Turbulence Ingestion Noise
of Open Rotors
Rosalyn A. V. Robison
Jesus College, Cambridge.
A Dissertation submitted for
the degree of Doctor of Philosophy
at the University of Cambridge
October 2011
Acknowledgements
Firstly, heartfelt thanks to my supervisor, Nigel Peake, who has always been generous
with his time, helpful, and never dismissive of my thoughts or ideas. I am also grateful
to Tony Parry, of Rolls-Royce, for our many useful conversations, and to Rolls-Royce and
EPSRC for funding this Ph.D.
Secondly, thank you to all the people who have helped me push through the most
difficult parts of the Ph.D. process, offering encouragement and advice. In particular,
Katy Richardson, Helen Arnold, Dawn Lake, Alice Moncaster, Jane Cooper, Jill Shields,
Susan Haines, Cally Roper and Pam Black. Also, huge thanks to all those who did a
lot of proof-reading in a short space of time!: Tim Hughes, Clare Robison, Peter Foord,
Brian Venner, Matthew Tinsley, Samantha Archetti and Hannah Dudley.
Thirdly, thanks to all the people at the Centre for Mathematical Sciences who have
made this such a good workplace over the past four years: my fellow PhD students, the
postdocs and academics who I’ve got to know, and also Mick Young, Zvezda Woodhouse
and Beth Sweet-Rosborough, who keep everything running. Particular thanks also to my
officemate, He´le`ne Posson, who read through drafts of Chapters 1 and 2 and gave very
helpful comments.
Finally, thank you to all my family and other close friends, and especially Tim, for
being such a source of support always.
Declaration
This dissertation is the result of my own work and includes nothing which is the
outcome of work done in collaboration except where specifically indicated in the text. In
particular, this work builds directly upon that of Majumdar (1996) and Cargill (1993).
All numerical calculations presented were computed using my own MATLAB code,
with the exception of a LINSUB routine written by Dr V. Jurdic at the Institute of
Sound and Vibration, University of Southampton. As indicated in the text, output from
industry code was used in two plots and this was provided by Dr. M. J. Kingan (Institute
of Sound and Vibration), Dr. P. J. G. Schwaller and Dr A. B. Parry (both at Rolls-Royce).
Some of the work presented in Chapter 4 has been published as Robison and Peake
(2010), but is given here in more detail.
Rosalyn A. V. Robison
Cambridge, October 2011
Summary of the dissertation
‘Turbulence Ingestion Noise of Open Rotors’
by Rosalyn A.V. Robison
Renewed interest in open rotor aeroengines, due to their fuel efficiency, has driven
renewed interest in all aspects of the noise they generate. Noise due to the ingestion of
distorted atmospheric turbulence, known as Unsteady Distortion Noise (UDN), is likely
to be higher for open rotors than for conventional turbofan engines since the rotors are
fully exposed to oncoming turbulence and lack ducting to attenuate the radiated sound.
However, UDN has received less attention to date, particularly in wind-tunnel and flight
testing programmes.
In this thesis a new prediction scheme for UDN is described, which allows inclusion of
many key features of real open rotors which have not previously been investigated theo-
retically. Detailed features of the mean flow induced by the rotor, the form of atmospheric
turbulence, asymmetries due to installation features, and the effect of rotor incidence are
all considered. Parameter studies are conducted in each of these cases to investigate their
effect upon UDN in typical static testing and flight conditions.
A thorough review of the technological issues of most relevance and previous theo-
retical work on all types of turbulence-blade interaction noise is first undertaken. The
prediction scheme is then developed for the case in which the mean flow into the rotor
is axisymmetric. This shows excellent qualitative agreement with previous findings, with
increased streamtube contraction resulting in a more tonal noise spectrum. The theoret-
ical framework involves using Rapid Distortion Theory to calculate the distortion of an
isotropic turbulence field (such as given by the von Ka´rma´n spectrum) by the mean flow
induced by the rotor (such as given by actuator disk theory), leading to an expression
for the velocity incident upon the leading edge of the rotor blades. Strip theory is then
used to calculate the pressure jumps across the blades, input as the forcing term in the
far-field wave equation.
Models are derived for open rotor-induced flow which account for the variation of
blade circulation with radius, and the presence of the rotor hub and rear blade row. An
investigation of appropriate turbulence models and realistic turbulence parameters is also
undertaken. A key finding is that the heights of the tonal peaks are determined by the
overall magnitude of the induced streamtube contraction (dependent on the total thrust
generated) whereas the precise form of distortion (affected by the detailed components
of the mean flow and the form of atmospheric turbulence present) alters the resulting
broadband level.
The prediction scheme is formulated in such a way as to facilitate extension to the
asymmetric case, which is also fully derived. The model is applied in the first instance to
the case of two adjacent rotors and then to the case of a single rotor at incidence. Under
flight conditions, when distortion is reduced but UDN can still contribute a significant
broadband component to overall noise levels, asymmetry is found to increase broadband
levels around 1 Blade Passing Frequency but reduce levels elsewhere.
To Clare
Contents
1 Introduction 5
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Technological context: the open rotor design . . . . . . . . . . . . . . . . . 6
1.2.1 Commercial development in the 1980s - the propfan . . . . . . . . . 7
1.2.2 Commercial development since 2001 - the Advanced Open Rotor . . 11
1.2.3 Technological points of most relevance to our work . . . . . . . . . 16
1.3 Academic context: review of the literature . . . . . . . . . . . . . . . . . . 17
1.3.1 Noise arising from turbulence . . . . . . . . . . . . . . . . . . . . . 19
1.3.2 Use of experimental measurements . . . . . . . . . . . . . . . . . . 22
1.3.3 Analytic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.3.4 Computational approaches . . . . . . . . . . . . . . . . . . . . . . . 30
1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 Axisymmetric Rotor Systems 33
2.1 Chapter outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Calculating the turbulent perturbation, u . . . . . . . . . . . . . . . . . . . 34
2.2.1 Rapid Distortion Theory (RDT) . . . . . . . . . . . . . . . . . . . . 35
2.2.2 The quantity X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2.3 Majumdar’s solution to the RDT equation, Aij . . . . . . . . . . . 41
2.2.4 Aij for zero azimuthal mean flow (when U · eˆφ = 0) . . . . . . . . . 43
2.2.5 Mean flow model - introducing the actuator disk . . . . . . . . . . . 47
2.2.6 Turbulence model - introducing the von Ka´rma´n spectrum . . . . . 49
2.2.7 Distorted turbulence spectrum . . . . . . . . . . . . . . . . . . . . . 52
1
2 Contents
2.2.8 Distorted turbulence plots . . . . . . . . . . . . . . . . . . . . . . . 54
2.3 Blade pressure jump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.3.1 LINSUB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.3.2 Calculating the blade pressures . . . . . . . . . . . . . . . . . . . . 60
2.3.3 Limiting the rk integral . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.3.4 Blade pressure plots . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.3.5 Unsteady force generated by a single rotor . . . . . . . . . . . . . . 72
2.3.6 Total force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.4 Far-field solution of the wave equation . . . . . . . . . . . . . . . . . . . . 74
2.4.1 Auto-correlation of pressure . . . . . . . . . . . . . . . . . . . . . . 79
2.4.2 Modal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.4.3 SP plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3 Alternative inputs: mean flow and turbulence models 85
3.1 Chapter outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.2 Mean flow model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.2.1 Vortex theory of a propeller . . . . . . . . . . . . . . . . . . . . . . 87
3.2.2 Variable circulation actuator disk . . . . . . . . . . . . . . . . . . . 88
3.2.3 Modelling the bullet . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.2.4 Co-axial propellers . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.2.5 Calculating ∂Xi/∂xj for new flows . . . . . . . . . . . . . . . . . . 99
3.3 Atmospheric turbulence models . . . . . . . . . . . . . . . . . . . . . . . . 101
3.3.1 The three-dimensional energy spectrum . . . . . . . . . . . . . . . . 101
3.3.2 Turbulence shed by installation features . . . . . . . . . . . . . . . 105
3.3.3 Integral lengthscale, L . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.4 Radiated noise results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.4.1 Adjusting the mean flow . . . . . . . . . . . . . . . . . . . . . . . . 112
3.4.2 Adjusting the turbulence spectrum . . . . . . . . . . . . . . . . . . 116
4 Generalisation to asymmetric rotor systems 121
4.1 Chapter outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
CONTENTS 3
4.2 Effects of asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.2.1 Previous work on asymmetry . . . . . . . . . . . . . . . . . . . . . 122
4.3 Asymmetric turbulence distortion . . . . . . . . . . . . . . . . . . . . . . . 124
4.3.1 A simple asymmetric mean flow . . . . . . . . . . . . . . . . . . . . 124
4.3.2 Turbulence at the rotor face . . . . . . . . . . . . . . . . . . . . . . 128
4.4 Blade pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.4.1 The general form of Aij . . . . . . . . . . . . . . . . . . . . . . . . 132
4.4.2 Input into LINSUB . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.4.3 Output from LINSUB . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.5 Far-field noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.5.1 Spectrum of radiated sound . . . . . . . . . . . . . . . . . . . . . . 139
4.5.2 SP plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.5.3 Sound from two adjacent rotors . . . . . . . . . . . . . . . . . . . . 141
5 Rotor at incidence: an important asymmetric case 147
5.1 Chapter outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.2 Effects of incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.2.1 Previous work on non-zero incidence . . . . . . . . . . . . . . . . . 148
5.3 Distortion of turbulence by rotor at incidence . . . . . . . . . . . . . . . . 150
5.3.1 Adapting the mean flow model . . . . . . . . . . . . . . . . . . . . 150
5.3.2 Distorted turbulence spectrum . . . . . . . . . . . . . . . . . . . . . 151
5.4 Far-field noise calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.4.1 Adjusting the force term . . . . . . . . . . . . . . . . . . . . . . . . 155
5.4.2 Adjusted Green’s function . . . . . . . . . . . . . . . . . . . . . . . 158
5.4.3 Correlation of far-field pressure . . . . . . . . . . . . . . . . . . . . 160
5.4.4 SP and SPL plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6 Summary and Conclusions 163
6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.2 Summary of key results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.2.1 Chapter 2 results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
4 Contents
6.2.2 Chapter 3 results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.2.3 Chapter 4 results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.2.4 Chapter 5 results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.3 Overall conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.3.1 Findings of most relevance to industry . . . . . . . . . . . . . . . . 171
6.3.2 Findings of most relevance to theoreticians . . . . . . . . . . . . . . 172
6.4 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
A Legendre functions and actuator disk streamfunction 175
B Definition of sound metrics 177
C Sound from non-identical adjacent rotors 181
D Stationary phase argument 183
Nomenclature 185
Bibliography 193
Chapter 1
Introduction
1.1 Background
The underlying motivation for this work is the improvement of aviation efficiency. Opti-
misation of the very widely used turbofan engine has been ongoing since the 1960s, and
has delivered both efficiency improvements and noise reduction within the commercial
aircraft sector. There is now considerable interest in the future of another engine, the
open rotor1 (see Figure 1.1), predicted to deliver fuel savings of at least 30% compared to
current turbofans (Smith, 1985), and 10-20% savings compared to next-generation tur-
bofans. Open rotor driven aircraft can operate efficiently at speeds up to Mach 0.8, and
are therefore suitable candidates for commercial short-haul flights2. However, the level
of noise generated by the open rotor was a significant factor in commercial development
being abandoned a decade after the technology first emerged in the 1980s. Research into
many areas of open rotor aeroacoustics has now been resumed by manufacturers with the
aim of meeting current and future noise certification criteria.
This Ph.D. project, in collaboration with Rolls-Royce, has involved analysing in detail
one particular source of both broadband and tonal open rotor noise, termed Unsteady
Distortion Noise (UDN). This arises from the interaction between unsteady perturba-
tions to the mean flow (turbulent eddies) which are generated upstream of the engine and
the large, unducted blades of the open rotor itself. As the turbulent eddies are drawn
1Also known as the Counter-Rotating Open Rotor (CROR), the Advanced Open Rotor (AOR), the
advanced turboprop, the Counter-Rotating Propfan (CRP), or simply the propfan.
2Higher speeds would be desirable for long-haul flights. The conventional turboprop, a close relative
of the open rotor, only delivers efficiency savings when travelling below Mach 0.6, although advanced
turboprops do exceed this.
5
6 1. Introduction
towards the engine they are stretched by the streamtube contraction which is induced as
air is sucked into the engine. Such incoming eddies arise naturally due to atmospheric
turbulence, but in addition any coherent fluid structures which are shed from parts of
the aircraft several rotor diameters upstream of the rotors may also undergo significant
distortion during their passage downstream. The subsequent interaction of the distorted
eddies with the front row of rotating propeller blades leads to high tonal3 noise at multi-
ples of the Blade Passing Frequency4 (BPF). Distortion is strongest at low flight speeds,
and thus the tonal contribution of UDN may be particularly significant at take off and
approach. These are also the situations in which strict noise certification criteria must
be satisfied. For operating conditions where the distortion is weaker, UDN may still
contribute a significant broadband component to the overall noise of the open rotor.
front
blade row rear
blade row
engine nacelle, ‘bullet’
jet
exhaust
intake
Figure 1.1: Illustration of a ‘pusher’ configuration of open rotor, with the blades toward
the rear of the engine. Reproduced with permission from Rolls-Royce.
1.2 Technological context: the open rotor design
This work is strongly motivated by a real-world application within the rapidly expanding
aviation sector. It is therefore desirable to be aware of the key technological issues which
surround open rotor aeroacoustics research. Noise generated by the external rotors far
outweighs noise from within the engine core, and is made up of many interacting com-
3Tonal noise is noise at a specific frequency, such as a note being played on an instrument, and arises
from periodic variations in sound pressure. Broadband noise is noise at a wide range of frequencies, such
as road noise heard when travelling in a car, and arises from more random variations in sound pressure.
