aqueous-saline solutions. (a) Front-lit dye visualisation in which the dye was injected at regular time intervals via the source of the plume. (b) Backlit visualisation showing the formation of a heavy layer at the base of the tank. (c)- (f) Fluorescence visualisations showing the rolling and slumping regimes as interpreted by the respective authors (see caption in figure). (g) Contours of density estimated from shadowgraph visualisations. (h) Barnett’s (1990) schematic interpreting the plume ‘breakdown’ process. Images re-printed with permission. displacement filling φ & 2.0 breakdown 0.15 . φ . 0.25 blocking 1.25 . φ . 2.0 slumping 0.66 . φ . 1.25 rolling 0.25 . φ . 0.66 breakdown & RT convection φ . 0.15 Figure 2.11: Schematics showing idealised patterns for filling-box flows that occur when a point- source plume forms in a cylindrical container. These flow patterns are further described in Chapter 5. 2. Fundamentals Equation (2.85) can be integrated, assuming that the container is relatively wide R2  b2, to obtain an analytical solution for the location of the first front: t = 3 2 R2 H2/3B1/30 CpQ ( 1 − (z f H )−2/3) . (2.86) Baines & Turner (1969) then employed an unsteady one-dimensional advection equation to model how the vertical profile of reduced gravity changed over time: ∂g′e ∂t = −U ∂g ′ e ∂z for z f ≤ z ≤ H. (2.87) Approximate solutions were deduced by the authors to the system of equations formed by (2.87) and the plume conservation equations (2.28) in the limit of when the first front approached the level of the source z f → 0, i.e. the so-called asymptotic state. In the asymptotic state, the vertical density gradients do not change and the whole reduced grav- ity profile proportionally increases in time as total buoyancy accumulates in the container. By assuming, that as a result of this, plume fluxes do not vary in time, while the reduced gravity of the environment varies linearly in time, equation (2.87) was reduced to: ∂g′e ∂t = B0 HR2 for 0 ≤ z ≤ H. (2.88) The equations governing the filling of the container, namely (2.85) and (2.87), can be extended to other configurations (e.g. Baines & Turner 1969 also examine line plumes and periodically released thermals). Germeles (1975), for example, examined the filling of a container by a horizontally-released forced plume. Analytical solutions were not possible in this case so the author developed a numerical scheme to solve for the density profiles. The solution was initialised by specifying that an initial layer formed of an infinitesimally small depth at the bottom of the container that was of reduced gravity corresponding to that of the plume as it impinges against the floor of the container. The solution scheme progresses in discrete time steps, tallied by a counter j for each time t( j = n), where n is the number of steps from the initial time t( j = 1). The density profile at time t( j = n) was represented by a step function: g′e = n∑ j=1 g′j ( χ(z − z j−1) − χ(z − z j) ) for 2 ≤ j ≤ n, (2.89) where g′j is the reduced gravity of the impinging plume at each time step j and χ is the unit step function: χ(z − z j) =  0 for z ≤ z j,1 for z > z j. (2.90) The depth of the individual layers were given by a discretised form of the advection equation (2.87): z j = z j−1 + U j [ t( j) − t( j − 1)] for 2 ≤ j ≤ n. (2.91) 38 2.5. Bounds of the current investigation Finally, a description was introduced to model how the buoyancy flux changes as the plume enters the stratified portion of the environment. The buoyancy flux changes due to the relative density of the stratification compared to that of the original environment: B = B0 − n∑ j=2 g′j [ Q j−1 χ(z − z j−1) − Q j χ(z − z j) ] for 2 ≤ j ≤ n. (2.92) Predicting the stratification pattern produced by a lazy plume in a container requires a similar numerical scheme (presented in Chapter 6). Analytical solutions are not avail- able for the plume itself when α = f (Γ), so no attempts were made to derive analytical solutions for the stratification of the box. The theoretical descriptions of Barnett (1991) for regimes in which the plume breaks down also relied on the plume conservation equations. Barnett modified the equation for the conservation of vertical momentum flux so as to include the effect of the return flow: d dz ( w2b2 + U2(R2 − b2) ) = g′b2, (2.93) where the return flow velocity U is given by the continuity argument of Baines & Turner (1969) prescribed in (2.84). The author showed that the solution to (2.93) in combination with the plume equations becomes singular as b/R → 1/√2, a limit which corresponds to when the mean plume velocity is equal and opposite in magnitude to the velocity of the environment w = −U. Above the level of breakdown, Barnett (1991) modelled the flow as one-dimensional unsteady turbulent diffusion flow, described by: ∂g′e ∂t = κT ∂2g′e ∂z2 , (2.94) and characterised by a turbulent diffusivity coefficient for which Barnett suggests a value of κT ≈ 1.5 m2/s. These models are extended in Chapter 5 and 6 to consider the stratification produced by a pure and lazy plume formed in containers of aspect ratio 0.25 ≤ φ ≤ 2.0. In these cases, the dynamics of the initial transient are far from being one-dimensional and over- turning instabilities effectively fill the container much quicker than what predicted by these models. There is currently no model that relates the behaviour of the early tran- sients to the gradual filling observed after these. Moreover, plumes from sources which are not small in size with the dimensions of the container can considerably affect the vari- ation of volume fluxes within the plume, which is proportional to the rate of fill of the container from (2.85). Both these effects can radically change the dynamics of filling-box flows. The theoretical framework of Baines & Turner (1969) can nevertheless be modified to investigate the role of these effects on stratification. 2.5 Bounds of the current investigation Reviewing previous work concerning lazy plumes and filling-boxes helped us define a set of research questions which are summarised in this section. 39 2. Fundamentals Plumes: At the onset of the investigation, we set out to review techniques to model the dynamics of lazy plumes. We discussed several elements of the debate of modelling near- source entrainment in increasingly lazy plumes and suggested that there are still several lingering questions concerning their behaviour, such as: • Over what range of plume scaled source Richardson numbers Γ0 (see equation 2.38) can constant-α models be employed to predict plume behaviour? • What is the appropriate description of entrainment in the adjustment region of a plume? • What are the reasons behind the apparent increase in entrainment in the near-source region? • Can we draw analogies with flows from distributed sources to better understand lazy plumes? Currents: Discussions on the plume are extended to its behaviour in an enclosed con- tainer, where the plume tends to stratify the space. It is noted that the impingement of a plume with a horizontal boundary forms a radial outflow current. Several questions were raised concerning these currents in regard of how to model their impingement and their dynamics, in particular: • What is the value of the impingement loss factor γi? • What is an appropriate description of entrainment in the current? Filling-box flows: Having compared descriptions of stratifying plume flows, we outlined a classification of filling-box flows. In the upcoming chapters, we further describe these flows, identify key features to differentiate them and test this classification experimentally to answer the following questions: • What flow pattern is observed during the plume breakdown, rolling, slumping and blocking regimes? • Can a definitive regime classification for filling-box flows be drawn? • Do existing models adequately predict the behaviour of filling-box flows? • Can certain features (e.g. currents, overturning and vortices) of filling-box flows be modelled? • How does the flow transition from the early transient of impingement to the gradual filling stage? • Can the classification of these flows be extended to the filling-box driven by a lazy plume? • How do the dynamics of filling-box flows change with the source conditions of the plume of increasingly large size? The investigation is restricted to turbulent pure and lazy (Γ0 ≥ 1) plumes in containers of aspect ratio 0.25 ≤ φ ≤ 2.0 with sources of relative size which are substantially larger than previous studies (0.05 . β0 := b0/H . 0.4). 40 chapter 3 Methodology This chapter overviews the experimental methods employed in the investigation. While Chapters 4-6 each dedicate a section to the exact configuration and parameters pertain- ing to the experiments conducted therein, here we outline the common design features (viz. flow delivery and visualisation systems) and basic principles of the measurement techniques used. An experimental campaign was launched to observe and measure the dynamics of buoyant lazy plumes. We inspected plumes ensuing from the release of an aqueous-saline solution through a circular nozzle in a freshwater-filled visualisation tank. The aim was to study the scalar, velocity and density fields of the plume and that of the flow ensuing from its impingement against the base of a cylindrical container; and to see how these fields change by varying the releases conditions of the plume and the container aspect ratio. The experimental configuration is further described in §3.1 to explain how these flows were formed in the laboratory setting. Table 3.1 summarises the parameter range investigated. The plume source conditions were varied in three ways: (i) by releasing saline solutions via nozzles of different radii b0, and by varying (ii) the flow rate Q0 and (iii) density ρ0 of the saline solution supplied by the nozzle. The container aspect ratio was varied by using two containers of different internal radii R and by varying the level H at which the nozzle was fixed above the base of the container. Flow visualisations and measurements were instrumental for the formulation of the models developed in Chapters 4-6. Our flows were interrogated using four different tech- niques: backlit dye visualisations, planar light-induced fluorescence (LIF), planar parti- cle image velocimetry (PIV) and simultaneous fluorescence and velocimetry (LIF+PIV). Sections §3.2.1-3.2.5 outline how these techniques were employed for the experiments described in §3.1. We restrict ourselves to highlighting the fundamental and bespoke aspects of their application to our experiments. Further information was extracted from these techniques by complementing them with measurements of flow rate and density of the saline solution released by the nozzle (§3.1.2). 3.1 Configuration of a typical experiment In each experiment, an aqueous-saline solution was released via a circular nozzle into a clear acrylic cylindrical tank (figure 3.1). The cylindrical tank was placed into a larger square-based tank of base dimensions 1000 × 1000 mm2 and depth 700 mm. Before the release, both visualisation tanks were filled with freshwater and allowed to quiesce over time. The solution was injected by means of a positive displacement gear pump at a measured flow rate from a reservoir of known salinity and dye/seeding concentration. 41 3. Methodology overflow supporting frame air vent tank saline reservoir gear pumpflow meter ball valves nozzle 1000 mm 70 0 m m H R supply circuit Figure 3.1: Schematic of a typical experimental configuration in which buoyant fluid is released in a cylindrical visualisation tank. The solid black lines represent a rigid PVC supply circuit. Chapter b0 (mm) Q0 (L/min) ρ0 (g/cm3) R (mm) H (mm) Technique ±0.1 mm ±1% ±0.00005 g/cm3 ±0.5 mm ±0.5 mm 4 22.5, 44.5 0.1 - 3 1.005 - 1.070 300 200 - 500 Dye, PIV 5 22.5 0.1 - 3 1.005 - 1.010 110, 300 200 - 500 LIF+PIV 6 10.8, 22.5, 44.5 0.1 - 10 1.005 - 1.070 300 200 - 500 Dye, LIF, LIF+PIV Table 3.1: Summary of experimental parameters and techniques used in each chapter. The release of the solution into the visualisation tank was activated at the start of each experiment and the flow was video recorded thereon. 3.1.1 Design of nozzle and supply system The flow delivery system by which the plume was formed (viz. the nozzle and supply circuit shown in figure 3.1) was designed to ensure that the source conditions of the plume were controlled and that its release was quasi-impulsive. The objective was to achieve a turbulent release with a uniform (top-hat) source velocity profile at a salinity that could be measured and that would not change over time. Nozzles were constructed with an outlet made of a microporous polyethylene disc. The outlet acted as a diffuser that homogenised the source velocity profile of the release by redistributing pressure radially across the nozzle. It also separated the inherently unstable interface between the fluid in the nozzle and that in the tank. This allowed for an experimental set up in which the supply circuit could be entirely filled with saline (of uniform salinity) before it was released into the visualisation tank. 42 3.1. Configuration of a typical experiment Source velocity profile: The uniformity of the source velocity profile was assessed by inspecting the advection of a dye tracer immediately after the activation of the release. Examples of image frames, acquired via LIF, which immediately succeed the release are shown in figures 3.2a-c for each of the nozzles built for the investigation (Table 3.1). In each case, the interface between the source fluid and the environment is located using an edge detection routine (described in §3.2.3). An ‘instantaneous’ source velocity profile w0 is estimated, following a Lagrangian argument, by measuring the level of this interface from the source ∆z over a time interval ∆t: w0  ∆z ∆t . (3.1) The estimate of source velocity based on (3.1) is shown in figure 3.2d-f for the three nozzles. Note that the average source velocity w0 was measured by means of a flowme- ter (see §3.1.2) and this should strengthen the confidence in the measurement of w0 by means of the dye advection argument presented in (3.1). In each case, the flow appears turbulent, with turbulence intensities (computed as the standard deviation of the profile between −b0 ≤ r ≤ b0) which range between 4-11%. The skewness of the velocity pro- file sw is computed to assess its uniformity. The magnitude of this statistic shows that, notwithstanding imperfections arising from uneven distributions of the outlet pores, the velocity profiles are reasonably uniform (sw/w0 < 1). Control of release conditions: The supply system was designed to ensure that fluid of the correct salinity was released from the nozzle at the start of each experiment (fig- ure 3.3). Before each experiment, the nozzle and preceding supply pipes were filled with saline solution and air bled by discharging reservoir fluid through the nozzle into an empty tank. Fluid in the nozzle was then locked in the pipes by closing an arrangement of valves. A bypass circuit preceding the nozzle was then run until the flow rate adjusted steadily to the required value. The release was then activated by manually diverting the flow from the bypass circuit to the nozzle. 3.1.2 Source flow rate and density measurements The flow rate was measured with three different commercial flow meters: for low to mod- erate flow rates (0.5 - 3 L/min) an Apollo LowFlo meter and an Atrato flow meter were used, for higher flow rates (3 - 10 L/min) an Apollo flow meter (model ET3/B) was used. The flow meters relied on two different physical principles for flow rate measurement. The Apollo flow meters are magnetic rotameters which measure flow rates by the Pelton wheel principle; fluid flows through revolving compartments of fixed volume at an elec- trically measured revolution rate. The relative error associated to the Apollo flowmeters is respectively of ± 1% and ± 2% of the full reading (i.e. ± 0.03 L/min and ± 0.2 L/min respectively). The Atrato flowmeter is an ultrasonic flowmeter which relies on the sonar principle; ultrasonic waves are emitted by the device and the reflected wave is examined for distortions due to the flow. The Atrato flow meter allowed for the logging of the sig- nal with a relative error of ± 1% on the actual reading. These measurements of flow rate are shown in figure 3.4 to assess the steadiness of the pump delivery. A drawback of the ultrasonic flowmeter was that it could not be used in conjunction with PIV techniques 43 3. Methodology −1 0 10 0.5 (a) b0 = 10.8 mm −1 0 10 0.5 (b) b0 = 22.5 mm −1 0 10 0.5 (c) b0 = 44.5 mm −1 0 10 0.5 1 1.5 (d) w 0/ w 0 z/ b 0 σw/w0 = 0.04 sw/w0 = +0.27 −1 0 10 0.5 1 1.5 (e) r/b0 σw/w0 = 0.09 sw/w0 = +0.28 −1 0 10 0.5 1 1.5 (f) σw/w0 = 0.11 sw/w0 = −0.50 Figure 3.2: Source velocity profiles of the nozzle estimated from the tracking the advection of an LIF tracer added to the plume. The solid black line shows the source velocity profile estimated by averaging ten successive image frames after the releases (individually shown in colour). The standard deviation σw and skewness sw of the estimated source velocity profile are shown in each figure as a ‘measure’ of turbulence intensity and uniformity. owing to the interference created by the seeding. The densities of the reservoir solution and the water in the visualisation tank were measured with an Anton Paar 5000M density meter. The density meter estimates the density of the solution by oscillating a sample of known volume to its natural frequency. A benefit of this density meter over optical refractometers is a substantial increase in accuracy: the absolute error associated to these density measurements is very low at ± 0.00005 g/cm3. The density of freshwater supplied from mains averaged at ρa = 0.998 ± 0.001 g/cm3 over 50 samples taken at different times during the experimental campaign. Errors on compounded quantities were estimated using the propagation rule (Taylor, 1997): q = XaYb ⇒ δq|q| = |a| δX |X| + |b| δY |Y| . (3.2) Upper bounds on error could be constructed for the worst case scenarios for the source velocity: w0 = Q0 b20 ⇒ δw0|w0| = δQ0 |Q0|︸︷︷︸ ≤6% +2 δb0 |b0|︸︷︷︸ ≤0.5% ≤ 7%, (3.3) and the source Richardson number: Γ0 = g′0b 5 0 Q20 ⇒ δΓ0|Γ0| = δg′0 |g′0|︸︷︷︸ ≤0.005% +5 δb0 |b0|︸︷︷︸ ≤0.5% +2 δQ0 |Q0|︸︷︷︸ ≤6% ≤ 15%. (3.4) The major source of error were the measurements of flow rate. Errors on container aspect ratio, δφ|φ| ≤ 0.5%, and relative source size, δβ0|β0 | ≤ 0.7%, were relatively low. 44 (a) Supply circuit filled with saline and bled of air (b) Nozzle filled with saline (empty tank) (c) Bypass circuit activated and flow rate set (d) Saline released from nozzle F P F P F P F P pumpflow meter air-vent tank saline reservoir visualisation tank gate valves Figure 3.3: Schematic showing the succession of steps required to set up an experiment so that the correct release conditions were achieved. The black line represent the PVC supply circuit. The red line shows the path the saline solution travels in the circuit after being drawn from the reservoir. 3. Methodology 0 10 20 30 40 50 60 70 0.99 1.00 1.01 t (s) Q 0/ Q 0 0.5 L/min 1.0 L/min 2.0 L/min 3.0 L/min Figure 3.4: Time record of the flow rate Q0(t) supplied by the gear pump, as logged by the Atrato flow meter, for varying degrees of time-averaged source flux Q0 (see legend to the right). A slight decay in volume flux can be observed in the time record, which may be attributed to the reduction in head experienced by the pump as the reservoir tank progressively depletes. It is nevertheless retained that this reduction is minimal (≤ 1%) and that the source flow rate is nominally steady for the timescales of interest. 3.2 Flow visualisation & measurement techniques 3.2.1 Backlit dye visualisations The dye visualisation technique is of simple implementation (figure 3.5a). The visuali- sation tank was backlit by a uniform LED panel. The saline solution released into the tank was stained with methlyene blue dye at a concentration of about 10-20 mg/L. The dye attenuated the light going through the visualisation tank so that a depth-integrated image of the dye could be acquired by a camera placed opposite the LED panel. Since backlit dye visualisations allowed for a relatively rapid acquisition of data, they were the primary technique used to inspect the parameter space. Images were acquired with a JAI Spark 5000-M USB monochrome scientific camera of 5 Megapixel resolution at varying acquisition rates (typically, 25-60 frames per second). 3.2.2 Light-induced fluorescence (LIF) visualisations Light-induced fluorescence visualisations relied on a similar implementation to the back- lit dye visualisations (figure 3.5b). A fluorescein sodium tracer was used to stain the saline solution which was then lit by a thin white light sheet crossing through a vertical section of the visualisation tank. The light sheet was formed by shining a 300 W Xenon arc lamp through light sheet forming optics. A rectangular aperture slot was placed on the outside wall of the tank to trim the light beam to a thickness of about 10 mm. The light sheet was of relatively low divergence, spreading by about 2 mm over the 1000 mm width of the tank. The LIF technique was used to inspect the internal structures of the flows examined. 3.2.3 Robust edge detection routines for flow visualisation Both backlit dye and LIF visualisations were coupled with image analysis techniques to extract quantitative information from the experiments. Edges between the dyed portion of the fluid and the environment could be detected from individual images frames. Multiple techniques were used to track these edges (thresholding, maximum gradients and a Canny 46 3.2. Flow visualisation & measurement techniques (a) (b) Figure 3.5: Example (a) backlit dye and (b) LIF flow visualisations of an experiment in which a stained saline plume is released in a cylindrical tank. algorithm). In each detection routine, each image frame was pre-processed by subtracting its background signal from it (i.e. an image of the field of view before the release) and by increasing the contrast so that the range of intensities of each image spanned the full bit depth, i.e. range of greyscale values. The pre-processed images were then analysed using the following three techniques. Thresholding: Each frame was scanned along vertical and horizontal arrays of pixels to locate an edge which was defined by a threshold value of pixel intensity. In the routine that was developed, the edge Emn of an m × n pixel image with a field of view denoted FOVmn was defined by a 95% increase/reduction relative in intensity Imn relative to the intensity of the stained solution I0: Emn := FOVmn (Imn = 0.05I0) . (3.5) In most simple cases (e.g. in the absence of PIV seeding in the environment), thresh- olding provides a very stable and intuitive means for edge detection. A drawback of the technique is the arbitrariness with which the magnitude of the threshold is selected. Maximum gradients: An alternative approach was to define the edge as the peak spa- tial gradient of intensity along each array of pixels: Emn := FOVmn ( max ( ∂Imn ∂x , ∂Imn ∂z )) . (3.6) The technique is less robust in ‘noisy’ pictures (e.g. where the electrical gain was in- creased due to low-intensity lighting or were seeding/residues were present in the tank). In these cases, Gaussian filtering was implemented on the image frame to remove the high spatial frequency information (i.e. ‘grainy’ textures and seeding) in the image and substantially increase the success rate of the detection of ‘longer’ edge features. To this end, a Gaussian filter Gmn of kernel size k × k and blur σG was applied to each image frame by multiplying each intensity matrix Imn by a mask corresponding to: Gmn := 1 2piσ2G exp ( − (m − (k + 1)) 2 + (n − (k + 1))2 2σ2G ) for 1 ≤ (m, n) ≤ 2k + 1. (3.7) 47 3. Methodology Edge detection algorithms: Other algorithms are readily available that in a similar way exploit discontinuities in brightness to detect a boundary in an image (Mathworks, 2018). The ‘Canny’ edge detection algorithm was adopted as an alternative to threshold- ing and maximum gradient detection. The algorithm estimates the gradient magnitude, G := √ G 2x + G 2 y , and direction, Θ := atan2(Gx,Gy), from a Gaussian filtered image and compares these to the surrounding pixels to reconstruct the ‘most likely’ edge in the image based on the length and similarity of neighbouring edges (Canny, 1986). The algorithm was implemented using the edge function in Matlab. Moving windows of interrogation: A substantial decrease in computation time and increase in robustness was achieved in the detection routines by only interrogating smaller windows in a succession of image frames. The interrogation windows was centred on the expected location of the edge that needed to be detected and then displaced in the subsequent image frames by a distance that was estimated by the expected velocity of the edge. Post-processing: A local median signal test was applied to the edge detected within each routine. The test is of simple implementation and is robust provided the signal is of high quality (≥ 90% correct). The signal was divided into windows (X) of sample size equal to 5 pixels. A test variable  was defined as the ratio of the standard deviation to the median (indicated by the tilde) of the values in the window:  := σX X˜ . (3.8) If the test variable fell below a threshold of  = 0.2, then the outlier (defined as the sample of largest relative error) was detected and replaced with the median of the window. The median statistic was chosen over the mean as the sample size is small. After applying this filter, the median value of each edge detected with different technique was chosen as the final detected edge. As a result, detection was more robust as the likelihood of two techniques failing at the same location was relatively low (≤ 1%). 3.2.4 Planar particle image velocimetry (PIV) measurements A relatively simple setup was employed for particle image velocimetry measurements. Solid polyamid seeding particles were uniformly mixed into both visualisation and reser- voir supply tanks. The seeding was then continuously illuminated during an experiment by a thin vertical white light sheet cutting across the visualisation tank. The light sheet was formed in the same way as in the LIF configuration (§3.2.2). Image pairs were ac- quired by a single camera perpendicular to the light sheet. Typical PIV fields of view ranged in size between 0.1 × 0.1 - 0.3 × 0.6 m2 and were recorded at a resolution of 5 Megapixels giving a spatial resolution of about 6-12 mm. The effective thickness of the light sheet was reduced by decreasing the depth of focus (DOF) of the image. This was done by increasing the aperture of the lens to an F-number (= focal length/aperture) as low as F# = 1.4; a commonly used trick in micro- PIV applications where the volume of illumination is typically that of the whole flow. The depth of field can be estimated using thin-lens approximations (Ray, 2002). For a 48 3.2. Flow visualisation & measurement techniques 5 µm length scale of focus, corresponding to the size of the pixels on the camera sensor and commonly referred to as the ‘circle of confusion’, c, the ‘hyperfocal’ distance yHF , that is, the distance at which this scale of focus is achieved, for a lens of focal length, FL = 75 mm, can be estimated to be: yHF = FL + F2L F#c ≈ 800 m. (3.9) The depth of field of an object at a working distance, yWD ≈1-2 m, which is smaller than the hyperfocal length, can be approximated to: DOF = 2yHF y2WD y2HF − y2WD ≈ 2-5 mm. (3.10) The in-plane depth of the measurement volume for PIV measurements was of 2-5 mm, a significant improvement on the thickness of the light sheet (10-12 mm). Fundamentals of the PIV technique In PIV, the instantaneous velocity field is estimated from the motion of particle tracers that are advected by the flow. The high particle density that is typically used in PIV makes it hard to track each individual particle. PIV algorithms instead rely on correlating an image pair, i.e. two images acquired in rapid succession, to estimate the velocity field along a regular grid of square interrogation areas (IAs). Each pair of IAs is cross-correlated to compute an estimate of mean velocity within the IA: the displacement of the correlation matrix peak from the centre of the IA corresponds to the most likely mean displacement of particles over the time interval. Additional manipulations are applied to the algorithm to increase speed and performance. A PIV routine can be divided into four stages. The first stage consists of the acqui- sition of the image frames. Important decisions have to be made at this point on the choice of seeding and rate of acquisition of the image frames. Following acquisition, the images are enhanced in pre-processing to improve the likelihood with which good corre- lation peaks are detected. The images are successively analysed using cross-correlation to retrieve a grid of velocity vectors. The vectors are then validated against a series of tests. Selection of seeding Before describing the PIV routine, we will discuss the choice of PIV seeding. There were two main criteria for selection: (i) the particles need to be advected by the flow field in such a way that their velocity was representative of the flow velocity, and (ii) be of an appropriate scatter size (i.e. effective size of particle in image) for the PIV algorithm to work accurately. The objective for the particle scatter size was to record particles which were about 3 pixels in diameter. Solid polyamide particles of 50 µm nominal size (PSP50 from Dan- tec Dynamics) of density 1.3 g/cm3 were selected to seed the flow field. Different parti- cles sizes were tested (2 µm TiO2 and 10 µm AlO2), before using the polyamide seeding, 49 3. Methodology which however resulted in a particle scatter size which was too small to be resolved with the camera/illumination setup. The responsiveness of the particles to the flow was assessed by estimating the set- tling (Stokes) velocity of the seeding. Assuming the particles are spherical, this velocity corresponds to: wS t = ρp − ρa 18µ D2p g ≈ 0.45 mm/s, (3.11) where the dynamic viscosity of water is taken to be µ ≈ 0.9 g/m s. The settling velocity was lower than the velocities typically measured in the flow field 10 - 100 mm/s. Acquisition Before acquiring images of the experiment, the camera was positioned in front of the visualisation tank so that the field of view could be adjusted to cover the region of interest. A calibration image was acquired of a calibration plate of regularly spaced dots so that a physical measure of a pixel was available during post-processing. The successful implementation of a PIV routine relied on a number of choices regard- ing the size of IAs and the time interval between image pairs. Good practice rules were employed to ensure the scheme was accurate. These rules are commonly referred to as the NIFIFO criteria, following Keane & Adrian (1992). These criteria aim at obtaining an optimal particle density (NI = number of particles / IA) and minimising the number of in- plane (FI) and out-of-plane (FO) lost particles out of an interrogation volume. FI increases when particles displace (in-plane) further than the bounds of the IA, while FO increases when particles displace out-of-plane further than the thickness of the light sheet δy. The following measures are adopted to address these effects (following from Anthoine et al. 2009 and Raffel et al. 2007): • particle displacement in IA is about 5-8 pixels (a smaller displacement increases the relative error, while a larger displacement increases the error associated to curvature of the travelled path); • particle scatter size of at least 2 pixel and optimally about 2.5 (with a smaller parti- cle size, the cross-correlation algorithm cannot differentiate between different parti- cles, as single pixel particles are identical; while using larger particles sizes creates a bias in the correlation peak towards bigger, brighter particles); • about 10 particles per interrogation area (to ensure correlation is statistically repre- sentative); • IA size about 4 times the displacement of particles (to minimise chances of out of window movement). While these could not be satisfied for each interrogation area in the measurement plane, the PIV settings were chosen with regard to the particular area of interest (e.g. for Chapter 4, the near-source region of the plume and for Chapter 5, the flow in the proximity of the corner of the filling box). While estimating the out-of-plane motions was difficult for our experiments, these are expected to be low as the flows investigated were primarily moving in-plane. 50 3.2. Flow visualisation & measurement techniques Image pre-processing The first step to prepare images for analysis was to subtract the background signal, just as in the dye visualisations (§3.2.1). A perspective error correction was applied to each image as a function of the focal length of the lens used using the undistortimage function in Matlab, a simple polynomial interpolation based on a calibration grid of known size. A second high-pass filter was then applied to reduce the unevenness of the lighting and to remove out-of-focus particles. The filter was applied by blurring the image using a Gaussian filter (3.7) of kernel size similar to that of the particle scatter size and subtract- ing this low-frequency information from the original image. The high spatial frequency information (i.e. the seeding) was thus retained in the image while the uneven greyscales and larger out-of-focus particles are removed. The same filter was instrumental for the implementation of the simultaneous LIF+PIV technique and is thus further discussed in §3.2.5. Analysis The analysis was conducted using a modified script based on the open-source software PIVlab which was developed by Thielicke & Stamhuis (2014). A multiple (two) pass interrogation scheme (implemented as described in Raffel et al. 2007, p. 146-147) was implemented with IAs typically reducing in size from (128 × 128) to (64 × 64) pixels. The size of the second pass IA was selected so that the number of particles in the IA was greater than four. A standard ‘three-point’ Gaussian peak fit (Raffel et al. 2007, p. 160) was implemented to achieve sub-pixel accuracy during the detection of the correlation matrix peak. Validation Sources of error arise from not satisfying the NIFIFO criteria, whose conditions often can- not be met everywhere in the field of view for a regularly spaced matrix of interrogation areas. A vector validation scheme which was here employed was based on a local median filter test (Raffel et al. 2007, p. 184). This resembles in intent the test employed in edge detection routines, described in §3.2.3, in that it assumes that large changes in velocity between neighbouring vectors are unlikely. For each vector, a test variable is defined as the ratio of the standard deviation to the median of the nine neighbouring vectors:  := σu ˜|u| . (3.12) If the test variable falls below a threshold of 0.2, the vector is replaced with the median of the neighbouring vectors. No other vector rejection tests were conducted on the data. Histograms of velocity magnitudes were inspected to check that measurements did not bias towards integer pixel shifts (as this clustering is symptomatic of peak locking, Raffel et al. 2007, p. 169). 