41 BPF = BΩ, where B is blade number and Ω is angular velocity of the rotor.
1.2 Technological context: the open rotor design 7
ponents. The aerodynamic and aeroacoustic performance of the open rotor’s unducted
blades are closely interlinked and it can be difficult to separate the optimisation of one
from the other. The way in which experimental, computational and (ultimately) certifi-
cation testing is carried out also affects which analytical results will be of most use.
1.2.1 Commercial development in the 1980s - the propfan
The current generation of open rotors are direct descendants of the propfan engine, first
developed by NASA in the 1980s. The global nature of oil supply and demand has involved
a number of ‘oil shocks’ in recent decades when oil prices have risen extremely rapidly. In
19735 fuel costs made up, on average, about 25% of the direct operating costs of aircraft,
but this rose to 44% in 1978, (Lange, 1986). This was a key driver in the search for
more efficient ways to fly and even led to the US Congress tasking NASA with identifying
concepts which had major efficiency saving potential (Hager and Vrabel, 1988). As a
result, the Advanced Turboprop Project ran at the NASA Lewis Research Center from
1976-1987 involving several large American manufacturers.
Much of the efficiency gains since the 1960s in aviation have been made by increasing
the bypass ratio6 of turbofan engines. This increases propulsive efficiency by reducing the
energy wasted in high velocity wakes (Stuart, 1986). An added advantage of increasing
the bypass ratio for turbofans is that noise is reduced. This is due to a combination of
factors including the reduced jet velocity and a shielding effect from the air which has
bypassed the core. However, for a turbofan engine, there is a bypass ratio limit beyond
which fuel burn begins to increase due to drag along the interior of the engine nacelle.
Today’s high bypass ratio turbofans operate at ratios of around 10. As the external rotors
of the open rotor are completely unducted, drag is no longer a limiting factor, and they
can achieve ‘Ultra High Bypass Ratios’ of 30 and above. Ducting liners, which play a
crucial role in specific tone reduction for turbofans, cannot of course be used to attenuate
noise from these rotors.
General Electric (GE) in the US began an in-house ‘UnDucted Fan’ (UDF) research
programme in 1983 and also collaborated with NASA. As the new engine design moved
further from the starting point of the single-propeller turboprop, a significant increase
5The year the Organization of Arab Petroleum Exporting Countries (OAPEC) declared an oil embargo
triggered by the Arab-Israeli conflict.
6Bypass ratio = mass flow rate of air drawn in by the engine which bypasses the engine core : mass
flow rate of air which passes through the core.
8 1. Introduction
in efficiency was made by adding a second counter-rotating blade row behind the first,
see Figure 1.2. The rear rotor removes some of the swirl introduced by the front row,
turning the flow back toward the axial direction and increasing the thrust generated very
effectively. However, several new noise sources arise from the interaction of the wakes and
tip vortices shed by the front blade row impinging upon the rear blade row (see recent
work by Kingan and Self (2009)). Other features which distinguish the propfan from the
turboprop include the distinctive blade shape (wide-chord and high-sweep7), and the use
of variable rotor speeds. In order to achieve high speeds at cruise, the open rotor has a
very large diameter and this leads to transonic tip speeds which are a source of significant
tonal noise.
(a) (b) (c)
Figure 1.2: a) Propeller swirl recovery testing at the NASA Lewis Research Center in
the 1980s, courtesy of nasaimages.org. The rear row of vanes are fixed. b) MacDonald
Douglas test flight. Image reproduced with permission from Niels Sampath. One turbofan
engine has been replaced by a propfan. c) Rolls-Royce Rig 140 testing in 1988. Image
reproduced with permission from Rolls-Royce. Note the large hub protruding upstream
of the rotors; due to its appearance this is called the ‘bullet’.
In the UK, Rolls-Royce also began a propfan research programme in the late 1980s.
Work was focused on Rig 140, a 1/5th scale 7x7 blade configuration which underwent low-
speed testing in 1988 (Kirker, 1990), seen in Figure 1.2c. This was then reconfigured as a
ducted version, Rig 141, which was tested in 1990 at the Aircraft Research Association’s
transonic wind tunnel in Bedford.
The noise spectra of 1980s propfans were dominated by very many close tones, con-
centrated toward the lower frequency range, see Figure 1.3. Both the front and rear blade
rows of the propfan experience a range of fluctuating forces due the unsteadiness of the
7Sweep denotes the translation of each radial section of the blade in the chordwise direction relative to
adjacent sections. Thus a blade whose leading edge curves back in the axial direction, as seen in Figure
1.4, has non-zero sweep.
1.2 Technological context: the open rotor design 9
Figure 1.3: Example of Rig 140 noise results. Image reproduced with permission from
Blandeau et al. (2009). Note there are the same number of blades in the front and rear
rows, so tones are doubled up (shown in zoomed-in box).
flow in their reference frame, leading to noise at multiples of the front and rear BPF, B1Ω1
and B2Ω2. In addition, rotor-rotor interaction gives rise to tones at integer combinations
of these two frequencies, nB1Ω1 + mB2Ω2. In contrast, turbofans generate fewer tones
and produce relatively smooth spectra.
Much early propfan aeroacoustics work was therefore focussed on the most prominent
tonal sources. As might be expected, propfan noise more closely resembled that of tur-
boprops8 than turbofans. Perhaps the most detailed publicly available propfan report
from that period is that by Hoff (1990), which described testing programmes run in col-
laboration between GE and NASA from 1984-1987. The testing concentrated on three
major sources of propfan tonal noise - steady loading noise, rotor-rotor interaction noise
and pylon wake interaction noise - and included rig testing, flight testing and calibration
of methods. Hoff included a diagram which illustrated the many different propfan noise
sources, see Figure 1.4. There was also a recognition that distortion is an important factor
in predicting the acoustics of propeller planes. Distortion is a broad term used by different
authors to describe a whole range of effects which are not measured in rotor alone wind
tunnel testing. It can therefore include both mean flow effects (such as the influence of
installation features, the airframe and angle of attack), and also unsteady effects due to
turbulence generated both in the atmosphere and by the aircraft itself. There was some
evidence that distortion noise dominated over rotor alone noise for certain single unducted
rotor configurations.
Another area of interest at the time was the potential response of the public to the
8Commercially active turboprops include Bombardier’s Q400 and ATR’s 72 series.
10 1. Introduction
Figure 1.4: Figure from Hoff (1990), detailing the major tonal noise source mechanisms.
There are additional broadband sources and in particular Hoff noted that broadband
levels would be related to inflow turbulence.
noise of the propfan. Studies have shown that changes in the tone-to-broadband ratio and
fundamental frequency (both of which are different for the propfan and the turbofan) affect
annoyance levels (McCurdy, 1990). Tonal noise is usually perceived as more irritating by
the human ear than broadband, and frequencies within the range 2-6 kHz are perceived as
louder (for a given Sound Pressure Level) than lower or higher frequencies. In addition,
the ‘noiseprints’ (the area around take-off/landing location which experiences noise above
a certain level) were predicted to change with the use of the propfan (Lange, 1986), with
the noiseprint at some levels increasing in area, and at other levels decreasing. High levels
of cabin noise were also identified as a potential problem.
Later studies found some evidence that people’s annoyance response to propfan noise
might not be significantly different to contemporary turbofan and turboprop noise (Mc-
Curdy, 1992). Certainly, in the mid-1980s, there was confidence that the propfan could
meet the Chapter 3 noise regulations in place at that time if blade design, blade num-
ber and rotor spacing were chosen appropriately. The joint GE/NASA rig testing had
produced configurations which would comfortably meet Chapter 3, see Figure 1.5, and
they had also successfully calibrated between rig testing and flight testing results. As
NASA’s Advanced Turboprop Project progressed, test flights were carried out between
1986-1988 with a GE36 UDF engine replacing one of the turbofans on both a Boeing
727 (Harris and Cuthbertson, 1987) and a McDonnell Douglas MD-80, as seen in Figure
1.2b. Commercial production of a propfan-powered McDonnell Douglas, the MD-94X,
was planned for 1994. However, a dramatic drop in oil prices had removed the major
driving factor for the technology by 1988. Other commercial issues, such as the replace-
1.2 Technological context: the open rotor design 11
ment of airport infrastructure and maintenance equipment which would be involved with
any major transfer to a new technology, meant the balance of the argument fell on the side
of continued use of the turbofan. Many manufacturers, including GE and Rolls-Royce,
had pulled their research programmes by the early 1990s. However, work on the propfan
did not stop completely; in particular, in the Ukraine development of the Antonov An-70
military transport aircraft has been ongoing.
Figure 1.5: Table from Hoff’s 1990 NASA report, showing predicted noise levels of the
propfan at the three certification points, based on their rig testing results. Three blade
configurations are shown, where F-7, A-7 etc. denote the particular blade models used
for the front and rear (aft) rows. The A-3 model has a reduced diameter compared to the
A-7. The results were obtained by scaling up the rig results to a 10-foot diameter engine,
and then adjusting the dB level to account for the 2 engines, ground reflections during
testing, core noise (at both cutback and approach) and airframe noise (at approach).
1.2.2 Commercial development since 2001 - the Advanced Open
Rotor
Reduction of carbon emissions is a new driver for the development of more efficient tech-
nologies, as well as the return of oil price rises, and there is pressure on all industries to
use less fossil fuels9. In addition, complaints over aircraft noise remain one of the most
9Any long term sustainable option for aviation, e.g. the use of biofuels, must be considered against a
backdrop of ever-increasing aviation demand. Since 1960 there has been growth of nearly 9% per year
of air passenger traffic, and the UK’s Department for Transport predicts 500 million air passengers in
2030, almost 3 times as many as in 2002. Transporting cargo by air to the UK doubled between 1989
12 1. Introduction
significant issues for airport operators and regulatory bodies. The Advisory Council for
Aeronautics Research in Europe (ACARE), has established ambitious targets for new
aircraft entering service in 2020, compared to those entering service in 2000, of a 50%
reduction in fuel consumption and CO2 emissions per passenger kilometre (of which 20%,
that is just under half of the reduction, should come from engine technology) and a 50%
reduction in Perceived Noise (Busquin et al., 2001). Although these are aspirational tar-
gets, they impose a tight timescale for manufacturers, and the expectation of some in the
field is that open rotors will be seen on airliners by the end of the decade, (Spalart et al.,
2010). For example, in 2007 easyJet announced plans to develop an ‘easyJet ecoJet’ by
2015 which would emit 50% less CO2 than their current fleet, using open rotor technology.
Figure 1.6: Rig 145 testing, which took place between 2008-2010. Image reproduced with
permission from Rolls-Royce.
There is general consensus that any future significant gains in efficiency must come
from a major shift to a new technology. The two most promising designs are Geared
Turbofans (Pratt & Whitney are now concentrating on this area with their PW1000G
engine) and open rotors10. The former is likely to be quieter, but the open rotor will
be better in terms of fuel efficiency. The huge financial investment needed to develop
these designs from concept to production has led to several collaborative programmes
between manufacturers, with significant government funding. These have included the
New Aircraft Concept Research in Europe (NACRE) programme (2005-2009), the UK-
led Omega project (2007-2009) which concentrated heavily on Advanced Open Rotor
technology, the DREAM project which Rolls-Royce coordinates with funding from the
European Commission, and the Clean Sky Joint Technology Initiative (2008-2013).
and 1999, Anderson et al. (2006). Efficiency savings must therefore play a crucial role in moving towards
sustainability.
10A third option which has received less attention is the Ducted Contra-Fan, essentially an open
rotor with casing. Rolls-Royce’s Rig 141 was used to test this concept in 1990. The Counter-Rotating
Integrated Shrouded Propeller (CRISP) is a separate, but related, concept which was investigated by
MTU and Snecma in the mid-1980s.
1.2 Technological context: the open rotor design 13
Under DREAM, Rolls-Royce and Airbus have carried out three sets of modern open
rotor testing. Rig 140, built for the first propfan tests in the late 1980s (and subsequently
rebuilt as Rig 141), has again been rebuilt as Rig 145 for this new round of testing
with different blade designs and blade number combinations, see Figure 1.6. Low-speed
0.2-0.25 Mach testing was undertaken in 2008, where different combinations of blade
number, blade row gap and rotor speeds were trialled. High-speed 0.8 Mach testing took
place in 2009, and further low-speed testing was carried out in 2010 using an optimised
blade design, with increased blade chord and thickness and improved sweep for high
performance. Throughout this thesis our primary interest is in predicting the impact of
Unsteady Distortion Noise on community noise levels on the ground near airports and
therefore we are particularly interested in the low-speed testing which is used to simulate
take-off/approach conditions. High speed testing, simulating cruise, is more relevant to
the assessment of cabin noise11.
Testing results have shown that advances made since the 1980s have led to reductions
in both the number of tones and the tone protrusion above broadband level seen in open
rotor noise spectra. Although this is a commercially sensitive area, and therefore exact
designs are not publicly available, broadly speaking noise reduction has been achieved
through adjustments in blade design (sweep, chord, thickness, camber12, tip shape), blade
numbers and diameters (mismatched between the front and rear rows, see Parry and
Vianello (2010)), adjustments in pitch13 and tip speed (increased blade number allows
a reduction in tip speed), and careful consideration of the installation, including using
‘pylon blowing’ to reduce peaks at harmonics of the front rotor BPF, see Figure 1.7. Thus
tonal noise is still of primary concern, but broadband sources are becoming an increasingly
important factor, Parry et al. (2011). There is confidence that noise regulations can be
met but it is also likely the open rotor will not be as quiet as next-generation geared
turbofans.