3.2.5 Simultaneous LIF & PIV Implementing the simultaneous LIF and PIV technique was a relatively straightforward extension to the LIF and PIV techniques described in §3.2.2-3.2.4. The flow was indeed 51 3. Methodology illuminated in the same way. PIV seeding was mixed into both the environment and reservoir fluid, while fluorescein sodium dye was only mixed into the reservoir fluid at a concentration of about 0.5 - 1 mg/L. An example raw image of one of these experiments is shown in figure 3.6a. By fine tuning the concentration of fluorescent dye and the camera acquisition settings, the dye tracer can be recorded at mid-grey (100-150) intensities while the seeding at saturated (255) intensities (figure 3.6b). The simultaneous implementation of both techniques re- lied on the image pre-processing techniques described in §3.2.4. The dye information is available in raw images and edges can be detected as described in §3.2.3. The seeding information can be isolated via a high-pass filter (cf. §3.2.4) in which each image frame is subtracting by a Gaussian-filtered (blurred) image of itself (figure 3.6c). Standard PIV analysis (§3.2.4) can then be applied to the pre-processed images to retrieve the velocity field. 3.2.6 Analysing PIV data The velocity vector field obtained by analysing the PIV images was then analysed. Results from the data are discussed in Chapters 4 - 6. Here we demonstrate how a number of the quantities discussed in these chapters were computed. Integration of fluxes The volume and momentum flux in regions of the flow can be estimated by integrating the velocity field with respect to the radial r and vertical coordinates z. The volume and momentum fluxes in the plume were estimated as: Q = ∫ b 0 wr dr, M = ∫ b 0 w2r dr. (3.13) The volume fluxes in the plume outflow currents were estimated as: Qc = ∫ r 0 ∫ h 0 u dz dr, Mc = ∫ r 0 ∫ h 0 u2 dz dr. (3.14) Simple trapezoidal schemes were employed to integrate the fluxes numerically. Vortex detection In Chapter 5, different filling-box regimes are discussed in light of the dynamics of large- scale rotational structures that form in the flow. Vortices are typically discussed in terms of the vorticity of a flow field, which can be computed as the curl of the velocity vector: Ω = ∇ × u. (3.15) Gradient operations performed on PIV data are known to be noticeably inaccurate due to the coarse spatial resolution and noisiness inherent to the method (Raffel et al., 2007)). To compensate for this, alternative techniques have historically been developed to describe vortex structures such as local schemes as the Q or λ2 criteria (see, for example, the 52 (a) 128 × 128px 64 × 64px slit reflections saturated wall reflections PIV window (b) 0 1 2 3 ·104 0 100 200 background dye seeding pi xe lc ou nt (c) 0 1 2 3 ·104 0 100 200 pi xe lc ou nt Figure 3.6: An example of the application of image pre-processing techniques used to isolate the seeding in simultaneous LIF and PIV image acquisitions. (a) Raw picture of a sample experiment showing the size of the PIV window and IAs. Sources of error are highlighted in red. (b) Close- up of the raw image, and (c) pre-processed (background subtracted, contrast-adjusted, high-pass filtered) image with respective histograms. 3. Methodology review of Morgan et al. 2009), or non-local schemes such as the γ2 criteria (Graftieaux et al., 2001). These techniques fundamentally rely on splitting the velocity gradient tensor into a symmetric and an asymmetric part to isolate contributions from rotation and shear respectively: du dx =  dudx dudzdw dx dw dz  = (3.16) = S + R =  dudx 12 ( du dz + dw dx ) 1 2 ( du dz + dw dx ) dw dz  +  0 12 ( du dz − dwdx ) 1 2 ( dw dx − dudz ) 0  . (3.17) The Q and λ2 criteria examine the eigenvalues of S2 + R2 to determine where rates of rotation exceed that of strain. The γ2 criterion of Graftieaux et al. (2001) instead examines the rotation of the ve- locity vectors without employing gradient operations. This is a useful property for the relatively coarse resolution of the velocity field obtained from the PIV measurements. Graftieaux et al. (2001) defined the weighted factor γ2 as: γ2(P) := 1 S ∑ L × (uM − uP)] · z ||L|| · ||uM − uP|| , (3.18) where velocities uP and uM are the velocities at P and M respectively and L is the dis- tance between the points (figure 3.7). The γ2 criterion is thus computed by inspecting the velocity vectors at each point M located at a distance L from point P over a pre-defined interrogation area S. It increases with the magnitude of velocity vectors which are orthog- onal to the interrogation point. As a result, it gives an indication of the location of a vortex locus. The interrogation area S is defined a priori and estimated from the likely physical scale of the vortices that need to be detected. The local velocity at the interrogation point, i.e. uP, is used to compensate for the fact that the vortex locus is being advected by the flow. The γ2 criterion is adopted in Chapter 5 owing to its relative simplicity and proven success. 54 P uP M uM L Figure 3.7: Schematic showing how the γ2 criterion is estimated at a point P in a 3 × 3 grid of velocity vectors of area S. The size of the grid S depends on the expected scale of the vortex. chapter 4 Rayleigh-Taylor lazy plumes In Chapter 2, previous work concerning the dynamics of lazy plumes was discussed. These plumes are formed by highly buoyant releases from relatively large-area sources at low flow rates. The gravitational acceleration experienced by them, when released at suf- ficiently high scaled source Richardson numbers (n.b. theoretically at Γ0 ≥ 5/2, Hunt & Kaye 2005), causes them to contract to a neck before they expand further away from the source. Our discussions highlighted the debate concerning dilution in this rapidly adjust- ing near-source region, where the convective flow shares similarities with that released from a distributed and that from a localised source of buoyant fluid (see figures 2.6a-i, p. 25). New data, acquired through an experimental programme of flow visualisations and planar particle image velocimetry, is presented in this chapter for the radial spread and the streamwise variation in volume flux of lazy plumes. A broad range of scaled source Richardson numbers is covered, namely 1 < Γ0 < 106. The data is compared to predic- tions from the integral models that were reviewed in §2.2. These comparisons reveal the extent of the applicability of existing entrainment models for scaled source Richardson numbers in excess of Γ0 & 103. An excess in dilution is observed for these plumes in their near-source region relative to that measured at elevations where the plume has reached a state of self-similarity. Observationally, the excess in dilution for Γ0 & 103 concurs with the development of Rayleigh-Taylor (RT) instabilities along the contraction of the plume, which are reasoned to cause the additional engulfment of ambient fluid into the near-source region. A new theoretical model is presented for the near-source entrainment that, based on RT-layer growth rates, allows a unified plume description over the inves- tigated range of source Richardson numbers and a discussion on the morphology of the mixing mechanism. 4.1 Introduction The dynamics of vertical flows that are driven by density differences have been exten- sively studied in two limiting cases: that of a distributed and that of a localised release of buoyant fluid. Convective flows ensuing from distributed sources are typically discussed in light of the growth and breakdown into turbulence of Rayleigh-Taylor (RT) instabili- ties, where the potential energy of an unstable vertical density gradient is converted into motion (kinetic energy) and dissipated during mixing to small-scale viscous interactions (Dalziel, 1993; Dalziel et al., 1999; Boffetta & Mazzino, 2017). Convective flows that ensue from localised releases of buoyant fluid into unbounded environments are typically described as buoyant plumes (see, for example, the reviews of Woods 2010 and Hunt & van den Bremer 2011). These are slender density-driven flows that primarily mix by lat- erally shearing with the environment. Little effort has been placed to contextually bring together these two fundamental classes of flows studied in turbulent convection. 57 4. Rayleigh-Taylor lazy plumes Plumes display an array of properties which enable relatively simple parameterisa- tions of the turbulence observed within them and thus of their dynamics. Sufficiently far from the source of such releases, fully turbulent plumes reach a state of local equilibrium where the dominant forces acting upon them, buoyancy and inertia, are in balance. Lo- cally, the relative contribution of these forces is typically characterised by a Richardson number: Ri := g′b w2 (4.1) (Turner, 1973), where w, b and g′ are, respectively, the time- and horizontally-averaged local values of the vertical velocity, radius and reduced gravity, g′ := g (ρa − ρ) /ρ, of the plume, g is the acceleration due to gravity, ρ is the density of the plume and ρa is the density of the environment, taken to be uniform herein. At the level at which this state of balance of forces is reached, the local value of the Richardson number in the plume equals a constant value Ri = Rip, concurrent with the turbulence within the plume attaining a state of local equilibrium or self-similarity (n.b. theoretically, the value Ri → Rip is approached asymptotically, Hunt & Kaye 2005). This enables the application of the most commonly-used modelling approaches, such as models relying on the entrainment hypothesis of Morton et al. (1956), also referred to as constant-α models, or second order closures which rely on the assumption of spatially-invariant turbulence (Craske, 2016). If the plume is released at a source Richardson number, Ri0 := g (ρa − ρ0) b0/ρaw20, which is out of balance from the value it approaches in the far field, there will be a region extending from the source over which the plume adjusts to this value (Morton, 1959). The adjustment is driven by how the buoyancy compensates for the local excess (Ri < Rip) or deficit (Ri > Rip) of inertia in the plume. In this chapter, we set out to describe the dynamics of a fully turbulent injected plume issuing from a circular source of volume, momentum and buoyancy fluxes into a uniform and otherwise quiescent environment, that based on the ratio of these fluxes has been de- scribed as ‘lazy’ within this adjustment region. A buoyant plume is said to be lazy when the flux parameter Γ0 = Ri0/Rip > 1, that is, the source Richardson number is greater than the local Richardson number to which the plume approaches as it distances itself from the source (Morton & Middleton, 1973). This subset of plume dynamics originates when highly buoyant fluid is released from relatively ‘large’ area sources at low flow rates (Hunt & Kaye, 2001). Alternative names have been used to describe lazy plumes includ- ing ‘mass-source’ (Lane-Serff et al., 1993), ‘distributed’ (Caulfield & Woods, 1995), or ‘large-area source’ plumes (Kaye & Hunt, 2009) - the term ‘lazy’ is retained herein as it is better suited to the definitions later presented. The term ‘injected’ is included to dif- ferentiate from lazy plumes originating from a heated disc. In this study, we individually set the ratio of source fluxes by varying the source area, velocities and densities of the release. A lazy plume, an example of which is illustrated in figure 4.1, can be readily recog- nised, provided Γ0 > 5/2, by a characteristic necking of the plume in the proximity of the source (Hunt & Kaye, 2005). To date, the phenomenology of entrainment within the region below this neck has seen much debate (Colomer et al., 1999; Epstein & Burelbach, 2001; Plourde et al., 2008; Kaye & Hunt, 2009; van Reeuwijk et al., 2016; Marjanovic et al., 2017). Experimental observations for plumes of scaled source Richardson number in excess of Γ0 & 103 indicate that, within the near-source region, the plume dilutes con- siderably more than what would be predicted by constant-α models. For example, mea- 58 4.1. Introduction (a) ρa ρ bw z Q0,M0, B0 plume (b) in st an ta ne ou s tim e av er ag e near-source region far-field neck db dz < 0 db dz > 0 source Figure 4.1: (a) An instantaneous (LHS) and time-averaged (RHS) image of a passive tracer il- luminated along the plume axis is shown. Characteristic features of a lazy plume are illustrated including: the neck of the plume, that is, the location of maximum contraction of the plume ra- dius; the near-source region, which extends from the source to the neck; the far-field of the plume, where the plume is nominally in a state of local (buoyant-inertial) balance and thus of invariance in local Richardson number. (b) A representative schematic of the plume is shown. Notation is introduced for the time- and horizontally-averaged local vertical velocity w, plume radius b, and density ρ of the plume, and for the density of the environment ρa. surements in a plume of Γ0 ≈ 104, which are presented later in §4.6.2, figure 4.11, reveal that the plume has diluted by a factor of five times more than constant-α predictions over the a vertical distance comparable to a single source radius; the relevance of this being that constant-α models have been relied upon in a significant number of theoretical stud- ies describing lazy plumes (Morton & Middleton, 1973; Hunt & Kaye, 2001; Fanneløp & Webber, 2003; Hunt & Kaye, 2005; Carlotti & Hunt, 2005; Michaux & Vauquelin, 2008; van den Bremer & Hunt, 2010). The first hints concerning an excess in dilution in the near-source region were flagged from observations that lazy plumes of very high source Richardson number tend not to contract as much as predicted by these models (Colomer et al., 1999; Epstein & Burel- bach, 2001). As suggested in Chapter 2, for these flows a decrease in spreading rate is indicative of an increase in the volume of ambient fluid entrained. The excess dilution of the plume was later measured in the experiments of Kaye & Hunt (2009) and evaluated more recently in the DNS studies of van Reeuwijk et al. (2016) and Marjanovic et al. (2017) strengthening this notion within subsets of possible lazy plume behaviour. The objectives of this study are thus to (i) measure the near-source spread and entrain- ment for lazy plumes of varying source Richardson number, (ii) compare this to existing experiments and integral models of turbulent lazy plumes, and (iii) reconcile the dis- 59 4. Rayleigh-Taylor lazy plumes agreement between experimental measurements on entrainment and classic plume theory by providing amendments to constant-α models. En route to the achievement of these objectives, we explore a wide range of source conditions (1 ≤ Γ0 ≤ 106) for injected plumes and identify in the development of Rayleigh-Taylor (RT) instabilities developing along the contracting portion of the plume an explanation for the increase in measured time-average spread and entrainment for lazy plumes of suitably high source Richardson number. 4.2 Background To date, only a few experimental studies have been conducted to quantify the near-source dilution of a lazy plume. Colomer et al. (1999) conducted a particle tracking velocime- try and a light-induced fluorescence (LIF) study on plumes formed from the release of saline from a microporous disc into a freshwater environment. They obtained direct mea- surements of time-averaged velocity and density profiles. However, they cite their data in such a way as to make it impossible to reconstruct the source Richardson number of the plume itself. The descriptions of Colomer et al. (1999) nevertheless provide valuable insight into the unconventional dynamics of lazy plume behaviour. They note that there is a physical limit to how much a lazy plume contracts for increasing source Richardson number. Epstein & Burelbach (2001) conducted experiments on lazy plumes by releasing a freshwater plume through a porous disc into a saltwater environment. They measured the spread of the plume by shadowgraph visualisation and observed that the extent of the contraction and its distance from the source was independent of source Richardson number for Γ0 & 103. The only known experimental measures of near-source volume flux variation with source Richardson number come from the works of Hunt & Kaye (2001) and Kaye & Hunt (2009). In their studies, Hunt and Kaye set up a displacement flow in a freshwater tank by extracting fluid from the tank while simultaneously allowing it to be filled by a saline plume. Their approach was inspired by the technique of Baines (1983), origi- nally conceived for small-source plumes where the tracking of the first front formed by the plume enabled the estimation of the volume flux at various heights. Kaye & Hunt (2009) noted that while plumes characterised by scaled source Richardson numbers of order 100 to 101 were suitably modelled by the constant-α descriptions of Morton et al. (1956) (as discussed in Hunt & Kaye 2001, 2005), plumes injected at higher scaled source Richardson number, ranging between 105 to 107, were not. More recently the DNS simulations of Marjanovic et al. (2017) shed new light on the dynamics of the near-source region of lazy plumes. Marjanovic et al. (2017) conducted DNS for Γ0 = 1, 2, 20, 200 and ∞. For the infinitely lazy case (Γ0 = ∞), they iden- tify a mechanism for the excess dilution, which they attribute to helical vortex structures that appear in the near-source region and that are swept by the rolling of vortices into the plume. These structures indeed closely resemble those observed in thermal plumes ensu- ing from a heated disc, which were studied experimentally by Pottebaum & Gharib (2004) when they performed a combined temperature and particle image velocimetry study us- ing thermochromatic crystals. Plumes that arise from heat exchanges, compared to in- jected plumes, represent the limiting case of an infinite scaled source Richardson number Γ0 = ∞ in that w0 ≡ 0, but whose flow immediately above the source is by definition 60 4.2. Background laminar. Similar structures were also observed in the DNS study of Plourde et al. (2008) which revealed the distinctive mixing features of these infinitely lazy plumes. Plourde et al. (2008) described the nature of the increased entrainment in the near source of a lazy plume as a pulsating mechanism, whereby, close to the source, vortices develop surround- ing the thin laminar stem of the plume. All existing studies fall into narrow bands of possible ‘laziness’ (laziness increasing with increasing Γ0) and our aim is to unify these within a broader framework in this chapter. Saline plumes released from a microporous disc into a freshwater environment are interrogated. The radial spread and volume fluxes of the plume are measured herein by conducting a series of dye visualisations (backlit and planar LIF) and planar parti- cle image velocimetry (PIV) measurements. These techniques are complemented with a modification to the Baines (1983) technique whereby the near-source volume flux is mea- sured by tracking the interface in a filling-box (Baines & Turner, 1969) arrangement. The latter technique is far simpler than the LIF/PIV experiments and enables the more rapid acquisition of data required to examine a larger number of source Richardson number cases. The experimental configuration and techniques are further described in §4.4. Our data is compared to existing theoretical models in §4.5, where we examine im- ages of the near-source region in order to visualise, via the dispersion of passive tracers, the contracting dynamics of lazy plumes. Observations in the near-field for Γ0 & 103 reveal the reason for the increase in radial spread and entrainment observed. As the plume contracts horizontally owing to the pronounced gravitational acceleration, it forms quasi-horizontal surfaces which are inherently unstable due to the vertical density gra- dients. Finger-like intrusions develop along the contracting portions of these plumes. These structures closely resemble those seen in the growth of the RT instabilities which are typically observed in the mixing of convective flows ensuing from distributed sources of buoyancy. First-order estimations of dilution using classic RT quadratic growth rates, developed in §4.3, find good agreement with the measured increase rate of entrainment presented in §4.6. A new entrainment model is proposed based on these results. The model enables the description of the near-source volume flux variation for varying de- grees of laziness. These new results are discussed in light of the way this increased entrainment modifies the morphology of near-source engulfment and the dynamics of the plume, in both near and far field, with increasing Γ0. Conclusions are drawn in §4.8. 4.2.1 Basic definitions & existing solutions Buoyant plumes can be categorised by the ratio of fluxes at source (Morton, 1959). These fluxes are that of volume (Q = wb2), specific momentum (M = w2b2) and buoyancy (B = g′wb2) at any given level z from the source. They are computed as: Q := ∫ ∞ −∞ wr dr, M := ∫ ∞ −∞ w2r dr, B := ∫ ∞ −∞ g′wr dr, (4.2) where r is a radial coordinate measured from the axis of symmetry of the plume. Inte- gral models for plumes have been developed by estimating the variation of these time- averaged quantities downstream of the source. The simplified governing equations of integral models are based on the conservation of mass, momentum and internal energy in 61 4. Rayleigh-Taylor lazy plumes the plume, alongside a number of crucial assumptions that characterise entrainment (Hunt & van den Bremer, 2011; van Reeuwijk & Craske, 2015). For a plume in an otherwise quiescent and uniform environment, the equations for the conservation of volume, specific momentum and buoyancy flux within the plume can be expressed as dQ dz = 2αM1/2, dM dz = BQ M , dB dz = 0; (4.3) here relying on the assumption of uniform (so-called ‘top-hat’) profiles of density and velocity (Morton et al., 1956; Hunt & Kaye, 2005; Woods, 2010). The entrainment pa- rameter α, introduced, for example, in Morton et al. (1956), relates the mean vertical velocity in the plume to a mean peripheral velocity that is induced into the plume by the turbulent entrainment of ambient fluid. More information of how these equations are derived was presented in Chapter 2. Solutions to the plume equations (4.3) are compared in this chapter to data acquired during the experimental campaign as a means to assess the entrainment behaviour of a lazy plume. Proposals for solutions to (4.3) fall under three broad categories: approximate solutions, typically in the form of virtual origin corrections (Hunt & Kaye, 2001; Kaye & Hunt, 2009), analytical solutions that assume a constant α (Fanneløp & Webber, 2003; Hunt & Kaye, 2005; Carlotti & Hunt, 2005; Michaux & Vauquelin, 2008; van den Bremer & Hunt, 2010), and entrainment parameterisations that assume α = f (Γ) (Priestly & Ball, 1955; Kaminski et al., 2005; van Reeuwijk et al., 2016; Carlotti & Hunt, 2017). Virtual origin solutions are based on a form where a vertical offset zv is applied to the original solutions of Morton et al. (1956) for a pure plume released from a point source of buoyancy flux B0 and are given by: b = 6αp 5 (z + zv) , Q = 6αp 5 ( 9αp 10 )1/3 B1/30 (z + zv) 5/3 , (4.4) M = ( 9αp 10 )2/3 B2/30 (z + zv) 4/3 , B = B0. (4.5) The entrainment parameter αp in equations (4.4) and (4.5) corresponds to the constant value of α measured in the far-field of a plume. Note that this coefficient was estimated to be of about αp ≈ 0.09 by Morton et al. (1956); Hunt & Kaye (2001), a reference value we adopt henceforth. Hunt & Kaye (2001) showed theoretically that for very lazy plumes Γ  1, if α is assumed to be constant in equations (4.3), the virtual origin is proportional to the acceleration length scale (La0) : zv = −Cδ6αp5 b0Γ −1/5 0 ∝ Q3/50 B1/50 (∝ La0) . (4.6) In (4.6), Cδ is a constant which approximates to Cδ ≈ 0.147 for Γ  1 (see definition in §4.7.3). Analytical solutions to equations (4.3) that assume α is constant are also discussed in §4.5, but not repeated herein. These solutions can be found in Hunt & Kaye (2005) and for the sake of reduction in computational expense, equations (4.3) are solved using a 4th order Runge Kutta scheme, which is faster than the numerical solution of the 62 4.3. Theoretical modelling hypergeometric functions within the analytical solutions. Entrainment functions that are a function of local flux parameter Γ were discussed in §2.2.5 and are further addressed in §4.7.1. In the upcoming sections, these solutions are compared to data for the necking of the plume and to measurements of volume flux in order to assess how entrainment varies in the near-source region with increasing Γ0. 4.3 Theoretical modelling A model is developed to estimate the dilution in the near-source region. In §4.3.1, pre- vious measurements for the growth rates of RT layers are used to quantify mixing in the region. Beyond contributing to our predictive capability of lazy plume dynamics (§4.6), we will see in §4.3.2 that this enables a discussion, presented in §4.3.3, on the role and morphology of the RT fingers and their dependence on Γ0. 4.3.1 A volume-flux-based correction In this section, a model is developed to estimate the volume fluxes in a plume whose source Richardson number is sufficiently high, i.e. Γ & 103, so that RT mixing occurs. Entrainment due to RT mixing is only possible in the presence of an unstable vertical density gradient, i.e. below the level of the plume neck, ζ < ζn, and outside the core of the plume. In an analogous manner to an origin correction, the proposed model is initially formulated to describe the behaviour in the far field of the plume. For their constant-α model, it was shown by Hunt & Kaye (2001) that the volume fluxes in the limit of an infinitely lazy plume, i.e. Γ0 → ∞, are equal to: Qp ≈ CpQB1/30 z5/3, where CpQ = 6αp 5 ( 9αp 10 )1/3 . (4.7) In other words, that the virtual origin zv in (4.4) tends to zero, i.e. zv → 0. The mea- surements of plume radius, presented in the previous section, suggest that an additional volume flux is entrained in the near-source region compared to that predicted by (4.7). We denote this additional volume flux, QRT , in reference to the fact we attribute it to RT mixing. For the sake of argument, this additional component of entrained fluid is assumed to be independent of the plume component Qp of entrainment. An expression for volume fluxes in the far field is therefore proposed in the form: Q(z > zn) := Q0 + QRT + CpQB 1/3 0 z 5/3︸ ︷︷ ︸ ≡Qp . (4.8) As QRT is attributed to the development of RT instabilities along the contraction of the plume, the theory of RT convection will consequently be used as a starting point by which QRT is quantified. Note that (4.8) is prescribed so that QRT is constant beyond the level of the neck, i.e. for z > zn, as RT mixing can only occur when the plume contracts (db/dz < 0). As a consequence, as we shall see in §4.6.2, this results in a constant offset of the plume volume flux in the far-field compared to solutions assuming a constant value of α. 63 4. Rayleigh-Taylor lazy plumes A description for the volume flux entrained within the near-source region is sought by considering the following arguments. As observed in experiments (§4.5), the plume outline is independent of Γ0. This allows for the simple model description of the plume radius prescribed later in § 4.5 by equation (4.51). The volume of the contracting portion of the plume is denoted VC. This volume can be estimated by integrating the volume of revolution enclosed by the plume radius and the plume central axis, from the source to the level of the neck, i.e. z = 0 to z = zn. This gives: VC = 2pi ∫ zn 0 b2 dz = pib30 ( 2 ∫ ζn 0 βˆ2 dζ ) ︸ ︷︷ ︸ ≡VˆC . (4.9) An analytical solution for the dimensionless volume of the contraction, VˆC, given by (4.9), is deduced. Consider the integration of (4.47) with respect to ζ to solve for the volume of revolution of the contracting region: VˆC = 2 ∫ ζn 0 βˆ2 dζ. (4.10) Upon integration, the dimensionless volume of the contracting region is: VˆC = 2 ( 1 + βˆ2nζnP1 + K 2ζ3n 3 + βˆnζnP2 +K βˆnζ 2 nP3 + ζ 2 n ) , (4.11) with coefficients: P1 = 1 − 2(expW − 1) W + exp 2W − 1 2W , (4.12) P2 = −2 + expW − 1 W , P3 = 1 − 2 ( (W − 1) expW + 1) W 2 , (4.13) where: W = W−1 (−6αp 5 ζn βˆn ) and K = 6αp 5 . (4.14) The value of VˆC approximates to VˆC ≈ 0.77±0.05, taking the approximate values of neck level and radius used in (4.52). The volume flux QRT is estimated by considering a dimensional argument. The ratio of the volume of this region to a characteristic time scale of mixing (tRT ) can be used to estimate a volume flux: QRT ∼ VCtRT . (4.15) In (4.15), tRT is deduced from classic arguments concerning the growth rate of RT layers. For two initially unstable layers of uniform densities ρ1 and ρ2, where ρ2 > ρ1, this growth rate is typically described as a function of the Atwood number: At := ρ2 − ρ1 ρ1 + ρ2 . (4.16) 64 4.3. Theoretical modelling (a) (b) z 2zRTzn 2b0 2b0 ρ1 = ρ0 ρ2 = ρa plume plume (4.52) Figure 4.2: Schematic of a simple model for RT finger growth in the near-field of a very lazy plume. In our configuration, idealised schematically in figure 4.2, it is assumed that the density of top layer is that of the ambient environment, ρ2 = ρa, and that the density of the bottom layer is that of the fluid supplied at the source, ρ1 = ρ0 = ρa − ∆ρ. For convenience, the Atwood number is expressed as a function of the reduced gravity of the source fluid, that is: At = ∆ρ 2ρa ( 1 − ∆ρ2ρa ) ≈ g′0 2g , (4.17) for small density differences ∆ρ/ρa  1. The growth rate of a layer of half-depth zRT (figure 4.2) is estimated by the quadratic scaling law of Fermi & von Neumann (1955), where zRT = αRT At gt2 2 = αRT g′0t 2 4 . (4.18) In (4.18), the coefficient for RT entrainment αRT is estimated from previous measurements in RT layers to be αRT ≈ 0.055 ± 0.015 (Boffetta & Mazzino, 2017). The RT layer is assumed to grow over a distance corresponding to the vertical height of the contracting region, i.e. to the level of the neck, z = zn. Hence, it is deduced from (4.18) that a characteristic timescale for layer growth is: tRT = ( 4ζnb0 αRT g′0 )1/2 . (4.19) Reverting now to expression (4.15), an estimate of the entrained portion of ambient fluid from RT instabilities is obtained: QRT ≈ VCtRT = VˆC 2 √ αRT ζn g′0b 5 0. (4.20) The form Q ∼ √gL5 is a common dimensional scaling for flows falling under gravity, such as spills over weirs, liquid jets falling through air and stack flows through buildings (e.g. Linden 1999), which is consistent with the description of a gravitationally-driven flow such as that of a RT lazy plume. In dimensionless form, q = Q/Q0 and ζ = z/b0, the 65 4. Rayleigh-Taylor lazy plumes expression for volume fluxes above the neck given by (4.8) takes the following form: q(ζ > ζn) = 1︸︷︷︸ q0 + RT Γ 1/2 0︸ ︷︷ ︸ qRT + pΓ 1/3 0 ζ 5/3︸ ︷︷ ︸ qp , (4.21) where the coefficients: RT = α 1/2 RT ζ 1/2 n VˆC 2 ( 8αp 5 )1/2 ≈ 0.033 ± 0.008, (4.22) p = 6αp 5 ( 9αp 10 )1/3 (8αp 5 )1/3 ≈ 0.024 ± 0.005. (4.23) In (4.22) and (4.23), the dimensionless volume of the contraction region was taken to be VˆC ≈ 0.77±0.05, from (4.9), and the RT entrainment coefficient to be αRT = 0.055±0.015 (Boffetta & Mazzino, 2017). This ‘volume flux offset’ model is later compared to volume flux data in §4.6.1 and §4.6.2. An alternative formulation is derived in the form of a virtual origin correction in §4.7.3 based on these scalings. Before this comparison, the model is further developed to obtain an expression for the variation of volume fluxes in the near-field. 4.3.2 Simple models for entrainment in the contracting region A limitation of the expression given by (4.21) which was deduced in the previous section, or indeed to any offset correction provided Γ0 , 1, e.g. (4.4), is that the variation of the volume fluxes in the near-source region is not described. To address this, proposals for a function to describe the entrainment over 0 ≤ ζ ≤ ζn are considered in this section. Following from the approach adopted to deduce (4.21), a function for the near-source volume fluxes is proposed in the following form: q(ζ) := 1 + pΓ 1/3 0 ζ 5/3 + qRT (ζ). (4.24) In (4.24), qRT (ζ) is an unspecified monotonic function, n.b. such that there is no de- trainment of fluid out of the plume, that describes the additional entrained volume flux. Physically, it needs to satisfy the following conditions:∫ ∞ 0 qRT (ζ) dζ = ∫ ζn 0 qRT (ζ) dζ = RT Γ 1/2 0 , qRT (ζ = 0) = 0, (4.25) that is, the additional entrained volume flux due to RT mixing in the near-source region qRT integrates to the value given by the RT volume flux offset (4.21) and does not increase further beyond ζ > ζn, and that it is equal to zero at source. In what follows, proposals for three models of increasing complexity are provided that meet these requirements. First, we consider a linear increase of entrained volume flux, resembling in intent an average, qRT (ζ) := RT Γ 1/2 0 min (ζ, 1) . (4.26) 66 4.3. Theoretical modelling A drawback of (4.26) is that q(ζ) is not continuous. Secondly, we consider an exponential function with constant growth, of growth rate λQ and given by: qRT (ζ) := RT Γ 1/2 0 ( 1 − e−λQζ ) . (4.27) To approximately satisfy the integral condition in (4.25), coefficient λQ > 0.57 so that 95% of increase in the volume flux to qRT > RT Γ 1/2 0 occurs below the level of the neck, ζ > ζn. The expression given by (4.27) results in a smooth function that predicts the near- field variation of volume flux reasonably well (see §4.6.2). The variation of the volume fluxes is, however, arbitrarily specified by the choice of the value of λQ. On average, the best-fit for λQ to the near-source region volume flux data, estimated using PIV data and later presented in §4.6.2, was of λQ = 2.4 ± 0.4. 4.3.3 A model for the morphology of the near-source region Finally, a third model is proposed to estimate the variation of the volume fluxes in the near-source region. Several assumptions are invoked concerning the shape of the flow so as to discuss how we anticipate the morphology of the RT mechanism to change with increasing Γ0. The entrainment function proposed here is based on the assumption that the outline between the source fluid and the environment is deformed according to a pre- scribed shape which instantaneously resembles that characterised by finger-like intrusions and described in §4.5.1, and that is consistent with (i) the observed increase in entrained volume flux given by (4.21) and (ii) the fixed time-average radius of the near-source re- gion given by (4.52). A proposal for the shape of the external outline of the source fluid as it is perturbed by the overlaying denser fluid is illustrated in figure 4.3. In this figure, the instantaneous shape of the source fluid ζ = ηˆ0(r/b0) is described by a Morlét wavelet, denoted ηˆRT , of amplitude a0, which is superimposed upon a sinusoid ηˆs, of wavelength 4b0 and amplitude as, that extends over the source of the plume −b0 ≤ r ≤ b0. This corresponds to a profile equivalent to: ηˆ0 := η0 b0 := a0 exp ( − r 2 2σ2 + 2pifi ) ︸ ︷︷ ︸ ηˆw + as exp ( ipir 2b0 ) ︸ ︷︷ ︸ ηˆs . (4.28) The shapes prescribed by (4.28) are chosen for their resemblance to the outline of the contracting region of a very lazy plume, where finger-like structures of variable length develop along its radius (figure 4.10). The (Gaussian) width of the wavelet is set to that of the source σ = b0/2 so that the oscillations are equally distributed over the ‘near- source region’ with shorter ‘fingers’ approaching the edge of the source and the longest lobe of light fluid centred over the axis of the plume. The spatial frequency of the fingers is fixed by f = 1/L; with L = b0/N being the wavelength of the instability, that is the crest-to-crest separation of the fingers, and N the number of fingers observed along the radius of the plume (figure 4.3). An estimate for the number of fingers that develop along the radius of the source is deduced from experimental observations where these have typically been displayed in sets of around N ≈ 2-3 at any given time (e.g. figures 4.9a- c). For simplicity, the radial coordinate is non-dimensionalised upon the source radius as rˆ = r/b0 henceforth. 67 4. Rayleigh-Taylor lazy plumes A further constraint is imposed to the amplitudes of the profile given by (4.28). This constraint enforces that the crests of the wavelet extend vertically to the level of the neck, i.e. a0 + as = ζn. (4.29) In other words, the amplitudes of the profiles given by ζ = ηˆ0 (rˆ), and ζ = ηˆs (rˆ), i.e. a0 and as, are fixed so that their sum corresponds to the level of the neck. This constraint (4.29) is imposed to model how RT fingers, that occur as the plume contracts in the near- field, leave behind a trace as they are swept into the plume, that matches in size that of the time-average near-source region (as discussed in §4.5.1). For convenience, this time- average size of the near-source region is approximated by a half-sinusoid that extends over the source: ηˆC := ηC b0 := (a0 + as) exp ( ipir 2b0 ) , (4.30) (see figure 4.3). In figure 4.3, examples are provided for the shapes outlined by (4.28), for preselected values of a0/as and N, to show how these coefficients affect this model for the instantaneous outline of the near-source region. Equation (4.28) allows one to vary the volume of the region representing the source fluid, given by the area enclosed by ζ = ηˆ0 (rˆ) and ζ = 0 revolved around rˆ = 0 (shaded red and denoted Vˆ0), with respect to the area representing the entrained fluid from the environment, given by the area enclosed by ζ = ηˆC (rˆ) and ζ = ηˆ0 (rˆ) revolved around rˆ = 0 (shaded blue and denoted VˆRT ), see figure 4.3. The relative volume of the regions can be related to the magnitude of the source and entrained volume flux. Over a fixed time interval, ∆t, a volume flux can be estimated by Q = V∆t, such that: VRT ∆t V0∆t = QRT Q0 . (4.31) The volume VRT can be estimated by: VC = V0 + VRT , (4.32) in such a way so that VRT is consistent with the time-average volume of the near-source region: The volume of the corresponding regions, Vˆ0, VˆRT and VˆC, are deduced by evalu- ating: V = 2pib30Vˆ , where Vˆ = ∫ ζn 0 ( ηˆ−1 )2 dζ, (4.33) where ηˆ−1 is the inverse function of ηˆ. Note that the integration of ηˆ0 is not necessary due to the periodicity of the wavelet, as Vˆ0 ≈ Vˆs and equal to: Vˆ0 ≈ Vˆs = 4 pi2 ∫ as 0 log2 ζ as dζ = 8asζn pi2 . (4.34) Despite the profile ζ = ηˆ0(rˆ) does not require integration, the periodicity in the wavelet ηˆ0 is introduced to model the fingers that develop in the near-source region. The volume of the contracting region is estimated as: VˆC = 4 pi2 ∫ ζn 0 log2 ζ dζ = 8ζn pi2 . (4.35) 68 4.3. Theoretical modelling −1 −0.5 0 0.5 10 0.5 1 1.5 2 a0/as = 1/6, N = 3 ζ/ ζ n −1 −0.5 0 0.5 10 0.5 1 1.5 2 a0/as = 1/3, N = 3 time-averaged ‘outline’ instantaneous ‘outline’ −1 −0.5 0 0.5 10 0.5 1 1.5 2 a0/as = 1/6, N = 2 N a0/as ζ/ ζ n rˆ −1 −0.5 0 0.5 10 0.5 1 1.5 2 a0/as = 1/3, N = 2 ηˆ0 ηˆs ηˆC βˆ(ζ) Vˆ0 (source) VˆRT (entrained) rˆ Figure 4.3: Examples of the near-source region profiles, ζ = ηˆ0 (rˆ), ζ = ηˆs (rˆ) and ζ = ηˆC (rˆ), given by equations (4.28)-(4.29), plotted for different combinations of amplitude ratios, a0/as, and wave number of fingers, N. The dotted line corresponds to the time-average radius expected for a plume with Γ0 ≥ −2RT as described by equation (4.52). The volume of these regions is now related to the volume fluxes. The volume flux offset correction (4.21), presented in §4.3.1, suggested that the ratio of entrained volume flux due to RT mixing relative to the source volume flux corresponds to qRT = RT Γ 1/2 0 . If one considers that for each time instance ∆t, a volume flux can be estimated by Q0 ≈ Vs∆t, such that QRT ≈ ∆tVRT , then it follows that: VRT V0 = 1 − as as = QRT Q0 = RT Γ 1/2 0 . (4.36) From (4.36), one can solve for the amplitudes of the wavelet a0 and as and express these as a function of scaled source Richardson number: a0 = RT Γ 1/2 0 1 + RT Γ 1/2 0 , as = 1 1 + RT Γ 1/2 0 . (4.37) The wavelet shape is used to discuss the morphology of the finger intrusions in re- sponse to the laziness of the plume. In figure 4.4, the profiles of ζ = ηˆ (rˆ) are shown for 69 4. Rayleigh-Taylor lazy plumes −1 −0.5 0 0.5 10 0.5 1 1.5 2 Γ0 = 1 ζ/ ζ n (a) −1 −0.5 0 0.5 10 0.5 1 1.5 2 Γ0 = 100 rˆ a0 (b) −1 −0.5 0 0.5 10 0.5 1 1.5 2 Γ0 = 10000 (c) 100 101 102 103 104 105 106 0 0.2 0.4 0.6 0.8 1 Γ0 a 0 (d) Figure 4.4: In (a)-(c), profiles of ζ = ηˆ0 (rˆ), ζ = ηˆs (rˆ) and ζ = ηˆC (rˆ) plotted for Γ0 = 1, 100 and 10000, respectively, as solved with amplitudes prescribed by (4.37). (d) The value of the predicted dimensionless amplitude of the RT finger a0 plotted against Γ0, as estimated by equation (4.37). amplitudes that vary with scaled source Richardson number, as given by (4.37). It can be seen that the relationship between V0 and VRT , and, as a result, of Q0 and QRT , has a profound effect on the relative vertical penetration of the intrusions. The two components of the volume flux become of comparable magnitude, i.e. QRT ≈ Q0, when: Γ0 =  −2 RT ≈ 918. (4.38) Interestingly, based on the scalings proposed in §4.3.1, the two components QRT and Q0 are expected to be of similar magnitude for a value of Γ0 ≈ 103, which is similar to that at which the appearance of fingers is observed physically. It follows that if Γ0  −2RT , that means that QRT  Q0 and that these fingers would not be expected to appear. On the other hand, if Γ0  −2RT , then the RT component of volume flux far exceeds that the source volume flux QRT  Q0. From figure 4.4, it appears that the amplitude a0 is relatively independent of Γ0 for Γ & 106, with the fingers that extend up to the level of the neck. The final step to the formulation of the model is to deduce an expression for qRT (ζ) . To this end, the profiles ηˆs and ηˆC are integrated up to a level ζ and again revolved around the rˆ = 0 axis, giving: Vˆ0(ζ) = 4 pi2 ∫ ζ 0 log2 ζ as dζ =  4ζpi2 ( log2 ζas − 2 log ζ as + 2 ) , for 0 ≤ ζ ≤ as, 4as pi2 , for ζ ≥ as; (4.39) 70 4.4. Experiments and VˆC(ζ) = 4 pi2 ∫ ζ 0 log2 ζdζ =  4ζpi2 ( log2 ζ − 2 log ζ + 2 ) , for 0 ≤ ζ ≤ ζn; 4ζn pi2 , for ζ ≥ ζn. (4.40) The volumes that range with the coordinate, given by (4.39) and (4.40), can be related to the plume volume fluxes by assuming that for a fixed time interval, the volume of each region is proportional to volume flux, qRT . This results in an expression for the entrained volume flux in the form: qRT (ζ) = RT Γ 1/2 0 ( 1 − Vˆ0(ζ) VˆC(ζ) ) . (4.41) As a result of the work presented above, four options are available for equations that describe the streamwise volume variation, namely, (4.3), (4.26), (4.27) and (4.41). These expression are compared to volume flux data in the upcoming sections. 4.4 Experiments A series of four separate experimental sets are presented in this chapter. The experiments are summarised in table 4.1. The experimental configuration which is depicted in figure 4.5 is shared between all these sets. In each of these, a plume is formed by the release of an aqueous-saline solution into a freshwater environment. The saline solution was extracted from a reservoir of known salt/dye/seeding concentration and injected through a microporous circular nozzle. The outflow that results from the impingement of the plume with the floor of the tank forms a stable layer that grows in depth, in similar fashion to the filling box of Baines & Turner (1969); an example of which is shown in figure 4.6. The slow filling of the container allows (i) the plume to reach a quasi-steady state above the first ascending front for sufficient time to record measurements from dye and particle tracers, (ii) the measuring of the volume flux at various levels using an interface tracking technique which is described in §4.4.2. Different optical techniques were used to visualise the plume flow and to extract data via image processing techniques. The plume flow was visualised in two ways: by (i) illuminating a fluorescein tracer with a white light sheet shining along the plume axis and (ii) by backlighting a methlyene blue dye tracer. By means of the interface tracking technique, volume fluxes in the plume were estimated by comparing container filling rates with predictions based on plume volume flux models. In a final set of experiments, particle image velocimetry was implemented to measure the velocity field along the axis of the plume and to compute the volume fluxes by integrating the velocity field directly. These techniques are discussed in further detail in the upcoming sections §4.4.2-4.4.3. The equipment used to generate and control the flow is described in §4.4.4. 4.4.1 Flow visualisation The outline of the plume was identified by video recording the flow which was visualised by the addition of a passive tracer to the plume source fluid. Two sets of experiments are presented: (i) fluorescence visualisations of the near-source region and (ii) backlit dye visualisations of the time-averaged spread of the plume. 71 Experiment b0 (mm) w0 (mm/s) g′0 (mm/s 2) Γ0 Re0 := w0b0/ν Gr0 := g′0b 3 0/ν 2 φ := R/H Filling-box 10.8 20-300 100-300 0.5-4 500-3000 1-4 × 105 0.7 22.5 10-100 100-700 10-2000 100-1500 1-8 × 106 0.7-1.4 44.5 10-100 100-700 1000-400000 50-800 1-7 × 107 0.7-1.4 LIF 22.5 10-80 100-600 10-1600 100-900 1-6 × 106 - 44.5 10-100 100-300 1000-300000 50-800 1-3 × 107 - PIV 22.5 10-80 50-200 1-300 100-900 0.5-2 × 106 - 44.5 10-100 50-200 500-200000 50-800 0.5-2 × 107 - Table 4.1: Summary of experimental parameters according to the different techniques used. Xenon arc lamp & light-sheet forming optics slit flowmeter gear pump stained/seeded saline reservoirvisualisation tanks overflow nozzle vertical light sheet Figure 4.5: The experimental setup used to perform LIF and PIV. Saline is injected by a positive displacement gear pump through a nozzle with a porous diffuser outlet into an acrylic visualisation tank. The central plane of the plume is illuminated by a thin light sheet produced by a Xenon arc lamp. The flow rate is monitored with an in-line magnetic rotameter. 4.4. Experiments z f H Q ( R2 − b2 ) dz f dt Figure 4.6: An example visualisation (shown in false colour) of a filling-box experiment where a plume stratifies the environment contained within a cylindrical tank. The outflow of the plume as it impinges with the base forms a stable layer that gradually deepens. The plume is visualised via the addition of dye and backlit with an LED panel. Light-induced fluorescence visualisations were conducted along the plume axis in order to inspect entrainment in the near-source region. The configuration for the lighting is illustrated in figure 4.5. A solution of fluorescien sodium dye of known concentration (2 mg/L) was added to the solution to illuminate the flow. A light sheet illuminated the central cross-section of the plume formed by a LUXTEL Xenon 300W arc lamp whose beam was projected through a thin vertical slot (2 mm wide). Backlit dye visualisations were conducted to obtain a measure of the time-average plume radius. This technique was selected for radius measurements over LIF to com- pletely remove any potential sources of error due to light-sheet misalignment. The flow was visualised using a methylene blue dye tracer that was backlit using an LED panel. The averaging time of the time-averaged radius measurements was chosen to exceed three minutes. Colomer et al. (1999) assessed the convergence of velocity measurements in a lazy plume showing that steady state velocities reached a constant value at around t ≈ 2 (2b0)4/3 /B1/30 – a value they observed to be independent of the source Richardson number. For the range of values used in our experiments, that corresponds to a minimum averaging time requirement which is between 80-180 s. We decided to abide to an av- eraging time of three minutes for each time-averaged plume picture. This proved to be sufficient in that the plume radius did not vary noticeably with greater averaging periods. 4.4.2 Tracking of the first front The rate of filling of a container by a lazy plume was measured in a configuration that resembles that of the filling-box experiment of Baines & Turner (1969). The comparison of the rate of filling, that is, the rise of the first ascending front of the deepening layer, with predictions based on plume volume flux descriptions enables the comparison with functions that describe the variation of volume fluxes with height. In these experiments, a cylinder of 600 mm internal diameter was placed below the plume, and allowed to fill with buoyant fluid until the layer formed in the cylinder reached the level of the source. Methlyene blue dye was added to the plume so as to trace the stratification that developed in the cylinder. An example visualisation is shown in figure 4.6. The ascending front was tracked using a combination of thresholding, gradients and edge detection algorithms applied to the regions external to the plume (see Chapter 3). The tracking of the interface enabled an estimation of the volume flux in the plume at 73 4. Rayleigh-Taylor lazy plumes different levels via a simple continuity argument. During the filling process, a stable layer forms at the bottom of the container (as seen in figure 4.6). There is no mixing through the density interface between the original ambient fluid and the newly-formed salt layer except for the penetration of the plume. The volume flux in the plume at the horizontally- averaged level of the interface z f corresponds to the volume flux of the rising layer such that: Q(z = z f ) = − ( R2 − b2 ) (dz f dt ) ; (4.42) where dz f /dt is the horizontally-averaged vertical velocity of the interface and R is the radius of the container. 4.4.3 Particle image velocimetry (PIV) measurements PIV measurements were taken with the same lighting setup as the LIF experiments de- scribed in §4.4.1. Polyamide seeding particles of nominal diameter of 50 µm and density 1.03 g/cm3 were used to seed the flow. PIV measurements were only conducted for plume release densities lower than 1.02 g/cm3 in order to minimise the optical distortion of parti- cles owing to changes in refractive index of the medium. Distortion was assessed visually upon inspection of the recorded images. The videos were recorded at a acquisition rate of 30 fps with a shutter speed of 10 ms. A high aperture (F/1.4) was used to reduce the depth of field to a depth shallower than the thickness of the light sheet. The PIV window was 180 × 360 mm2 (width by height) recorded with a resolution of 5 Megapixel (2048 × 2560). Particle scatter size roughly ranged between 3-5 pixels. A 15 pixel highpass filter was used to improve uneven lighting and a 5 pixel Wiener denoise filter was applied to reduce ghosting due to noise and unfo- cused particles. A two-pass interrogation scheme was adopted. Interrogation windows of area 128 × 128 and 64 × 64 pixels2 were used with a 50% overlap for each window. 4.4.4 Flow control & measurement Historically, experiments on ‘injected’ lazy plumes have been conducted by discharging fluid through a porous disc source (Colomer et al., 1999; Epstein & Burelbach, 2001; Kaye & Hunt, 2009). In our configuration, a polyethlyene disc was fixed as an outlet diffuser. The disc was 6 mm ±0.15 mm thick and of mean and maximum pore size of 85 µm and 105 µm, respectively (SPC Technology PE10060). The pore size was selected as a compromise between increasing the pressure drop at the diffuser outlet, needed to homogenise the velocity profile at the source, and to prevent filtering of the PIV seeding. Two separate nozzles were used to cover the desired source conditions of exit diameters 2b0 = 44.5 and 89.5 mm (±1%). The nozzle outlet was preceded by a straight length of pipe of about ten exit diameters (460 mm and 700 mm in each case) to prevent swirling effects from pipe bends. The saline solution was supplied from a reservoir of known salinity and injected into the cylinder at a known flow rate. The density of the injected was measured with an Anton Paar density meter (DMA 5000 M), accuracy ± 5 × 10−4 g/cm3, and ranged from 1.01- 1.07 g/cm3. The density of freshwater was measured to be 0.998 ± 0.001 g/cm3. The flow in the plume was supplied with a positive displacement gear pump made by ISMATEC 74 4.5. Observations & measurements of the contracting region (model ISM 198A). The flow rate was measured with an Apollo LowFlo flowmeter to an accuracy of ±1% of the full reading (i.e. ±0.03 L/min). 4.5 Observations & measurements of the contracting region The necking behaviour of a lazy plume is examined and discussed in light of its impli- cations for near-field entrainment. Measurements of the neck level zn and radius bn are employed as a diagnostic to assess whether solutions to the plume equations (4.3) under the assumption of constant α suitably describe the spreading rates observed experimen- tally. Comparisons of these diagnostics with predictions based on different descriptions of α are presented in figures 4.7 and 4.8. Note that the observed increase in spreading rates, relative to these solutions, are indicative of an increase in entrained volume flux. In Chapter 2, it was shown that as the source Richardson number of a lazy plume increases, so does its tendency to contract. An approximate constant-α solution to equa- tions (4.3) for Γ0  1 was deduced by Hunt & Kaye (2005) for the neck level and radius which revealed that they both scale with Γ −1/50 : zn b0 ≈ 1.57Γ −1/50 and bn b0 ≈ 1.4Γ −1/50 . (4.43) This is equivalent to expressing the neck level and radius as directly proportional to the source acceleration length scale: zn La0 ≈ 6 25 ( 3 2 )−4/5 ≈ 0.1735 and bn La0 ≈ 0.168, (4.44) where: La0 = ( 5 6αp ) ( 8αp 5 )3/10 Q3/50 M1/50 . (4.45) This scaling is helpful for graphical representations (see figure 4.7c). For 10 . Γ0 . 103, the approximate solutions compare well with the data presented in figure 4.7a-c for the time-averaged plume radius. The plume contracts to a smaller radii, located closer to the source for increasing Γ0. Scaling the streamwise coordinate on the source acceleration length scale z/La0, as given in (4.44), yields the appropriate collapse for the far-field plume radius. This confirms that, for Γ0 . 103, constant-α solutions can predict the behaviour in the near-source region. The trend of an increasing necking of the plume with Γ0 ceases to be representative for Γ0 & 103. Upon inspection of the data for radius and level of the neck which is presented in figure 4.8a-b, it can be observed that necking is relatively independent of Γ0 and on average: zn b0 ≈ 0.92 ± 0.05 and bn b0 ≈ 0.62 ± 0.07 for Γ0 > 103. (4.46) The result is in line with past experimental studies (Colomer et al., 1999; Epstein & Burelbach, 2001; Kaye & Hunt, 2009) both in terms of the Γ0 invariance for the spreading and the magnitude of the neck level and radius (figure 4.8a-b). The increased rates of 75 −1 0 10 1 2 3 4 b/b0 z / b 0 z / b 0 00.51 0 0.5 1 (a) (b) (4.16) increasing βˆn with Γ0 −0.5 0 0.50 0.5 1 1.5 2 b/La0 z /L a 0 0 0.5 1 1.5 2 2.5 3 z /L a 0 log10 Γ0 (c) (4.3) ︸ ︷ ︷ ︸ cl as si c sc al in gs fo rl az y pl um es – 1≤ Γ 0. 10 3 −2 −1 0 1 20 1 2 3 4 5 6 7 b/b0 z / b 0 3 3.5 4 4.5 5 5.5 6 z / b 0 00.51 0 0.5 1 constant βˆn with Γ0 (d) (e) (4.16) log10 Γ0 ︸ ︷ ︷ ︸ Γ 0− in va ri an ts ca lin gs fo rl az y pl um es – Γ 0& 10 3 Figure 4.7: Dimensionless radius of the plume plotted against streamwise coordinate from data ac- quired through backlit dye visualisations. In (a)-(c), data is presented for lazy plumes of Γ0 ≤ 103, while in (d)-(e), for lazy plumes of Γ0 > 103. The graphs (b) and (e) are closeups of the near- source region of the data presented in (a) and (d), respectively. Colourbars indicate the value of Γ0 on a logarithmic scale. In (a),(b),(d) and (e), the solid green line (-) shows the profile of plume radius proposed in equation (4.52). In (c), the expression for plume radius on the virtual origin correction (4.4) is shown as a blue dashed line (- -), taking zv = −CδLa0 where Cδ ≈ 0.147 (a result adapted from Hunt & Kaye 2001). The solution based on the virtual origin correction is plotted to show how constant-α solutions retain the constant-α source acceleration length scaling for 1 < Γ0 < 103. 4.5. Observations & measurements of the contracting region 101 102 103 104 105 106 0 0.2 0.4 0.6 0.8 1 1.2 Γ0 β n (a) α = cst α = f (Γ) βˆ n 101 102 103 104 105 106 0 0.2 0.4 0.6 0.8 1 1.2 Γ0 ζ n (b) α = cst α = f (Γ) Col99 Eps01 Figure 4.8: Dimensionless (a) radius and (b) level of the neck of the plume plotted against stream- wise distance for varying Γ0 from data acquired through backlit dye visualisations . The solid black (–) line represents the analytical solutions (4.43) of Hunt & Kaye (2005). The dashed (- -) line represents the experimentally-averaged neck radius and level for 103 ≤ Γ0 ≤ 106. The dashed-dotted (- · -) line shows the numerical solution to the neck as solved from the plume con- servations (4.3) using the entrainment function of van Reeuwijk et al. (2016) presented in §4.7.1. The data of Epstein & Burelbach (2001), denoted EPS01, is indicated by the green square mark- ers. The range of data of Colomer et al. (1999), denoted Col99, is shown as a blue band drawn over their measurements indicating βˆn = 0.55± 0.05 and ζn = 0.56± 0.26, n.b. it was not possible to reconstruct their values of Γ0 from the work presented therein. time-averaged radius for plumes whose scaled source Richardson numbers exceed Γ0 & 103 suggest an increase in α in the near-source region. The invariant necking behaviour of plumes of high Γ0 can be used to develop a simple expression for the radius with distance from the source for Γ0 > 103. To this end, a function is proposed which is composed of an exponentially decaying near-field and a linear far-field component in the form: b = b0 − bn ( 1 − exp ( −λnz bn )) + ( 6αp 5 ) z. (4.47) The decay constant λb is introduced so that (4.47) can be defined such that the neck, defined by db/dz = 0, occurs at z = zn. It follows that as the vertical gradient in plume radius corresponds to: db dz = 6αp 5 − λn exp ( −λbz bn ) , (4.48) the decay constant can be evaluated as the roots of: λn exp ( −λnzn bn ) = 6αp 5 . (4.49) The solution for λn can be expressed analytically as the -1 branch of the Lambert-W product log function (Corless et al., 1996), which approximates to: λn = −bnzn W−1 (−6αp 5 zn bn ) . (4.50) 77 4. Rayleigh-Taylor lazy plumes When implementing (4.50) in a computational solver it should be noted that the 0-branch solution of the lambert-W function gives the trivial solution db/dz = 0 for all z and all other branches yield solutions that are complex and as result, not physical. In dimension- less form, denoting βˆ = b/b0 and ζ = z/b0, this profile for plume radius can be expressed as: βˆ = 1 − βˆn [ 1 − exp ( ζ ζn W−1 (−6αp 5 ζn βˆn ))] + ( 6αp 5 ) ζ. (4.51) For αp = 0.09, βˆn = 0.62 and ζn = 0.92, equation (4.51) approximates to: βˆ = 0.38 + 0.62 exp (−3.2ζ) + 0.11ζ for Γ0 > 103. (4.52) In figure 4.7d-e, this expression is compared to the data for the spread of the plume for Γ0 > 103 with good agreement. In §4.7.2, simple matchings of solutions for the neck radius and level are proposed so that (4.51) can be used to estimate the spread of the plume for all Γ0. The relative independence of the plume radius with Γ0 is exploited further in §4.3.1- 4.3.2 to model near-field entrainment. Before delving into the modelling of this entrain- ment, however, we will qualitatively describe this mixing as observed in fluorescence visualisations of the near-source region. 4.5.1 Visualisations of the near-source region Flow visualisations are shown in figure 4.9a-c of the central cross-section of a lazy plume for varying degrees of plume laziness, Γ0, which range from O(102)-O(104). Each figure shows an instantaneous and time-averaged image of the plume visualised by means of a fluorescent tracer. Observationally, the increase of scaled source Richardson number past Γ0 & 103 coincides with the appearance of a different mixing mechanism in the near- field. For these very lazy plumes , fingers of ambient fluid penetrate vertically into the contracting region. Examples of this behaviour are shown in figure 4.9a-c and magnified in figure 4.10. These structures are subsequently advected into the plume by the rolling of plume vortices; leaving a trace in the time-averaged picture which manifests as an increased spread of the plume outline. The engulfment of the fingers into the vertical flow dilute the plume. The occurrence of this mixing mechanism can be interpreted as follows. The plume is expected to contract more strongly for increasing Γ0 and, as a result, has a tendency to neck to a smaller radii and closer to the source for increasing Γ0 as prescribed by (4.43). Based on these theoretical predictions, quasi-horizontal density interfaces would be expected to form, which are inherently unstable in vertical gradients. Above the threshold value of scaled source Richardson number observed, i.e. Γ0 & 103, the gradients are sufficiently unstable to promote additional density-driven mixing. The unsteady structures observed along the contraction of these plumes resemble those of classic RT convection (see, for example, figures 2.6g-i on p. 25) in that multiple slender vertical fingers grow and break down into turbulence before being swept into the plume. These structures are expected to only occur during contraction, i.e. where vertical density gradients are unstable, and to increase the entrainment occurring in the near-field as a result of the fact they increase plume spreading rates relative to solutions based on height invariant entrainment. 78 −1−0.5 0 0.5 10 1 2 3 x/b0 ζ ... z/ b 0 Γ0676 2346 25881 (a) −1−0.5 0 0.5 10 1 2 3 x/b0 ζ x/b0 (b) −1−0.5 0 0.5 10 1 2 3 x/b0 ζζ (c) Figure 4.9: Fluorescence visualisations showing an instantaneous (LHS) and time-averaged (RHS) views of the central cross-section of a plume of scaled source Richardson number: (a) Γ0 = 676, (b) Γ0 = 2346 and (c) Γ0 = 25881. Note that, while in (a), only a small semblance of a finger can be seen in the contracting region, in (b) and (c), distinct finger-like penetrations of the contracting region can be observed. The vertical scale of the penetrations increases with Γ0. 00.20.40.60.81 0 0.2 0.4 0.6 z /b 0 00.20.40.60.81 00.20.40.60.81 0 0.2 0.4 0.6 x/b0 z /b 0 00.20.40.60.81 x/b0 t0 t0 +0.4 s t0 +0.8 s t0 +1.2 s Figure 4.10: Fluorescence visualisations of the near-source region of the plume shown in figure 4.9(c) of scaled source Richardson number of Γ0 = 25881 showing the evolution of RT fingers in time. 4. Rayleigh-Taylor lazy plumes 4.6 Volume flux measurements Measurements of volume flux by filling rates, described in §4.4.2, and by PIV, described in §4.4.3, are presented in §4.6.1 and §4.6.2, respectively. 4.6.1 Volume flux measurements by filling rates The volume flux in the plume was indirectly measured by tracking the interface of the deepening layer in the filling-box experiments (§4.4.2). To avoid the inherent noisiness of taking gradients of the interface position to track its velocity (cf. equation (4.42)) an alternative method was adopted. Equation (4.42) was integrated numerically to predict the location of the interface by assuming that the volume flux in the plume after the initial layer has formed varies as the volume flux in the plume at steady state. In figure 4.12, the location of the interface is compared to predictions based on nu- merical solutions of the plume equations (4.3) and the new entrainment model presented in (4.41). Sample experiments of increasing Γ0 are shown. The time, t, is scaled on a (modified) filling-box timescale: tF := HR2 Qi , (4.53) where we recall H and R are the height and radius of the container and Qi is the volume flux at z = H predicted by (4.41). It is shown in figures 4.12a-f, that for increasing Γ0, the filling rates are concurrently increased relative to the constant-α solutions, and that the new entrainment model successfully predicts filling rates. 4.6.2 Volume flux measurements by particle image velocimetry The validity of (4.26) and (4.41) for the description of the volume flux variation is in- vestigated. These expressions are compared to the volume flux data estimated from par- ticle image velocimetry measurements. The results of three example experiments are shown in figures 4.11a-c for scaled source Richardson numbers Γ0 = 245, Γ0 = 9395 and Γ0 = 60681, respectively. The centre plane velocity fields are shown in the left column of figure 4.11. Measurements of volume flux obtained by integrating the vertical velocities along horizontal cross-sections, as given by equation (4.2), are shown in the right column. The volume flux data is compared to predictions estimated from the numerical solution of equations (4.3), (4.26), (4.27) and (4.41). Above the neck, i.e. ζ & 0.92, the volume fluxes adjust to a classic ‘5/3’ power law variation whereby q ∼ ζ5/3. The volume flux variation below the neck is approximately linear, q ∼ ζ (n.b. in agreement with the data of Kaye & Hunt 2009). A constant offset in volume flux is observed compared to predic- tions based on a constant value of α. The predictions based on the revised entrainment functions agree well with the data, consolidating the hypothesis that entrainment is not only increased in the near-source region, but also offset by a fixed volume flux that scales on Q0Γ 1/2 0 . 80 −2 −1 0 1 20 2 4 6 8 x/b0 ζ 0 2 4 6 Γ0 = 245 (a) w/w0 z/ b 0 0 2 4 6 8 10 0 2 4 6 8 Q/Q0 ζ α = cst (4.41)z /b 0 −1 0 10 1 2 3 x/b0 ζ 0 5 10 15 20 Γ0 = 9395 (b) z/ b 0 w/w0 0 2 4 6 8 10 12 14 0 1 2 3 Q/Q0 ζ α = cst z/ b 0 (4.41)qRT −1 0 10 1 2 3 x/b0 ζ 0 20 40 60 Γ0 = 60681 (c) z/ b 0 w/w0 0 5 10 15 20 25 0 1 2 3 Q/Q0 ζ α = cst z/ b 0 (4.41) qRT Figure 4.11: Comparison between near-field volume flux measurements, the plume theory of Morton et al. (1956) with α = cst and the lazy plume theory presented herein, (4.26), (4.27) and (4.41). In the left column, the velocity field, superimposed on contours of velocity magnitude, plotted for three sample experiments of increasing Γ0, (a)-(c). In the right column, the volume flux plotted from the respective velocity measurements, computed by the integration of the velocity field with respect to r. The green dashed (- -) lines show the solutions to the conservation equation based on the assumption of a constant entrainment coefficient αp = 0.09. The dotted (· · · ) lines show the predictions based on a linear increase in volume flux up to the bulk RT entrainment predicted by the volume flux offset prescribed by (4.26). The solid black (–) line indicates a prediction based on the exponential growth model prescribed by (4.27) assuming λQ = 2.4. The solid blue line (–) shows the prediction developed in §4.3.2 given by (4.41). 4. Rayleigh-Taylor lazy plumes 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Γ0 ≈ 34 (a) z f /H 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Γ0 ≈ 701 (b) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Γ0 ≈ 3988 (c) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Γ0 ≈ 8114 (d) z f /H 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Γ0 ≈ 12374 (e) (t − tinitial) /tF 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Γ0 ≈ 479563 (4.41) α = cst (f) Figure 4.12: Level of the first front of the deepening layer formed by a plume in a container of radius-to-height aspect ratio R/H ≈ 1. Separate experiments are shown for increasing Γ0 (a)-(f), the value of which is given above each figure. Markers indicate the location of the interface as tracked in each experiment. The solid line (-) indicates the prediction for the front level based on a constant-α entrainment model. The dashed line (- -) shows the prediction for the front level that uses the revised expression for volume flux presented in equation (4.41). The initial time, tinitial, was identified for a given experiment as the time at which the layer was first formed. 4.7 Discussion In this section, we first compare the theoretical models developed in §4.3 to the entrain- ment models proposed by other authors. Then, we develop an alternative description for the plume in terms of a virtual origin correction. 4.7.1 Comparison to other entrainment models In figures 4.13a-c, in which predictions of volume flux are plotted against streamwise distance, different entrainment models are compared to the model developed in §4.3.1 for additional RT mixing. The volume flux offset model (4.26) is compared to: (i) the constant-α model of Morton et al. (1956); (ii) the entrainment function of van Reeuwijk & Craske (2015) that assumes that α = f (Γ) ; and (iii) the entrainment model deduced by Kaye & Hunt (2009) from their measurements in saline plumes over 105 ≤ Γ0 ≤ 107. Further comparisons are shown in figures 4.13d-e, of predictions of volume flux at the level of the neck ζ = ζn and at the level comparable in height with the radius of the plume, ζ = 1, as a function of Γ0. Van Reeuwijk & Craske (2015) generalised the plume conservations equations by retaining the turbulent terms from the shear layer equations. Van Reeuwijk et al. (2016) showed that, for a steady plume, the entrainment function corresponds to: α = − δg 2γg + ( 1 β˜g − θm γg ) Ri, (4.54) 82 4.7. Discussion where δg = −0.184, γg = 1.391, β˜g = 1.076 and θm = 1.011 are coefficients that were resolved from their DNS data (see van Reeuwijk et al. 2016 for more details). Agree- ment with the entrainment function of van Reeuwijk et al. (2016) is shown in figure 4.13 over the range of 103 ≤ Γ0 ≤ 105 shown. The entrainment function of van Reeuwijk et al. (2016) also suggests that there is an increased entrainment in the near-source region of a very lazy plume. The added benefit from the approach presented herein is that the underlying physical mechanisms responsible for increasing entrainment has been identi- fied. It should be noted nevertheless that, as shown in figure 4.8, the entrainment function (4.54) does not give appropriate predictions of the neck radius and level. As a result, there remain questions on the suitability of an α-approach for the prediction of necking behaviour. Kaye & Hunt (2009) developed an origin correction where the volume flux varied according to: q =  1 + 31.5ζΓ1/30 , for 0 ≤ ζ ≤ 0.51,Γ1/30 (9.26ζ − 0.31)5/3 for ζ ≥ 0.51. (4.55) Their expressions are modified in (4.55) from those presented in Kaye & Hunt (2009) for consistency with the notation presented in the chapter, taking αp = 0.09. The Kaye & Hunt (2009) entrainment model agrees well with their data and our data over the range of scaled source Richardson numbers investigated 105 . Γ0 . 