Little wind tunnel testing has been done to date specifically addressing Unsteady
Distortion Noise, attempting to measure key turbulence parameters at the rotor face or
the UDN contribution to overall noise levels. Wind tunnel operators aim to achieve very
11The relatively low fundamental frequencies of open rotor noise means ‘active control’ (cancelling noise
out using loudspeakers) may be used within the cabin. Such systems have been used in Saab 340 and
2000 aircraft, Peake and Crighton (2000)
12When slicing through a blade at a particular radial station, high camber corresponds to a profile
which is significantly curved.
13Pitch denotes the rotation of a blade about its pitch-change axis, which runs in the radial direction
but whose axial position must be carefully chosen.
14 1. Introduction
Figure 1.7: Figure reproduced from Ricouard et al. (2010). Installed = with the pylon
in place, isolated = rotor alone. The large pylon, which connects the open rotor to the
fuselage (shown in Figures 1.4 and 1.6), reduces the velocity of the air hitting the rotor in
the region directly behind it. A system which blows air from the rear of the pylon can be
effective in reversing this effect, thereby producing a more uniform flow onto the blades
and reducing the level of certain tones.
low levels of natural turbulence and since atmospheric turbulence lengthscales do not scale
with the rig, it may not be until full-scale outdoor testing takes place that atmospheric
UDN will become an important factor (although the effect of distortion on turbulence
generated by the airframe may appear in rig testing).
Wind tunnel testing is only one part of the current design phase of the advanced
open rotor, there are also significant computational modelling programmes being run,
and the development of better analytic models to predict noise levels is ongoing. In
the second half of this introductory chapter we will review numerical, experimental and
analytic work addressing noise specifically due to turbulence. Certain noise mechanisms
are better understood than others, as they correspond to aspects of turbofan noise which
have undergone decades of study. Distortion noise is expected to be more significant for
the open rotor than the turbofan, due to the large rotor diameters (and therefore greater
upstream streamtube radius), the lack of ducting, and the complex interaction between
flow shed from structures upstream of the rotors and the rotor blades. It is therefore of
interest to revisit past analytic work on UDN and extend it; this is the premise of this
Ph.D.
As in the 1980s, key design questions for the open rotor include: blade number, rotor-
1.2 Technological context: the open rotor design 15
rotor spacing, rotor diameter, blade shape, position of pitch change axes and tip speed.
Traditionally, propeller engines set one blade speed and then use pitch change to vary
the thrust generated, whereas a key design feature of the open rotor is the ability to vary
tip speed. There are two different configurations possible to achieve variable tip speed:
Geared open rotors have greater tip-speed control, whereas Direct Drive open rotors avoid
having a complex gearbox placed in the middle of the engine airflow. Besides the rotor
itself, positioning of the engine and prediction of installation effects are crucial, and will
affect noise directivities, (Ricouard et al., 2010). It is even possible for installation effects
to be used to reduce noise in certain directions, for example careful positioning of engine
ducts could help reduce cabin noise. Key installation design questions include: pusher vs.
puller configuration (see Figure 1.8), blade-off/failure testing compliance, pylon position
and shape, position of exhaust ducts, and the use of gearboxes.
Figure 1.8: The puller configuration, shown above, has the rotor blades towards the front
of the engine and is attached to the airframe downstream of the blades. Image reproduced
with permission from Rolls-Royce. The rotor blades experience a cleaner inflow than in
the pusher configuration, shown in Figure 1.1, which has the pylon upstream. Other
upstream structures in the pusher configuration include the wing wake and exhaust flow
(Pagano et al., 2009), making the rotor harder to optimise. Advantages of the pusher
include potentially reduced cabin noise as the blades are further back on the aircraft.
Today, in 2011, rotor alone and rotor-rotor interaction noise can be fairly well quanti-
fied, but it is the interaction effects when the rotor is placed into atmospheric conditions,
with the pylon and fuselage in close proximity, and with varying angle of attack, which
are less well known. These effects are expected to mainly affect the tones generated by
the front rotor, rather than the rear rotor. In this Ph.D. we address each of these features
in the context of UDN, in particular by incorporating asymmetry into the model.
16 1. Introduction
A note on noise certification
Noise regulations for airports in the UK are set at a number of levels. The International
Civil Aviation Organisation (ICAO) has set increasingly strict certification conditions
over time, as the frequency of flights increases. New aircraft today must comply with
Chapter 4 regulations which mark a 10dB cumulative difference14 from Chapter 3 which
was in place for the 1980s propfan. Chapter 5 is expected to come in before long, and
manufacturers are designing with this in mind.
As previously noted, tonal noise is more irritating than broadband and there are sev-
eral different ‘weightings’ which attempt to take into account the increased annoyance
level caused by pure tones. Each of these weightings only gives a representation, there is
no one objective measure (Young et al., 2010) and since the 1980s standards have been de-
veloped primarily for turbofans. EPNL is currently the standard measure for certification,
but McCurdy found that an alternative metric - A-weighted with tone/duration correc-
tions - reduced the standard deviation between calculated results and people’s reaction
to propfans.
In addition to the ICAO certification conditions, community noise contours around
airports are also regulated, but a different metric is used for assessment of these contours
(dBA). The difference between the dBA and EPNL metrics for typical turbofan spectra,
and the difference for typical open rotor spectra are not the same, and this could be a
source of discrepancy for future comparisons between engines, (Young et al., 2010).
It is not yet known exactly what regulations open rotors will be required to meet.
For example, they could either be judged as a propeller or as an open fan, each of which
is regulated differently. However, it seems that regulatory bodies would be keen not to
exclude open rotors with the Chapter 5 target, due to their efficiency advantages.
1.2.3 Technological points of most relevance to our work
To summarise, open rotor technology has huge potential to address the need for the
aviation industry to reduce its use of fossil fuels, and if oil prices continue to rise the
business case for their use will continue to get stronger. Noise is seen as a potential
barrier to full open rotor deployment; another new engine, the geared turbofan, is likely
14The ICAO regulations limit the total found when the measured noise at the three testing positions
- takeoff, sideline and approach - is summed together. The Chapter 4 limit is 278.8 dB, Chapter 3 was
288.8 dB.
1.3 Academic context: review of the literature 17
to be quieter but less fuel efficient. As part of the ongoing research effort in this area
it is useful to develop quick analytical models for the many different aspects of open
rotor noise. One aspect which has received less attention is that of Unsteady Distortion
Noise, and in this thesis we focus on this particular mechanism. Little work has been
done on the significance of UDN relative to other open rotor sources, but manufacturers
are particularly interested in UDN generated at approach, when there is high distortion
(i.e. a strong streamtube contraction into the rotors), and lower tonal levels from other
sources.
Technological issues which we consider in the context of Unsteady Distortion Noise in
this thesis are
• the presence of the ‘bullet’, and how this affects the mean flow onto the rotors.
• installation features which lie upstream of the rotors, including the pylon and the
wing, and the use of appropriate upstream turbulent spectra to model these different
sources.
• the introduction of asymmetries in the mean flow when the engine is installed (rather
than in isolated testing conditions), including asymmetries due to the presence of
the fuselage.
• angle of attack15, which varies between about -/+3◦ for the pusher AOR configura-
tion, but can be up to 12◦ for the puller configuration due to the close proximity of
the wing.
Although we concentrate on the specific application of open rotors in this work, much is
also relevant to turboprops, contra-rotating ducted fans and wind turbines.
1.3 Academic context: review of the literature
Within turbomachinery aeroacoustics, periodic interactions between the air flow and solid
surfaces give rise to tonal noise. Analytic investigations of the most significant tonal
sources for open rotors include Hanson (1985), Parry (1988) and Whitfield et al. (1990).
In addition, the interaction of turbulence with solid surfaces is a source of broadband
15Angle of attack, also known as incidence, denotes the angle between the flight direction and the axis
of the rotor.
18 1. Introduction
noise, due to the random unsteady pressure field induced via the condition of zero normal
velocity there. However, when turbulence is distorted, for example by a strong streamtube
contraction, the eddies can become long enough to give rise to a correlation between the
forces exerted on adjacent blades, and it is this which can then lead to a quasi-tonal
noise component in addition to the broadband, see Figure 1.9. As noted by Hanson
(1974), the peaks which arise due to correlated turbulence are ‘narrow-band random
noise’, somewhere between pure harmonic and broadband. An infinitely long turbulent
eddy would produce a tonal spike, but real eddies produce peaks with non-zero width,
and it is the area under each peak which is of most interest. Increased blade-to-blade
correlation (e.g. due to longer, thinner eddies) will lead to higher level, narrower peaks
(Paterson and Amiet, 1982).
U∞eˆx + u∞
U + u
Figure 1.9: Illustration of the Unsteady Distortion Noise mechanism. A turbulent eddy
can be thought of as a spherical-ish blob of moving fluid which has angular momentum
(Davidson, 2004), and such structures naturally occur in the atmosphere. As turbulent
eddies undergo a streamtube contraction, such as that induced by an open rotor, they
are stretched. The long thin eddies then provide coherent forcing upon the rotor blades
as they are ‘chopped’ many times by adjacent blades at the same location, leading to
both tonal and broadband noise. The velocities denoted in this figure are upstream axial
velocity, U∞, upstream turbulent velocity, u∞, mean flow velocity near the rotor, U, and
turbulent velocity near the rotor, u.
There is much active research into aeroengine noise arising from turbulence, and in
this half of the introductory chapter we give an overview. We note that distortion effects
are often neglected in aeroacoustic research and testing. When undertaking this review,
two approaches were used in order to get as complete a picture as possible. Firstly, papers
from the recent AIAA/CEAS Aeroacoustics Conferences (2008-2011) and the 2010 CEAS-
ASC workshop on open rotors were examined to see which areas of turbulence interaction
noise are currently receiving the most attention. Secondly, a thorough literature search
was done for publications concerning the interaction of blades with turbulence, which
1.3 Academic context: review of the literature 19
yielded in excess of 90 papers from the last 40 years. Of these, 18 were considered to be
of the most direct relevance to this thesis, and these were analysed in detail.
1.3.1 Noise arising from turbulence
As already described, the tonal component of Unsteady Distortion Noise arises from
blade loading pulses which exhibit coherence between blades. Hanson (1974) was the first
to show the significance of the stretching of eddies which can take place during static
testing, due to streamtube contraction. In experiments, he found streamwise turbulent
length scales of over 100 rotor diameters, and correspondingly the forces on the rotor
blades were found to be correlated for up to a second, see Figure 1.10. UDN was at that
point recognised as a potentially significant noise source and could be used to explain
results which had previously been attributed to fixed inflow distortion.
Figure 1.10: Figure reproduced from Hanson (1974). The dark line indicates a very long
eddy which interacts with the rotor blade at the same azimuthal position for hundreds of
revolutions.
Although axial stretching is much reduced in flight (due to the reduced speed differ-
ential between the upstream velocity and the velocity at the rotor face) which leads to a
reduction in tonal UDN levels, a broadband component is still present. In addition, the
20 1. Introduction
effects of transverse stretching due to asymmetries of the mean flow have not been fully
quantified previously; most studies restrict themselves to the axisymmetric case. There
are many potential sources of UDN, including externally generated turbulence, secondary
inflow distortions and the boundary layers of ducts and blade wakes. However, we note
that boundary layer and blade wake turbulence is less likely to exhibit coherence (Hanson,
1974), as it may not have time to undergo significant stretching.
These several sources of turbulence also give rise to other types of turbulence-solid
body interaction noise which are distinct from UDN, but nonetheless can be examined
to give insight into the variety of analytical, computational and experimental approaches
commonly used. Firstly, the creation of turbulence as flow passes over structures on the
airframe, for example high-lift flaps and slats, is itself a source of noise (Terracol and
Kopiev, 2008). Such structures which lie several rotor diameters upstream of the rotor
may produce turbulence which is significantly distorted as it travels towards the rotor,
and therefore could, in addition, form a source of UDN. In Chapter 3 we indeed find that
a significant proportion of the distortion induced by an idealised rotor takes place within
an axial distance of 10 radii from the rotor face. Secondly, turbulent self-noise arises
when turbulence produced by one part of a rotor interacts with another part of the same
rotor. For example Turbulent Boundary Layer Trailing-Edge Interaction (TBL-TE) is a
dominant noise source for wind turbines (Kamruzzaman et al., 2008), as well as being a
subject of active research within aeroengine aeroacoustics (Pagano et al., 2009). Towards
the trailing edge of a blade the boundary layer plays an important role, and Carolus
and Stremel (2002) found that in this region the precise structure of ingested turbulence
becomes less of a factor. Thirdly, when considering open rotor rather than turbofan
engines, rotor-rotor interaction noise is generated when wakes and tip vortices from the
front rotor hit the rear rotor (Redmann et al., 2010, Kingan and Self, 2009). As well as
the mean flow component of the wakes, which produces tones when it interacts with the
second row, the turbulent part gives rise to broadband noise (Blandeau et al., 2009). The
corresponding phenomenon within ducted turbofans is the turbulent aspect within both
fan-vane and rotor-stator interaction, although in both those cases one row is stationary
and thus the velocity differential between the two rows is much smaller than for open
rotors. Fourthly, an issue primarily within ducted turbofan aeroacoustics is interaction
between the turbulent casing boundary layer and the rotor tips. Correspondingly for an
open rotor is the issue of the interaction between the hub boundary layer and the roots of
the rotor. Of these four mechanisms, rotor-rotor interaction noise from turbulent wakes
1.3 Academic context: review of the literature 21
involves a similar mechanism to UDN - unsteady forces impinging along the entire leading
edge of a blade (rather than solely the root or tip) and giving rise to unsteady lift forces
across it.