107 by Kaye & Hunt (2009), but does not extend to lower values of Γ0 (understandably as the model relied on the invariant necking dynamics of plumes of Γ0  1). Carlotti & Hunt (2017) proposed an entrainment function of the form: α := αpΓω, (4.56) for constant ω. The exponent that best fit the data presented in Kaye & Hunt (2009) was a constant of value ω = 0.64. We observe that predictions for volume fluxes are in good agreement with this formulation for the range observed by Kaye & Hunt (2009) 105 < Γ0 < 106, however for Γ0 < 105, lower values of ω would be required to obtain predictions of volume flux which would agree with our data. 4.7.2 Matchings for βˆn and ζn solutions Solutions for the dimensionless neck level ζn and radius βˆn are available for the two ranges of lazy plume behaviour: for 1 ≤ Γ0 . 103, analytical solutions have been deduced assum- ing α is constant (Hunt & Kaye 2001, 2005) and for Γ0 & 103, experimental observations suggest that ζn and βˆn approach constant values independent of Γ0. A simple matching be- tween these two behaviours is proposed in the next section to develop a single description over the whole range of Γ0. The radius at the neck can be estimated, for low values of Γ0, using (i) the analytical constant-α solution to the plume equations derived by Hunt & Kaye (2005): βˆn = bn b0 = 51/2 33/1021/5 (Γ0 − 1)3/10 Γ 1/2 0 ≈ 1.4 (Γ0 − 1) 3/10 Γ 1/2 0 , (4.57) which approximates, with 95% accuracy for Γ0 > 88, to: βˆn ≈ 1.4Γ−1/50 for 88 ≤ Γ0 < 103; (4.58) 83 100 101 10−3 10−2 10−1 100 101 Γ0 = 1000 (a) ζ 100 101 10−5 10−3 10−1 101 Γ0 = 10000 (b) q 100 101 102 10−7 10−5 10−3 10−1 101 KH09 MTT vR16 (3.14) Γ0 = 100000 (c) (4.26) 100 101 102 103 104 105 106 10−1 100 101 102 Γ0 q n (d) 100 101 102 103 104 105 106 10−1 100 101 102 Γ0 q (ζ = 1 ) (e) Figure 4.13: Different entrainment models are compared to the RT scaling for near-source vol- ume flux (4.26) (–) for varying Γ0. (a)-(c) Predictions of volume flux plotted against streamwise distance for Γ0 = 103, 104 and 105. (d) Predictions for the volume flux at the neck plotted against Γ0. (e) Predictions of volume flux at a vertical level equivalent to a single source radius, q(ζ = 1), plotted against Γ0. In (a)-(e), KH09 (–) denotes the model of Kaye & Hunt (2009) specified by (4.55); MTT (–) denotes solutions to the plume equations (4.3) with a constant-α model, taking α = 0.09; and vR16 (- -) denotes solutions to the plume equations (4.3) with the entrainment function of van Reeuwijk et al. (2016) given by (4.54). Note that in (d), the levels of the neck are lower for the predictions of vR16, cf. figure 4.8 compared to the scaling for increasing Γ0, and as a result, the volume fluxes at the neck are also lower. 4.7. Discussion or (ii), for higher values of Γ0, from the results presented in §4.5, where it has been established that the contraction is independent of Γ0: βˆn ≈ 0.62 for Γ0 > 103. (4.59) A simple matching of (4.58) and (4.59) is proposed to transition between these two be- haviours in the form: βˆn = 0.62 1 + ( 1.40.62 )5 1 Γ0 1/5 . (4.60) Similarly, the level of the neck from analytical solutions (Hunt & Kaye, 2005) is given by: ζn = 5 6αp 5 4CΓ (32 )−4/5 − (Γ0 − 1)−4/5  for Γ0 < 103, (4.61) where for a given plume, the constant CΓ = 10 3 Γ 1/2 0 (Γ0 − 1)3/10 . (4.62) Equation (4.61) can be approximated as: ζn ≈ 2.53Γ−1/50 for 88 < Γ0 < 103. (4.63) The neck level for RT lazy behaviour is taken to occur approximately at a vertical distance equivalent to a single source radius, i.e. ζn ≈ 1 for Γ0 > 103. (4.64) The approximation from the constant-α model (4.63) is matched to (4.64) by: ζn = ( 1 + 1.365 Γ0 )1/5 . (4.65) These matched expressions, namely (4.60) and (4.65), are shown in figure 4.14 and are used in §4.7.3 to deduce origin corrections. 4.7.3 A revised virtual origin correction In this section, an approach to derive an origin correction for a lazy plume that includes the effects of RT mixing in the near-source region is presented. Consider a contracting plume, which originates from a source located at z = 0 with fluxes Q0,M0, and B0, that necks at some level z = zn above the source. The local scaled Richardson number at the neck of the plume is expected to be sufficiently close to unity, i.e. Γn ∼ O(1), such that a classic origin correction is applicable at this level (see the review of Hunt & Kaye 2001). The proposed origin correction idealises the neck of the plume as a new ‘source’ from which to apply a classic Hunt & Kaye (2001) origin correction: zv,rev = zn − zavs. (4.66) 85 4. Rayleigh-Taylor lazy plumes 101 103 105 0 0.5 1 Γ0 β c (a) (4.57) (4.58) (4.59) (4.60) βˆ n 101 103 105 0 1 2 Γ0 ζ c (b) (4.61) (4.63) (4.64) (4.65) ζ n Figure 4.14: Matched solutions for (a) the neck radius and (b) the neck level. The dotted (· · · ) line plots the analytical solutions of Hunt & Kaye (2005). The dash-dotted (- · -) line plots the approximate solutions of Hunt & Kaye (2005). The dashed (- - -) line plots the constant values of neck radius and level measured for highly lazy plumes. The solid line (—) plots the matched solutions. See legend in figure. ζavs ζv ζ (a) physical source (q0 = 1,m0 = 1) βˆ0 idealised sourceβˆn ζn (qn,mn) βˆ = 6αp (ζ − ζv) /5 neck R T pl um e fa r- fie ld db/dz < 0 db/dz > 0 101 103 105 −8 −6 −4 −2 0 Γ0 ζ v (b) (4.70) (4.73) Figure 4.15: (a) Schematic for the revised origin correction where a classic virtual origin cor- rection is applied to an idealised ‘source’ at the neck of the plume. The volume and momentum fluxes at the neck level are estimated using (4.71)-(4.72). (b) A comparison between the classic virtual origin correction of Hunt & Kaye (2001) shown as a solid black (–) line compared to the revised virtual origin correction estimated from (4.73) shown as a dashed (- -) line. A schematic of the origin correction configuration is shown in figure 4.15a. Recall that the vertical coordinate is scaled on the radius of the source, ζ = z/b0. The neck is chosen as the source at which a classic Hunt & Kaye (2001) origin correction is applied to, as above the neck, the additional contribution to mixing provided by RT instabilities ceases to occur as the vertical density gradients that develop along the edge of the plume are stable. The analytical solution for the Hunt & Kaye (2001) origin correction is based on the 86 4.8. Summary assumption of α = αp = cst and can be expressed in the form: ζavs = 5 6αp 1 −Cδ Γ 1/5 n for Γn > 1 2 , (4.67) where Cδ = 3 5 ∞∑ i=1  ϕi5i−1i!(10i − 3) i∏ j=1 1 + 5( j − 1)  and ϕ = Γn − 1Γn . (4.68) If the scaled Richardson number at the neck is pure, i.e. Γn = 1, the origin correction is: ζavs = 5βˆn 6αp for Γn = 1. (4.69) If the scaled Richardson number at the neck is 5/2, as deduced theoretically by Hunt & Kaye (2001), the origin correction is: ζavs = 5 6αp · 1 − 0.0624 (5/2)1/5 · 1.4 (Γ0 − 1) 3/10 Γ 1/2 0 for Γn = 5/2. (4.70) Alternatively, an estimate for the local value of Γn can be computed as: Γn Γ0 = q2n m5/2n = βˆ5n q3n . (4.71) Following from the simplified model presented in §4.3.1 that takes into account the RT phenomenology of entrainment in the near-source region, an estimate of the volume, mo- mentum and buoyancy fluxes at any elevation above the neck (ζ > ζn) of the plume is given by: qn = Qn Q0 ≈ 1 + RT Γ1/20 + pΓ1/30 ζ5/3n , mn = Mn M0 ≈ ( qn βˆn )2 , Bn B0 = 1. (4.72) The revised origin correction (4.66) then can be computed as: ζv,rev = ζn − 56αp · 1 −Cδ Γ 1/5 0 q3/5n βˆn . (4.73) The origin correction (4.73) is shown in figure 4.15 as function of Γ0. The solutions show how the length of the adjustment region to pure-like behaviour is increased relative to constant-α predictions consistent with the appearance of the RT mechanism. 4.8 Summary The dynamics of turbulent lazy plumes required investigation in order to answer funda- mental questions concerning the mismatch of theoretical descriptions and experimental 87 4. Rayleigh-Taylor lazy plumes evidence. A few flags were raised throughout the years by researchers suggesting that the behaviour of this subset of plume dynamics was unconventional (Colomer et al., 1999; Kaye & Hunt, 2009; Marjanovic et al., 2017). The first hints came from observations regarding the radial spread of the plume. For lazy plumes of scaled source Richardson number Γ0 < 103, the neck of the contraction becomes thinner and moves closer to the source for increasing Γ0. This is due to the increasingly pronounced accelerations expe- rienced by more buoyant releases over a wider area. For higher source Richardson num- bers, we observe in good agreement with historical data that the location and extent of the contraction exhibited by a lazy plume is relatively independent of source conditions. This has far-reaching implications on the dynamics of the plume itself. The increased time-averaged spread indicates that the plume is diluting more rapidly than anticipated in the proximity of the source, that is, engulfing additional ambient fluid from the surrounding environment. The increased spread also suggests that the mixing dynamics that characterise near-source mixing must be quite distinct from the conven- tional roller mechanism that characterises mixing in small-source axisymmetric jets and plumes and whose dynamics are suitably predicted by constant-α models. This realisa- tion was further reinforced by observations of necking behaviour where for suitably high scaled source Richardson numbers (Γ0 & 103), a different mixing mechanism appeared. It was observed that finger-like intrusions develop along the contraction (|db/dz| < 0) of the plume that resemble the growth and breakdown into turbulence of Rayleigh-Taylor insta- bilities. These fingers arise due to the inherently unstable vertical density gradients that develop in very lazy plumes as these contract inwards. The appearance of this mechanism results in an invariance of the neck level and radius with Γ0, which, within this regime, both scale on the radius of the source. As a result of this change in behaviour, the onset of the dynamics of the plume towards pure-like behaviour, that is, the classic 5/3 power law of Morton et al. (1956), is offset further away from the source compared to the levels suggested by constant-α theory, (i.e. above a distance of a source acceleration length from the source). This suggests that the dynamical variability of RT lazy plumes is changed as a result of this increased mixing. Measurements of volume flux data reinforce these observations. As a result, we can revise our understanding of the ‘laziness’ of a buoyant plume based on the increased entrainment generated by the RT mechanism relative to plumes travelling at a constant Richardson number. This enables the identification of a range of scaled source Richardson numbers (Γ0 > 103) in which the source radius of a lazy plume becomes important to take into account. This has far-reaching implications on many modelling formulations which rely on the use of point-source plume solutions (e.g. the filling-box model of Baines & Turner 1969) which indeed cease to work within this regime. An entrainment model is proposed here that based on Rayleigh-Taylor layer growth rates quantifies this additional dilution and successfully predicts volume fluxes in both the near and far field of the plume. 88 chapter 5 A classification for filling-box flows The filling-box problem concerns the stratification produced by a localised source of buoyancy in a container. In Chapter 2, previous work on the filling box problem was discussed. We re-visit this work in this chapter by studying the early transient associated with the original filling-box configuration of Baines & Turner (1969), whereby a turbulent plume develops from a source which is positioned centrally at the bottom of a cylindrical container and is small in width relative to height H and radius R of the container. Several authors highlighted different behaviours for these flows depending on the radius-to-height aspect ratio (φ := R/H) of the container (Barnett 1991; Hunt et al. 2001; Kaye & Hunt 2007; Ezzamel 2011, see figure 2.10 on p. 36). A classification for filling-box regimes is presented by examining this dependency. Simultaneous light-induced fluorescence vi- sualisations and particle image velocimetry measurements of dense brine plumes shed new light on the structure of filling-box flows: the ‘plume breakdown’ regime (φ . 0.25) originally observed by Barnett (1991); the ‘rolling’ (0.25 . φ . 0.66) and ‘slumping’ (0.66 . φ . 1.25) regimes described by Kaye & Hunt (2007); and the ‘blocking’ regime (1.25 . φ . 2.0) described by Ezzamel (2011). The bulk flow patterns of the four regimes are described, including a discussion on the new insights that velocimetry measurements provide compared to the traditional descriptions based on the transport of a scalar tracer added to the plume source fluid. We find that the dynamics of the different regimes are closely linked to the plume outflow current vortices, and that the different flow regimes can be distinguished by inspecting how these current vortices interact with the plume and the corners of the container. Vortex detection is applied to this class of flows and shows that the classification of filling-box flow regimes is intrinsically linked to how vorticity is generated in these flow structures. 5.1 Introduction The turbulent plume that ensues from a localised release of buoyant fluid into a con- fined space tends to stratify the environment with vertical density gradients. Amongst those studying turbulent convection, this class of flows is commonly referred to as the ‘filling-box’ (Turner, 1973). In its original configuration (Baines & Turner, 1969), the filling-box problem examines theoretically the evolution of the stable stratification pro- duced by a point-source turbulent plume with constant buoyancy flux, whose source is located centrally at the base of a sealed cylindrical container. Baines & Turner (1969) applied the solutions developed by Morton et al. (1956) for the variation of volume and momentum fluxes in a turbulent plume to describe the evolution of the density profile in the container. The basis of the theoretical model was formulated upon the assumption of infinitesimally thin currents ensuing from the impingement of the plume with the base that rapidly became part of the quiescent environment into which the plume rises. The 89 5. A classification for filling-box flows classic description of the density profiles relied on assuming that these thin layers were of uniform density corresponding to that of the plume at impingement. Solutions, both numerical (Baines & Turner, 1969; Germeles, 1975) and successively analytical approx- imations (Worster & Huppert, 1983), were developed based on this formulation resulting in the extensive use of the filling-box model in the study of turbulent convection. The stratification pattern formed in containers that are not short and wide, i.e. of radius-to-height aspect ratio φ := R/H ≤ 2.0, differs from the classic description of Baines & Turner (1969) (as noted by Barnett 1991; Kaye & Hunt 2007 and Ezzamel 2011, see a review in §2.4). These authors identified distinct filling-box regimes for cylindrical containers of radius-to-height aspect ratio ranging between 0.25 ≤ φ ≤ 2.0 (figure 5.1). The different evolutions of the filling mechanisms are primarily due to the increasing importance, as the container becomes more tall and narrow, of (i) the outflows ensuing from the impingement of the plume with the base and sidewalls (overturning processes) and (ii) the return flow surrounding the plume (confinement). The relevance of these effects, prevailing within the investigated range of geometries, are contained within the early transient of filling and profoundly influence the early and late development of the stratification. A study on the tall cylindrical filling box (φ ≤ 0.25), was conducted by Barnett (1991). Barnett (1991) investigated the increasing significance of the return flow that developed around the plume as the container was selected to be more tall and narrow. He showed by modifying the plume conservation equations of Morton et al. (1956) that the plume could induce locally a return flow of comparable velocity magnitude to that of the plume itself. He demonstrated that a critical return velocity exists whereby the shear between the plume and return flow causes the flow to breakdown into a turbulent mixing region. Observations made in cylindrical filling-box experiments on containers ranging be- tween 0.25 ≤ φ ≤ 1.0 revealed the importance of the plume outflow currents in the evolution of the stratification (Baines & Turner, 1969; Kaye & Hunt, 2007). If the later- ally spreading plume outflow contains sufficient inertia when it collides with the sidewall to overcome the buoyant and viscous dissipative forces acting upon it, fluid is forced to propagate vertically against its buoyancy. This process is known as ‘overturning’ (Baines & Turner 1969 or ‘upwelling’ in oceanography, Condie & Kawase 1992). Baines & Turner (1969) alluded to overturning insofar as it limits the validity of their stably strati- fied filling-box model. To this end, they balanced the outflow inertia and buoyancy forces, I andB, to obtain a characteristic Richardson number that is a function of container as- pect ratio: B I = 10φ2 9αp , (5.1) where αp is the plume entrainment coefficient introduced by Morton et al. (1956), here as- sumed to be constant. Experimental observations led Baines & Turner (1969) to conclude that overturning was not significant in enclosures of aspect ratio φ & 1. Kaye & Hunt (2007) investigated overturning in a more focussed treatment on its dynamics. They identified two regimes that were dictated by the development of the outflow current, from a forced buoyant wall jet towards that of a purely buoyancy-driven gravity current. Forced impingement of the current against the sidewall of the cylinder, occurring within the range 0.25 ≤ φ ≤ 0.66, causes the ‘rolling’ of the current back up the wall and its re-entrainment into the plume flow. For shorter and wider containers (φ > 90 5.2. Experiments 0.66), they observed that the current reaches an approximately constant depth and Froude number as it impinges against the sidewall. This caused the rise of the outflow current to a level of approximately twice its depth at impingement and the subsequent ‘slumping’ of the overturned fluid as it collapses under its own weight. Kaye & Hunt (2007) modelled the early transient of these two regimes to estimate the maximum penetration depth of the intrusion of the overturning motion into the originally quiescent environment in the container. Their data indicates that the functional dependence of the non-dimensional penetration depth, zL/H, on the aspect ratio φ, evolves as zL/H ≈ 0.33φ−1/3 for rolling and is independent of aspect ratio for slumping zL/H ≈ 0.38. Thus, in each case, the depth of the corner intrusion was observed to be of significant magnitude relative to the height of the container which calls into question the assumption of thin layers at the early stages of filling adopted in the Baines & Turner (1969) model. Ezzamel (2011) conducted a series of (non-simultaneous) fluorescence visualisations and velocimetry filling-box experiments on boxes ranging between 0.54 ≤ φ ≤ 2.05. He identified a new regime referred to as ‘blocking’ whereby the slumped overturning intru- sion was re-entrained into the current, forming an accumulation of buoyant fluid along the in the proximity of the corners of the container. Admittedly, according to the author, there was not enough information about the nature of the new regime to fully ascertain its existence. The new experiments presented herein seek to address the ‘blocking’ regime and describe its evolution. We report on a series of simultaneous flow visualisation and velocity measurement experiments on the filling-box flow that cover a range of box aspect ratios between 0.23 ≤ φ ≤ 1.6. Experiments, which are described in §5.2, are conducted with dense saline plumes released in freshwater environments. In §5.3, we briefly touch upon the fundamentals of the Baines & Turner (1969) formulation to elucidate on the differences between filling-box descriptions. The four filling-box regimes are then described with a particular focus on the role of large-scale vortices on the nature of the overturning regimes. The early-stage filling is compared to the description of Baines & Turner (1969) revealing that over the investigated range of geometries, containers fill much more rapidly. The motions of a passive tracer added to the plume source fluid are tracked to describe the transient dynamics of the dispersion of a scalar in a filling-box flow. In § 5.4, diagnostics are developed to highlight the key differences between each regime. Non-local vortex de- tection methods are implemented to outline distinct paths taken by the plume and current vortices in each of the filling regimes. Conclusions drawn from the experimental work are discussed in §5.5. 5.2 Experiments Experiments were conducted using saline plumes released in a freshwater environment. The setup for the experiments is depicted in figure 5.2a. Simultaneous light-induced flu- orescence (LIF) and particle image velocimetry (PIV) were conducted to visualise the dispersion of a scalar within the plume and to measure the velocity field along the cen- tral cross section of the plume axis. The release of buoyant fluid into the cylinder was activated at the start of each experiment and recorded by video until the first ascending front of the developing stratification reached the level of the source (in some cases re- stricted by requirements on maximum field-of-view size). Table 5.1 summarises the key 91 5. A classification for filling-box flows rolling slumping blocking 0.25 0.66 1.25 φ H R Figure 5.1: Visualisations of the three regimes of overturning for a small-source plume filling a cylindrical container. It is observed that, for a ‘pure’ Γ0 ≈ 1 plume of source radius much smaller than the container (b0  R), the regimes are solely dependent on the radius-to-height aspect ratio, φ := R/H, of the container. (a) 70 0 m m H Xenon arc lamp& light sheet optics nozzle filter portion R 1050 mm overflow bypass circuit saline reservoir gear pumpflowmeter air vent angle slit (b) PE disc (outlet) PE disc gravel layer O-ring 21.5 mm 40 m m Figure 5.2: (a) Representative schematic for a filling-box experiment in which a saline solution is injected into a freshwater cylindrical tank. A white light sheet is shone through the central vertical plane of the cylindrical tank to illuminate a fluorescein sodium tracer added to the source solution and to illuminate seeding particles seeding added to both the source solution and the environment. (b) A schematic of the diffuser of the plume nozzle. The nozzle was designed to create a turbulent and uniform outflow velocity profile. A layer of gravel is sandwiched by two coarse polyethlyene discs and glued into PVC socket casings. experimental parameters investigated. A circular source of diameter 21.5 mm was rigidly fixed at the centre of the cylinder at a variable height of 190 mm ≤ H ≤ 472 mm from the tank base. Two cylinders, of radii R = 110 and 300 mm respectively, were used to span the desired range of aspect ratios. A solution of saline (ρ0 ≈ 1.008 g/cm3) stained with 0.5-1.0 mg/l of fluorescein sodium and seeded with polyamide seeding particles of 50 µm nominal size was injected through the nozzle at known volume flow rate and salinity. These release conditions were 92 5.3. Filling mechanisms set so that the plume was approximately pure at source level (i.e. the source Richardson number Γ0 = 5g′0b0/8αpw 2 0 ≈ 1, Morton 1959). The flow rate was supplied with a positive displacement gear pump manufactured by ISMATEC (model ISM 198A) and measured with an Apollo LowFlo flowmeter. The nozzle was manufactured with a filter (diffuser) section at the outlet. A schematic of the nozzle is shown in figure 5.2b. The outlet of the nozzle was constructed with a pair of coarse polyethylene (PE) discs (that were 6 mm ±0.15 mm thick, and of mean and maximum pore size 154 µm and 196 µm, respectively, SPC Technology PE16060), sandwiching a 40 mm deep layer of aggregate gravel of median size 5 mm. The central cross section of the tank was illuminated by a 300 W Xenon arc lamp and light sheet forming optics. A high aperture (f/1.4) was used to reduce the depth of field. The camera settings were adjusted for each experiment such that the seeding was of sufficient size to be seen with a resolution of at least 4-6 pixels (larger scatter sizes were used than what typically recommended owing to the presence of the dye) and the fluorescent dye was visible, but not saturated in intensity, so that it could be removed in image post-processing with the seeding retained. Intensity saturation in the fluorescent dye portions within the plume in some instances could not be avoided to obtain full details of the more dilute overturning motions. A high-pass filter was used to remove the dye portion of the flow and to the isolate the seeding during PIV image pre-processing. The PIV window was of a maximum of 600 by 300 mm2 (width by height). The near-source of the plume was not recorded in the experiments for which H > 300 due to limitations in the vertical spread of the light sheet. A two-pass interrogation scheme was adopted with windows of area 128 × 128 and 64 × 64 pixels2 were used with a 50% overlap for each window. Optical distortion effects owing to the refraction of light through the saline solutions of varying density were min- imised by using a relatively low density difference between the plume and environment. Changes in refractive index, n, were estimated to be of less than ∆n/n . 0.002 (Daviero et al., 2001). 5.3 Filling mechanisms The Baines & Turner (1969) filling-box model is formulated for a point-source plume of constant buoyancy flux (B0). The dynamics of the plume are described by the plume equations developed by Morton et al. (1956). The solutions of Morton et al. (1956) for the volume, Q = wb2, and specific momentum flux, M = w2b2, for a plume of source buoyancy flux, B0 = g′wb2, whose source is at z = 0, are given by Q = 6αp 5 ( 9αp 10 )1/3 B1/30 z 5/3, M = ( 9αp 10 )2/3 B2/30 z 4/3, (5.2) where ‘top-hat’ (uniform) profiles of velocity w and reduced gravity g′ := g(ρ−ρa)/ρa are assumed with b defined as the plume radius. Baines & Turner (1969) use these solutions to describe (i) the ascent of the first front between the dense deepening layer and the original ambient environment, and (ii) the horizontally-averaged density profile in the container at late time (the so-called ‘asymptotic state’). The filling time of Baines & Turner (1969) 93 5. A classification for filling-box flows Expt R (mm) H (mm) w0 (mm/s) g′0 (mm/s 2) Γ0 φ b0/H zv/H regime 1 110 462 74.2 102.0 1.23 0.238 0.02 0.06 breakdown 2 110 370 68.4 92.2 1.31 0.297 0.03 0.10 rolling 3 110 309 68.4 92.4 1.31 0.356 0.04 0.15 rolling 4 110 283 67.2 73.1 1.08 0.389 0.04 0.16 rolling 5 110 224 66.5 73.1 1.10 0.491 0.05 0.21 rolling 6 300 472 76.4 102.0 1.16 0.636 0.02 0.10 slumping 7 300 372 46.5 103.9 3.20 0.806 0.03 0.09 slumping 8 300 366 78.0 102.1 1.12 0.820 0.03 0.13 slumping 9 300 310 30.5 95.5 1.85 0.968 0.04 0.09 slumping 10 300 275 58.4 103.4 2.02 1.091 0.04 0.15 slumping 11 300 264 72.2 97.2 1.24 1.136 0.04 0.17 slumping 12 300 234 61.6 95.1 1.67 1.282 0.05 0.18 blocking 13 300 227 43.6 90.6 3.17 1.322 0.05 0.15 blocking 14 300 210 29.2 96.5 2.54 1.429 0.05 0.14 blocking 15 300 190 40.0 92.8 3.86 1.579 0.06 0.18 blocking Table 5.1: Summary of key experimental parameters for the filling-box experiments conducted with a source of radius b0 = 10.75 mm. The source Richardson number Γ0 = 5g′0b0/8αpw 2 0 was chosen to be as close as possible to that of a pure plume Γ0 ≈ 1. The origin correction zv was evaluated using the solutions presented by Hunt & Kaye (2001), for these refer to equation (4.67) in Chapter 4. is defined by: tF = 5 6αp ( 10 9αp )1/3 R2 B1/30 H 2/3 . (5.3) It is characteristic of the time over which the first front asymptotes to the level of the source. It is also indicative of the time over which the whole volume of the environment is recirculated through the plume. An alternative definition is, indeed, that of the time scale formed by the ratio of the volume of the container (∼ HR2) over the volume flux of the plume at impingement (Qi ∼ B1/30 H5/3 with subscript (.)i denoting the predicted value at impingement). Dimensionless time is henceforth expressed as: τφ−2 = t H2/3B1/30 R −2, (5.4) such that it is consistent with the scaling of Baines & Turner (1969), where τ := t B1/30 H −4/3, (5.5) for consistency with later chapters. Baines & Turner (1969) showed that first front evolves as a ‘2/3’ power law decay on the filling time scale: t tF = 3 2 5 6αp ( 10 9αp )1/3 ( ζ−2/3f − 1 ) , (5.6) 94 5.3. Filling mechanisms 0:2 0 0:2 0:6 0:4 0:2 (a) τ/φ2 = 2.97 1 −z /H (b) 0:2 0 0:2 0:6 0:4 0:2 (c) τ/φ2 = 4.18 (d) 0:2 0 0:2 0:6 0:4 0:2 0 (e) τ/φ2 = 6.74 1 −z /H (f) 0:2 0 0:2 0:6 0:4 0:2 0 (g) τ/φ2 = 10.5 (h) Figure 5.3: The ‘plume breakdown’ regime (φ = 0.24) visualised as the instantaneous velocity field (vector resolution reduced by a factor of 2 from measured) and the corresponding image of the dye (shown in false colour) with which the plume is stained, visualised along the central plane of the cylinder for experiment 1 (see Table 5.1). The estimated filling time is tF ≈ 67 s. Note that images are flipped vertically from what was observed experimentally. where ζ f := z f H , (5.7) is the distance between the front and the source divided by the height of the container. We will see in §5.4.1 how filling rates compare to predictions based on these solutions during the early transient of each regime. Solutions for the evolving density profile in the layer are described in Worster & Huppert (1983). We find that horizontal inhomogeneities in density, which are persistent throughout the filling process over the range of experimen- tal conditions investigated, make comparisons with classic density profile solutions of limited relevance. 5.3.1 Breakdown The onset of the breakdown of the plume owing to the self-induced return flow was pre- dicted by Barnett (1991). He modified the conservation equations of Morton et al. (1956) 95 5. A classification for filling-box flows to include the return velocity (U) of the environment surrounding the plume: d dz ( wb2 ) = 2αpwb, d dz ( w2b2 + U2(R2 − b2) ) = g′b2. (5.8) The return flow velocity was estimated by applying volume conservation for any given cross section of the enclosed container, U(R2 − b2) = wb2, and assuming U was uniform along the cross section. Barnett (1991) showed that there is a singularity in the equation for the gradient of the vertical momentum flux at b/R = 1/ √ 2, which corresponds to the level at which the mean plume velocity is equal and opposite to the mean self-induced velocity in the environment, w = −U. He suggested that the breakdown of the governing equations would concur with that of the plume flow, and via experimental observations, observed that it would form a turbulent mixing region above the plume. Barnett (1991) performed depth-averaged shadowgraph visualisations in a relatively small container (R = 20 mm) compared to the one used herein. In figure 5.3, we show what we observed in our own experiments when the plume broke down in a cylinder of aspect ratio of φ = 0.23. In the initial transient, it is evident upon inspection of figures 5.3a-d that as the starting plume approaches the ceiling, the width of the plume becomes comparable to that of the container. In agreement with Barnett’s (1991) description, a return flow of similar order of magnitude to the velocities in the plume is quickly established in the direction opposing buoyancy. The plume increases in width as it feels the effect of confinement. A vortex ring formed by the re-entrainment of the outflow of the plume into the plume itself establishes in the upper portion of the container, a feature that is similar to what is observed in the rolling regime. Axial symmetry is however not retained within the flow and the layer that develops is characterised by vigorous turbulent mixing with the plume meandering in and out of the field of view as it enters the turbulent region. 5.3.2 Rolling In figures 5.4a-h, the initial transient of the rolling regime is visualised as a series of instantaneous images of the velocity and scalar fields. The observed flow is characterised by a vortex ring that is far better established that what is observed in the breakdown regime in that it persists as a structure even for large time. This primary vortex, formed after the plume impinges against the ceiling, grows as it propagates outwards towards the walls and gains vorticity from eddies that are subsequently formed and that merge into it. The rotation of the primary vortex is maintained by the shear from the plume and the overlying radial outflow. There is only a limited collapse of the overturned buoyant fluid in the rolling regime as opposed to previous descriptions of the process (Kaye & Hunt, 2007). The mecha- nism is dominated by the primary vortex ring that establishes in the top of the container. Buoyancy would be expected to generate vorticity in the sense opposite to the rolling cir- culation, inhibiting the growth of the vortex. In the rolling regime, the baroclinic vorticity seems to be negligible compared to the entraining effect of the plume, in contrast with what observed in the slumping regime described in the upcoming section. 5.3.3 Slumping The slumping regime is characterised by a bulk collapse of the overturned fluid after it has reached an initial peak in penetration depth. The flow is visualised in the series of images 96 5.3. Filling mechanisms −0.2 0 0.2−1 −0.8 −0.6 −0.4 −0.2 0 (a) τ/φ2 = 4.34 1 −z /H (b) −0.2 0 0.2−1 −0.8 −0.6 −0.4 −0.2 0 (c) τ/φ2 = 6.76 (d) −0.2 0 0.2−1 −0.8 −0.6 −0.4 −0.2 0 (e) τ/φ2 = 11.9 1 −z /H (f) −0.2 0 0.2−1 −0.8 −0.6 −0.4 −0.2 0 (g) τ/φ2 = 21.8 (h) Figure 5.4: The ‘rolling’ regime (φ = 0.38) visualised as the instantaneous velocity field and the corresponding image of the dye (shown in false colour) with which the plume is stained, visualised along the central plane of the cylinder for experiment 4 (see Table 5.1). The estimated filling time is tF = 568 s. Note that images are flipped vertically from what was observed experimentally. shown in figure 5.5a-j. As the plume impinges against the ceiling, a current is formed that radially spreads towards the walls. A vortex ring is formed at the nose of the current which is displaced down the sidewalls as the current impinges against these boundaries. After the vortex is displaced vertically, it unrolls and forms a current that overlays the original plume outflow and converges towards the centre of the container, figure 5.5e. The newly-formed current impinges against the plume, causing it to overturn against the direction of buoyancy as if the current was impinging against a rigid boundary, fig- ure 5.5g. A third current, formed by the ‘splashing’ of fluid down the plume, intrudes horizontally into the environment to then establish itself as the first ascending front, fig- ure 5.5i. A stable interface is thus formed between the developing layer and the original ambient environment which gradually rises as the layer is continuously fed by the plume. Contrary to the description of Baines & Turner (1969) however, motions within the layer are not dissipated by viscosity, but distinguished by a series of shearing currents, fig- ure 5.5i. After the impingement of the plume outflow current with the sidewall, the persistence of the leading vortex at the nose of the current is expected to depend on two mechanisms. On the one hand, the inertia of the current drives circulation in the vortex. On the other hand, after the current has impinged against the sidewall, the gradients in density are expected to drive the circulation in the vortex in the opposite direction. The production 97 −0.5 0 0.5−1 −0.8 −0.6 −0.4 −0.2 0 (a) τ/φ2 = 2.07 1 −z /H (b) −0.5 0 0.5−1 −0.8 −0.6 −0.4 −0.2 0 (c) τ/φ2 = 2.68 1 −z /H (d) −0.5 0 0.5−1 −0.8 −0.6 −0.4 −0.2 0 (e) τ/φ2 = 3.66 1 −z /H (f) −0.5 0 0.5−1 −0.8 −0.6 −0.4 −0.2 0 (g) τ/φ2 = 5.12 1 −z /H (h) (i) τ/φ2 = 9.75 1 −z /H (j) Figure 5.5: The ‘slumping’ regime (φ = 1.1) visualised as the instantaneous velocity field (vector resolution reduced by a factor of 2) and the corresponding image of the dye (shown in false colour) with which the plume is stained, visualised along the central plane of the cylinder for experiment 8 (see Table 5.1). The estimated filling time is tF = 986 s. Note that images are flipped vertically from what was observed experimentally. White portions (1−z/H > 0.6) are out of the field of view. Note that in the LIF images, the unevenness in the intensity of the dyed layer is attributed to the attenuation of the light sheet due the fluorescence of the dye and not to horizontal inhomogeneities in dye concentration. 