Once the turbulence source of interest has been specified the next key question in any
analysis is deriving an appropriate model for this impinging turbulence, and central to
this is the issue of isotropy and homogeneity vs. anisotropy and/or inhomogeneity. On
the one hand, turbulence in turbomachinery is clearly not always perfectly isotropic and
homogeneous, for example Hanson and Horan (1998) have used CFD to plot the turbu-
lent kinetic energy behind a rotor, to illustrate the inhomogeneities present. Turbulence
generated in the atmosphere is also not truly isotropic as it arises from a shear flow (Si-
monich et al., 1990), and certainly varies with height over topographical features. The
assumption of homogeneity in each horizontal plane will also not hold in all conditions.
However, it is very usual to use isotropic, homogeneous models for simplicity, and there
are often reasonable justifications for doing so. For example, between the front and rear
rotors it is believed there is not sufficient distance for significant distortion to take place,
and it is therefore possible to use homogeneous, isotropic models for rotor interaction
noise (Blandeau, 2011). Grid generated turbulence is often found to exhibit properties
accurately described by an isotropic spectrum (Paterson and Amiet, 1982, Mish and De-
venport, 2006a, Devenport et al., 2010) and this allows validation of isotropic analytical
models through comparison with experiment. However, there are instances where grid
turbulence has been seen to deviate from isotropic (Amiet, 1975).
Thus certain effects are often neglected, for example anisotropy can lead to ‘haystack-
ing’ around tones (Stephens et al., 2008). Devenport et al. (2010) note that ‘wind tunnel
grid-turbulence studies and isotropic turbulence calculations may significantly underesti-
mate angle of attack effects seen in practical applications where anisotropic inflow turbu-
lence is common’, and this is of particular relevance to our work on incidence in Chapter
5. Anisotropy plays a key role in UDN generation, and much of the work in this thesis
is concerned with deriving a model which accounts for the stretching of turbulence be-
fore it hits the rotors. Different methods for modelling inhomogeneity and anisotropy are
discussed in more detail in Section 1.3.3.
When assessing turbulence, and the noise arising from turbulence, there are a range
of difficulties inherent whichever method is chosen: experimental, analytical or computa-
tional. Experimental measurement is typically time-consuming and expensive, in addition
to which quantities which you might wish to observe in order to compare to theoretical
22 1. Introduction
models may be impractical to measure, or require a very large number of measurements.
It is challenging to formulate sufficiently accurate, and yet sufficiently simple, analytic
models which perform well for a range of real-life situations. Computational Fluid Dy-
namics (CFD) calculations are associated with long run-times and it is still necessary to
determine appropriate turbulence models to input into any numerical model. Next we
discuss each of these approaches.
1.3.2 Use of experimental measurements
As the specific structure of ingested turbulence is a critical question in all of the tur-
bulence noise problems listed above, ideally we would gain a more detailed knowledge
of that structure from experiment. Hot-wire anemometry can be used to measure ve-
locity correlations, for example. In addition, both computational and analytic work can
overpredict tonal levels when compared to experiments, as not all real-life aspects of the
flow-field can be included within any model and this tends to lead to the purely tonal
components being over- or under-estimated. Mish and Devenport (2006a) noted a lack of
experimental work available to validate the range of theories which have been developed
around turbulence-blade interaction. This is partly due to a problem inherent in turbu-
lent analysis, namely the lack of general models applicable to a wide range of situations
(Davidson, 2004). Often each individual situation must be considered separately.
In addition to measurements of the turbulence structure itself, experimental measure-
ments of the noise arising from turbulence are scarce, due to difficulties in separating
this from the other rotor sources of varying strength, frequency and directivity (Pater-
son and Amiet, 1982, Simonich et al., 1990). Commercial sensitivity has also restricted
experimental measurements of open rotor noise available in the open literature to some
extent. The vast majority of testing is static, in wind tunnels, rather than in flight, and
thus UDN is not often considered. Wind tunnel experiments do not include noise from
ingested turbulence from external sources as a system of honeycomb grids is typically used
to remove all turbulence and very low levels of turbulence can be successfully achieved,
(Kamruzzaman et al., 2008). Attempts are also made to minimise incoming turbulent
intensity in most flat plate studies (Carolus and Stremel, 2002).
Experimental measurements of leading edge noise arising from impinging turbulence
tend to consider grid-generated turbulence only, where the aim is to create isotropic,
homogeneous turbulence, (e.g. Staubs et al. (2008)). These grids are themselves noisy,
1.3 Academic context: review of the literature 23
and Devenport et al. (2010) recently used a special anechoic facility to overcome this and
other methodological problems, such as the distortion of turbulence by angle of attack
corrections.
We now highlight some key studies of turbulence-blade interaction noise spanning the
last four decades which include experimental measurements, often in order to validate an
accompanying theory. As already described, Hanson (1974) first observed the stretching of
atmospheric turbulence eddies, which improved understanding of the mechanisms which
lead from turbulence to noise. The aim was to compare theoretical lift to measured
pressure for calibration, and then predict the sound. These experiments showed UDN to
be a dominant noise source for subsonic blades in static testing. We note that a more
complete prediction scheme for UDN is therefore of particular importance in extrapolating
from static testing data to flight test results.
Elsewhere, much research has been concerned with a single airfoil immersed in turbu-
lent flow, but the majority of investigators have considered isotropic turbulence. Amiet
(1975) and Paterson and Amiet (1982) developed turbulence-airfoil interaction theories
and then compared these to measurements of the Sound Pressure Level (SPL) due to an
airfoil in a stream of grid-generated turbulence. They compared both high and medium
turbulent intensity producing grids and found their inviscid, flat-plate theory gave rea-
sonable agreement with measurements. No adjustable constants were used to fit their
predictions to the results. Key quantities were shown to be the spanwise correlation
length and the Power Spectral Density (PSD) of vertical velocity correlations. Carolus
and Stremel (2002) also compared the turbulence produced by several different grids and
honeycomb structures. The honeycomb was found to reduce the integral lengthscale and
turbulent intensity somewhat. In contrast, the square grids led to high turbulent intensi-
ties and small values of the turbulence integral length scale. Devenport et al. (2010) also
looked at an airfoil in grid-generated turbulence, but as mentioned above, using novel ex-
perimental techniques to improve results. They found little change in leading edge noise
as angle of attack was varied in the case of isotropic turbulence, but predicted greater
effects for the anisotropic case. Mish and Devenport (2006a) have also considered angle
of attack effects experimentally and found differences between their results and variable
angle of attack theory. They attributed this to the difference between a flat-plate model
and the distortion induced by a real blade’s leading edge.
Next we turn to three studies of ducted configurations, but where the stretching of
turbulence is taken into account or anisotropic turbulence considered.
24 1. Introduction
Hanson (2001), using Glegg’s 3D cascade theory, further developed theory for both
isotropic, homogeneous turbulence (specifically the Liepmann spectrum) and axisymmet-
ric but axially stretched turbulence, convected into a cascade of stators. He compared his
theoretical results with NASA experimental data of rotor noise - a Pratt & Whitney 22
inch fan model, an Allison 22 inch fan with swept and leaned stators, and a Boeing 18
inch fan. (We note that these sets of data are commonly used as benchmarks, for example
see Cheong et al. (2006).) Of particular interest in Hanson’s work was the effect of lean
and sweep, as well as the possibility of accounting for inhomogeneities in the turbulence
by appropriate averaging. Turbulent intensity and integral length scale were used as free
parameters when fitting the data in that case, and agreement was good.
Considering the issue of anisotropy, Atassi and Logue (2009) recently looked at the
interaction between anisotropic turbulence and a ducted fan, using a uniform distortion
model. They compared their results with low-speed experiments (Stephens et al., 2008),
however they found that the isotropic expressions fitted the data best, perhaps because
their theory assumed turbulence across the whole span of the blades whereas the source
of turbulence in the experiments was solely the duct wall boundary layer.
Finally, Koch (2009) undertook an experimental study of fan inflow distortion noise.
The effect of turbulence shed from upstream radially orientated cylinders upon the noise
spectra was considered with a honeycomb used to remove other incoming turbulence. The
asymmetric Strouhal shedding led to extra peaks in the noise spectra, as well as raising
the broadband level.
1.3.3 Analytic models
The open rotor design is younger than the turbofan. Especially during these early stages,
fast analytic or semi-analytic methods which compare well to full CFD are advantageous,
but necessarily will involve some degree of approximation. Here we review the develop-
ment of theoretical turbulence-blade interaction modelling.
As noted by Paterson and Amiet (1982), the first analytical models of airfoil acous-
tics concentrated on relating a given set of blade forces to the resultant noise. From the
mid-1970s, researchers widened their scope and began to consider in more detail how
an impinging gust gives rise to forces on a blade in the first place, and thus to noise.
A steady progression has then been seen whereby extra effects have been added, to de-
termine more and more sophisticated airfoil response functions. To the incompressible,
1.3 Academic context: review of the literature 25
2D model of an isolated airfoil in rectilinear motion (Sears, 1941) have been added com-
pressibility, skewed gusts, non-convected gusts, quasi-3D strip theory and other cascade
models (such as Smith (1973)), and rotation of sources. More realistic blade geometries
have been considered including effects of camber, thickness, angle of attack, skewed 3D
gusts, and non-compactness. The recent review in Mish and Devenport (2006a) gives de-
tailed references for these various extensions. Pushing back the boundary in one of these
directions may lead to sacrifices in other areas, or restrictions in the range of validity. For
example Glegg’s 3D cascade theory, as used by Hanson and Horan (1998), requires the
approximation of infinite span.
In our present work we consider the surface pressure in response not just to a single gust
but to a complete turbulent inflow field. Thus we aim to relate the distorted turbulence
spectrum to the upwash spectrum.
Turbulence distortion
As previously mentioned in section 1.3.2, Amiet (1975) and Paterson and Amiet (1982)
developed a foundational model for the far-field noise due to a single airfoil in terms of
the turbulent energy spectrum and an airfoil’s response function, and then compared this
to experiment. Like other turbulence interaction theories previously developed (Sharland,
1964, Mani, 1971, Homicz and George, 1974) stretching of turbulence is not accounted
for within their framework; the turbulence is considered isotropic.
Once the experiments of Hanson, and also Cumpsty and Lowrie (1974), had shown
the importance of turbulent stretching, several models for this mechanism emerged. Han-
son used a pulsing model to represent discrete, radially compact eddies, and found an
expression for the average energy spectral density due to each eddy. He used a statistical
distribution of discrete eddies rather than considering the complete turbulence spectrum.
This involves assuming the turbulence takes the form of discrete eddies, whose character-
istics are random variables, giving a ‘pulse’ theory formulation. Hanson assumed eddies
acted as point forces in the radial direction. In this framework models must be input for
the lift pulse due to an eddy and the joint probability density functions for turbulent eddy
characteristics (lengths/widths etc.). Hanson (2001) has also developed a method whereby
random offset times within turbulent velocities can be used to represent inhomogeneity.
In a pair of papers (Simonich et al., 1990, Amiet et al., 1990) a full method of UDN
prediction was laid out which comprised several self-contained blocks, of which one is
the distortion of the incoming turbulence. The analysis in this thesis follows a similar
26 1. Introduction
structure, as sketched within the Chapter 2 outline on page 33. They modelled a flat
plate in rectilinear motion, and thus cascade effects were not treated. Crucially though,
differential drift between particles on neighbouring streamlines was taken into account,
thus allowing the consideration of non-uniform distortions. They were concerned with
the case of a helicopter rotor in hover, vertical ascent and forward flight. An assumption
of their model is that fluid planes upstream of the rotor remain fluid planes downstream.
This becomes more valid as the scale of turbulence decreases, hence they treated only
turbulence of lengthscale less than the scale of contraction. They plotted the PSD given
different values of the distortion tensor. When comparing the distorted and isotropic
cases they found widths of peaks in the noise spectra and depths of troughs varied most
significantly, whereas the heights of peaks were comparable. In addition to the total
amount of distortion, they also considered in some detail the effect of the direction of
distortion. When distortion acts primarily along the axis of the rotor, the upwash velocity
on the blades is reduced, leading to lower overall noise and reduced width peaks compared
to the isotropic case within their model.
Majumdar (1996), in part formalising work by Cargill (1993), developed Simonich et
al.’s analytic framework further. He used an approximate solution for the full distorted
velocity field (rather than the assumption that fluid planes remain planes) and included
cascade effects. The significant quantities within the expression for far-field radiated
sound were found to be the distorted turbulence spectrum at the rotor face, and the
Bessel functions which govern the radiation. Non-uniform distortions were considered
(although still with some restrictions, such as axisymmetry) by applying Rapid Distortion
Theory to an isotropic upstream turbulent field. This is the approach we extend, and will
be laid out in detail in Chapter 2. Wright (2000) added effects of upstream swirl in
the mean flow field to this model. In an alternative approach, Atassi and Logue (2009)
have treated uniform turbulence distortions using an anisotropic spectral density tensor,
with three constant distortion coefficients, and we discuss the direct use of anisotropic
turbulence models further in Chapter 3, §3.3.2. Within their uniform distortion model,
increased axial stretching was found to lead to a shift in acoustic energy towards the lower
frequencies.
As mentioned in §1.3.2 in recent experiments Mish and Devenport (2006a) considered
an airfoil immersed in isotropic turbulence for various angles of attack, and concluded
that significant distortion takes place near to a blade’s leading edge. They noted differ-
ences between the theoretical distortion found with a flat plate model (with large velocity
1.3 Academic context: review of the literature 27
gradients) used in the analytic model of Reba and Kerschen (1996), and real distortion
by a blade with thickness. They went on to develop an analytic model for the observed
distortion by modelling the leading edge as a sphere of the same radius. Similarly, Deven-
port et al. (2010) highlighted the important role of vorticity just upstream of the leading
edge.
Turbulence properties
Whether modelled before or after distortion, the properties of the upstream turbulence
will affect the UDN generated by a rotor. Increased turbulent intensity produces an
increase in SPL at all frequencies (Carolus and Stremel, 2002, Hanson, 2001) whereas
varying the turbulent integral lengthscale, L, has a more complex effect. SPL is found to
increase as L increases only until a critical value is reached.
Turbulence, by its nature, will always contain a very wide range of lengthscales but L
gives some measure of the average size of eddies. The role of L has been considered by
Majumdar and Peake (1998), Hanson and Horan (1998) and Atassi and Logue (2008), and
varying L appears within the latter’s model to have greatest effect in the lower frequency
range. The critical value of L, above which SPL does not increase, is believed to be
related to the lengthscale of the streamtube radius. Eddies much larger than this will not
appear as eddies to the rotor but rather as a varying background flow. However, Hanson,
looking at noise and turbulence data from Boeing, believed it was not the mean eddies
but those which were larger than average which dominated noise generation. In the high
distortion case, it may be that it is the large eddies which give rise to tonal noise, whereas
the smaller eddies have a similar effect as at low distortion (Majumdar and Peake, 1998)
as they of course do not become very long even after stretching.
Regarding tonal noise, peaks are found to have higher magnitude and narrower width
for increased L. At high frequency the results of Atassi and Logue showed little L depen-
dence. Carolus similarly concluded that the influence of inflow turbulence characteristics
diminishes in the high frequency range.
Finally we note the role of the intersection area between an eddy and a rotor blade,
(Amiet et al., 1990). It is in part through changes to this quantity that we expect
differences to arise in the asymmetric cases we consider, compared to the axisymmetric
case.
28 1. Introduction
High and low frequency limits
Certain effects become more important depending on whether a high or low frequency
regime is being considered16. As one example, Carolus and Stremel (2002), looking at
broadband noise due to incident turbulence, found the blade surface pressures of a ro-
tating fan were dependent not only on the turbulent parameters but also on chordwise
position, which side of the blade was being considered and on frequency range. From an
analytical modelling point of view, as frequency is increased noise can be more and more
accurately determined by representing the blade forces by point sources at the leading
edge (Amiet, 1975, Evers and Peake, 2002). Also, long wavelength/low frequency distur-
bances will essentially see rotor blades as flat plates, whereas effects of blade thickness
and geometry as well as boundary layer pressure fluctuations will be significant for short
wavelength/high frequency eddies, see Mish and Devenport (2006a) and Atassi and Logue
(2009) respectively. In the experiments of Mish and Devenport, a previously unseen effect
was found; at low frequencies there was a reduction in spectral level as angle of attack
was increased. However, certain effects, such as non-zero camber, diminish in significance
when results are integrated over the full range of wavenumbers contained in an incoming
turbulent field, as shown by Evers and Peake (2002).
Interaction between blades (cascade effects) will be critical for larger disturbances,
whereas at high frequencies turbulence ingestion noise is caused by small eddies to which
each blade appears isolated; the resultant noise in that case is expected to scale with
blade number B. The critical frequency separating these two regimes was considered by
Cheong et al. (2006) for the case of a 2D cascade of flat plates. Thus, for small eddies,
both decreased chopping and reduced coherence between blade forces leads to broadband
rather than tonal noise.
Wavenumber and frequency considerations come into our analysis in several ways. In
Chapter 2 an expression for the stretched turbulence spectrum at the rotor face is derived,
and requires the modulus of the wavevector ≫ 1/(measure of distortion), and we show
this is indeed a reasonable assumption. We also use a flat plate approximation in our
blade response model.
16For a fixed sound speed, c0, increasing frequency, ω, will lead to decreasing wavelength, 1/|k|, since
c0 = ω/ |k|.
1.3 Academic context: review of the literature 29
Three-dimensional effects
Efforts are continually being made to extend existing 2D theories to better include 3D
effects, such as including radial variation (e.g. Majumdar and Peake (1996)) or modelling
annular rather than rectilinear geometries (e.g. Atassi and Logue (2008)). Within a
lengthy analysis 3D effects are often not included at every stage; Glegg’s harmonic cascade
theory uses a 2D mean flow and geometry, but a 3D unsteady input excitation and
response and thus the spanwise wavevector component is included.
Dividing the blade into 2D strips is a common approach whereby we neglect radial
gradients when calculating the blade response (Majumdar and Peake, 1998, Cheong et al.,
2006). There are several variations of strip theory, and we follow the analysis of Smith
(1973) in our work. However, the strip theory approximation breaks down in certain
situations. For example, when duct modes are near to cut-off17 one expects larger unsteady
amplitudes, leading to spanwise motion. In addition, for large radial wavenumber (i.e.
very small radial lengthscales) radial gradients can become large. In the analysis of
Chapter 2 it will be necessary to impose limits on the integral over radial wavenumber to
account for this.
Stators and ducted fans
We note, for completeness, that much work has also been done in the field of turbulence
impinging upon stators. A key difference in that case is that the circumferential mode
number is given by nB − kV (Tyler and Sofrin, 1962) where B is the blade number, V
is the number of stator vanes, and n and k are integers. Thus the modes are spaced out,
and it is possible to choose B and V so that the first mode is cut-off18. In contrast, open
rotors, with their counter-rotating blade rows, produce modes of order m+ lB, which can
take all integer values.
The interaction of turbulence with ducted rotors is of interest since as the bypass ratio
of turbofans is increased, fan noise becomes more significant, (Dieste and Gabard, 2009).
Atassi and Logue (2008) looked at the broadband noise of a fan within an annular duct
caused by ingested isotropic turbulence, where distortion was not accounted for. They
17A cut-off mode is one where the axial wavenumber contains an imaginary component, and thus the
mode decays away from the source.
18By fixing the circumferential mode number at a given tonal frequency, for example a multiple of the
BPF, the axial wavenumber is forced to take imaginary values for disturbances of that frequency. Thus
any particular tone can be cut-off.
30 1. Introduction
investigated varying both the turbulence lengthscale and flight speed. The change in
noise level as the turbulence lengthscale was varied depended on both frequency range
and Mach no., with the largest differences being observed for low to medium frequencies
at high Mach no.s. They also found that as the tip speed approached the flight speed the
difference in height between peaks and troughs of the spectrum diminished.
Majumdar also treated the ducted rotor case, adapting his open rotor analysis by
adding a condition of zero penetration at the duct radius. A simplifying assumption of
parallel flow at the rotor face was used in that case. The overall spectrum was found not
to be significant affected, but the directivity was altered.
1.3.4 Computational approaches
Today, effects of turbulence are often tackled using computational simulation. Extensive
open rotor studies are currently running at Germany’s DLR research centre, the French
aerospace lab ONERA, and NASA in the US. Although powerful computing tools are
now available, there are still limitations as to what can be achieved within a reasonable
timeframe, and many studies restrict themselves to a few wavenumbers and frequencies
only, as noted in Mish and Devenport (2006a). Disadvantages of CFD include the long
run times needed for low frequencies, and the small grid and step sizes needed for high
frequencies. Our focus is on developing a quick analytic framework but here we highlight
some commonly used computational methods. Wang et al. (2006) provides a detailed
review of current computational techniques used to predict flow-generated noise.
Full simulation of all the fluid dynamics and acoustics can be achieved with Direct
Numerical Simulation (DNS), but this is restricted to fairly low Reynolds number and
simple geometries. Large Eddy Simulation (LES) may also be used in this way, but in
such simulations scales smaller than the mesh are modelled, rather than fully resolved.
Hybrid methods take a full non-linear approach near the source, but then use a simpler
analytic model for the far-field noise, for example Lighthill or Linearized Euler (Deconinck
et al., 2010). Thus, for example, LES can be used near the source and then coupled to an
acoustic analogy model or Computational AeroAcoustics (CAA) code for the radiation.
Finally, the simplest computational codes model only the aerodynamics and acoustic
sources and then compute the radiation using the acoustic analogy. Reynolds-Averaged
Navier Stokes (RANS) is one such method, used to provide statistical properties only.
In all cases very careful consideration of boundary conditions is required to avoid
1.4 Summary 31
spurious reflections. Resolving all scales and implementing non-reflecting boundary con-
ditions can be particularly tricky for turbulent flow fields (Gloerfelt and Berland, 2009).
Issues can also arise when trying to decide upon a realistic form for the turbulence input.
Appropriate models for different forms of turbulence, such as within the boundary layer,
can be input into CFD either using an analytical method (where the key input parameters
are determined either from experiment or other simulations) or using a numerical method.
Approximations may be made, such as using a flat plate boundary layer turbulence model
rather than the full boundary layer over a real geometry (Carolus and Stremel, 2002). The
anisotropy of turbulence is also difficult to treat as well as more computationally intensive.
Many computational studies therefore also assume an isotropic/homogeneous turbulent
flow - Kamruzzaman et al. (2008) recently looked at coming up with a simple way to
account for anisotropy within RANS.
The optimum positioning of the engines of an open rotor on the airframe is still a
somewhat open question (Envia, 2010) and ONERA have found within their CFD work
that installation effects are of particular importance (Boisard et al., 2010). To study
these effects in more detail, Oma¨ıs and Ricouard (2010) at Airbus have implemented a
RANS k−ω model for the pylon wake and found that the front BPF tones and azimuthal
directivity were most affected. The role of the pylon wake is a motivating factor in our
work considering flow asymmetries.
1.4 Summary
In summary, much work has been done on turbulence interaction noise to date but the
distortion mechanism is not always modelled, with many studies restricted to the isotropic
case. Since the illuminating experiments of Hanson in 1974, as far as we are aware,
there have been no further experimental studies where the stretching of real atmospheric
turbulence by a rotor has been quantitatively assessed. Certainly, this mechanism is not
routinely considered during flight tests where there are many competing sources.
There has also been less attention given historically to non-axisymmetric flows. Our
review in this chapter has highlighted many reasons why it is useful to consider the asym-
metric case. Simonich et al. (1990) noted the importance of the direction of distortion,
in addition to its strength. Using Rapid Distortion Theory they calculated distorted vor-
ticity fields but did not produce full sound spectra. Secondary inflow distortions due to
asymmetric installation features are also of much current interest.
32 1. Introduction
This thesis builds in particular upon the work of Cargill (1993) and Majumdar (1996).
Both these authors highlighted areas which could be revisited within their analytic frame-
work which we now address. Cargill neglected radial variation of the distortion at the
rotor face and differences in drift time, both of which can be significant (Paterson and
Amiet, 1982, Hunt, 1973). Majumdar incorporated these effects in the axisymmetric case,
but neglected the azimuthal angle dependence within the amplitude of the distorted ve-
locity at the rotor face. We have now corrected this, allowing extension to the asymmetric
case. Majumdar also noted the need to incorporate more realistic flow fields as well as
non-uniform cross-section effects.
Our research aims are therefore as follows:
1. To re-evaluate Majumdar’s analysis for the calculation of UDN, and to extend it to
the asymmetric case.
2. To consider the important case of rotor incidence, which commonly occurs during
flight. What effect will this particular form of asymmetric distortion have on UDN
radiated to the far-field?
3. To incorporate realistic features of the mean flow into a next generation AOR, such
as due to the hub (bullet) and fuselage, and installation features.
4. To undertake a thorough investigation of appropriate turbulence models and integral
lengthscales to use in a prediction scheme for UDN, in order to allow manufacturers
to successfully calibrate between rig testing and flight data.
In Chapter 2 we address point 1 above and find that, in order to extend to the
asymmetric case, a reformulation of the analysis is required. Next, in Chapter 3 we
improve the mean flow field model, but still within the axisymmetric framework. We
consider the effects of the bullet and rear blade row as well as variable blade circulation
along the span. We also address point 4 in the second half of Chapter 3. In Chapter 4
we extend the analysis to the asymmetric case and incorporate non-axisymmetric mean
flow features for the first time, modelling the presence of the fuselage or a second rotor
adjacent to the first. In Chapter 5 we consider the case of turbulence distortion into a
rotor at incidence, and the effect this has on the far-field radiated sound.
Crucially, our work now allows for the use of any, not necessarily axisymmetric, mean
flow within a fully detailed UDN prediction scheme.
Chapter 2
Axisymmetric Rotor Systems
2.1 Chapter outline
In this chapter we detail the key components of our analytic model for calculating the
Unsteady Distortion Noise of an Advanced Open Rotor. This model extends previous
work by Majumdar and Peake (1998), clarifying the role of extra terms which arise due
to the dependence of the distortion amplitude upon azimuthal angle, φ; these terms were
neglected in their work. The expressions for the distorted turbulence spectrum, blade
pressures and radiated noise are all corrected. Our aims are always two-fold. Firstly,
to gain insight into the problem by carefully examining quantities found analytically
at intermediate stages on the way to the full noise calculation. Secondly, to produce
computed results for the full noise spectrum, to allow comparison with other experimental
and theoretical results. The schematic below gives the structure of our analysis. Different
blocks are adjusted in each chapter, and where different models are used or expressions
found in later chapters these are indicated in red.
Inputs Outputs
Mean flow model, U:
constant circulation
actuator disk
Upstream turbulence
model, u∞:
von Ka´rma´n spectrum
Turbulence spectrum at
rotor face: Sij
Blade pressures: Pˆblade
Far-field pressure: Pˆ
leading to spectral power
Distortion of upstream
turbulence, u∞, by
mean flow, U, using
Rapid Distortion Theory.
Extension: previously
neglected terms shown to
be non-zero, azimuthal
components of distorted
wavevector, lφ, and dis-
torted amplitude tensor,
eˆφ · A and A · eˆφ.
33
34 2. Axisymmetric Rotor Systems
2.2 Calculating the turbulent perturbation, u
Our method of UDN calculation involves calculating several different components of the
fluid velocity at different stages. Under suitable assumptions, described at each stage,
the non-linear interactions between these components can be neglected. In addition,
different physical assumptions are appropriate to each component, in particular concerning
incompressibility. At all stages the flow is considered inviscid.