5.3. Filling mechanisms (a) plume current primary vortex current primary vortex zL ,R ≈ 0 .33H φ − 1 /3 wi wi zL 2 zL 4 g′i rolling (b) wi zL 4 g′i slumping zL ,S ≈ 0 .38H Figure 5.6: Schematic showing a model for the circulation in the corner vortex in (a) the rolling and (b) the slumping regime. For the rolling regime, both the plume and current feed the circula- tion in the vortex. For the slumping regime, only the current feeds the circulation. of the circulation by a density gradient is referred to as baroclinic torque (Turner, 1973). A dimensional argument is now proposed to relate the two mechanisms, viz. the inertially-driven circulation and the counteracting baroclinic torque, to their dependency on container aspect-ratio. An idealised schematic is shown in figure 5.6 to assist this discussion. The vortex is fed by the continuous influx of inertia supplied by the plume. We estimate the fuelling rate for this inertially-driven circulation (KI , with subscript (.)I denoting the inertially-driven component) as the product of the velocities feeding the circulation and their distance from the centre of the vortex. For rolling and slumping, denoted by subscripts (.)R and (.)S , this corresponds to: KI,R = 3wi zL,R 4 , KI,S = wi zL,S 4 , (5.9) respectively. The influx of circulation into the primary vortex is fed by the plume and is thus expected to scale on the mean velocity of the plume upon impingement with the base: wi = Mi Qi ∝ (B0 H )1/3 . (5.10) Note that the velocity in the current is expected to be of similar magnitude. The vertical depth of the vortex (∼ zL) is restricted by the maximum vertical penetration depth of the overturning intrusion. The measurements of Kaye & Hunt (2007) are used to estimate penetration depth for the two regimes and thus, the maximum depth of the vortex, zL. We assume that the feeding rate of circulation into the primary vortex corresponds to: ∂KI ∂t ∼ KI tV , where tV := VV Qi , (5.11) is a formation time scale analogous to the time scale used to describe vortex rings devel- oped by Gharib et al. (1998). The time scale is estimated by the ratio of the volume of 99 5. A classification for filling-box flows 0.4 0.6 0.8 1 1.2 0 1 2 3 4 R/H ∂ Kˆ ∂ t (a) ∂KˆI,R ∂t ∂KˆI,S ∂t ∂KˆB,R ∂t ∂KˆB,S ∂t (b) rolling slumping Figure 5.7: (a) The dimensionless rate of growth of circulation in the vortex, denoted by a hatted indicator and scaled as ∂Kˆ/∂t = ∂K/∂t (H/B0)2/3, plotted against container aspect ratio. The scalings given by (5.14), (5.15), (5.21), and (5.22) are plotted and labelled in figure: the solid (–) lines are the scalings for the inertial component and the dashed lines (- -) are for the baroclinic component. Blue is used to denote the scalings for slumping and black is used to denote those for rolling (as idealised in figure 5.6). (b) The difference between the inertial and baroclinic com- ponents of dimensionless growth rate of circulation plotted against container aspect ratio. These growth rates, n.b. acting to drive circulation in opposing directions, are compared to identify for which range of aspect ratios the inertial contribution to the rate of growth of circulation (∂KI/∂t) exceeds that of the counter-rotating baroclinic torque (∂KB/∂t) . the vortex ring, assumed to be a toroid of outer radius R and depth zL, given by: VV = 2pi2 (zL 2 )2 ( R − zL 2 ) , (5.12) to the volume flux of the plume at impingement Qi. As a result, the growth rate of the circulation produced by the inertia of the current is expected to scale as: ∂KI ∂t ∼ KI tV ≈ MizL VV ≈ 2zL pi ( 9αp 10 )2/3 B2/30 H4/3 zLR ( 1 − zL2R ) ; (5.13) which for rolling and slumping corresponds to: ∂KI,R ∂t ∼ 0.576 1 − 16 ( H R )4/3 (B0H )2/3 ( R H )−2/3 , (5.14) ∂KI,S ∂t ∼ 0.168 1 − 0.19 ( H R ) (B0 H )2/3 ( R H )−5/12 , (5.15) respectively. As the vortex is displaced down the sidewall, a horizontal density inhomogeneity is created. A hydrostatic pressure gradient develops as a result of this inhomogeneity which 100 5.3. Filling mechanisms acts towards the centre of the container. The pressure gradient is expected to unroll the vortex as it generates circulation in the direction opposite that of the pre-existing rota- tion. In other words, a counter-rotating baroclinic torque acts on the vortex due to the hydrostatic pressure that develops from these horizontal inhomogeneities of density. To estimate the growth rate of the circulation due to baroclinicity we revert to the vorticity equation (see, e.g. Turner 1973). The generation of vorticity (Ω), in the direction oppos- ing that provided by the inertia of the current, can be estimated by considering only the baroclinic component of the vorticity equation: ∂Ω ∂t = ∣∣∣∣∣∣ ∣∣∣∣∣∣∇ρ × ∇Pρ2a ∣∣∣∣∣∣ ∣∣∣∣∣∣ . (5.16) The pressure gradient in (5.16) is assumed to be hydrostatic and given by: ∇P = dP dz = ρag. (5.17) The density gradient (∇ρ) is approximated by the difference in density between the vortex, ρV = ρag′i , and ambient density (ρa) over a characteristic length LB over which the gradient is acting: ∇ρ ∼ ∣∣∣∣∣ρV − ρaLB ∣∣∣∣∣ = ρag′iLB . (5.18) In (5.18), the reduced gravity is taken to be that of the current at impingement, g′i = B0/Qi. The dissipation of vorticity can be estimated as: ∂Ω ∂t ∼ g ′ i LB . (5.19) The characteristic vertical scale is chosen as the half-width of the vortex (LB ∼ zL/2) which was estimated as half the typical penetration depth of the intrusion. The dissipation of the circulation by the baroclinic torque is thus expected to scale as: ∂KB ∂t ∼ ∂Ω ∂t z2L 4 ≈ g ′ izL 4 ≈ zL 4 5 6αp ( 10 9αp )1/3 B2/30 H5/3 , (5.20) the subscript (.)B reading as the contribution due to baroclinicity. Using the estimates of penetration depth from the two regimes of Kaye & Hunt (2007), this component is estimated, for both the rolling and slumping scenarios, to respectively be equal to: ∂KB,R ∂t ∼ 1 12 ( R H )−1/3 5 6αp ( 10 9αp )1/3 (B0 H )2/3 , (5.21) ∂KB,S ∂t ∼ 0.38 4 5 6αp ( 10 9αp )1/3 (B0 H )2/3 . (5.22) The vortex is expected to dissipate if the dissipation rate exceeds the fuelling rate, such that a transition can be identified by matching the two rates, defined by equations (5.13) and (5.20), ∂KB ∂t & ∂KI ∂t . (5.23) 101 5. A classification for filling-box flows The expressions for ∂K/∂t deduced in (5.14), (5.15), (5.21), and (5.22) are plotted in figure 5.7 as a function of container aspect ratio. The growth rate of circulation in the vortex due to the inertia of the plume increases with decreasing aspect ratio. For the rolling scalings, the portion of the inertially generated vorticity exceeds the dissipation by baroclinicity for aspect ratios smaller than φ . 0.6. This threshold agrees well with experimental observations for the value at transition between rolling and slumping. While the scaling should nevertheless be interpreted as an order of magnitude estimate, it points to some of the dependencies of the two mechanisms on the container aspect ratio. It sug- gests that the transition between the rolling and slumping regimes is expected to depend on geometric quantities alone and to be independent of the source buoyancy flux. The rel- ative production and dissipation of circulation increase with decreasing container aspect ratio in different ways. Interestingly, it can be inferred from these scalings, that it is the contribution of the plume in sustaining the circulation in the primary vortex that enables the existence of the vortex at large time. 5.3.4 Blocking Ezzamel (2011) suggested that blocking is a sub-regime of the slumping mechanism and upon inspection of figures 5.5 and 5.8, it becomes evident that the two regimes share some similarities. What differentiates the blocking regime is the behaviour of the plume outflow as it approaches the sidewall of the container. Turbulence within the plume outflow is suppressed by buoyant and viscous dissipation as the current approaches the wall. The current does not form a vertical overturning current along the sidewall (such as in the slumping regime), but instead causes the accumulation of buoyant fluid in the proximity of the corner. The subsequent incoming fluid from the plume outflow is displaced vertically by the fluid that has accumulated in the corner (figure 5.8c-f). Motions in the layer are thus said to be ‘blocked’ by the corner fluid. The plume outflow feeds the blocked portion of the fluid up to a critical point in which hydrostatic pressure gradients, which develop from the horizontal density inhomogeneity, cause the accumulated corner fluid to slump back towards the centre of the container. The converging current ‘splashes’ up the plume and slumps back to form a stable layer in similar fashion to the slumping regime. Motions at late time in the layer are considerably dissipated by viscosity and the stability of the buoyant layer. The late stages of filling within the blocking regime resemble much more closely the description (Baines & Turner, 1969) of a quasi-laminar layer that deepens as it is fed by the plume. 5.4 Diagnostics A series of diagnostics were developed to distinguish between the flow regimes. The over- turning regimes (namely rolling, slumping and blocking) each trace distinctive signatures during the early transient that can help further aid our understanding of the dynamics of these flows. Initially, focus is on diagnostics based on the detection of the edges between the scalar tracer and the original environment. Then, vortex detection-based diagnostics are implemented to determine the distinctive features internal to the buoyant layer. 102 1 0:5 0 0:5 1 1 0:8 0:6 0:4 0:2 0 (a) τ/φ2 = 2.69 1 −z /H (b) 1 0:5 0 0:5 1 1 0:8 0:6 0:4 0:2 0 (c) τ/φ2 = 3.49 1 −z /H (d) 1 0:5 0 0:5 1 1 0:8 0:6 0:4 0:2 0 (e) τ/φ2 = 4.46 1 −z /H (f) 1 0:5 0 0:5 1 1 0:8 0:6 0:4 0:2 0 (g) τ/φ2 = 7.03 1 −z /H (h) 1 0:5 0 0:5 1 1 0:8 0:6 0:4 0:2 0 (i) τ/φ2 = 11.0 1 −z /H (j) Figure 5.8: The ‘blocking’ regime (φ = 1.3) visualised as the instantaneous velocity field (vector resolution reduced by a factor of 2) and the corresponding image of the dye (shown in false colour) with which the plume is stained, visualised along the central plane of the cylinder for experiment 13 (see Table 5.1). The estimated filling time is tF = 1200 s. Note that images are flipped vertically from what was observed experimentally. 5. A classification for filling-box flows 0 0.1 0.2 0.3 0.38 0 0.2 0.4 0.6 0.8 1 x/H z /H 0 5 10 15 20 25 (a) τ/φ2 0 0.5 1.1 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 (b) τ/φ2 0 0.5 1 1.3 0 0.2 0.4 0.6 0.8 1 0 1 2 3 (c) τ/φ2 Figure 5.9: The interface between the tracer added to the plume and the original environment, detected using a Canny edge detection algorithm, plotted for half of the central cross-section of the container. The colour of the lines indicates how the interface progresses in time. The graphs show the detected edges for sample experiments in the (a) rolling (φ = 0.38), (b) slumping (φ = 1.1), and (c) blocking (φ = 1.3) regimes. 5.4.1 Sidewall penetration A commonly described feature of the early filling-box transient is the penetration of a passive tracer added to the plume as its disperses into the ambient environment (Baines & Turner, 1969; Kaye & Hunt, 2007). The tracer was detected from the LIF visualisations along the central cross section of the container using a ‘Canny’ (1986) edge detection algorithm with a relatively high Gaussian blur to prevent contamination of the signal by the PIV seeding. This allowed for the tracking of the interface between the stained fluid and the initial environment, and the definition of the penetration depth of the tracer as the plume outflow overturned against the sidewalls. Figures 5.9a-c reveal the edge between the tracer and the environment detected using the Canny method as a function of time for the rolling, slumping and blocked regime. Details of the nibbling scales at which mixing occurs in the plume and currents are lost within these images due to the introduction of the blurring effect. A picture of the bulk motions of the stained fluid is nevertheless retained. For the rolling regime, shown in figure 5.9a, one can distinctly observe the growing of the vortex as it fills the space between the sidewalls and the subsequent formation of a horizontal interface. On the other hand, the slumping and blocking regimes, shown in figures 5.9b-c are instead characterised by a sequence of horizontal currents that fill the space. One may be forgiven here for not being able to distinguish between these two regimes simply from the examination of the external outline of the flow (which indeed was the sole technique previously used to examine these flows). In the slumping regime, the plume outflow overturns against the sidewall and forms a current that moves towards the centre of the container. In the blocking regime, the plume outflow is being displaced 104 5.4. Diagnostics 0 2 4 6 0 0.2 0.4 0.6 0.8 1 φ = 0.30 φ = 0.39 φ = 0.49 Rolling B&T69 z L /H (a) 0 2 4 6 0 0.2 0.4 0.6 0.8 1 φ = 0.64 φ = 0.81 φ = 0.97 φ = 1.09 φ = 1.28 B&T69 Slumping (τ − τinitial) φ−2 (b) 0 2 4 6 0 0.2 0.4 0.6 0.8 1 φ = 1.28 φ = 1.32 φ = 1.43 φ = 1.58 B&T69 Blocking (c) Figure 5.10: The peak penetration depth (ζL) of the scalar added to the injected fluid, tracked along the central plane of the container in regions external to the plume. Data points are shown for: (a) the rolling, (b) slumping and (c) blocking regimes. The data is compared to the solution for the first ascending front of Baines & Turner (1969), shown as a solid black (–) line. The time τinitial is defined by the time required for the plume outflow current to first reach the sidewall. by the blocked fluid until the space is filled horizontally. This results in a similar external outline of the flow with different internal motions. This can be noticed in figures 5.9b-c by the difference of steepness of the edge as the stained fluid appears to move towards the centre of the container. The internal structure of these two regimes will be addressed in §5.4.2 to provide further diagnostics to differentiate between the flows. In figure 5.10, the tracking of the peak value for the penetration depth of the tracer into the environment is plotted as a function of dimensionless time. Kaye & Hunt (2007) deduced the maximum peak penetration depth for the rolling and slumping regimes, de- fined as the highest peak penetration of the tracer preceding the slumping of the fluid into a newly-formed stable layer. They established the scalings, zL/H ≈ 0.33φ−1/3 for the rolling regime and zL/H ≈ 0.33 for the slumping regime, for the peak penetration depth of the intrusion. There is good agreement for this initial peak penetration depth for the slumping regime, where a peak is clearly identified at zL/H ≈ 0.34 ± 0.03. The measure- ment of the peak penetration depth is less clearly defined for the rolling case (relative to the slumping) as fluid does not collapse under its own weight. The peak penetration depth of the scalar during blocking averages at ζL = 0.4 ± 0.04 across all experiments. The fact that the peak penetration during blocking is higher than in slumping is somewhat unexpected. As noted by Kaye & Hunt (2007), the early peak pen- etration depths of the intrusion are closely related to the horizontal distance the plume out- flow travels before impinging against the sidewall. The transients for the rolling case thus exhibit larger penetration depths compared to the slumping as less buoyancy-generated inertia is dissipated during the (shorter) horizontal travel. Counter-intuitively, blocking seems to exhibit higher penetration depths than slumping regardless of the relatively in- creased horizontal distance dictated by a higher aspect ratio. At the very early stage, however, the plume outflow is blocked by the accumulated corner fluid and causes over- turning to occur closer to the plume, in a region where the current is more inertial. This results in an increase in the penetration of these overturning motions (due to the momen- tary effective ‘reduction’ of the container aspect ratio). The predictions for interface height of Baines & Turner (1969), (5.6), grossly under- 105 5. A classification for filling-box flows 0 2 4 6 8 10 12 14 16 18 20 22 24 0 0.2 0.4 0.6 0.8 1 1 −z /H τ/φ2 (i) (ii) (iii) (iv) (v) (vi) Rolling no apparent collapse in layer (i) (ii) (iii) (iv) (v) (vi) Figure 5.11: An example time series for the rolling regime (φ = 0.38) showing key frames in the filling process. The solid line shows the horizontally-averaged penetration depth of the scalar. estimate the penetration depth of the intruding buoyant fluid. From a practical viewpoint, the implication of the penetration depth of these intrusions is that buoyant fluid, and a contaminant released at the plume source, penetrate a significant vertical distance into the enclosure comparatively early. As a final attempt to discern differences in the external outline of the flow, we con- struct time series in figure 5.11-5.13 for the three overturning regimes. Each vertical band in the time series corresponds to the horizontally-averaged intensity of an image frame over the regions external to the plume. A clear distinction that can observed is that, dur- ing rolling, the buoyant layer deepens monotonically; as opposed, to the slumping and blocking regimes, which both exhibit an initial peak caused by overturning followed by the establishment of the layer. The latter two regimes can instead be distinguished by the peakedness of the initial horizontally-averaged penetration depth. In slumping, the plume outflow current rises relatively suddenly as it impinges against the corner and marks a clear peak. In blocking, the peak is instead less narrow as it is formed by the progressive horizontal filling of the accumulating blocked fluid. 5.4.2 Vortex paths Vortex detection is used as a diagnostic tool to further differentiate between the rolling, slumping and blocking regimes. The path of coherent rotational structures is examined during the early transient of each regime. The vortex location algorithm, referred to as the γ2 criterion, proposed by Graftieaux et al. (2001), is used to locate these vortices (see definition in Chapter 3, equation (3.18) ). In figures 5.14-5.16, the time-average of the γ2 criterion is shown to map the path of vortex loci along the central cross section of the container. Solid markers indicate the location of individual vortex loci for progressive frames within the velocity data. These techniques outline the trajectories taken by the vortices in the plume and currents as well as their relative direction of spin. 106 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 0 0.2 0.4 0.6 0.8 1 1 −z /H τ/φ2 (i) (ii) (iii) (iv) (v) (vi) clear peak in overturning plume up-splash Slumping (i) (ii) (iii) (iv) (v) (vi) Figure 5.12: An example time series for the slumping regime (φ = 1.1) showing key frames in the filling process. The solid line shows the horizontally-averaged penetration depth of the scalar. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 0 0.2 0.4 0.6 0.8 1 1 −z /H τ/φ2 (i) (ii) (iii) (iv) (v) (vi) blocked fluid progressively accumulates plume upslash Blocking (i) (ii) (iii) (iv) (v) (vi) Figure 5.13: An example time series for the blocking regime (φ = 1.3) showing key frames in the filling process. The solid line shows the horizontally-averaged penetration depth of the scalar. −0.3 −0.15 0 0.15 0.3 0.2 0.4 0.6 0.8 1 x/H z /H −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 primary vortex ring plume vortices γ2 τ 0 2.5 5.0 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30 ︸ ︷︷ ︸ ︸ ︷︷ ︸ +ve -ve (i) (ii) Figure 5.14: Vortex detection applied to the velocity field for a sample experiment in the rolling regime (φ = 0.38). Vortices are detected from the instant the plume outflow impinges against the sidewalls to when a stable layer is formed. The contours spatially map the value of the time- averaged γ2 criterion along a vertical cross section of the container, coloured as indicated in colour map (i). The γ2 factor is time-averaged over the early transient period to show the trajectory of the vortex paths. Markers shows the loci of these vortices located at progressive time instances, the colour of which is shown in colour map (ii). Note that a mask has been applied below z/H ≤ 0.45, i.e. below the level of the first front after the layer is formed, to accelerate the vortex detection algorithm. −1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 1 x/H z / H −1 −0.5 0 0.5 1 vortices unroll by baroclinic torque γ2 τ (i) (ii) ︸ ︷︷ ︸ ︸ ︷︷ ︸ +ve -ve Figure 5.15: Vortex detection is applied to the velocity field for a sample experiment in the slump- ing regime (φ = 1.1). The contours spatially map the value of the time-averaged (over the early transient period) γ2 criterion along a cross section of the container, coloured as indicated in colour map (i). Markers shows the loci of these vortices located at progressive time instances, the color of which is shown in colour map (ii). Note that a mask has been applied below z/H ≤ 0.42. 5.5. Summary In the rolling and slumping regimes, the trajectories of the vortices are relatively sta- tionary in time. Figure 5.14 reveals how in the rolling regime, vortices within the plume feed the primary vortex ring that fills the space. Below the primary vortex ring, the envi- ronment is quasi-laminar. For the slumping regime shown in figure 5.15, one can clearly observe how the plume vortices feed the current. The current vortices intrude horizontally into the environment until these reach the sidewall. As these vortices are displaced down the sidewall, baroclinic torque unrolls them, reversing the direction of spin, and forming a current towards the centre of the container. The process repeats as this current overturns against the plume. The final current, which goes on to form the density interface between the newly-formed layer and the original environment, appears to be quasi-laminar. The vortex loci that are detected in the blocking regime do not follow paths that are stationary in time. In figures 5.16a-c, three instantaneous images of an example flow in the blocking regime are shown. The dye visualisations on the left hand side show how the plume outflow current impinges against the block corner fluid and overturns against it. The current vortices are displaced below the blocked portion of the flow and then intrude into it from below. The vertical displacement of the vortices due to blocking causes them to unroll after they have reached a peak height and change direction of spin (figures 5.16b-c). 5.5 Summary Observation of the early transient in filling boxes over the range of container aspect ra- tios ranging 0.25 ≤ φ ≤ 2.0 do not agree with the thin, laminar layer of fluid suggested by Baines & Turner (1969) in their filling-box model. Three different filling mecha- nisms (rolling, slumping and blocking) were identified within this range of geometries and described by simultaneous fluorescence visualisation and velocimetry measurements. The experimental technique allowed unprecedented insight into the formation of these regimes. The aspect-ratio dependency of the regimes was confirmed with good agree- ment with the experimental observations of Kaye & Hunt (2007) and Ezzamel (2011). In slender geometries (φ . 0.66), a large toroidal vortex occupies the space between the sidewall and the plume. The flow regime distinguished by this feature is referred to as the ‘rolling’ mechanism. The data presented herein calls into question the modelling of the sidewall flow as a two-dimensional line fountain as suggested by Kaye & Hunt (2007) in that the corner flow manifests as a large vortex ring rather than a slender flow along the sidewall. Slumping occurs in larger box aspect ratios (0.66 . φ . 1.25). This corner flow is similar in many ways to a wall fountain as suggested by Kaye & Hunt (2007). Due to the hydrostatic pressure gradient which arises due to the accumulation of buoyant fluid in the corner, the accumulated fluid collapses and flows back towards the plume. As the returning current overrides the initial outflow, vigorous mixing between the two currents occurs. As this returning current impinges against the plume, it is projected vertically upwards as if overturning against a rigid boundary. The up-splash along the plume then forms a further diverging current that thereon forth constitutes the stable interface between the layer and the original environment. Blocking occurs in wider containers (1.25 . φ . 2.0) and is distinguished by the dis- sipation of the plume outflow current into a quasi-laminar regime. This transition occurs 109 5. A classification for filling-box flows −1 −0.5 00 0.2 0.4 0.6 0.8 1 y /H 0.5 1 −0.5 0 0.5 vortices are displaced by blocked fluid and unroll (a) −1 −0.5 00 0.2 0.4 0.6 0.8 1 y /H 0.5 1 −0.5 0 0.5 (b) γ2 +ve -ve −1 −0.5 00 0.2 0.4 0.6 0.8 1 y /H 0.5 1 −0.5 0 0.5 (c) x/H z/ H Figure 5.16: Vortex detection is applied to the velocity field for a sample experiment in the block- ing regime (φ = 1.3). The contours spatially map the value of the γ2 criterion along a cross section of the container, coloured as indicated in colour map. Note that a mask has been applied below z/H ≤ 0.45 because the radially spreading current needs to thin out and slow down to conserve vol- ume flux. The accumulation of buoyant fluid along the corner of the container causes the subsequent plume outflow to be displaced below the blockage and fill the layer from be- low. This causes the accumulation of more and more fluid along the corner, until enough buoyant fluid is collects in the corner for it to suddenly slump back as a current converging towards the centre of the container, in similar fashion to the slumping regime. 110 chapter 6 The filling of a container by a turbulent lazy plume in the slumping regime A number of filling-box processes (breakdown, slumping, rolling and blocking) have been described in Chapter 5. In this chapter, focus is on modelling the slumping regime (see figure 5.5, p. 98). Slumping occurs when the outflow current, formed by the impingement of the plume with the base of the container, impinges against the corner to rise up the sidewall. The flow up the sidewall collapses to form a horizontal current which converges back towards the centre of the container. Once the current reaches the centre, it forms a stable layer of buoyant fluid which gradually deepens as it is fed by the plume. A theoret- ical model is developed herein to predict the bulk velocities and densities formed by the plume in the container under this filling-box regime. The model idealises regions within the flow (plume, currents, corner upflows and layers) so that their dynamics and interac- tions can be described. Solution are presented for different combinations of plume source conditions and container geometry in terms of three dimensionless governing groups: the scaled source Richardson number of the plume (Γ0), the container radius-to-height aspect ratio (φ := R/H) and the relative size of the plume source radius to the height of the container (β0 := b0/H). An experimental programme of flow visualisations is conducted to observe these flows at laboratory-scale by the release of a saline solution in a fresh- water environment (see Chapter 3). The data acquired from these experiments allows us to address important idealisations of the model regarding entrainment and impingement. The initial motions that form the buoyant layer result in the relatively rapid dispersal of plume fluid in the space compared to the gradual filling predicted in the original filling- box model. The layer formed by the early slumping motions establishes at a depth of about one third of the container height (zS /H = 0.33 ± 0.08), within a time from the activation of the plume of tS /B 1/3 0 H −4/3 ≈ 0.8 + 1.0φ2 + 1.3φ4/3 and with a mean reduced gravity of g′S = B0 tS H/zS R 2, where B0 is the source buoyancy flux of the plume. The implications of recent findings regarding entrainment in lazy plumes (Γ0 > 1, see discus- sion in Chapter 4) are discussed in regards to their effect on the pattern of stratification, revealing how Γ0  1 sources can also significantly increase predicted filling rates even during the displacement filling stages. 6.1 Introduction The stratification of a container by a turbulent plume was theoretically described by Baines & Turner (1969). Their model predicted the density stratification formed by a turbulent plume originating from an idealised point source of buoyancy flux located at the base of a cylindrical container. In their experiments, an outflow current ensuing from the impingement of the plume formed along the base. For sufficiently short and wide 111 6. The filling of a container by a lazy plume containers, of aspect ratio (radius-to-height, φ := R/H & 1), the currents slowed down and thinned out to such an extent as to inhibit the rise of the current along the sidewall. The stratification subsequently evolved as a buoyant layer which gradually deepened as it was progressively displaced from the bottom by the impinging plume fluid. In light of this distinctive feature, this type of stratification is here referred to as displacement filling. Theoretically, the filling of the container was idealised as the gradual deepening of infinitesimally thin and quiescent layers of density corresponding to that of the plume at the base of the container. This idealisation enabled the successful prediction of how the stratification evolved in time (Baines & Turner, 1969; Germeles, 1975; Worster & Huppert, 1983) in what is now referred to as the original ‘filling-box’ model (after Turner 1973). The fidelity of the Baines & Turner (1969) model in practical scenarios in which it was commonly applied has been called into question (Barnett, 1991; Kaye & Hunt, 2007; Giannakopoulos et al., 2013). The pattern of stratification radically changes when filling- box flows occur in containers of radius-to-height aspect ratio of less than φ . 2.0. The change in flow dynamics from the displacement filling case is primarily due to two predominant physical effects which are exacerbated by a decrease in aspect ratio below a value of two: the reversal of the flow against the direction of buoyancy due to its impinge- ment against the boundaries of the box (so-called overturning, Kaye & Hunt 2007) and the shearing of the plume with the return flow it induces in the environment (confinement, Barnett 1991). In practice, overturning and confinement effects give rise to multiple flow regimes for a filling-box flow: breakdown, rolling, slumping, blocking and displacement filling (these regimes are described in Chapter 5). Provided the source of the plume is smaller in width relative to the size of the container, a condition that herein is charac- terised by the ratio of the plume source radius to the container height, β0 := b0/H  1, these regimes are bounded within ranges of φ, as shown in figure 6.1. In the chapter, we examine the stratification produced by a turbulent plume that is formed from a circular source of buoyant fluid which is centrally located at the base of a cylindrical container in what is referred to as the slumping regime. A schematic of the configuration considered is shown in figure 6.2. Detailed observations of the slumping regime are presented in §6.2. The modelling of the dynamics of the slumping regime are generalised in §6.3 for a turbulent plume supplied by a non-idealised source of radius b0 > 0 of buoyant fluid released at a time- and horizontally-averaged velocity and density w0 and ρ0. Three gov- erning dimensionless groups can by obtained by analysing the configuration examined in figure 6.2: the container aspect ratio φ, the plume source Richardson number Ri0 and the relative source size β0. The source Richardson number of the plume is defined as: Ri0 := g′0b0 w20 , (6.1) where g′0 := g(1 − ρ0/ρa), is the mean source reduced gravity of the plume, and ρa is the density of the original environment. The source Richardson number characterises the relative ratio of buoyancy to inertia provided at source; as opposed to the far-field, where a plume asymptotes to a state of balance of buoyancy and inertia which corresponds to a constant value of local Richardson number Rip = g′b/w2. It is thus common to express the imbalance from this self-similar (so-called ‘pure’ plume) state as the ratio of the source 112 displacement filling φ & 2.0 breakdown 0.15 . φ . 0.25 blocking 1.25 . φ . 2.0 slumping 0.66 . φ . 1.25 rolling 0.25 . φ . 0.66 breakdown & RT convection φ . 0.15 Figure 6.1: Schematics showing the patterns of filling-box flows that occur when a point-source plume is positioned within a cylindrical container. 6. The filling of a container by a lazy plume (a) plume source original environment cylindrical container deepening layer (b) R z f H − z fb w ρ b0w0 ρ0 U = dz fdt z ρa Figure 6.2: (a) An example visualisation of a filling-box flow during the late stage of the slumping regime. A stable buoyant layer forms which gradually deepens as it is fed by a buoyant plume. (b) A schematic representation of the filling process is drawn to introduce notation. The source fluid is injected at a uniform velocity w0 from a circular source of radius b0 at a density of ρ0 in a initially uniform (density ρa) and quiescent cylindrical environment of height H and radius R. and far-field Richardson numbers, denoted Γ0 = Ri0/Rip (Hunt & Kaye, 2005). Here we focus on fully turbulent plumes which are passively released. These are commonly referred to as pure (Γ0 = 1) and ‘lazy’ (Γ0 > 1) plumes. Bounds on φ,Γ0 and β0 for which the slumping regime is expected to occur are outlined in §6.6.4. Flow visualisations are combined with tracking techniques to observe slumping dur- ing the release of a dyed saline solution in a freshwater environment. These experiments are described in §6.4 and their results discussed in §6.5. A separate suite of particle im- age velocimetry measurements are drawn upon to validate key assumptions concerning impingement and entrainment presented in the theoretical model. A discussion on the dy- namics of slumping is presented in §6.6 to showcase solutions for different combinations of φ, Γ0 and β0. Conclusions are drawn in §6.7. 