We begin by determining an expression for the distorted turbulent perturbation found
at the rotor face, due to an isotropic turbulent field far upstream. Accurately modelling
a turbulent velocity field while preserving an analytically tractable set of equations of-
ten poses a significant challenge. For this reason several, fairly limiting, assumptions are
often made when considering a turbulent field. The most common of these assumptions
is isotropy, which also implies homogeneity (Batchelor, 1953). In particular, atmospheric
turbulence is often modelled as isotropic, and we will look in more depth at this approx-
imation in Chapter 3. However, in certain situations, for example when the presence of
solid boundaries provides a nearby source of vorticity, anisotropy of turbulence can play a
significant role and accurate results may depend upon taking this into account. Unsteady
Distortion Noise is a clear example of this, where the elongation of eddies in the axial
direction leads to far higher tonal noise than would be experienced were the turbulence
isotropic at the rotor face (Hanson, 1974).
Instead of attempting to model the turbulent field at the rotor face where experimental
measurements are scarce, Majumdar and Peake (1998), building upon work by Cargill
(1993), developed a method to assess the distortion of turbulent eddies as they travel
downstream towards the rotor using Rapid Distortion Theory (RDT). RDT is concerned
with calculating the change in a small velocity perturbation (in this case, the upstream
turbulent field) as it is convected by a dominant, distortive, potential mean flow. In this
way it is possible to quantify turbulent anisotropy at different points in the flow, while
still using the more straightforward approximation of isotropy far upstream. Majumdar
and Peake thus derived an expression for the spectrum of turbulence at the rotor face in
terms of the spectrum of turbulence far upstream. A discussion of other approaches to
modelling UDN was given in §1.3.3.
The work of this Ph.D. has incorporated new axisymmetric features into Majumdar’s
method (Chapter 3) as well as further extending the theory to asymmetric systems (Chap-
ters 4 and 5). In re-examining the details in order to make these extensions, certain terms
2.2 Calculating the turbulent perturbation, u 35
which were originally neglected, such as the contribution of the azimuthal wavenumber
kφ, have been shown to be non-zero, even in the axisymmetric case. Thus in this chapter
we present the full analysis for calculating Unsteady Distortion Noise in an axisymmetric
system.
2.2.1 Rapid Distortion Theory (RDT)
Firstly we describe Rapid Distortion Theory, which is the method we use to calculate the
velocity field at the rotor face, given the upstream turbulence field.
Background
Batchelor and Proudman (1954), building on earlier work by Taylor, considered the effect
of a uniform distortion on an isotropic turbulence field (in fact, uniform pure strains
only as the effect of rotation is straightforward). The turbulent eddies are assumed to
be distorted by the mean flow before they can exchange energy with each other, hence
the term ‘rapid distortion’. The restriction of this analysis to uniform distortions only
means explicit expressions for the downstream perturbation velocities can be determined
without integrating over the entire history of the flow. The approximation of uniformity
will be valid only over restricted portions of any real background mean flow. For our
purposes the distortion of a turbulent eddy with a lengthscale comparable to that of the
rotor is of particular relevance, as these are believed to contribute most to tonal levels.
Distortion of these eddies will certainly not be uniform at different axial locations and
will also vary between the interior and the edge of the eddy.
Hunt (1973) went on to extend RDT to situations of non-uniform distortion induced
by the presence of a solid body, and to satisfy the condition of no-penetration on that
body. Hunt starts with the full Navier-Stokes equations and proceeds to justify neglecting
viscosity, the body’s boundary layer and the possible wake behind the body through
scaling arguments. The assumptions used are also examined a posteriori and found to be
valid as long as the turbulent wavenumber being considered isn’t too large or the scale of
the turbulence too small, as the boundary layer must then be taken into account. Hunt’s
analysis also highlights the two different effects at play; that of distortion by the mean
flow and the separate effect of the presence of solid boundaries upon which the condition
of zero normal velocity must be satisfied. These two effects can be made explicit by
splitting the perturbation velocity into two separate terms, as shown in the re-working
36 2. Axisymmetric Rotor Systems
of RDT undertaken by Goldstein (1978). Goldstein showed that Hunt’s analysis could
be simplified by reducing the number of equations to be solved. He also extended the
analysis to compressible flows, and included the possibility of entropy fluctuations. Over
the next three pages we review the major results contained in Goldstein, who gave a very
clear formulation for the application of RDT. For our purposes we assume zero entropy
fluctuations throughout.
Theory
The development of a small unsteady vorticity disturbance, introduced far upstream
of a steady potential mean flow, as it is convected downstream by that mean flow is
calculated. The mean flow is assumed to tend to a uniform value far upstream, as would
be the case for the potential flow induced around an object placed in a uniform, inviscid
flow.
Mean flow: We split the flow variables into mean and perturbation quantities: velocity
v = U (x) + u (x, t), pressure p = p0 (x) + p′ (x, t) and density ρ = ρ0 (x) + ρ′ (x, t). (We
note that the mean flow must be steady in order to apply RDT.) The steady mean flow
quantities then satisfy the steady flow equations
ρ0U · ∇U = −∇p0, (2.1)
U · ∇ρ0 + ρ0∇ ·U = 0, (2.2)
with boundary conditions
U → U∞eˆx as x→ −∞, (2.3)
nˆ ·U = 0 on the solid surface of the object, (2.4)
corresponding to uniform flow at upstream infinity and zero penetration on the object (in
our case the blades of the rotor). Note that throughout this thesis we work in a reference
frame in which the engine is stationary, and so U∞ corresponds to the forward flight speed
of the aircraft.
We assume that the turbulence is of a low-level compared to the mean flow, by which
we mean of a low enough intensity for linearisation to be justified (|u| / |U| ≪ 1) and
of a large enough scale to make the approximation that the eddies are distorted before
2.2 Calculating the turbulent perturbation, u 37
exchanging energy with each other. Thus we require
LUδu
LδU
≪ 1, (2.5)
where LU is a typical lengthscale of the mean flow, L is the turbulent integral lengthscale,
and δu and δU are typical perturbation and mean velocity changes respectively.
For our application the mean flow U is the flow induced by the engine. It is assumed
to be steady and incompressible. The flight speeds of interest in this work are subsonic,
take-off Mach numbers typically being in the range 0.2-0.25. As the mean flow tends to
a uniform flow far upstream it must be irrotational there, and as vorticity is conserved in
inviscid, axisymmetric flows, it follows that U is irrotational everywhere.
Upstream boundary conditions: Considering the full (mean plus perturbation) flow
equations, linearising and subtracting out the mean flow equations (2.1) and (2.2) we find
ρ0
(
D0u
Dt
+ u · ∇U
)
+ ρ′U · ∇U = −∇p′, (2.6)
D0ρ′
Dt
+ ρ′∇ ·U+∇ · (ρ0u) = 0, (2.7)
where D0/Dt = ∂/∂t+U ·∇. Far upstream, where ρ0 → ρ∞ = constant, these equations
become
ρ∞
D∞u
Dt
= −∇p′, (2.8)
D∞ρ′
Dt
+ ρ∞ (∇ · u) = 0, (2.9)
where D∞/Dt = ∂/∂t + U∞∂/∂x. We can deduce various properties of the perturbation
velocity if we restrict attention to harmonic disturbances of the form
u = (u, v, w) exp {i (αx+ βy + γz + ωt)} . (2.10)
Then D∞/Dt = i (ω + U∞α) and, considering equation (2.8), we find two possibilities:
1. ω + U∞α = 0 and hence p′ = 0 and, from equation (2.9), ∇ · u = 0, i.e. the
disturbance is incompressible; or
2. ω + U∞α 6= 0, in which case p′ 6= 0 and equation (2.8) gives us u/α = v/β = w/γ,
38 2. Axisymmetric Rotor Systems
i.e. the disturbance is irrotational.
We consider disturbances whose pressure fluctuations tend to zero far upstream as we
do not wish to impose incoming acoustic disturbances (as in Goldstein). The turbulent
disturbance, u∞, is ‘non-acoustic’ and vortical far upstream, i.e. it is a type 1 disturbance.
This is purely convected with the flow far upstream and so its x and t dependence are
coupled. We therefore find the following upstream boundary conditions for the full velocity
and pressure fields
v (x, y, z, t) → U∞eˆx + u∞ (x− U∞t, y, z) , (2.11)
and p (x, y, z, t) → p∞, (2.12)
as x→ −∞.
Full flow equations: We now turn our attention to the main region of the flow, near
the rotor. Using equation (2.1) we can rewrite (2.6) as
D0u
Dt
+ u · ∇U = − 1
ρ0
∇p′ + ρ
′
ρ20
∇p0. (2.13)
For an ideal gas at constant entropy, we have the relation p/ργ = C where γ = cp/cv is
the ratio of specific heats and C is a constant. Using
p′ ≃ γCργ−10 ρ′ = c20ρ′, (2.14)
where c0 is the speed of sound, we rewrite the last two terms of (2.13) as
D0u
Dt
+ u · ∇U = −∇
(
p′
ρ0
)
. (2.15)
Form of solution: We can distinguish between two different effects at play by writing
the induced perturbation velocity as the sum of two components: u = ∇ψ + uvort. The
first term, ∇ψ, gives the non-acoustic1 pressure perturbation and ensures the right hand
side of (2.15) is satisfied, but it is irrotational and so must tend to zero far upstream.
Therefore the second term, uvort, (which satisfies the homogeneous form of (2.15)) is
needed to meet the upstream boundary condition.
1By examining the phase of ψ0, given later in equation (2.27), it can be seen that ∇ψ is indeed
non-acoustic.
2.2 Calculating the turbulent perturbation, u 39
The novel part of Goldstein’s approach, as compared to Hunt’s analysis, is the ex-
pression found for uvort. As will be shown below, we can always define a quantity
X = X eˆx + Y eˆy + Zeˆz with the following properties
D0
Dt
(X− U∞teˆx) = 0, (2.16)
and X → x as x→ −∞. (2.17)
In the key step we then find that the combination of
uvorti = u
∞ (X− U∞teˆx) ·
∂X
∂xi
, (2.18)
and − ρ0
D0ψ
Dt
= p′, (2.19)
satisfies (2.15) and the upstream boundary conditions. (We have used the fact that U
is irrotational and so ∂Ui/∂xj = ∂Uj/∂xi.) Finally the mass conservation equation (2.7)
must be satisfied and so the following inhomogeneous wave equation for ψ is found
D0
Dt
(
1
c20
D0ψ
Dt
)
− 1
ρ0
∇ · (ρ0∇ψ) =
1
ρ0
∇ · ρ0uvort. (2.20)
Applying RDT to any given situation therefore comes down to two mathematical prob-
lems, finding X and hence uvort, and then solving (2.20) for ψ. At this stage, we can
further split ψ into two components, ψ0 satisfying the inhomogeneous RHS of the wave
equation above and ψ′ being the upwash velocity needed to satisfy the no penetration
boundary condition on the blades (satisfying the homogeneous version of (2.20)). By def-
inition, it is ψ′ which induces an unsteady pressure jump across the blades of the rotor,
and we will calculate this later using the LINSUB theory in §2.3.1.
The problem of calculating ψ0 can be simplified by considering the incompressible
limit (as Majumdar did) for which (2.20) reduces to
−∇2ψ0 = ∇ · uvort =
∂
∂xi
{
u∞j (X− U∞teˆx)
∂Xj
∂xi
}
. (2.21)
Thus, within this formulation, certain components of the turbulent perturbation,
uvort + ∇ψ0, are assumed to be incompressible. At this stage of the analysis we are
primarily concerned with approximating the velocity found immediately upstream of the
rotor, and the assumption of incompressibility allows us to calculate this velocity ana-
40 2. Axisymmetric Rotor Systems
lytically. Later, when calculating the far-field pressure, the compressible component of
velocity, ψ′, is re-introduced.
Much of the work in this thesis is concerned with formulating a non-axisymmetric
UDN theory, and we note that the formulation of Rapid Distortion Theory we have given
in this section does not depend on whether the flow is axi- or a-symmetric. However, as
we will see in §2.2.4, the existence of a streamfunction greatly simplifies the problem of
calculating X.
2.2.2 The quantity X
We previously claimed that, for a flow U which tends to a uniform velocity far upstream
U∞eˆx, it is always possible to define a quantity X (x, y, z) = X eˆx + Y eˆy + Zeˆz which
satisfies conditions (2.16) and (2.17). Condition (2.16) is a statement that the three
quantities X −U∞t, Y and Z remain constant when moving with the flow. We can label
each streamline uniquely according to its y and z coordinate far upstream and then, for
every point along that streamline, define Y and Z to be these upstream values. The third
quantity, X, is defined via the drift function (Lighthill, 1956) which gives the time taken
by individual particles to move along streamlines from some reference position2. The drift
function ∆(x) is defined as
∆(x) ≡ x
U∞
+
∫ x
−∞
[
1
(U∞ + Ux)
− 1
U∞
]
dx′, (2.22)
where (U∞ + Ux) is the total axial velocity, including the uniform upstream component,
and the path of integration is taken along a streamline. The difference in ∆ at two points
x1 and x2 on the same streamline gives the time taken by a fluid particle to travel from
x1 to x2. Thus the quantity (∆ − t) remains constant when moving with the flow. We
use X = U∞∆ as the axial component of X in Rapid Distortion Theory and note that
∆ → x/U∞ as x→ −∞.
For the most general three-dimensional mean flow we have
X (x) = X(x, y, z)eˆx + Y (x, y, z)eˆy + Z(x, y, z)eˆz
= X eˆx +R cos (Φ− φ) eˆr +R sin (Φ− φ) eˆφ (2.23)
2The integral expression for drift will converge for an irrotational three-dimensional flow field induced
around an object, but would not necessarily converge for a two-dimensional flow field, e.g. along a
streamline which leads to a stagnation point.