6.2 Observations The filling of the container in the slumping regime is characterised by two distinct phases which we will refer to as the ‘initial’ and ‘late’ stages of filling. The features observed in the initial stages are shown in the visualisations presented in figures 6.3a-e and idealised schematically in figures 6.4a-f. Regions in the flow are established during the initial stage and persist in structure as the stratification process progresses into the late stage. The structure of the theoretical model presented in §6.3 is based on these regions which we proceed to describe in this section. All descriptions presented in this chapter are presented in the laboratory frame of reference, i.e. of saline released to propagate downwards in a freshwater environment. Buoyant fluid was released through a circular nozzle and formed a turbulent plume that falls under gravity towards the base of the container, figure 6.4a. The impingement of the plume against the floor formed a radial outflow, figure 6.4b. The outflow current was led by a vortex ring which propagated towards the sidewalls. The current resembled the behaviour of a turbulently entraining gravity current in that mixing with the overlaying environment can be observed all along its stem (see figure 6.6). The impingement of the 114 (a) (b) (c) (d) (e) τ = 0.8 τ = 1.5 τ = 2.1 τ = 3.1 τ = 4.9 (f) (g) (h) (i) (j) t = 11 s t = 22 s t = 31 s t = 47 s t = 74 s Figure 6.3: Sample image frames of a typical slumping flow during the initial stages of filling. Flow visualisations are shown for the filling process using two different visualisation techniques for experiments with similar source conditions and container geometry. A cylindrical container of radius R = 300 mm and height H = 323 mm was used. In the left column, (a)-(e), backlit images of the flow are shown as visualised by means of a methlyene blue tracer. The source conditions for the ensuing plume are b0 = 44.5 mm, w0 = 6.96 mm/s, g′0 = 265 mm/s 2 and Γ0 = 1690. In the right column, (f)-(j), planar LIF visualisations are shown of the flow along the central cross- section of the container. The source conditions for the plume are b0 = 44.5 mm, w0 = 6.80 mm/s, g′0 = 240 mm/s 2 and Γ0 = 1604. Dimensionless time τ := tB 1/3 0 /H 4/3 is measured after the activation of the release. 6. The filling of a container by a lazy plume (a) starting plume (b) outflow current (c) overturning (slumping) (d) inflow current (e) plume upflow (f) layer formation Figure 6.4: Schematics for the initial stage of filling showing the motions that develop during slumping: showing (a) the starting plume, (b) the starting forced gravity current ensuing from the plume outflow, (c) the overturning of the current along the lower edge of the cylinder, (d) the slumping back of the current towards the centre of the container, (e) the rise of the current up the plume, (f) the collapse of the plume upflow and the resulting formation of the initial stable layer. current against the lower corners of the cylinder resulted in the rise of a slender vertical intrusion up the sidewall (Kaye & Hunt, 2007). These intrusions, traditionally referred to as the overturning of the flow (Baines & Turner, 1969), penetrated vertically into the original environment reaching a peak depth before collapsing under gravity, figure 6.4c. The relative density of overturning motions inhibited their rise and caused them to slump back down towards the base. The downflow ensuing from the overturning motions formed a second horizontal current which converged towards the centre of the container and in doing so, overlaid the original plume outflow current, figure 6.4d. We refer to this second current as the inflow current, a distinctive feature of the slumping regime. The inflow current upon approaching the centre of the container, impinged and rose up the plume, figure 6.4e. The shear between the current and the plume drained inertia from the current. The current flow thus collapsed under gravity and settled into a stable layer as a buoyancy-viscous current, figure 6.4f. The viscous current displaced the original environ- ment without further mixing via a filling mechanism which at this stage closely resembled the displacement filling description of Baines & Turner (1969). We observed, by inspect- ing the videos of the flow visualisations, that the counterflowing currents at the bottom of the layer continue to turbulently exchange fluid for the entire filling period. This internal flow structure was also confirmed in the filling of a container by a small-source plume during the slumping regime in Chapter 5 using simultaneous LIF+PIV measurements. 6.3 A theoretical model for slumping The observations presented in §6.2 suggest that distinct regions exist in the flow field that once established, persist in structure throughout the entire filling period. Extending 116 6.3. A theoretical model for slumping 1. plume (B) 2. outflow current (I-B) 3. corner upflow (I-NB) 4. inflow current (B) 5. displacement filling (B-V) 1 2 3 4 5 Figure 6.5: The internal flow structure of the buoyant layer is subdivided into five regions which are labelled in figure (RHS). Dark grey areas between these flow regions represent impingement regions. The letters used in the image caption indicate the dominant forces acting in each region. The letters read as: (B) for purely buoyancy-driven, (I-B) for inertia-buoyancy-driven, (I-NB) for inertia-driven with buoyancy acting in opposite direction and (B-V) for buoyancy-viscous. the approach taken by Kaye & Hunt (2007) for the description of the initial filling-box flow formed by a point-source plume, we divide the flow into regions: the plume, the outflow current, the corner upflow, the inflow current and the displacement filling layer. These regions are illustrated in figure 6.5 to show how they are interconnected, with the flow moving predominantly downstream of the source through regions one to five. In the spirit of a simplified description of the formation of the buoyant layer and its internal dynamics, the behaviour in the plume and ensuing flows are modelled using equations based on the conservation of volume, momentum and buoyancy fluxes in each region. Idealised impingement zones are used to relate the dynamics of each region based on the conservation of these integral quantities. In this section, the governing equations for each flow region are presented. Their solutions are discussed in §6.6. Additionally, the experimental measurements required to support the theoretical model are identified here and are subsequently addressed in §6.5. 6.3.1 Plume The conservation equations for a turbulent plume of Morton et al. (1956), also adopted by Baines & Turner (1969) in the original filling-box formulation, are extended to in- clude physical effects which are seen to be important during the observed filling-box flow following from the observations presented in §6.2. These include (i) the return flow sur- rounding the plume (Morton, 1961; Barnett, 1991), and (ii) modified near-source entrain- ment effects that arise due to the variation of source Richardson number (van Reeuwijk et al., 2016). The plume issues from a source located at a level of z = 0 with source ra- dius, velocity and reduced gravity: b0,w0, and g′0 (see figure 6.2 for schematic). Uniform (top-hat) cross-stream profiles of velocity w and reduced gravity g′ for a plume of radius 117 6. The filling of a container by a lazy plume b are assumed. The profile choice is chosen for the sake of simplicity, whose necessity can be better appreciated when confronted with profile choices in other regions of the flow. We nevertheless acknowledge that the approach could be extended to any choice of profile for these quantities. Conservation of volume flux in the plume dictates that vertical volume flux is increased by the turbulent entrainment of fluid from the environment: ∂wb2 ∂z = 2αb(w − U). (6.2) In (6.2), the velocity U is assumed to surround the plume uniformly and is included so as to consider the effect of the relative velocities of the return flow on the plume (following the approach developed by Morton (1961) for co- and counter-flowing jets, see Chapter 2, §2.2.7). Further discussions on return flow effects are presented in §6.6.4. Despite our assumption that the plume and the ensuing flows behave as if they were quasi-steady, owing to the fact the plume source conditions are steady, partial derivatives are used henceforth to indicate that the flow parameters are time-dependent due to the changing stratification. To satisfy continuity along any horizontal plane within the container, the return flow volume flux has to correspond to that of the plume at any given level: U ( R2 − b2 ) = −wb2. (6.3) Turbulence within the plume is modelled using the entrainment parameter α := ue/w, which relates the local mean vertical velocity in the plume to the induced horizontal ve- locity at the plume edge. For turbulent plumes, it is commonly assumed that the entrain- ment parameter α is constant (also assumed by Baines & Turner 1969) provided the flow is sufficiently downstream of the source. Here we take a reference value for this con- stant which is equal to αp = 0.09, following from Morton et al. (1956) and Hunt & Kaye (2001). There is ample evidence for α , cst in the adjustment region of non-pure (Γ0 , 1) plumes such as in forced (Kaminski et al., 2005) and lazy plumes (Kaye & Hunt, 2009). Van Reeuwijk et al. (2016) have shown that the entrainment parameter for a steady plume can be expressed as a function of local Richardson number: α = − δg 2γg + ( 1 βg − θm γg ) Ri, (6.4) where coefficients δg = −0.184, γg = 1.391, βg = 1.076 and θm = 1.011 were resolved from their DNS data (further details in van Reeuwijk & Craske 2015; van Reeuwijk et al. 2016 and a discussion on the near-source entrainment dynamics of lazy plumes is pre- sented in Chapter 4). We apply this description by assuming that it largely holds within a stratifying environment and note that stratification exists in the near-field only when the first front of filling reaches the proximity of the source, that is, at late filling time. Vertical momentum flux in the plume is generated by the buoyancy forces acting on the plume fluid and exchanged with the return flow. To incorporate these two effects, the conservation of vertical specific momentum flux can be expressed as: ∂ ∂z ( w2b2 + U2(R2 − b2) ) = g′b2 − 2αbwU. (6.5) 118 6.3. A theoretical model for slumping The momentum flux exchange approach of Morton (1961) is adopted to model the tur- bulent exchange of inertia with the return flow (further examples of the approach in- clude the modelling for coaxial jets in Morton 1962 and turbulent fountains in Hunt & Debugne 2016). The buoyancy flux of the plume changes due to the stratification of the environment. The vertical reduced gravity gradient is denoted N2 = ∂g′e/∂z where the reduced gravity of the environment is defined by g′e := g(ρe − ρa)/ρa; ρe being the horizontally-averaged density of the environment, and assumed to be horizontally homo- geneous. Buoyancy flux is also exchanged between the plume and the environment due to entrainment of plume fluid into the return flow. To model these effects, a conservation equation for the buoyancy flux in the plume is deduced and given by: ∂g′wb2 ∂z = −wb2N2 − 2αg′ebU. (6.6) Numerical solutions to equations (6.2) , (6.5) and (6.6) subject to source conditions: b = b0, w = w0 and g′ = g′0, at z = 0, (6.7) can be readily deduced with suitable descriptions of the stratification g′e(z, t), as further discussed in §6.3.6. To simplify the solution procedure, it is useful to express equations (6.2)-(6.6) in terms of volume, specific momentum and buoyancy flux: Q := wb2, M := w2b2, B0 := g′wb2. (6.8) Note that these fluxes are divided by a factor of pi relative to their actual physical value. Equations (6.2)-(6.6) are re-expressed in terms of fluxes and in dimensionless form, by substituting: ζ := z H , q := Q Q0 , m := M M0 , f := B B0 , gˆ′e := g′e g′0 , Nˆ2 := ∂gˆ′e ∂ζ , (6.9) into (6.2)-(6.6), to give: ∂q ∂ζ = 2αm1/2 β0 ( 1 + 1 R2 − 1 ) , (6.10) ∂ ∂ζ [ m ( 1 + 1 R2 − 1 )] = Ri0 β0 q f m − 2αm 3/2 qβ0 1 R2 − 1 , (6.11) ∂ f ∂ζ = −qNˆ 2 β0 − 2αgˆ′em1/2 ( 1 R2 − 1 ) . (6.12) To simplify notation, a dimensionless parameter R is introduced in (6.10)-(6.12) that is defined as the ratio of the radius of the container to that of the plume: R := R b = φ β0 m1/2 q . (6.13) 119 6. The filling of a container by a lazy plume (a) current nose / leading vortex ring current stem plume impingement region (b) wi hc0 uc0 r0 r rl Figure 6.6: A visualisation is compared to a schematic of the model of the impingement of the plume with the base of the container and the resulting gravity current outflow. In (a), an instan- taneous image of the flow visualised via LIF. In (b), a schematic of the impingement region that serves as the idealised source of the outflow. The governing dimensionless groups that follow from the analysis are the source Richard- son number (Ri0), the relative source radius (β0) and the aspect ratio of the container (φ): Ri0 := B0Q20 M5/20 , β0 := b0 H , φ := R H . (6.14) These groups characterise all the subsequent regions in the filling-box model (§6.6). The system of equations (6.10)-(6.12) can be readily integrated numerically as a first order initial value problem with source conditions q(0) = m(0) = f (0) = 1 from ζ = 0 to ζ = 1. 6.3.2 The plume impingement region The outflow current ensuing from the impingement of the plume with the base is mod- elled as an axisymmetric gravity current issuing from a cylindrical source, of height hc0 and radius r0. The subscript (.)c is used henceforth for parameters associated to the out- flow current. The current is supplied continuously with volume, specific momentum and buoyancy fluxes: Qc := ucrhc, Mc := u2crhc, Bc := g ′ cucrhc, (6.15) respectively, where uc and g′c := g(ρc − ρa)/ρa are the vertically-averaged velocity and reduced gravity in the current, ρc being the vertically-averaged density of the current, over a depth hc (figure 6.6). This impingement model is equivalent to that of Kaye & Hunt (2007), where the source fluxes of the current are dictated by the fluxes of the plume upon impingement. These impingement fluxes are denoted with a subscript (.)i and are estimated from numerical solutions to the plume equations (6.10)-(6.12) at z = H − bi. The fluxes of volume and buoyancy are conserved owing to the conservation of mass through the control volume. On the other hand, while momentum flux in the flow is not expected to be conserved as the plume impinges against the boundary, it may be expected 120 6.3. A theoretical model for slumping that the momentum flux supplied to the current scales on the momentum flux of the plume prior to impingement. Following the approach of Kaye & Hunt (2007), a scaling factor γi is introduced based on this argument. The factor γi is expected to be γi ≤ 1, as pointed out by Kaye & Hunt (2007), owing to the mean kinetic energy flux loss experienced by the flow at impingement. The source fluxes of the current are thus estimated from the predicted fluxes of the plume at impingement: Qc0 = Qi, Mc0 = γiMi, Bc0 = B0. (6.16) Kaye & Hunt (2007) suggest a value of γi ≈ 0.7 based on an analogy with flows turning in pipe bends. Ezzamel (2011) investigated the γi factor by inspecting the impingement of a thermal plume against a horizontal boundary. For a purely-momentum driven jet, Ezzamel (2011) measured γi = 0.8 and suggested that the value is between 0.8 ≤ γi ≤ 1 for forced and pure plumes. Measurements of γi are discussed in §6.5.4. Following Kaye & Hunt (2007), the conservation of mean kinetic energy fluxes is applied to determine the initial depth of the current, where the flux of kinetic energy ‘in’ matches the flux of kinetic energy ‘out’ of the control volume shown in figure 6.6: γ2i (pibi) 2w3i = (2pibihc0)u 3 0. (6.17) Equation (6.17) suggests that the height of the idealised source is hc0 = bi/2γi. The Richardson number for a gravity current is defined as Ric := g′chc/u 2 c (Turner, 1973). A source value for the Richardson number of the current can be expressed, as a result of (6.16), as a function of the impingement plume Richardson number: Ric0 = Bc0Q3c0 M3c0r0 = B0Q3i γ3i M 3 i bi = Rii γ3i . (6.18) In (6.18), Rii := g′ibi/w 2 i is the local Richardson number of the plume at impingement. 6.3.3 Outflow current A model for the plume outflow current is proposed which relies on similar conserva- tion equations to those presented for the plume in §6.3.1. The model is similar to those deduced by Ezzamel (2011) or Slim & Huppert (2011), where the fluxes of volume mo- mentum and buoyancy are modelled using an entrainment parameter αc and assuming quasi-steady behaviour (∂/∂t = 0), and that there is no transfer of buoyancy to the con- tainer and no wall friction experienced along the stem of the current. Conservation of volume flux dictates that current volume flux, Qc := ucrhc, is conserved radially: ∂ ∂r (ucrhc) = αc (uc − ur) r. (6.19) In (6.19), the counterflow velocity ur is included to model the effect of the overlaying inflow current. The subscript (.)r is used henceforth for parameters associated to the inflow current. Entrainment is modelled, in an analogous way to that used for the plume, that is, by using an edge velocity proportional to the mean velocity of the current αc := wce/uc, where wce is the vertical velocity induced at the upper edge of the current by the 121 6. The filling of a container by a lazy plume engulfment of fluid overlaying the current. Entrainment in gravity currents (reviews in Simpson 1982, 1999; Ungarish 2009) is non-trivial to model simplistically as it depends on the relative depth of the current and the degree of forcing. A popular parameterisation for current entrainment is that of Ellison & Turner (1959) (applied, for example, in Slim & Huppert 2011). The function was deduced from early measurements for planar gravity currents by Ellison & Turner (1959). It suggests that the entrainment parameter varies as: αc = max [ 0.08 − 0.1Ric 1 + 5Ric , 0 ] , where Ric := g′chc u2c . (6.20) This entrainment function is employed for the sake of discussing solutions to the model presented in this chapter. We acknowledge that (6.20) underestimates entrainment con- siderably (see §6.5.5). The reduced gravity of the inflow current is defined by g′r := g(ρr−ρa)/ρa, ρr being its vertically-averaged density. The conservation of momentum flux in the outflow current, Mc := u2crhc, is given by: ∂ ∂r ( u2crhc ) = − r 2 ∂ ∂r [( g′c − g′r ) h2c ] − αchcucur. (6.21) In (6.21), the turbulent exchanges between the currents are also idealised using the ap- proach of Morton (1961). Buoyancy flux, Bc := g′ucrhc, is conserved in the current except for the buoyancy flux detrained out of the current due to shear, i.e. ∂ ∂r ( g′cucrhc ) = −αcurhrg′c. (6.22) In other words, in equation (6.22), a sink of buoyancy introduced to compensate for the entrainment of buoyant fluid out of the outflow and into the inflow current. 6.3.4 Corner intrusion As in Kaye & Hunt (2007), the corner upflow is modelled as a forced planar wall fountain. The fluxes in the fountain are defined, assuming ‘top-hat’ profiles of velocity and density, per unit length of the circumference of the cylinder (2piR), such that: Qu := wubu, Mu := w2ubu, Bu := g ′ uwubu. (6.23) The subscript (.)u is used henceforth for parameters associated to the corner upflow. As- suming an annular impingement zone along the lower edge of the cylinder, as shown in figure 6.7, the conservation of volume and buoyancy fluxes are invoked to relate the source fluxes of the fountain to the fluxes in the current as it impinges against the corner: Qci = 2Rwubu, Mci = 2Rw2ubu/γu, B0 = 2Rg ′ uwubu. (6.24) The subscripts (.)ci denote values estimated in the current at a distance of r = R− hci from the plume axis. In (6.24), a momentum flux scaling coefficient for the fountain, denoted γu, is introduced to accommodate for momentum flux losses due to the flow turning 90 degrees. The value of γu is assumed to be equal to that of the plume turning (i.e. γi) and 122 6.3. A theoretical model for slumping (a)peak penetration fountain top fountain stem outflow current impingement region (b) zut zut − hci bu0 bu wu0 hchci uci Figure 6.7: A visualisation is compared to a schematic of the model for the corner upflow. In (a), an instantaneous image of the flow visualised via LIF. In (b), a conceptualised schematic of the flow. taken to be equal to γu = 0.8 (see discussions in §6.5.4). The source Richardson number of the fountain, defined as: Riu0 := g′u0bu0 w2u0 = BuQ3u M2u = B0Q3ci (2R)2 (γuMci)3 = Rici γ3u , (6.25) can be estimated by the conservation of these fluxes. The conservation equations for a turbulent line fountain are given by: ∂wubu ∂z = αuwu, ∂w2ubu ∂z = −(g′u − g′e)bu, ∂g′uwubu ∂z = 0, (6.26) (e.g. Hunt & Coffey 2009). The entrainment parameter for the fountain is denoted αu and is taken to correspond to that of a forced line fountain, i.e. αu = 0.0515, following Hunt & Coffey (2009). The conservation equations (6.26) become singular at the rise height of the fountain as flow reverses direction (as shown by Morton 1959). Equations (6.26) can be used to estimate the initial rise height of the intrusion, as discussed in Hunt & Coffey (2009), which in an unstratified environment is expected to scale as: zut − hci bu0 ∝ Ri−2/3u0 . (6.27) The approach circumnavigates the necessity to include more detailed modelling of en- trainment in the fountain cap and of exchanges with the counterflow, as modelled, for example, for a round fountain in Hunt & Debugne (2016). 6.3.5 Inflow current Conservation equations can be deduced for the inflow current, following from (6.19)- (6.22), that characterise the conservation of volume, Qr := urrhr, specific momentum, 123 6. The filling of a container by a lazy plume Mr := u2r rhr, and buoyancy, Br := g ′ rurrhr, fluxes: ∂ ∂r (urrhr) = αc (ur − uc) r, (6.28) ∂ ∂r ( u2r rhr ) = r 2 ∂ ∂r ( (g′r − g′S )h2r ) − αchrucur, (6.29) ∂ ∂r ( g′rurrhr ) = αcuchc(g′r − g′S ), (6.30) where the reduced gravity of the environment overlaying the inflow current is given by the reduced gravity in the lower portion of the deepening layer, defined by g′S := g(ρS−ρa)/ρa; ρS being the horizontally- and vertically-averaged density of the layer up to a height zS (see §6.3.6 for the definition of zS ). The initial conditions of the inflow current are estimated from the solutions to the outflow current as it impinges with the corner: Qr0 = Qci, Mr0 = γuMci, Br0 = Bci. (6.31) The system prescribed by (6.28)-(6.30) is an initial value problem which is to be solved from r = R − 2bu0 to r = hc0 (figure 6.8). Given the interdependence between the outflow current equations (6.19)-(6.22) and the inflow current equations (6.28)-(6.30), the two systems of ordinary differential equations are coupled and need to be solved iteratively. A similar approach to that developed for round fountains in Hunt & Debugne (2016) is adopted to solve the equations for the two currents. An initial solution to the outflow equations is solved assuming there is no counterflow, that is, hr = ur = g′r = 0, which then yields the initial conditions for the inflow current. The two systems of equations are then solved iteratively until solutions converge within a threshold percentage. Typically between 3-5 iterations are required for different values of Γ0, β0 and φ to reduce the mean relative difference between iterations to within 1%. Note that the convergence of the current towards the centre of the container, r = 0, causes substantial increases in current height as the fluid enters a smaller volume (this indeed causes the solutions for hc to become singular at r = 0; solutions which are shown in §6.6). 6.3.6 Displacement filling At the onset of the late stage, defined in §6.2 as the time at which the buoyant layer is established, a stable front gradually deepens which does not mix with the original overlaying ambient environment. The displacement filling region above the region of slumping is modelled using a formulation analogous to that of Baines & Turner (1969), with the exception that the region of slumping is idealised so that the lower portion of the container is instantaneously uniformly mixed up to a uniform depth of zS , level above which displacement filling occurs (see figure 6.9). Crucial to the formulation of the model is an estimate of the formation time tS required to form a uniform layer of depth zS . At the time the buoyant layer forms, measured after the time of release t = 0 and denoted: tS := τS (φ, β0,Γ0) H4/3B −1/3 0 , (6.32) 124 6.3. A theoretical model for slumping (a) (b) hc hr uc0uci ur0 R g′c g′r Figure 6.8: A visualisation is compared to a schematic of the model for the outflow and inflow currents. (a) An instantaneous LIF visualisation of the flow in the proximity of the corner. Contour lines of vorticity, coloured in yellow for clockwise and blue for counterclockwise motions, are superimposed onto the image to (qualitatively) highlight the turbulent exchanges between the two currents. (b) A schematic to show how the flow is idealised in the models presented in §6.3.3 and §6.3.5. it is readily estimated, by the conservation of total buoyancy in the layer, that for a layer of depth: zS := ζS (φ, β0,Γ0) H, (6.33) the horizontally- and vertically-averaged reduced gravity corresponds to: g′S = tS zS B0 R2 . (6.34) The dependency of the formation time, τS , and depth, ζS , of the initial buoyant layer on the three governing dimensionless groups φ, β0 and Γ0 is discussed further in §6.5.2. The environment in the container at the start of the displacement filling stage t = tS is then composed of the original environment and the buoyant layer: g′e(t = tS ) =  0 for 0 ≤ z ≤ H − zS ,g′S for H − zS < z ≤ H. (6.35) After the layer forms (t > tS ), the buoyant layer deepens at a rate governed by the volume flux of the plume as it enters the layer via the first front: z f = zS + ∫ t tS ∂z f ∂t dt, (6.36) where the rate of deepening of the first front is governed by volume flux of the plume as it enters the layer: ∂z f ∂t = Q f R2 − b2f . (6.37) In (6.37), Q f and b f are estimated by solving the plume equations (6.10)-(6.12). 125 6. The filling of a container by a lazy plume A numerical integration scheme was developed to describe this late displacement fill- ing stage. The formulation is equivalent to the staircase scheme developed by Germeles (1975) except for the specification of the initial layer depth to correspond to that formed by the slumping. A benefit of specifying this depth, compared to the assumption of an in- finitesimally thin initial layer suggested by Germeles (1975), is indeed that the dynamics of formation for the initial layer are no longer arbitrarily specified, but set by the dynam- ics of the early transient which are encapsulated within the dimensionless functions for formation time and initial layer depth given in (6.32)-(6.34). A schematic of the numerical scheme is presented in figure 6.9 to assist the discussion. For each fixed time step ∆t, a time counter j is defined so that the density profile in the environment at time t j forms k steps in the density profile. The depth of each step layer ∆z j corresponds to that dictated by the volume of plume fluid entering the layer over the time interval ∆t: ∆z j ∆t = Q f , j R2 − b2f , j . (6.38) Each step layer forms directly above the slumping layer and maintains the same density as it is displaced upwards by the subsequent step layers that form between it and the slumping layer. The reduced gravity of each newly-formed step layer is given by the density of the slumping layer at the previous time-step. The profile of reduced gravity thus corresponds to: g′e, j H − zS − j∑ n= j−k+2 ∆zn < z ≤ H − zS − j∑ n= j−k+3 ∆zn  = g′S ( j− k + 1), 2 ≤ j ≤ n. (6.39) In (6.39), the mean reduced gravity of the slumping layer is estimated assuming the plume fluid entering the layer mixes instantaneously with the layer itself. A conservation of total buoyancy in the layer dictates that at any given time, the reduced gravity in the bottom slumping layer is equal to: g′S (t j) = g ′ S (t j−1) + ∆t zS R2 ( QpS , j g′pS , j − Q f , j g′S , j−1 ) , (6.40) where g′pS and QpS are respectively the mean reduced gravity and volume flux of the plume at a level it enters the well-mixed slumping layer. Solutions to the five regions presented in this section are discussed in terms of combi- nations of φ, β0 and Γ0 in §6.6. To enable these discussions, key assumptions within the model are addressed via an experimental programme of flow visualisations and particle image velocimetry. In summary, to close the solutions to the modelled presented herein, the following is required: (i) an estimate of the impingement factors γi and γu, (ii) an estimate of entrainment in the plume and currents and (iii) an estimate for the formation time τS and depth ζS of the initial buoyant layer. 126 6.4. Experiments z R zS ∆z j ∆z j−1 ∆z3 ∆z2 ∆z1 g′0,Q0 g′f , j,Q f , j g′pS , j,QpS , jg ′ S , j−1,Q f , j g′S , j−1 g′S , j−2 g′S ,3 g′S ,2 g′S ,1 g′S , j Figure 6.9: A schematic showing how the displacement filling model is implemented into a nu- merical scheme. A layer of depth zS forms at the bottom of the container which is assumed to be instantaneously well mixed by the slumping motions and whose depth remains constant through- out the entire filling process. Above the slumping layer, filling occurs incrementally, deepening at a rate governed by the volume flux entering the layer Q f and in which the reduced gravity of each incremental layer corresponding to that of the slumping layer at the previous time step. 6.4 Experiments Two separate sets of experiments were conducted: backlit flow visualisations of filling- box experiments (§6.4.1) and simultaneous LIF+PIV experiments of the impingement region of a plume impinging against a horizontal boundary (§6.4.2). 6.4.1 Flow visualisation Experiments were conducted by injecting a saline solution through a circular nozzle into a cylindrical acrylic tank containing a quiescent freshwater environment (figure 6.10a). To minimise optical distortions, the cylindrical container, of height 500 mm and radius 300 mm, was placed into a water-filled square-based tank of 1000 × 1000 × 700 mm3. The plume was continuously supplied from a saline reservoir until the time the first as- cending front of the buoyant layer reached the level of the source. The injected fluid was drawn from a reservoir of known salinity that was stained with 10-20 mg/L (depending on expected dilution rates) of methlyene blue. Two pumps were used in separate sets of experiments to achieve different ranges of flow rates. For lower flow rates (0.1-3 L/min), the plume was supplied by a positive displacement gear pump manufactured by ISMATEC (model ISM 198A). The flow rate was measured with an Apollo LowFlo flowmeter, a magnetic rotameter with operating range from 0.1-3 L/min with an accuracy of ±1% of the full reading, i.e. ±0.03 L/min. For higher flow rates (3-10 L/min), the plume was supplied by a gear pump manufactured by Pompe Cucchi (model NCX41 / 80CACFS). The flow rate was measured with an 127 6. The filling of a container by a lazy plume Apollo flow meter (model ET3/B) with an accuracy ± 2% of the full reading, i.e. ±0.2 L/min. The nozzle and supply system via which the plume was injected was specifically manufactured to ensure impulsive start-up conditions. A porous disc, made of a medium grade polyethlyene disc (6 mm ±0.15 mm thick, of mean and maximum pore size 85 µm and 105 µm respectively, SPC Technology PE10060), was fixed to the outlet of the nozzle. Two nozzles were used of exit diameters 2b0 = 44.5 and 89.5 mm (±1%) – preceded by a straight length of pipe of about 10 exit diameters (460 mm and 700 mm in each case). A 200 mm layer of glass marbles of 10 mm diameter was placed above the porous disc within the nozzles to improve the uniformity of the outflow. Six sets of experiments, for a total of 96 experiments, equally spaced in flow rate and density, were conducted with the following parameter ranges: 1. b0 = 44.5 mm, H = 421 mm, Q0 = 3 − 10 L/min (Cucchi NCX41 / 80CACFS), 2. b0 = 44.5 mm, H = 348 mm, Q0 = 3 − 10 L/min (Cucchi NCX41 / 80CACFS), 3. b0 = 44.5 mm, H = 303 mm, Q0 = 0.1 − 3 L/min (ISMATEC ISM 198A), 4. b0 = 89.5 mm, H = 380 mm, Q0 = 0.1 − 3 L/min (ISMATEC ISM 198A), 5. b0 = 89.5 mm, H = 315 mm, Q0 = 0.1 − 3 L/min (ISMATEC ISM 198A), 6. b0 = 89.5 mm, H = 228 mm, Q0 = 0.1 − 3 L/min (ISMATEC ISM 198A), all with R = 300 mm and release densities ranging between 1.01-1.07 g/cm3. The pa- rameter space was varied according to the following ranges of relative source radius 0.1 . β0 . 0.4, container aspect ratio 0.7 . φ . 1.3 and plume source conditions 1 . Γ0 . 106, where for the scaling of Γ0 = Ri0/Rip it has been assumed that Rip = 8αp/5 with αp = 0.09. The experiments were video recorded with a CMOS-based camera (model JAI Spark 5000M) with a resolution of 5 Megapixels (2560 by 2048) at an acquisition frame rate of 12.5 fps and shutter speed of 1/50 s. The tank was backlit by a uniform LED white light panel. Image processing was conducted to track features within the flow field (i.e. leading fronts of the starting plumes and currents, and the depth of the buoyant layer) by applying a combination of thresholding and edge detection algorithms (see Chapter 3). 6.4.2 Velocimetry measurements in the near-impingement region The setup implemented to measure the momentum loss experienced by a plume as it impinged continuously against a horizontal plate by means of particle image velocimetry is shown figure 6.10b. The current spilled over a false floor and collected at the bottom of the tank so that sufficiently long recording times could be achieved. The dispersion of a fluorescent dye added to the plume source fluid was simultaneously recorded using the same LIF+PIV technique described in Chapter 5 to illustrate both the instantaneous (figure 6.6a) and time-averaged (figure 6.16a) outline of the flow. Illumination of the flow section was achieved by shining a white light beam generated by a 300W Xenon arc lamp through light sheet forming optics. A PIV window was used of 600 × 300 mm2 in size, recorded at a resolution of 5 Megapixels, and post- processed with two-pass interrogation areas of 64 × 64 and 32 × 32 pixels with 50% overlap. Optical distortion effects owing to the change in light refraction of the saline solution were minimised by using a relatively low density difference between the plume 128 (a) 70 0 m m 50 0 m m H LED panel nozzle supporting frame PE disc 600 mm 1050 mm 6000 mm camera overflow bypass circuit saline reservoir gear pumpflowmeter air vent (b) freshwater tank H saline source plume outflow false floor light sheet optics over spill PIV window Figure 6.10: (a) Schematic of the experimental setup for the filling-box experiments. A solution of saline is injected through a circular microporous nozzle into a cylindrical container containing an initially fresh and quiescent environment. (b) Schematic for the simultaneous fluorescence and particle image velocimetry measurements of the impingement of a plume against a horizontal boundary. The plume outflow spilled over a false floor so that quasi-steady state measurements could be taken of the impingement region and current. 6. The filling of a container by a lazy plume and its environment, maintaining a maximum change of refractive index of about ∆n/2n ≈ 0.002 (Daviero et al., 2001). 6.5 Experimental results We initially show in §6.5.1 how data is recorded from the flow visualisations by analysing the position of the tracer which was added to the plume before its release and show how this analysis helps one understand how the plume disperses in the environment. The tracking of certain features in the flow (starting plume, current fronts, corner intrusions, density interfaces) enables an estimate of the time required to form the initial layer, its initial depth (§6.5.2) and how the layer deepens in time (§6.5.3). We show how this tracking can be used to validate simple dimensional scalings for the formation time and depth of the initial slumping layer and explore deviation from these scalings by comparing them to solutions based on the model presented in §6.3. Filling rates are then compared for different plume entrainment models (§6.5.3). The section finishes with a discussion on impingement losses (§6.5.4) and entrainment in the plume outflow current (§6.5.5). 6.5.1 Stages of filling To extract data from the flow visualisations, each experiment was analysed by means of a time series of the intensities of each image frame. An example time series is shown in figure 6.11. The time series was constructed so that each vertical line in the time series corresponds to the horizontally-averaged intensities of the regions external to the plume, conservatively bounded by b0 + 6αpH/5 ≤ r < R. The signal for the peak vertical penetration of stained plume fluid that is shown in figure 6.11 was recorded by locating a threshold reduction in intensity 95% of a background subtracted image with a highly enhanced contrast. A peak-finding algorithm was implemented on this signal to system- atically identify key frames and features of the filling process, including: (i) the time at which the plume impinged against the base tp, (ii) the time at which the outflow current impinged against the corner tc, (iii) the peak penetration depth of overturning ζut and (iv) initial layer depth ζs, and the time required for the initial layer to form tS = tp + tc + tr (see captions in figure 6.11). After time ts, the signal records the horizontally-averaged loca- tion of the first front. Measurements extracted from the time series signal are discussed in the following sections. 6.5.2 Formation time and depth of initial layer This section is dedicated to deducing expressions for the formation time τS and depth ζS of the initial layer introduced in §6.3.6. The formation time of the layer is assumed to be composed of three parts: the time required for (i) the plume to reach the floor τp, (ii) the current to reach the sidewall τc, and (iii) the inflow current to reach the plume τr: τS = τp + τc + τr. (6.41) For (6.41), recall that time is non-dimensionalised on τ = tB1/30 H −4/3. Dimensional scal- ings are presented to estimate τp, τc, and τr as a function of φ. We use scalings that are equivalent to those of a point-source pure (β0 = 0, Γ0 = 1) turbulent plume. The role of 130 6.5. Experimental results 0 20 40 60 80 100 120 140 160 180 200 0 0.2 0.4 0.6 0.8 t (s) 1 − ζ pe ak ov er tu rn in g ini tia l la ye r interface oscillates plu me ups pla sh sta ble lay er tp tc tr • tS ,m ζut ζS ( t = tS ,m ) ζ f (t) Figure 6.11: An example time series of a filling-box flow used to identify key frames in the process. Each vertical band in the time series corresponds to the horizontally-averaged intensity of an image frame over the regions external to the plume. The time tS ,m corresponds to the time at which the horizontally-averaged front reaches a minimum after the first peak caused by the overturning motions. β0 > 0 and Γ0 > 1 are subsequently discussed by comparing these scalings to solutions based on the equations developed in §6.3. A description of the penetration depth of the starting cap zt(t) is sought to estimate the time required for the plume to reach the base of the container τp. Theoretical approaches to estimate the penetration of a starting plume principally followed Turner (1962) who modelled a starting plume by the matching of conservation equations for a plume (6.2)- (6.6) to those of a buoyant thermal starting cap (Middleton, 1975; Delichatsios, 1979; Law et al., 2011; Bhamidipati & Woods, 2017). A simple example of these matchings, which nevertheless predicts well the penetration of the plume starting cap, is: dzt dt = wt, (6.42) where zt is the level of the leading front of the cap and wt is an estimate of the mean velocity of the plume stem at the level of the cap which was estimated from the steady plume equations. Turner (1962) showed with a simple dimensional argument that for a point-source plume, the position of the starting cap is given by: zt = CpB 1/4 0 t 3/4, (6.43) where Cp is a constant. The same argument is expected to hold for lazy plumes and consequently, the data presented in figure 6.12a is scaled using a power law of the form: zt b0 := Cp  tB1/30b4/30 ωp , (6.44) 131 6. The filling of a container by a lazy plume 10−1 100 101 10−1 100 101 τ/β0 ζ t / β 0 2 3 4 5(a) log10 Γ0 4 3 10−2 10−1 100 10−1 100 τ r l / H 2 3 4 5(b) 4 3 2 1 log10 Γ0 Figure 6.12: (a) The vertical penetration depth zt/b0 of the starting plume plotted against dimen- sionless time. The solid black line (-) plots the relationship ζt/β0 = Cp(τ/β0)3/4. (b) The radial position of leading edge rl of the plume outflow current plotted against dimensionless time. The solid black line (-) plots the scaling for a purely momentum-driven current (6.48). The dashed red line (- -) plots the scaling for a purely buoyancy-driven current (6.49). Data points in (a) and (b) are colour-coded according to the plume scaled source Richardson number and cover a range of 1 ≤ Γ0 ≤ 105. where the exponent ωp = 3/4. The measurements of Diez et al. (2003) suggest that for a small-source pure (β0 . 0.01 and Γ0 ≈ 1) plume the coefficient Cp = 1.50 ± 0.03. Using data acquired by tracking the cap of the plume, a linear regression on log zt against log t is presented in figure 6.12a. A best-fit gives a value for the exponent of ωp = 0.77 ± 0.10 where the overbar indicates that the value is averaged over all the experiments. Imposing an exponent of ωp = 3/4 to a linear regression on our data, we obtain for the whole range of lazy plume behaviour, a coefficient for the power law of Cp = 1.3± 0.2. We then retain that the time required for the plume to reach the base in (6.41) corresponds to: τp ≈ C−4/3p = 0.8 ± 0.1. (6.45) Accordingly, an estimate is deduced for the time required for the leading front of the current to reach the sidewall. A common approach to determine the spreading of a gravity current is to impose a drag coefficient CD at the front, here located at a distance r = rl from the plume axis (figure 6.6), in the form drl dt = CD √ g′lhl, (6.46) where g′l := g(ρl − ρa)/ρa and hl are the vertically-averaged reduced gravity and height of the current at the front (Simpson, 1982); ρl being the vertically-averaged density. Ben- jamin (1968) showed analytically that CD = √ 2 under the assumption of no energy losses, while Simpson & Britter (1980) showed experimentally that CD = 0.91, a value we hence- forth adopt following suit from other studies (Ungarish, 2009; Slim & Huppert, 2011). A similar expression for the spreading of the first front of a radial isodensity wall jet is: drl dt = ul, (6.47) 132 6.5. Experimental results where ul is the mean velocity in the stem of the current at the front. The scalings that can be deduced from dimensional analysis for a wall jet and a gravity current are the following. For a radial isodensity wall jet, the position of the leading front follows a power law of the form: rl = CJ M 1/4 c0 t 1/2, (6.48) (e.g. Witze & Dwyer 1976; Tanaka & Tanaka 1977), where CJ is a constant and the sub- script (.)J denotes that the coefficient describes a quantity related to a purely-momentum driven current. The source momentum flux of the current is expected to scale on that of the plume at impingement Mc0 ∝ B2/30 H4/3. Kaye & Hunt (2007) measured, for the near- impingement region of the gravity current ensuing from the impingement of the plume with a horizontal boundary, a coefficient of CJ = 1.15 ± 0.1. Accordingly for a purely buoyancy-driven current, the position of the front is a function of: rl = CgcB 1/4 0 t 3/4, (6.49) (e.g. Britter 1979) where Cgc is a constant and the subscript (.)gc denotes that the co- efficient describes a quantity related to purely-buoyancy driven gravity current. Britter (1979) measured for a purely buoyancy-driven current that Cgc = 0.84 ± 0.06. The outflow current observed in the slumping regime is relatively forced near the im- pingement region as Ric0 = Rii/γ3i < 0.75 for β0 < 0.4. The currents are thus expected to transition from an inertially-dominated to a buoyancy-dominated regime. The transition between the two regimes is expected to occur at a distance proportional to the jet length of the current as measured radially from the plume axis, which can be shown, provided β0  1, to scale on the height over which the plume has fallen: rJ ∝ M3/4i B1/20 ∝ ( B2/30 H 4/3 )3/4 B1/20 = H. (6.50) In figure 6.12b, we show measurements acquired from tracking the leading front of the current from flow visualisations. Following from (6.48) and (6.49), we investigate a power law in the form: φl := C τω, (6.51) to which we obtain a best linear regression fit for the exponent of ω = 0.6 ± 0.2 averaged over all experiments. The value of ω lays between 1/2 and 3/4 and thus suggests that the flow is indeed transitioning between wall jet and pure gravity current spreading rate power laws. This is also evident in the plume impingement measurements shown later in figure 6.16 in which measurements of the momentum flux in the current show how the current transitions from a constant value of momentum flux (wall jet-like) to an increase of momentum flux (gravity current-like) sufficiently downstream of the impingement region. Imposing ωJ = 1/2 and ωgc = 3/4 to the power laws shown in (6.51), yields coefficients for the linear regression respectively of CJ = 1.0 ± 0.4 and Cgc = 0.8 ± 0.3. The reasons for the relatively large error on the scalings will become readily apparent soon as we compare scalings to the model and are compounded within a combination of effects due to the transition of the current from inertia to buoyancy-driven regimes, deviations arising from increasing β0 and errors associated to estimating τp for the starting plume. 133 6. The filling of a container by a lazy plume We estimate that transition between inertially and buoyancy dominated behaviour oc- curs over a distance of φJ = rJ/H ≈ 0.5 − 1.0 by noting the change in gradient of data trends in the log-log plot and the velocimetry measurements presented later. Kaye & Hunt (2007) suggest that φJ = 7.1γ6i , a value which is hard to pin down due to the sensitivity on the accuracy of γi, as based on (6.57), φJ = 1.6±0.7. We here retain that the time required for the current to reach the corner can be well approximated by a prediction based solely on a wall jet-like power law, i.e. taking ωJ = 1/2, and that this time approximates to: τc ≈ ( φ CJ )2 = (1.0 ± 0.4)φ2. (6.52) Despite our best efforts, tracking a leading front for the inflow current proved to be unreliable in that it was difficult to discern systematically whether structures were part of the outflow or inflow currents. The scaling for the buoyancy-driven current (6.49) is arguably more applicable for the inflow current and we thus suggest that the time required for the inflow current to reach the plume and thus to form the initial layer is: τr ≈ ( φ Cgc )4/3 = (1.3 ± 0.1)φ4/3. (6.53) The sum of the time scalings deduced in (6.45), (6.52) and (6.53) indicates that the aver- age time required to form the initial layer corresponds to: τS = 0.8 ± 0.1 + (1.0 ± 0.4)φ2 + (1.3 ± 0.1)φ4/3. (6.54) The measured formation time obtained from the time series analysis, denoted τS ,m, is compared to (6.54) in figure 6.13 to show the predictive capabilities of this expression. Measurements of the initial depth of the buoyant layer shown in figure 6.13b suggest that this depth is relatively independent of φ, β0 and Γ0 and was measured to be propor- tional to the height of the container at: ζS = 0.33 ± 0.08. (6.55) There is a slight increase in initial layer depth with increasing β0, which is to be expected as β0 affects the depth of the outflow and inflow currents forming the layer. An interesting comparison can be drawn here to the original theory of Baines & Turner (1969) to see how the time required to fill the layer to one third of the container height τH/3 compares between slumping and a classic displacement filling description. Using the expression for the prediction of the deepening of the buoyant layer deduced by Baines & Turner (1969), an estimate for this time is given by: τBT,H/3 ≈ 32 5 6αp ( 10 9αp )1/3 (31 )2/3 − 1  φ2 ≈ 34.7φ2. (6.56) It is evident that if the theory of Baines & Turner (1969) is applied to filling in the slump- ing regime, a prediction for front level would be considerably underestimated, here for 134 6.5. Experimental results 101 102 103 104 105 106 0 0.5 1 1.5 2 Γ0 τ s ,m /τ s (a) τ S ,m /τ S Γ0 101 102 103 104 105 106 0 0.2 0.4 0.6 0.8 1 Γ0 ζ s 0.8 1 1.2 (b) ζS = 0.33 ± 0.08 φ ζ S Γ0 Figure 6.13: (a) Measured dimensionless formation time of initial layer τS ,m estimated by the time at which the minimum peak in the time series occurs (cf. figure 6.11), divided by the prediction for the formation time deduced in (6.54). (b) Initial dimensionless layer depth estimated by the characteristic trough that succeeds the peak overturning recorded in each time series signal. ζ f = 1/3, by a factor between 15-54 depending on φ. The scalings in (6.54) are now compared to numerical solutions based on conservation equations to further explore their dependence on β0 > 0, Γ0 > 1 and φ. In figure 6.14a, the dimensionless time for the plume to reach the base, τp (6.45), approximated here by the scalings of a point-source turbulent plume, is compared to solutions to the plume con- servation equations (6.2)-(6.6) with the front conditions specified by (6.42). An increase in τp may be observed for increasing β0. This increase in τp with β0 can be understood in terms of the increase in source buoyancy flux B0, on which τp is scaled, required to achieve sufficiently high Γ0 for the plume to be pure or lazy (Γ0 ≥ 1). For larger sources, β0 & 0.01, these solutions become dependent on the scaled source Richardson number Γ0. The time for the plume to reach the floor, τp, increases with Γ0 up to about Γ0 ≈ 103 and then slightly reduces from this value owing to modified near-source entrainment effects. In figure 6.14b, solutions for the dimensionless time for the current to reach the side- wall, τc (6.52), which are based on the front conditions drl/dt = CD √ g′lhl, i.e. (6.46), and drl/dt = ul, i.e. (6.47), are compared with the scalings for a wall-jet (6.52) and a gravity current (6.53) for a point-source plume (β0 = 0). The front condition for the gravity current specified by (6.47) reveals to be equivalent to the gravity current scaling (6.53). The front condition for the wall jet specified by (6.46) underestimates the wall-jet scaling (6.52) and this is mostly due to the fact that the outflow current transitions to the buoyancy-driven power law at sufficient distances from the plume axis. In figure 6.14c, predictions are shown based on the front condition for the gravity current (6.46) for increasing β0 and Γ0. The time for the current to reach the corner, τc, maintains a power-law dependence of τc ∼ φ4/3 for varying β0 and Γ0; τc generally in- creases with β0, and decreases with Γ0 for 1 ≤ Γ0 . 103 and increases for Γ0 & 103 (in an analogous manner to the solutions for τp). Following this discussion, it becomes clearer why the data in figure 6.12b does not simply scale on the point-source plume scalings. We conclude the discussion by noting that simple dimensional scalings are now avail- 135 10−4 10−3 10−2 10−1 0 0.5 1 1.5 2 β0 τ p Γ0 = 1 Γ0 = 10 Γ0 = 100 Γ0 = 1000 Γ0 = 10000 Γ0 = 100000 Γ0 = 1000000 (a) starting plume 0.6 0.8 1 1.2 0.5 1 φ τ c (??) (??) (??) (??) (b) (6.52) (6.45) (6.46) (6.47) φ2 φ4/3 0.6 0.8 1 1.2 0.4 0.6 0.8 1 φ τ c β0 = 0.3 β0 = 0.1 β0 = 0.01 β0  1 (c) outflow current Figure 6.14: Plots used to discuss the dependencies of a point-source plume scalings on β0, φ, and Γ0. In (a), the time required for the starting plume to reach the base plotted against β0. In (b)-(c), the time required for the plume outflow current to reach the sidewall plotted against φ. In (b), the scalings for wall jet-like spreading (6.52) are shown as a dotted line (- -) and the scalings for a purely buoyancy-driven gravity current (6.53) are shown as a solid line (-). These scalings are compared to solutions based on the conservation equations (6.19)-(6.22) with front conditions specified by a ‘stem-velocity’ condition (6.47) and ‘drag-coefficient’ condition (6.46), shown as diamond () and circle (◦) markers, respectively. (c) Solutions to the conservation equations with front conditions specified by the drag-coefficient condition (6.46) are plotted for β0 = {0.01, 0.1, 0.3} and Γ0 = {1, 100, 10000} to show the variation from the scalings for a point-source plume. 6.5. Experimental results 0 10 20 0 0.2 0.4 0.6 0.8 1 Γ0 ≈ 30 (a) ζ f 0 10 20 0 0.2 0.4 0.6 0.8 1 Γ0 ≈ 700 (b) 0 10 20 0 0.2 0.4 0.6 0.8 1 Γ0 ≈ 4000 (c) 0 10 20 0 0.2 0.4 0.6 0.8 1 Γ0 ≈ 8000 (d) ζ f 0 10 20 0 0.2 0.4 0.6 0.8 1 Γ0 ≈ 12000 (e) (τ − τS ) φ−2 0 10 20 0 0.2 0.4 0.6 0.8 1 Γ0 ≈ 480000 (6.4) α = cst (f) Figure 6.15: The horizontally-averaged interface level plotted against dimensionless time for dif- ferent experiments of increasing source flux parameter Γ0 in which the container was of equivalent radius-to-height aspect ratio φ ≈ 1. The solid line (-) indicates the prediction for the interface level based on a constant-α entrainment model, taking α = 0.09. The dashed line (- -) shows the pre- diction for the interface level based on the entrainment function (6.4). able to estimate τS and ζS for β0  1. These scalings agree with descriptions based on the conservation equations presented in §6.3. We have explored the extent of this agree- ment with the relative size of the source radius, i.e. β0, noting how for β0 & O(10−1) the solutions with a specified front behaviour deviate from the dimensional scalings and can be used as substitutes for the prediction of τS . 6.5.3 Filling rates At the end of the initial stage of filling, a dense layer is formed in the lower portion of the container. A stable density interface separates the layer from the original ambient en- vironment. The plume continuously penetrates through the density interface introducing a volume flux into the layer. The ascent of the interface is therefore dictated, following a simple continuity argument (6.3), by the volume flux of the plume entering the layer relative to the cross-sectional area of the layer itself (6.37). Figure 6.15 shows data for the level of the interface plotted against dimensionless time for six sample experiments of increasing Γ0. The data is compared to predictions based on solutions to the plume conservation equations (6.10)-(6.12). Two entrainment models are compared for the predictions: that of Morton et al. (1956) which assumes a constant value of α = 0.09 and the more generalised form of van Reeuwijk et al. (2016), (6.4), which assumes α = f (Γ). It is evident that for increasing Γ0, predictions based on constant-αmodels significantly underestimate the filling rate and that the revised entrainment function is able to predict the level of the interface reasonably well. We 137 6. The filling of a container by a lazy plume further discuss the role of β0 and Γ0 on the rate of deepening of the layer in §6.6.3. 6.5.4 Impingement This section regards the source momentum flux of the plume outflow current which was introduced in §6.3.2 (via γi) . Coefficient γi is estimated from the measurements of the impingement region acquired through particle image velocimetry (described in §6.4.2). Data from these measurements is presented in figure 6.16 for an example experiment. The volume and momentum fluxes in the plume and current were measured by directly integrating the velocity field. In figure 6.16c, the volume and momentum fluxes in the plume and current are compared to show how these are conserved passing through a cylindrical control volume of a depth and radius comparable to the radius of the plume at impingement bi. Measurements of volume fluxes in the plume are here compared to the analytical solutions for pure point-source plumes developed by Morton et al. (1956) to validate the velocimetry data. Only small values of β0 = O(10−2) with Γ0 ≈ 1 were measured to simplify validation. The magnitude of γi is estimated by comparing the momentum flux in the plume at a distance of bi/H from the base to that of the current at the same distance bi/H from the plume axis. The ratio of the momentum fluxes measured was γi = 0.78 ± 0.05, (6.57) averaged over five experiments where the distance from the source H was varied system- atically so that 0.01 ≤ β0 ≤ 0.06 (figure 6.16e). A discussion regarding the kinetic energy losses due to the impingement of the current against the sidewall, encapsulated by γu and introduced in §6.3.4, is presented here. The penetration depth of the corner intrusion is plotted in figure 6.17a-b. For relatively small sources β0  1, the penetration depth of intrusions scales on the height of the container (figure 6.17a), in agreement with the result of Kaye & Hunt (2007), increasing in value for larger β0. In an attempt to estimate γu, in figure 6.17b, the penetration depth of the corner upflow is plotted against the predicted local Richardson number of the current at impingement with the sidewall. The power law for the rise height of two-dimensional forced fountains rise heights (6.27) is compared to the penetration depth data taking dif- ferent values γu. These comparisons suggest that, given the uncertainty associated with the impingement head loss of the plume and in the characterisation of entrainment in the current, there appears to be no obvious way for us to discern, accurately, the value of γu. We retain here that we expect γu to maintain a similar value to that of plume impingement (i.e. γu ≈ γi ≈ 0.8) and range between 0.7 . γu ≤ 1. 6.5.5 Entrainment in the plume outflow current Measurements of entrainment in the plume outflow are presented in figure 6.16f. These measurements suggest higher values by αc = 0.15 ± 0.05, in closer agreement with the data acquired for thermal plume outflows of Ezzamel (2011). The increased entrainment measured in radial plume outflows, as compared to the measurements of Ellison & Turner (1959), may have been anticipated in that, observationally, the largest scale vortices of the plume as it impinges against the floor appear to persist in structure as they transform into the concentric vortex rings that are observed along the stem of the current. This is 138 −0.9 −0.8 −0.7 z /H 0.02 0.04 0.06 0.6 0.7 0.8 0.9 1 β0 γ(a) (e) −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 −0.9 −0.8 −0.7 r/H z /H (b) 0 1 2 3 4 5 6 −1 −0.8 −0.6 −0.4 −0.2 0 Q/Qi z /H 0 0.2 0.4 0.6 0.8 1 r/ H (c) plume current bi/H 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 M/Mi z /H 0 0.2 0.4 0.6 0.8 1 r/ H (d) plume bi/H current 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0 0.05 0.1 0.15 0.2 (f) α c r/H ◦ Ciriello (2018) ∇ Ezzamel (2011) - - E&T(1959) Figure 6.16: Measurements for an example LIF+PIV impingement experiment (illustrated in fig- ure 6.10b). For the data shown, the source conditions were b0 = 10.7 mm, w0 = 63.6 mm/s, g′0 = 90.5 mm/s 2 and Γ0 = 1.65 for a plume that impinged against a rigid horizontal plate at a distance of H = 323 mm. Time-averaged picture of (a) the scalar field and (b) the velocity field superimposed on contours of velocity magnitude (the vector spatial resolution shown in figure was reduced by a factor of form from that measured). (c) Measured volume fluxes in the plume and current compared to the point-source solutions of Morton et al. (1956). (d) Measured momentum fluxes in the plume and current. Note that the data that is presented in (c) and (d) for the fluxes in the current is calculated for both sides of the current. (e) Measurements of γi := Mc0/Mi. (f) Measurements of the current entrainment coefficient, αc := dQc/dr |Qc/Mcr|, compared to data from (Ezzamel, 2011) – see legend. 6. The filling of a container by a lazy plume 101 102 103 104 105 106 0 0.2 0.4 0.6 0.8 1 Γ0 z u t /H (a) zut/H ≈ 0.38 (KH07) 0.02 0.04 0.06 0 5 10 15 20 Riu0 (z u t − h c i) /b u 0 0.8 1 1.2 γu ︸︷︷︸ 0.7 0.8 0.9 1.0 φ (b) (6.27) Figure 6.17: Peak penetration depth of overturning motion plotted against (a) the plume scaled source Richardson number (Γ0) and (b) the estimated source Richardson number (Riu0) of the corner upflow. The size of the marker is proportional to the relative size of the plume source (0.1 ≤ b0/H ≤ 0.4). The colour of the marker indicates the aspect ratio of the container as specified by the colourbar. analogous to what reported in free-shear radial outflows, that is, radial jets in the absence of a floor. Witze & Dwyer (1976), for example, showed that entrainment was increased by a factor of two in a ‘freely-impinging’ radial jet formed by the impingement of two round jets due to these structures, as opposed to that measured in ‘constrained’ radial jets which ensue from a radial nozzle (entrainment in constrained radial jets was also measured by Tanaka & Tanaka 1977 and agreed well the constrained radial wall jets measured by Witze & Dwyer 1976). 6.6 Discussion The model developed in §6.3 gives predictions for the bulk mean velocities and densities in the container at any given time and for different combinations of β0, Γ0 and φ. To breakdown the physics governing the many interactions occurring in the container, we uncover the structure of these flows in three ways. We initially discuss a single example of fixed β0, Γ0 and φ to showcase the information that can be extracted from the model and better understand how the stratification evolves. Then, we explore how the structure of the slumping layer changes with varying β0, Γ0 and φ by inspecting solutions which, to simplify the comparison, neglect the effect of a time-varying stratification. Finally, we discuss how the stratification evolves after the formation of the initial layer, i.e. during the displacement filling stage, for varying β0, Γ0 and φ. 6.6.1 The evolution of the stratification We discuss how the flow evolves in the container by examining an example case with β0 = 0.1, φ = 1 and Γ0 = 100. Solutions to the model presented in §6.3 are shown in figures 6.18 and 6.19 as contours of dimensionless mean reduced gravity and velocity magnitude. In figure 6.18, these quantities are scaled on the value at source, while in figure 6.19, these are scaled on the value predicted in the plume at the level of impingement with the 140 6.6. Discussion base. Key frames in the filling process are shown: (a) the peak penetration of the corner intrusion, (b) the formation of the initial slumping layer, and (c)-(d) times at which the container is 50% and 75% full. The main points to draw from figures 6.18 and 6.19 are as follows. If the plume source is lazy (Γ0 > 1), then the plume accelerates in the proximity of the source. Following the accelerating region, the mean plume velocity decays as w ∼ z−1/3 and the plume spreads linearly (b ∼ z). The plume outflow current also accelerates and contracts in the proximity of the impingement region. The current flows at approximately constant depth (h ∼ r0) in which the velocities decay linearly with radial distance (u ∼ r−1). The initial corner upflow rises higher than the subsequent upflow. This is due to the exchanges that arise as the inflow currents form, where fluid is drained from the outflow current thereby reducing its velocity and depth. The inflow substantially increases in depth as it converges towards the centre of the container. After the formation of the initial layer, the internal structure of the flow does not radically change as the stratification develops. The velocities marginally decrease as the plume enters the layer, with the layer becoming progressively denser. 6.6.2 The internal structure of the slumping layer Solutions for the plume and currents are shown in figures 6.20-6.22 for β0 = {0.01, 0.1, 0.3} and φ = {0.66, 1, 1.25} respectively for Γ0 = {1, 100, 10000}. These figures show the con- tours of dimensionless mean velocity and reduced gravity in each flow region. Different ranges are used for the velocity contours as larger accelerations are experienced in the near-field of a plume for increasing Γ0 (for Γ0 = 1, the plume source velocity is the largest velocity in the container, while for Γ0 = 10000, the velocity increases by a factor of more than 20 compared to the source value). The large accelerations experience by the plume concur with an increase in dilution. It is evident by comparing figures 6.20 and 6.22 that, for fixed β0 and φ, the reduced gravity in the plume and current at impingement are substantially reduced from their source value with increasing Γ0. Beyond the change in dynamics (understood here as velocity and reduced gravity variations), it becomes interesting to discuss the relative width and depth of the regions as opposed to the radius and height of the space. It is evident that increasing radius of the plume source β0, increases the size of each region; for example, increasing portion of the space occupied by the two currents (a discussion on the rise height of the corner upflow is provided in §6.6.4). 6.6.3 Late stage filling rates Figure 6.23 shows predictions based on the displacement filling description presented in §6.3.6 for the location of the first ascending front between the buoyant layer and the original ambient for times exceeding the initial layer formation time. Figures 6.23(a) and (d) show that, provided the source is small β0 < O(10−2), predictions for the front collapse onto a single curve solution. This solution corresponds to an offset, based on (τs, ζs), to the solution of Baines & Turner (1969): τ − τS = 5 αp ( 10 9αp )1/3 φ2 1 − ( ζ f − ζS )2/3 . (6.58) 141 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 |u|/w0 (scaled on source values) 0 0.2 0.4 0.6 0.8 1 (a) reduced gravity velocity βu singular at ζut 1.5 2.8 5.2 10.3 τ/φ2 peak corner intrusionζ 0 0.2 0.4 0.6 0.8 1 (b) initial layer formation 0 0.2 0.4 0.6 0.8 1 (c) 50% full −1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 1 (d) 75% full r/H 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 g′/g′0 Figure 6.18: Solutions to the model presented in §6.3 showing how the flow evolves in time for an example case with β0 = 0.1, φ = 1 and Γ0 = 100. The solutions are shown as contours of mean reduced gravity (LHS, bottom colourbar) and velocity (RHS, top colourbar) scaled on their source value (i.e. w0 and g′0, respectively). White lines outline the five regions. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 |u|/wi (scaled on values at impingement) 0 0.2 0.4 0.6 0.8 1 (a) reduced gravity velocity 1.5 2.8 5.2 10.3 τ/φ2 peak corner intrusionζ 0 0.2 0.4 0.6 0.8 1 (b) initial layer formation 0 0.2 0.4 0.6 0.8 1 (c) 50% full 0 0.2 0.4 0.6 0.8 1 (d) 75% full r/H 0 1 2 3 4 5 6 7 8 9 10 g′/g′i Figure 6.19: Solutions to the model presented in §6.3 showing how the flow evolves in time for an example case with β0 = 0.1, φ = 1 and Γ0 = 100. The solutions are shown as contours of mean reduced gravity (LHS, bottom colourbar) and velocity (RHS, top colourbar) scaled on their predicted value in the plume when it impinges against the base (i.e. wi and g′i , respectively). White lines outline the five regions. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 |u|/w0Γ0 = 1 −0.5 0 0.50 0.2 0.4 0.6 0.8 1 β00.01 0.1 0.3 φ 0.66 1.00 1.25 reduced gravity velocity −0.5 0 0.50 0.2 0.4 0.6 0.8 1 (ζut > φ) −1 −0.5 0 0.5 10 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 10 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 10 0.2 0.4 0.6 0.8 1 ζ −1 0 10 0.5 1 −1 0 10 0.5 1 −1 0 10 0.5 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 g′/g′0 Figure 6.20: Solutions to the conservation equations describing each region in the flow (plume, currents, and corner upflow), taking N2 = 0, for Γ0 = 1 and different combinations of β0 = {0.01, 0.1, 0.3} and φ = {0.66, 1, 1.25}. Solutions are shown as contours of mean re- duced gravity (LHS, bottom colourbar) and velocity (RHS, top colourbar) scaled on their source values. The plot for β0 = 0.3 and φ = 0.66 (top-right) is not shown, as slumping is not expected to occur for these parameters. White lines outline the plume, currents and corner up and downflows. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 |u|/w0Γ0 = 100 −0.5 0 0.50 0.2 0.4 0.6 0.8 1 β00.01 0.1 0.3 φ 0.66 1.00 1.25 reduced gravity velocity −0.5 0 0.50 0.2 0.4 0.6 0.8 1 −0.5 0 0.50 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 10 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 10 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 10 0.2 0.4 0.6 0.8 1 ζ −1 0 10 0.5 1 −1 0 10 0.5 1 r/H −1 0 10 0.5 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 g′/g′0 Figure 6.21: Solutions to the conservation equations describing each region in the flow (plume, currents, and corner upflow), taking N2 = 0, for Γ0 = 100 and different combinations of β0 = {0.01, 0.1, 0.3} and φ = {0.66, 1, 1.25}. Solutions are shown as contours of mean re- duced gravity (LHS, bottom colourbar) and velocity (RHS, top colourbar) scaled on their source values. White lines outline the plume, currents and corner up and downflows. 0 2 4 6 8 10 12 14 16 18 20 |u|/w0Γ0 = 10000 −0.5 0 0.50 0.2 0.4 0.6 0.8 1 β00.01 0.1 0.3 φ 0.66 1.00 1.25 reduced gravity velocity −0.5 0 0.50 0.2 0.4 0.6 0.8 1 −0.5 0 0.50 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 10 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 10 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 10 0.2 0.4 0.6 0.8 1 ζ −1 0 10 0.5 1 −1 0 10 0.5 1 −1 0 10 0.5 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 g′/g′0 Figure 6.22: Solutions to the conservation equations describing each region in the flow (plume, currents, and corner upflow), taking N2 = 0, for Γ0 = 10000 and different combinations of β0 = {0.01, 0.1, 0.3} and φ = {0.66, 1, 1.25}. Solutions are shown as contours of mean reduced gravity (LHS, bottom colourbar) and velocity (RHS, top colourbar) scaled on their source values. White lines outline the plume, currents and corner up and downflows. 6.6. Discussion The filling rate collapses onto (6.58) on decreasing the relative size β0 as these conditions correspond to the point-source limit. In the original Baines & Turner (1969) formulation, the interface level asymptotes to the level of the source due to the idealisation of the plume source as one of zero volume flux. Filling rates significantly increase with β0. For the solutions presented in figure 6.23, the interface level rises more quickly than what predicted by the solution of Baines & Turner (1969). The interface also rises above that of the source due to the non-zero source volume flux. These effects are prevalent for relatively larger sources, β0 & O(10−1). Note that solutions based on classic virtual origin offsets (e.g. Hunt & Kaye 2001) are expected not to work for β0 & O(10−1), owing to the limited validity of volume flux predictions in the proximity of the source (a near-source region which now occupies a substantial portion of the space). A crucial result, presented in Chapter 4 and discussed in §6.5.3, is that constant-α models do not suitably describe the entrainment in the near-source region of sufficiently lazy plumes (Γ0 & 103). In figures 6.23d-f, we show how substantially this increased near-source entrainment changes the filling rate relative to a constant-α description. An example of the evolution of the horizontally-averaged density profiles in the en- vironment, and as a result in the plume, is shown in figure 6.24a-c. For β0 = 0.01, by noting the extremes of the log-scaled axis in figure 6.24a, it becomes evident that the plume dilutes considerably before it has reached the base. The stratified environment has thus little effect on the plume itself. For larger magnitudes of β0, figures 6.24b-c, the solutions suggest that there are more pronounced interactions between the plume and the environment, as the plume entrains increasingly denser fluid as it enters the layer. It is remarkable how different the patterns of stratification can be, owing to how the variation of volume fluxes changes with β0 and Γ0 over the vertical extent of the plume. 