2.2 Calculating the turbulent perturbation, u 41
where R2 = Y 2 + Z2 and tanΦ = Z/Y (R is therefore the far upstream r value for
the streamline running through x, and Φ is its far upstream azimuthal angle). Note that
R 6= X · eˆr in general. For a mean flow with no eˆφ component (U · eˆφ = 0), each streamline
will maintain a constant azimuthal position, that is Φ = φ everywhere, and it is this case
we consider in this chapter.
The distortion tensor, ∂Xi/∂xj , gives a measure of the distortion which has taken
place in the flow. Values larger than 1 are found when the flow has been compressed
while travelling downstream, and values less than 1 are found when streamlines have
spread apart. As x→ −∞ we have ∂Xi/∂xj → δij .
Finally we note that ∣∣∣∣
∂X
∂x
∣∣∣∣ =
ρ0
ρ∞
, (2.24)
(Goldstein, 1978). For an incompressible mean flow, we therefore have det (∂Xi/∂xj) = 1
which gives a useful check on the numerics when calculating the distortion tensor.
2.2.3 Majumdar’s solution to the RDT equation, Aij
We can decompose any upstream turbulent field, u∞, into harmonics of separate wavevec-
tor k. This is a common approach (Atassi and Logue, 2008, 2009) and will later allow
us to input specific forms for the upstream spectra. As the upstream turbulence is being
convected at the flight speed we consider individual gusts of the form
u∞i (x) = ai exp {ik · (x− U∞teˆx)} , (2.25)
(where the ai are constant factors). Substituting into equation (2.18), this gives
uvorti (x) = u
∞
j (X− U∞teˆx)
∂Xj
∂xi
= aj
∂Xj
∂xi
exp {ik · (X− U∞teˆx)} . (2.26)
(summation convention assumed). By essentially neglecting terms which come from dif-
ferentiating the amplitude of uvort and ψ0, Majumdar found the following approximate
42 2. Axisymmetric Rotor Systems
solution to equation (2.21)
ψ0 (x, t) ≈
ilm
|l|2
∂Xj
∂xm
aj exp {ik · (X− U∞teˆx)} , (2.27)
where
li = km
∂Xm
∂xi
. (2.28)
This result is valid in the range |k| ≫ 1/ |X|, since we neglect terms of order ∂2Xj/∂xi∂xi
in favour of terms of order kl(∂Xj/∂xi)(∂Xl/∂xj). The justification for this assumption is
given in §2.4.2, after integral expressions for quantities of interest (such as the turbulent
spectrum at the rotor face) are found, based on the fact that the dominant contribution
to these integrals comes from the region |k| ≫ 1/ |X|.
Thus we find the full distorted velocity field, uvort +∇ψ0, is given by
ui (x, t) =
(
δim −
lilm
|l|2
)
∂Xj
∂xm
aj exp {ik · (X− U∞teˆx)} , (2.29)
to leading order in k. We thus define the distortion amplitude
Aij =
(
δim −
lilm
|l|2
)
∂Xj
∂xm
. (2.30)
Note that Aji 6= Aij in general, one involves derivatives of Xi, the other derivatives of
Xj. Also, |l| is always non-zero (and thus we avoid a vanishing denominator) since if |l|
were zero this would imply either k = 0 or one whole row or column of ∂Xi/∂xj is zero,
which cannot be the case from equation (2.24). For the case of zero distortion we have
∂Xj/∂xm = δjm.
Majumdar assumed Aij was independent of azimuthal angle φ when the mean flow
was axisymmetric. As we will see, when the flow is axisymmetric, it is more precise to
state that Aij only depends upon φ through the particular wavevector k of interest. More
specifically, it is the components of k in the y− z plane which bring in φ dependence and
thus, if Aij is integrated over ky and kz, it does indeed become independent of φ.
2.2 Calculating the turbulent perturbation, u 43
Exact solution
We note that an analytic exact solution to equation (2.21) can be written down, as the
integral solution to a Laplacian equation
ψ0 (r0) =
1
4pi
∫
V
∇ · uvort
|r0 − r|
dV. (2.31)
However, as this involves an infinite volume integral, it is likely to take a long time
to numerically compute a convergent solution using this expression. This will not be
considered further.
2.2.4 Aij for zero azimuthal mean flow (when U · eˆφ = 0)
If the eˆφ component of the mean flow is zero during distortion, as is the case for the mean
flow considered in this chapter, the form of X in (2.23) simplifies to
X (x) = X (x, r) eˆx +R (x, r) eˆr. (2.32)
Further, R can be easily found if the streamfunction Ψ (x, r) is known. Since Ψ = U∞r2/2
and r = R far upstream, we see R =
√
2Ψ/U∞ everywhere. In the expression for
distorted velocity (equation (2.29)), the quantity X is involved via k ·X (in the phase)
and l = ∇ (k ·X) and ∂Xi/∂xj (in the amplitude). At this point care is needed; although
X and R have no φ dependence, kr does and so l · eˆφ is non-zero:
k ·X = kxX + krR = kxX + (ky cos φ+ kz sinφ)R, (2.33)
l = ∇ (k ·X)
=
[
kx
∂X
∂x
+ kr
∂R
∂x
]
eˆx +
[
kx
∂X
∂r
+ kr
∂R
∂r
]
eˆr +
kφR
r
eˆφ, (2.34)
where kr = k · eˆr and kφ = k · eˆφ, and the vectors eˆx, eˆr and eˆφ are based on x. The final
term of equation (2.34) was not included within Majumdar’s analysis. Since kφ is given
by
kφ = −ky sin φ+ kz cosφ, (2.35)
we see that the final term of equation (2.34) is zero when the vectors (k− kxeˆx) and
(x− xeˆx) lie in the same radial direction (as illustrated later in Figure 2.1). In general
however, this will not be the case. The contribution of the kφ term to l is also minimised
44 2. Axisymmetric Rotor Systems
when ky and kz are small compared to kx, i.e. when k lies primarily in the axial direction.
However, as we see later in §2.4.2, we will ultimately choose to sum over modes which
correspond to small kx, due to the sharp drop off of the upstream turbulence spectrum
as kx increases. Thus kφ is a potentially significant term in the regions of interest.
We note briefly that the most general axisymmetric flow could, in addition, have a
constant swirl component, leading to
X = X eˆx +R cosΦ0eˆr +R sin Φ0eˆφ, (2.36)
where Φ0 is constant. This would lead to extra terms within l. This example was treated
by Wright (2000), who found swirl in the same direction as the propellers reduces spectral
peaks and counter-rotating swirl increases them, and we take Φ0 = 0 here.
In Chapters 4 and 5, where the eˆφ component of the mean flow is non-zero, all three
components of X depend upon φ as well as upon x and r (or equivalently, upon x, y and
z). The general, non-axisymmetric expression for l is given in equations (4.11) - (4.13) in
Chapter 4.
Returning to the form for X given in (2.32), components of the tensor A in the polar
coordinate directions3 of x are then given by
eˆx ·A · eˆx =
(
1− lxlx|l|2
)
∂X
∂x
− lxlr|l|2
∂X
∂r
,
eˆx ·A · eˆr =
(
1− lxlx|l|2
)
∂R
∂x
− lxlr|l|2
∂R
∂r
,
eˆx ·A · eˆφ = −
lxlφ
|l|2
R
r
,
eˆr ·A · eˆx = −
lrlx
|l|2
∂X
∂x
+
(
1− lrlr|l|2
)
∂X
∂r
,
eˆr ·A · eˆr = −
lrlx
|l|2
∂R
∂x
+
(
1− lrlr|l|2
)
∂R
∂r
,
eˆr ·A · eˆφ = −
lrlφ
|l|2
R
r
,
eˆφ ·A · eˆx = −
lφlx
|l|2
∂X
∂x
− lφlr|l|2
∂X
∂r
,
3Note that, strictly speaking, a tensor’s components cannot be expressed with polar suffices, e.g. Axr.
If we dotted Aij with two different vectors, their polar bases would differ in general, and we would need
to decide on a common basis to express the two vectors and A in before proceeding. By Axr we therefore
mean eˆx ·A · eˆr.
2.2 Calculating the turbulent perturbation, u 45
eˆφ ·A · eˆr = −
lφlx
|l|2
∂R
∂x
− lφlr|l|2
∂R
∂r
,
eˆφ ·A · eˆφ =
(
1− lφlφ|l|2
)
R
r
. (2.37)
Particular care is needed when calculating the components involving contraction with eˆφ.
We see that, even in the case where lφ = 0 as assumed by Majumdar, eˆφ ·A·eˆφ is non-zero.
From equation (2.34) we have
lx = kx
∂X
∂x
+ kr
∂R
∂x
= kx
∂X
∂x
+ (ky cosφ+ kz sin φ)
∂R
∂x
,
lr = kx
∂X
∂r
+ kr
∂R
∂r
= kx
∂X
∂r
+ (ky cosφ+ kz sin φ)
∂R
∂r
,
lφ =
kφR
r
= (−ky sinφ+ kz cos φ)
R
r
, (2.38)
|l|2 = l2x + l2r + l2φ. (2.39)
Note that |l|2 will vary with φ for a particular k = (kx, ky, kz), which makes the integration
of Aij over φ non-trivial analytically. We wish to do this later, when calculating the Fourier
components of Aij .
Polar representation of k
For numerical calculation of later expressions it is more convenient to re-express k in
polar form, as then any integrals over the ky − kz plane can be reduced from two infinite
integrals to a finite integral and a one-sided infinite integral. As illustrated in Figure 2.1,
we define
rk =
√
k2y + k2z , (2.40)
φk = tan−1
(
kz
ky
)
. (2.41)
We can then re-express kr, kφ for use in the above expressions for Aij as
kr = k · eˆr = rk cos (φk − φ) , (2.42)
kφ = k · eˆφ = rk sin (φk − φ) . (2.43)
46 2. Axisymmetric Rotor Systems
φφk
k− kxeˆx = kreˆr + kφeˆφ
x− xeˆx = reˆr
y
z
rk
eˆφ
Figure 2.1: Re-expressing k in polar form: kyeˆy + kzeˆz → kreˆr + kφeˆφ, where polar
coordinates are taken with respect to x. Integrating over dkydkz → rkdrkdφk.
This leads to
k ·X = kxX + rk cos (φk − φ)R, (2.44)
and
l =
(
kx
∂X
∂x
+ rk cos (φk − φ)
∂R
∂x
)
eˆx +
(
kx
∂X
∂r
+ rk cos (φk − φ)
∂R
∂r
)
eˆr
+ rk sin (φk − φ)
R
r
eˆφ. (2.45)
Note on alternative Fourier decomposition
Our approach differs from that of some authors, for example Blandeau (2011), who de-
compose the turbulent velocity into components with constant kr, kφ components, as
opposed to constant ky, kz components. Thus, in Blandeau’s framework, kr is defined to
be the component of k in the eˆr direction on a particular blade, and the following strip
theory approximation for the phase
ei(krr+kφφr+kxx) (2.46)
is valid on nearby blades, for which φ is small. For our purposes, and in particular in
Chapter 4 where we specifically wish to account for extra φ dependence within the velocity
field, we need to obtain general expressions valid for all φ between 0 and 2pi, hence our
current approach.
2.2 Calculating the turbulent perturbation, u 47
In order to apply our newly found expression for Aij to the case of Unsteady Distortion
Noise of an Advanced Open Rotor, we need appropriate models for the mean flow and
the upstream turbulence. We consider these models in the next two subsections.
2.2.5 Mean flow model - introducing the actuator disk
Here we introduce a simple streamtube contraction model, to simulate the flow into a rotor.
An ‘actuator disk’ is a mathematical tool widely used to capture the prominent features
of a streamtube contraction. Recently Yin and Stuermer (2010) compared an actuator
disk model to a uRANS (unsteady RANS) calculation for propeller thrust loading and
found that the actuator disk captured well the qualitative features of the more detailed
calculation. As our present task of calculating the turbulence distortion relies only on
the flow upstream of the rotor, the detailed flow around the blades is of less significance.
The blade passage region is re-examined more closely later when calculating the pressure
jumps using strip theory. In this chapter we outline a simple constant circulation actuator
disk model. In Chapter 3 we will give the derivation of the model in more detail and then
adapt it to incorporate more realistic features such as variable blade circulation.
An actuator disk is defined to be a surface over which certain flow properties vary
discontinuously due to the force exerted by the rotor, such as the axial and tangential
velocities (Horlock (1978), Cooper and Peake (1999)). An actuator disk may be thought
of as an infinitely bladed propeller, in the limit of which we retain steady velocities only.
For a model of the detailed velocity components and, crucially for our purposes, the
streamlines of the flow, we turn to the classical vortex theory of a propeller, as given in
Hough and Ordway (1965), where each blade is modelled as a vortex line, providing us with
relatively simple analytic expressions. The details of this method are given in Chapter
3, §3.2.1. The steady component of this solution reproduces the results of actuator disk
theory, and is therefore a reasonable physical model.
For the case of a constant circulation actuator disk of radius rd positioned at x = 0,
we take the circulation along the blade span to be given by Γ(r) = Γ for r < rd and zero
otherwise. The induced axial and radial actuator disk velocities upstream of the disk are
48 2. Axisymmetric Rotor Systems
then as follows
Ux =
Udx
2pir
3
2
∫ rd
0
1√
r′
Q′− 12
(ω′)dr′ for x < 0, (2.47)
Ur = −
Ud
2pi
√
rd
r
Q 1
2
(ωd). (2.48)
(Hough and Ordway, 1965). Here we have defined
Ud =
T
pir2dρ0U∞
, (2.49)
where T is total propeller thrust. The only input parameters into this actuator disk mean
flow model are therefore upstream velocity, total thrust generated and radius. The axial
and radial velocities involve Legendre functions Q 1
2
, Q′− 12
, see Appendix A for the full
expressions for these. The arguments of the Legendre functions used above are defined as
follows
ω′ = 1 +
x2 + (r − r′)2
2rr′
, (2.50)
ωd = 1 +
x2 + (r − rd)2
2rrd
. (2.51)
We denote the total axial velocity at the disk face by Uf . At x = 0, Hough and
Ordway’s model gives Ux = Ud/2, and thus we have Uf = U∞ + Ud/2. In Figure 2.2 we
show the axisymmetric streamlines induced by a constant circulation actuator disk.