6.6.4 Expected regime bounds In this section, simple theoretical arguments are proposed to delimit bounds for the dif- ferent regimes of filling identified in Chapter 5 in terms of the dimensionless parameters: β0, φ and Γ0. Plume breakdown Breakdown occurs for small-source (β0  1) pure or lazy plumes for aspect ratios of φ ≤ 0.25 (Barnett 1990). The regime can be readily recognised by the formation of a turbulent mixing region at a level of z/R ≈ 4 which ‘plugs’ the container. As explained by Barnett (1991), the mixing region forms due to the shear the plume experiences from the return flow, which indeed increases in magnitude when the container becomes taller and narrower. Barnett showed that by including confinement effects in the plume conservation equations, these equations become singular at a level zb below the source and that this level observationally coincided with the top of the mixing region. Below the level of the mixing region, Barnett (1990) showed that the filling-box flow could be described as a one-dimensional turbulent convective flow: ∂g′e ∂t = κT ∂2g′e ∂z2 for z ≥ zb, (6.59) 147 6. The filling of a container by a lazy plume 0 20 40 0 0.5 1 (a) β0 0.01 0.1 0.3 ζ f α = cst 0 20 40 0 0.5 1 (b) 0 20 40 0 0.5 1 (c) 0 20 40 0 0.5 1 (d) ζ f α = f(Γ) 0 20 40 0 0.5 1 (e) (τ − τs)φ−2 0 20 40 0 0.5 1 (f) increased filling rates 0 1 2 3 4 5 6 7 8 9 10 log10 Γ0 Figure 6.23: Comparison between first front predictions based on constant-α models, shown in (a)-(c), and those based on the revised entrainment parameterisation (6.4), shown in (d)-(f). The time coordinate is divided by φ2 to obtain a collapse for predictions of varying φ. The predictions are shown for β0 = {0.01, 0.1, 0.3} and for different values of scaled source Richardson number Γ0, denoted by the colourbar. where according to his own measurements, the turbulent diffusivity, κT = 1.5 m2/s. We here extend when we might anticipate breakdown to occur for larger values of β0. To do so, we revert to the plume conservation equations and deduce numerically the level at which these become singular. The level at which the singularity occurs is denoted zb and is evaluated as a function of β0, φ and Γ0. The location of the breakdown level zb based on this criterion is shown in figure 6.25. We expect that if the equations are singular at source, i.e. ζb = 0, the plume is not expected to develop. Filling is expected to resemble a one-dimensional turbulent convective flow. For solutions which became singular at a level between the source and the base, i.e. 0 < ζb ≤ 1, the plume is expected to break down into a turbulent mixing region. For solutions that did not become singular, it is expected that the plume reaches the base of the container and can thus exhibit either rolling, slumping, blocking or displacement filling behaviours. Rolling Rolling occurs for small-source (β0  1) plumes over aspect ratios of 0.25 ≤ φ ≤ 0.66. We here extend when we might anticipate rolling to occur for larger values of β0. Rolling 148 10−4 10−3 10−2 10−1 100 0 0.2 0.4 0.6 0.8 1 environment (–) plume (- -) β0 0.01 0.1 0.3 Γ 0 = 1 10−1 100 0 0.2 0.4 0.6 0.8 1 (ζut > φ) 10−5 10−4 10−3 10−2 10−1 100 0 0.2 0.4 0.6 0.8 1 ζ Γ 0 = 100 10−3 10−2 10−1 100 0 0.2 0.4 0.6 0.8 1 10−1 100 0 0.2 0.4 0.6 0.8 1 10−6 10−4 10−2 100 0 0.2 0.4 0.6 0.8 1 Γ 0 = 10000 10−4 10−3 10−2 10−1 100 0 0.2 0.4 0.6 0.8 1 g′/g′0 10−3 10−2 10−1 100 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 (τ − τS )φ−2 Figure 6.24: Predicted reduced gravity profiles based on (6.39) for different combinations of β0 = {0.01, 0.1, 0.3} and Γ0 = {1, 100, 10000}. The solid and dashed lines plot the reduced gravity of the environment and plume as a fraction of the source reduced gravity, respectively. 6. The filling of a container by a lazy plume 0.25 0.66 1.25 2 0 0.1 0.2 0.3 0.4 0.5 Γ0 = 1 β0 turbulent diffusion .2.4.6.81 no breakdown φb = 0.25 for β0 → 0 (Barnett 1990) 0.25 0.66 1.25 2 Γ0 = 100 no breakdown turbulent diffusion φ 0.25 0.66 1.25 2 Γ0 = 10000 no breakdown turbulent diffusion Figure 6.25: Contours of breakdown level, ζb := zb/H = {0.2, 0.4, 0.6, 0.8, 1.0} obtained as the level from the plume source at which equations (6.10)-(6.12) become singular (n.b. taking N2 = 0), plotted for varying φ, β0 for Γ0 = {1, 100, 10000}. is expected to occur if zut ≈ R, that is, if the vertical intrusion of the corner upflow is comparable to the radius of the container. This condition is suggested as one might expect that if zut ≈ R, then there would not be sufficient space for the toroidal vortex, the distinctive feature of rolling, to unwind and form the inflow current. In figure 6.26, we estimate how zut/R varies for varying degree of β0,Γ0 and φ using solutions to the model presented in §6.3. Given the uncertainty associated with the scaling of rise height, we extend the parameter space for which rolling is expected to the values suggested in the limit of the point-source plume. With regard to the modelling of this regime, the filling observed during rolling can be idealised in the same manner as is presented in this chapter for slumping. The initial layer formation time is expected to correspond to the time required for the current to reach the corner τS = τp +τc. The initial layer depth ζR is expected to correspond to the penetration depth of the intrusions, which varies for β0  1 varies as ζR = ζut = 0.33φ−1/3 (Kaye & Hunt 2007) or which alternatively could be estimated as done for figure 6.26. Blocking Blocking occurs for small-source (β0  1) plumes over aspect ratios of 1.25 ≤ φ ≤ 2.0. Blocking occurs when buoyant fluid from the outflow current accumulates in the bottom corner. The accumulated corner fluid then intrudes underneath the original plume outflow until it has propagated back to the centre of the container. Admittedly, a consensus is yet to be reached to agree on what governs the dynamics of blocking. It stands to reason nevertheless that slumping does not occur if the corner upflow does not develop. We revert to figure 6.26 and note that that the predicted penetration depth of the corner intrusion for a small-source plume is comparable to the thickness of the current ζut ≈ βi ≈ 0.1 at the transition between slumping at blocking φ. We extend this result to large-area sources by suggesting that slumping cannot occur if the penetration depth of the corner intrusion is comparable to the size of the current at impingement, zut ≈ hi. Following from this argument, it appears, from figure 6.26, that blocking is expected to occur only for relatively smaller plume source radii, β0  1. 150 6.7. Conclusion contours of zut/H .1 .2 .3 .4 .5 .6 .7 .8breakdown 1 100 10000 Γ0 β0 .1 .2 .3 .4breakdown φ .1 .2 .3 breakdown blocking contours of zut/R breakdown .1 .3 .5.7 β0 breakdown φ .1 .2.3 .4 breakdown .1 .2 .3 rolling slumping blocking Figure 6.26: Contours of dimensionless corner upflow penetration depth scaled as (a) zut/H and (b) zut/R plotted against container aspect ratio φ and relative source size β0 for Γ0 = {1, 100, 10000}. 6.7 Conclusion Filling-box models describe the stratification formed by a turbulent plume in a confined space. This chapter focussed on a single regime of filling-box behaviour known as slump- ing. During slumping, two counterflowing currents form at the bottom of the container af- ter the plume impinges against the base. A model was developed to describe the dynamics of these flows and to predict how these dynamics change for different plume source con- ditions and container geometry. The latter are encapsulated by the plume scaled source Richardson number Γ0, the radius-to-height aspect ratio of the container φ := R/H and the relative size of the plume source of radius b0 to the height of the container, H: denoted β0 := b0/H. The filling-box flow was understood as being composed of two phases in time: the initial phase in which regions (plume, outflow current, corner upflow, inflow current and displacement filling layer) were formed and the late stage, where all regions were es- tablished. This allowed us to isolate key physics within each regions and model their interactions. A model was developed to predict the bulk velocities and densities in each region as the flow evolves with time. The modelling of turbulent mixing into each region of the flow was discussed alongside the appropriateness of choices regarding entrainment functions for each region. The following scalings were deduced as the key assumptions in the model were ad- 151 6. The filling of a container by a lazy plume dressed experimentally. The initial depth of the slumping layer is approximately one third of the height of the container and is observed to be weakly dependent on β0, φ and Γ0. This layer forms in a time which can be estimated by: τS = 0.8 + 1.0φ2 + 1.3φ4/3. (6.60) Detailed measurements on impingement enabled the use of key assumptions presented within the model. The idealised momentum flux of the plume outflow current corresponds to approximately 80% of the plume impingement momentum flux. An estimate of the impingement head losses at the corner suggest that these losses are similar in magnitude to those measured in the plume as it impinges with the base. With regards to applications, there are several elements of the work presented in this chapter which are worth highlighting. A reduced-order model, such as the one devel- oped in §6.3, can explain what governs the dynamics of a flow and can help predict how the flow may be controlled at source. In engineering, it is common to need to predict how velocities, densities and temperatures vary within confined spaces in applications such as ventilation (Linden, 1999), fire egress design (Zukoski, 1978; Hurley, 2016), the modelling of geophysical and environmental flows (Turner, 1986; Woods, 2010). An important practical implication of this study is related to the spatial dispersal of plume fluid within an environment. During slumping, this dispersal is enhanced com- pared to the displacement filling description of Baines & Turner (1969) and is further enhanced when the sources of the plume may not be considered small relative to the base area or height of the container. Sources of convective flows are often spread over an area of finite extent. Examples include flows originating from the heat exchange between air and a floor patch that is warmed by sunshine (Hunt & Kaye 2005) or those observed when buoyant fluid is driven through an open vent (Hunt & Holford 2000, Michaux et al. 2017). The implications of an enhanced mixing of the flow are particularly significant in terms of pollutant dispersal within these contexts. If the plume is carrying pollutants, then these may spread relatively quickly over larger portions of the space. It follows for example that if this was a smoke plume ensuing from a fire, smoke particulates would fill a room more rapidly than expected and thus imminently increase the risk of asphyxiation (Zukoski, 1978; Cooper, 1982, 1983; Hurley, 2016). As a result of the work presented herein, it becomes evident that within these contexts, classic models need to be revised so that they do not underestimate the spatial extent over which pollutants disperse. 6.A A theoretical model for a non-idealised two-dimensional ‘line’ plume in a filling box during the slumping regime Following from the theoretical framework presented herein, the model for the slumping can be readily extended for a two-dimensional configuration for a plume issuing from a rectangular slot source that extends a length L over the depth of the container and placed at a distance of W/2 from the other two sidewalls and H from the base. The conservation equations for each flow region can be deduced following the same process explained in §6.3. These would be for the plume: dwb dz = α(w − U), (6.61) 152 6.A. A theoretical model for a non-idealised two-dimensional ‘line’ plume in a filling box during the slumping regime d dz ( w2b + U2(W − 2b) ) = g′b − αwUb, (6.62) dg′wb dz = −wb2N2 − αwg′e; (6.63) and for the outflow and inflow currents: duchc dx = αc(uc − ur), (6.64) du2chc dx = −1 2 d dx ( (g′c − g′r)h2c ) − αcuruc, (6.65) d dx ( g′cuchc ) = −αcurg′c, (6.66) durhr dx = αc(ur − uc), (6.67) du2r hr dx = −1 2 d dx ( g′rh 2 r ) − αcucur, (6.68) d dx ( g′rurhr ) = −αcuchcg′r. (6.69) For the corner upflow, the conservation equations are equivalent to those presented in (6.26). 153 chapter 7 Conclusions Each previous chapter concludes with a summary of the relevant findings presented therein. In this final chapter, these are brought together to form a complete summary before the final conclusions are drawn. 7.1 Summary of research undertaken The dynamics of a turbulent axisymmetric lazy plume (viz. highly buoyant releases typi- fied by a high source Richardson number Γ0) in a confined space were investigated. The investigation was motivated by the lack of information available concerning the stratifi- cation produced by a lazy plume. The flow patterns that occur in the space were studied both theoretically and experimentally. These flows were interrogated at laboratory scale in a series of experiments observing the release of aqueous saline solutions in freshwater environments. The observations and measurements acquired throughout the experimental campaign guided the development of: (i) a classification for filling-box flows, (ii) a theo- retical model for a lazy plume and (iii) a theoretical model for the plume in a filling box. In each case, the theoretical models were validated by comparisons with the measured data. En route to the formulation of a more comprehensive understanding of these flows, several questions were raised that pertain to our wider understanding of convection. Chapter 4 discussed the behaviour of lazy plumes that are formed in an unbounded environment. The focus of the chapter revolved around the mixing dynamics of lazy plumes in their near-source region. The lazy plumes of interest contract to a neck in the proximity of the source due to a pronounced gravitational acceleration. Two sub-regimes for contracting lazy plumes were identified. First, this campaign revealed that lazy plumes formed by a source whose Richardson number ranges between 1 < Γ0 . −2RT ≈ 900 (RT being a dimensionless volume flux coefficient associated to Rayleigh-Taylor layer growth, see equation (7.1)), entrain in a similar manner to the reference case of a pure plume (Γ0 = 1). Despite the significant departure from the reference Γ0 = 1 plume, the classic structure of a shear layer between the plume and the environment, composed of turbulent billows of a size comparable to the local plume radius, was observed. Based on comparisons with data acquired herein for the spread and streamwise volume flux of aqueous saline plumes, the dynamics for this range of Γ0 are well-described by the traditional integral models (Hunt & Kaye, 2005) which assume a height invariant entrainment parameter (referred to herein as constant-α models). Observations of lazy plumes whose source Richardson numbers exceed Γ0 & −2RT revealed that the high degree of laziness triggers the onset of a mixing mechanism which is visibly different from that at lower Γ0. The inherently unstable vertical density gradient formed by the contraction results in the development of vertical finger-like intrusions in the proximity of the source that resemble the growth and breakdown into turbulence of 155 7. Conclusions Rayleigh-Taylor (RT) instabilities. For plumes that display this mechanism, (i) the time- averaged radius and level of the plume neck are relatively independent of Γ0, both scaling on the plume source radius b0, and (ii) the volume flux entrained into the plume within the contracting region increases relative to that of a reference case pure plume by a factor of: QRT = RT Q0Γ 1/2 0 , (7.1) where Q0 denotes the source volume flux. The estimates of volume flux that were deduced in Chapter 4 based on classic quadratic RT layer growth rates (Fermi & von Neumann, 1955) successfully predicted the magnitude of dilution in this region and agreed well with both past experimental and numerical measurements (Kaye & Hunt, 2009; Marjanovic et al., 2017), and the generalised entrainment function developed by van Reeuwijk et al. (2016). The good comparison between this function, that was derived using the Navier- Stokes equations as a starting point and without explicit appeal to RT mixing, and the RT plume scaling developed herein, further supports the notion that the RT mechanism is the primary driver of mixing in the contracting region. Chapter 5 examined the stratification produced by a pure plume formed from a source whose radius was much smaller than the height and radius of the container. Experimental observations indicate that the flow patterns that can occur in the container can be remark- ably different compared to the pattern of filling proposed by Baines & Turner (1969). To better understand these flows, a classification of filling-box flows was proposed in Chap- ter 5 that was guided by these observations and the work of other researchers (viz. Baines & Turner 1969; Barnett 1991; Kaye & Hunt 2007; Ezzamel 2011). Five distinct flow patterns were identified: breakdown, rolling, slumping, blocking and displacement filling (see the supporting cartoon, figure 6.1 on p. 113). For fixed source conditions, the flows could be categorised within ranges of container aspect ratio (radius-to-height, φ := R/H). Instrumental to this classification were diagnostics based on the tracking of scalar tracers and of vortices which were enabled by the implementation of a simultaneous fluorescence and velocimetry technique. The bulk flow patterns were described qualitatively to guide the models developed in Chapter 6. In Chapter 6, the filling-box theories developed by Baines & Turner (1969) and Kaye & Hunt (2007) were extended to describe the stratification produced by a pure or lazy plume whose source could not be considered to be small in size relative to the height and/or radius of the container. While the focus was on developing a simplified theoretical model for a single regime of filling (viz. slumping), it was shown that the model can be straightforwardly extended to other filling-box regimes and also to other plume-container geometries (e.g. planar plumes whose slot length spans the width of a rectangular-based container, see Appendix 6.A). Solutions to the model for the velocities in the plume and currents, and for the density profiles in the container, were presented in terms of the rel- ative size of the source β0 := b0/H, the source Richardson number Γ0 and the container aspect ratio φ := R/H. These solutions relied on coefficients deduced from measurements of the velocity field in the plume impingement region, characterised by the momentum loss coefficient γi ≈ 0.78 (see Chapter 2), and by measurements of the formation time tS ≈ B−1/30 H4/3(0.8 + 1.0φ2 + 1.3φ4/3), and initial depth zS ≈ 0.33H (see Chapter 5) of the buoyant layer. The predictions for the filling rate and density profiles in the limit of 156 7.2. The physics of lazy plumes and filling boxes a point-source plume, β0 → 0, were equivalent to those obtained from Baines & Turner (1969). Predictions for the model show that increasing the scale of the plume source (β0) can radically increase the filling rate and change the horizontally-averaged density dis- tributions that form in the container relative to point-source descriptions. Moreover, the increased entrainment rates that occur in the near-field of RT lazy plumes also substan- tially increase filling rates. These predictions suggest that filling-box flows that develop from the release of a buoyant fluid by means of an area source can create patterns of strat- ification which are significantly different from the accepted pattern of filling proposed by Baines & Turner (1969). 7.2 The physics of lazy plumes and filling boxes The observations that are reported herein pertain to many different properties of lazy plumes. The investigation in the near-source contracting region, for example, allowed us to identify, based on the development of unsteady finger-like structures the underly- ing reasons, behind the apparent enhancement of entrainment which had been observed with increasing plume laziness. While it is well established that, in the proximity of the source, plumes behave differently to their self-similar state, the magnitude by which the entrainment is enhanced in very lazy plumes might surprise many readers. The mismatch between the volume flux data reported in Chapter 4 and the predictions based on classic plume models which rely on a height invariant description of entrainment is stark. For ex- ample, for the case of a plume of source Richardson number of Γ0 = O(104), predictions of volume flux at the level of the plume neck differ by a factor of over five. The relevance of the appearance of RT structures in the near-source region of plumes has broader scope within our understanding of convection. It points at lazy plumes as a hybrid between two canonical classes of flows: the slender buoyant plume and Rayleigh-Taylor convection. The properties of very lazy plumes, now understood to be for Γ0 & 103, depend on the region of interest: close to the source the flow resembles Rayleigh-Taylor convection, while further from the source, the flow transitions towards plume-like behaviour. The behaviour of a lazy plume in a container also has interesting properties. Regard- less of the laziness of the plume, the pattern of flow changes considerably for relatively large plume source sizes, β0 & O(10−1), compared to that for smaller sources. The possi- bility of creating a variety of different density profiles in the container, in circumstances of fixed container and plume source geometry, i.e. fixed R, H and b0, implies that a de- gree of control can be achieved over the stratification process by means of controlling the plume source conditions. This has particular relevance in engineering practice, for example, in conditioning systems, in which heating or cooling devices could be designed to create climates solely around room occupants, without wasting energy on conditioning unoccupied spaces. 7.3 Choice of methodology The observations presented herein described the behaviour of flows created in the labora- tory by means of aqueous saline solutions released in freshwater tanks. The acquisition of measurements for these flows were, in several instances, challenging. Some of these 157 7. Conclusions challenges were inherent to the problem that was investigated. For example, the veloci- ties that were measured typically varied by more than an order of magnitude at any given time. Furthermore, the scales of mixing ranged from the size of the container to the size over which viscous dissipation occurs. These problems of scale are prevalent in descrip- tions of entrainment. They often imply that measurements of entrainment require some degree of ingenuity. An established way to bypass these problems is to take measurements of entrainment that can be deduced from indirect measurements of volume flux variations. The technique of Baines (1983) which relies on the controlled extraction of fluid from an empty-filling box configuration, is an example of this approach. While this technique has proven highly successful for plumes (Baines, 1983), fountains (Baines et al., 1990; Burridge & Hunt, 2016) and indeed also lazy plumes (Kaye & Hunt, 2009), the acquisition of volume flux data by means of this experimental setup requires substantial time investments – primarily due to the slow asymptotes to steady state. Adopting a similar approach would have been prohibitive for the exploration of such a wide parameter space. An alternative approach was developed in Chapter 4 to measure volume fluxes indi- rectly. This approach was inspired by the filling-box model of Baines & Turner (1969). The continuity argument of Baines & Turner (1969) (i.e. zero net volume flux across each horizontal section), was used in Chapter 4 to relate the volume fluxes in the plume to the motion of the first filling front. The tracking of the flow from the dye visualisations was inherently noisy, and coupled with small oscillations of the front, meant that results had to be presented in terms of predictions of interface level rather than of volume fluxes. Nonetheless, the approach proved to be robust in that measurements of volume fluxes from particle image velocimetry were in good agreement. Validating the theoretical model for the filling box (Chapter 6) was arguably the most challenging part of the investigation. Experimental measurements were used to resolve a number of scaling coefficients deduced from dimensional analysis. These coefficients were essential to the predictions of the model. In some circumstances, the accuracy of these measurements had important implications on the interpretation of the solutions that were presented. For example, for the coefficient γi, the length rJ of the transition region of a plume outflow current is expected to vary as rJ ∝ γ6i H. The expected location of the transition is, as a result, very sensitive to the accuracy in the measurement of γi. To a lesser extent, but nevertheless in a significant manner, predictions of density profiles formed in the container are sensitive to the measurement of the formation time and initial layer depth. The real challenge, however, laid in validating the predictions of velocity and den- sity within the box. Ideally, if unrestrained by the reality of a limited amount of time and resources, a detailed validation could be conducted by repeating, for each combi- nation of the parameter space formed by Γ0, β0, φ, Re0 and Pe0, a sufficient number of ensemble measurements of the velocity and density field at each instance in time. The turnover time for a single experiment, at the scale used herein, however, ranged between 1-2 hours; additionally, the time required to process the images and extract data was of similar magnitude. These constraints make a parametric investigation of this problem practically infeasible. Similar time constraints would be encountered if a computational approach was adopted, compounded by the need for numerical quality control. These discussions do help to make a case for the value of the methodology chosen, that of sim- plified theoretical models, in terms of the predictive capability they offer. 158 7.4. Modelling This approach, coupled with experimental modelling, has scope to be adopted in other studies concerned with filling-box flows, and more generally, with the study of convec- tion. To this end, an experimental technique which is worth highlighting in this con- clusion is the simultaneous PIV+LIF technique discussed in Chapters 3 and 5. Recent developments in CMOS (camera sensors) and LED (light) technology have substantially increased the scope for a revival in techniques for flow visualisation and measurement in relatively low speed flows. These technologies are likely to become increasingly popular as they bypass the need of operating more expensive and often impractical laser systems. A word of warning is due nevertheless as the PIV+LIF technique requires a good deal of fine tuning that is experiment specific. For example, the visualisations presented in Chapter 5 required multiple repeats to achieve an acceptable compromise of lighting and camera settings. Potentially, multiple cameras could be used to interrogate the flow to achieve better control for regions of strongly varying light intensity. Visual distortions due to changes in refractive indices also limit the extent for which the technique may be applied to measurements of saline plumes in freshwater. As a rule of thumb, for round plumes observed at the scale investigated optical distortions were acceptable provided ρ0 ≤ 1.01 g/cm3. For larger density differences, we advise resorting to refractive index matching techniques to reduce these distortions. 7.4 Modelling In an age of increasingly sophisticated fluid mechanics software packages (CFD codes and alike), simplified theoretical models remain of value to both scientific and engineering practice for a number of reasons. The models presented in Chapters 4 and 6, for example, can be used to obtain near real time predictions of the bulk properties of lazy plumes and filling-box flows. Moreover, simplified models which can describe the bulk features of a flow, isolate the contributions that govern the dynamics of the flow, and as a result, can help identify the fundamental mechanisms that are being observed. Attempts to validating simple models for the near-field entrainment behaviour of a lazy plume in Chapter 4, for example, provided a number of physical insights into the be- haviour of these flows. Particular focus was directed towards models that assume a height invariant entrainment behaviour of the plume as analytical solutions for these models are readily available. The generalised model of van Reeuwijk et al. (2016), on the other hand, predicts the variation of volume flux for the whole range of source Richardson numbers investigated (1 ≤ Γ0 . 106), but requires the numerical integration of the plume conserva- tion equations. Solving these equations numerically is computationally inexpensive, but there is, nonetheless, a lack of direct insight when inspecting the equations prior to solv- ing them. The division of lazy plumes into two sub-regimes as recommended in Chapter 4 allows one to make an informed decision on the applicability of these solutions. Inter- estingly, the model of van Reeuwijk et al. (2016) fails to predict the necking behaviour, and as a result, the local plume radius at any level, correctly. As an alternative to the nu- merical solutions and to improve predictions of local plume radius for very lazy plumes, simple semi-empirical expressions are presented in Chapter 4 that can predict well both necking behaviour and volume flux. These are based on simple dimensional scalings that helped identify the RT mixing mechanism. 159 7. Conclusions Further insights were acquired by inspecting the impingement of a plume with a hor- izontal boundary and the resulting current that formed along it (Chapters 2 and 6). The idealisation of the flow into regions, which were interconnected by means of an impinge- ment control volume, was instrumental to understanding the rather complex structures formed in the filling-box flows that are driven by a lazy plume. Interestingly, measure- ments of volume flux in the plume outflow current indicate that entrainment is higher by as much as a factor of two compared to that recorded for radial wall jets or that predicted by the model of Ellison & Turner (1959) (originally deduced for two-dimensional gravity currents). The increased entrainment was observationally attributed herein to the persis- tence of the plume vortical structures that became part of the current after impingement. This suggestion challenges the notion that mixing in each region is independent of the behaviour of the other regions. It would be of interest to investigate how interactions between regions affect entrainment. Admittedly, further work is required to parameterise entrainment in buoyant currents, both when they are freely allowed to propagate or when they exchange turbulently with a counterflow (such as in slumping) – another avenue for future work? 7.5 Future work The features displayed by the filling-box flow driven by a lazy plume are encountered in many practical situations. The work was intentionally presented without reference to application. This was done to address the fundamental physics of these flows. It might be of interest to examine specific applications and test the limitations imposed by these. The use of filling-box studies in fire-egress design, for example, is of particular relevance. The filling model of Baines & Turner (1969) has been used to predict the filling of a room with fire smoke (Cooper, 1983; Hurley, 2016). It is clear, based on the findings presented herein, that predictions based on these models are unsafe in circumstances in which overturning occurs (e.g. in tall and narrow rooms) or in the cases where the source of the fire is of a size comparable to the floor area of the room. The theoretical framework proposed in Chapter 6 can be extended to other scenarios involving the filling of containers by plumes. The most direct example that may be of interest would be to investigate the filling produced by a forced plume. The model pre- sented in Chapter 6 for lazy plumes can be extended to this plume regime without further modification. It follows that the model can also be extended to other geometries, e.g. the planar plume (see Appendix 6.D). It remains to be seen how these models will test against experimental observation. 7.6 Closing remarks Early studies of buoyant plumes and convection date as far back as the early-to-mid decades of the previous century. A number of now famous fluid dynamicists worked on these subjects including the likes of Lord Rayleigh, Lewis F. Richardson, Walter Tollmien, Sir G. I. Taylor and George K. Batchelor. Much progress was made in the field of convection in the latter part of the century. The work of Bruce R. Morton, W. Douglas Baines and J. Stewart Turner, in particular, truly laid the foundations to the investigation presented herein by conceptualising a theoretical framework for the modelling of plumes 160 7.6. Closing remarks and stratification. Only more recently however there has been sufficient progress in the development of plume theory to allow for the prediction of the stratifying dynamics of plumes from lazy sources. This progress was made possible by the renewed spark of in- terest in lazy plumes that occurred over the last twenty years (Colomer et al., 1999; Hunt & Kaye, 2001; Fanneløp & Webber, 2003; Hunt & Kaye, 2005; Plourde et al., 2008; Kaye & Hunt, 2009), and in particular, over the years in which this investigation was conducted (van Reeuwijk et al., 2016; Carlotti & Hunt, 2017; Marjanovic et al., 2017). Building from these studies, this dissertation provides two major contributions to our understanding of convection. First, an apparently new regime of plume dynamics is iden- tified. Buoyant plumes have been studied for several decades and one might be surprised that there still are new fundamental insights that can be acquired about their behaviour. Rayleigh-Taylor lazy plumes could assert themselves as a canonical class of flow in flu- ids, whose study could draw from both the plume and instability research communities. 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