−20 −15 −10 −5 0
20
15
10
5
0
5
10
15
20
x, axial direction
r,
r
a
di
al
d
ire
ct
io
n
U
∞
= 1 m s−1 Uf = 100 m s
−1
High Distortion
−20 −15 −10 −5 0
3
2
1
0
1
2
3
x, axial direction
r,
r
a
di
al
d
ire
ct
io
n
U
∞
= 85 m s−1 Uf = 100 m s
−1
Low Distortion
Figure 2.2: Streamlines induced by a constant circulation actuator disk located between
y = −2 and y = 2 at x = 0. Both a high distortion (Uf/U∞ = 100) and low distortion
(Uf/U∞ = 1.18) example are shown.
2.2 Calculating the turbulent perturbation, u 49
2.2.6 Turbulence model - introducing the von Ka´rma´n spectrum
Upstream turbulence spectrum
Davidson (2004) provides a good reference for the derivation of the quantities introduced
in this section. The majority of turbulence models are specified via the Fourier transform
of the spatial velocity correlation tensor, known as the spectrum tensor. That is
S∞ij (k) =
1
(2pi)3
∫
ℜ3
R∞ij (η) exp (−ik · η) d3η, (2.52)
where
R∞ij (η) = 〈u∞i (x′, t) , u∞j (x′ + η, t)〉. (2.53)
Angle brackets here denote an ensemble average, an average over many realisations, as
if an experiment to measure u∞i and u
∞
j had been run many times. As we assume u
∞ is
both statistically stationary and homogeneous, R∞ij will be independent of both t and x
′.
Thus, in this case, the ensemble average is equivalent to a long-time average or volume
average (Batchelor, 1953).
When working with turbulent quantities, correlation tensors, such as R∞ij , are often
used. These are statistical quantities which give a measure of the average product of
velocities at two different points in the flow, which are separated either by a given distance
in space (η) or time (τ) or both. A value of a correlation tensor which is far from zero
indicates strong correlation, a value close to zero indicates weak correlation and we expect
a value of zero for velocities which have no influence on each other. In terms of spatial
separation, we expect high correlations up to the size of an eddy, and low correlations
beyond that. Of course, turbulence contains eddies of many sizes, so there will not be a
sudden cut-off.
We can make use of Taylor’s ‘frozen turbulence’ hypothesis to remove explicit time
dependence from equation (2.53). The quantity u∞ (non-averaged) represents a turbulent
velocity which will vary in time. Its t dependence has two aspects. Firstly, u∞ is convected
by a uniform mean flow at upstream infinity and we can write x′ = x−eˆxU∞t, to explicitly
show this (thus η could indicate a spatial or temporal separation, or a combination).
Secondly, u∞ has a t dependence which is not tied to the x dependence, that is if U∞ = 0,
the turbulent velocity at a particular point in space would still vary with t. Taylor’s
hypothesis gives conditions under which a convected eddy can be assumed to remain a
coherent structure for an infinitely long time. If these conditions are met then correlations
50 2. Axisymmetric Rotor Systems
across the eddy remain constant when convecting with the mean flow speed. Essentially
this implies that if turbulent intensity is small,
√
〈u2〉/U∞ ≪ 1, then when a correlation
is taken the second of these time dependencies can be neglected. Thus we can write
R∞ij (η) = 〈u∞i (x′) , u∞j (x′ + η)〉. (2.54)
For isotropic incompressible turbulence, the most general form for the spectrum tensor is
S∞ij (k) =
E (k)
4pik4
(
k2δij − kikj
)
, (2.55)
where E (k) = k3
d
dk
(
1
k
d
dk
Θ∞xx (k)
)
. (2.56)
Here Θ∞xx is the one-dimensional energy spectrum
4 of eˆx · u∞ (x, 0, 0),
Θ∞xx (k) =
1
pi
∫ ∞
0
R∞xx(reˆx) cos(kr)dr. (2.57)
Θ∞xx is easier to measure experimentally than the energy spectrum, E(k), (Davidson, 2004)
and can be related back to the axial component of S∞ij
Θ∞xx (kx) =
∫
ℜ2
S∞xx (k) dkydkz. (2.58)
We have used this analytic expression for the one-dimensional spectrum to validate our
numerics for the spectra at the rotor face.
von Ka´rma´n model
In his 1999 paper, Wilson summarised the assumptions commonly used by acousticians
about the form of atmospheric turbulence and compared these to the findings of atmo-
spheric scientists. In particular, he indicated that the von Ka´rma´n spectrum fitted the
measured atmospheric spectrum well, see Figure 2.3.
The primary turbulence model we use therefore is the von Ka´rma´n spectrum, given
4The one-dimensional energy spectrum is denoted by F11 in Davidson’s notation.
2.2 Calculating the turbulent perturbation, u 51
Figure 2.3: Figure from Wilson et al. (1999) showing the atmospheric turbulence one-
dimensional spectrum, together with a range of different models. The Gaussian model is
mathematically the simplest, whereas the von Ka´rma´n model approximates the energy
containing range better, and gives the k−
5
3 decay observed for small scale atmospheric
turbulence in the inertial subrange. Note that the drop off at high k is steeper in the actual
spectrum than in the von Ka´rma´n model, hence our use later of the Liepmann spectrum,
which falls off as k−2. The ‘acoustic filter’ region indicates those lengthscales which play
the most significant role in the scattering of sound as it travels through turbulence.
by
E (k) =
55g1u2∞,1L
5k4
9 (g2 + k2L2)
17
6
, (2.59)
where g1 ≈ 0.1955 and g2 ≈ 0.558, u2∞,1 is the mean square speed of the axial component of
turbulent velocity and L is the integral lengthscale. As ky, kz → ∞, we have S∞ij ∼ k−11/3,
ensuring our integrals converge. This is one of the most commonly used spectra (e.g. Lloyd
(2009), Blandeau (2011)) and allows ready comparison to other models.
The integral lengthscale, L, is often taken by acousticians to be around 1 metre,
however this is far lower than many observed measurements, which often find L to vary
proportionally to height off the ground. In Chapter 3 we revisit the question of appropriate
values for L, as well as considering turbulence models other than the von Ka´rma´n model.
52 2. Axisymmetric Rotor Systems
2.2.7 Distorted turbulence spectrum
We are now in a position to calculate the distorted turbulence spectrum at the rotor face
for any particular mean flow and turbulence model. As we are ultimately interested in the
frequency domain for the acoustics, the spectrum we calculate is the Fourier transform
of the correlation between velocities at the same point in space, separated in time by τ .
That is
Sij (x, ω) =
∫ ∞
−∞
Rij (x, τ) exp (iωτ) dτ, (2.60)
where the velocity correlation tensor for velocities separated in time but not space, Rij,
is given by
Rij (x, τ) = 〈ui (x, t) , uj (x, t+ τ)〉. (2.61)
We wish to compute Sij (x, ω) in terms of S∞ij (k). Note that Sij is a temporal Fourier
transform whereas S∞ij is a spatial Fourier transform. The appropriate comparison of Sij
to the undistorted spectrum is found by substituting ∂Xi/∂xj = δij into the expression
for Sij .
Velocity correlation, Rij
The full distorted velocity, integrated over all wavenumbers, from equations (2.25), (2.29)
and (2.30), is given by
ui (x, t) =
1
(2pi)3
∫
ℜ3
Aij (x,k) exp {i [kx (X (x)− U∞t) + kyY (x) + kzZ (x)]}
[∫
ℜ3
u∞j (y, t) exp (−ik · y) d3y
]
d3k, (2.62)
(real part understood). This expression for ui does not depend upon φ for an axisymmetric
flow field, due to integration over ky, kz. However, we note that this is for one particular
realisation of the upstream turbulent velocity field u∞i and, as we will never be in a
position to input this exactly, the only meaningful expressions are ones in which we have
averaged over many possible realisations.
We wish to input the form of the upstream turbulence via its velocity correlation in
wavevector space, S∞ij , as we have analytic models for this. Thus in the next section
we will treat each k component separately. Due to this decomposition, φ dependence
will be reintroduced in the expressions for blade pressure jump etc., as the difference in
2.2 Calculating the turbulent perturbation, u 53
k · x between blades must be accounted for. Note that t dependence appears via the
exp (−ikxU∞t) term (t dependence in u∞ will be removed when the autocorrelation is
taken, see discussion of Taylor’s hypothesis above). Thus the frequency of the distorted
velocity for a particular wavenumber component k is ω = kxU∞.
Substituting into (2.61) we have
Rij (x, τ) =
〈 1
(2pi)3
∫
ℜ3
Aik (x,k) exp {i [kx (X (x)− U∞t) + kyY (x) + kzZ (x)]}
[∫
ℜ3
u∞k (y) exp (−ik · y) d3y
]
d3k,
1
(2pi)3
∫
ℜ3
Ajl (x,k′) exp
{
i
[
k′x (X (x)− U∞t− U∞τ) + k′yY (x) + k′zZ (x)
]}
[∫
ℜ3
u∞l (y
′) exp (−ik′ · y′) d3y′
]
d3k′
〉
=
1
(2pi)6
∫
ℜ3
∫
ℜ3
∫
ℜ3
Aik (x,k)Ajl (x,k′) exp (−ik′xU∞τ)
exp
{
−i
[
(kx − k′x) (X (x)− U∞t) +
(
ky − k′y
)
Y (x) + (kz − k′z)Z (x)
]}
[∫
ℜ3
〈u∞k (y) , u∞l (y′)〉 exp (ik · y) exp (−ik′ · y′) d3y
]
d3k′d3y′d3k. (2.63)
The ensemble average acts upon the u∞ terms only. We make the substitution y′ = y+η
and note that, due to homogeneity, 〈u∞k (y) , u∞l (y + η)〉 is independent of y. Performing
the d3y integral of the last line above brings out a factor of (2pi)3δ(k − k′), and setting
k′ = k when performing the d3k′ integral allows us to cancel most of the exponential
terms.
We now have
Rij (x, τ) =
1
(2pi)3
∫
ℜ3
Aik (x,k)Ajl (x,k) exp (−ikxU∞τ)
[∫
ℜ3
〈u∞k (y) , u∞l (y + η)〉 exp (−ik · η) d3η
]
d3k
=
∫
ℜ3
Aik (x,k)Ajl (x,k) exp (−ikxU∞τ)S∞kl (k) d3k. (2.64)
Finally we Fourier transform in τ
Sij (x, ω) =
2pi
U∞
∫
ℜ2
Aik (x,k)Ajl (x,k)S∞kl (k) dkydkz, (2.65)
54 2. Axisymmetric Rotor Systems
and now kx = ω/U∞. In contrast to Majumdar’s result, this expression includes all
azimuthal order terms and has no φ dependence in the axisymmetric mean flow case.
2.2.8 Distorted turbulence plots
We have written a MATLAB program to calculate the distorted turbulence spectrum, as
given in equation (2.65). Figure 2.4 shows how the turbulent energy at the rotor face
increases with distortion level, at all lengthscales.
−2 −1 0 1 2 3 4
−12
−10
−8
−6
−4
−2
0
2
4
6
log(k
x
)
lo
g(S
xx
)
High distortion, near tip
High distortion, near hub
Medium distortion, near tip
Medium distortion, near hub
Low distortion, all r vals
Figure 2.4: Distorted turbulence spectrum (axial component) at different positions along
the rotor blade, for varying distortion levels. L = 1 in all cases. High distortion corre-
sponds to Uf/U∞ = 100, medium to Uf/U∞ = 10 and low to Uf/U∞ = 1.18. The radial
position chosen near the hub is r/rd = 0.4 and near the tip is r/rd = 0.9.
We find that for low levels of distortion the resulting turbulence spectrum does not vary
with r position, that is the streamlines which intersect with each radial position along the
blade have experienced approximately the same distortion. In contrast, for high levels of
streamtube contraction there are significant differences experienced between points near
the rotor hub versus the rotor tip. We see that the effect of increasing the distortion level
is to shift the spectrum upward at all frequencies. As can be seen from the definition of
Sxx (in equation (2.65)) and Axk (in equation (2.30)), all nine components of the ∂Xi/∂xj
2.3 Blade pressure jump 55
tensor appear within Sxx. The spectrum shift is thus due to a combination of both the
change in magnitude of individual components, and the change in the ratios between
components. When the distortion level is increased, but the radial blade position is kept
constant, it appears that the primary mechanism for the spectrum shift is the increase
in the magnitude of the ∂X/∂x term. This term corresponds to axial stretching of the
turbulent eddies, and a higher value of ∂X/∂x means increased axial stretching, and thus
a shift to larger lengthscales i.e. lower kx values. In Chapter 3 we calculate ∂Xi/∂xj as
functions of r at the rotor face, and examine how changes in these quantities translate
directly to changes in the distorted turbulence spectrum.
Positioning the actuator disk
Note that, within our model, the position chosen for the actuator disk relative to the
leading edge of the blades will affect the velocities within the model at the leading edge
of the blade. Throughout this work, we place the actuator disk at x = 0. The blade’s
leading edge should lie just ahead of the jump in flow properties, and we usually place the
leading edge at x = −0.005rd in our calculations. In turbofan tests a bellmouth inlet is
sometimes used to better simulate the streamtube contraction, in which case positioning
the actuator disk somewhat upstream of the leading edge of the blades may be a more
appropriate modelling assumption. However bellmouth inlets are not used for open rotor
tests.
2.3 Blade pressure jump
Unsteady distortion noise is generated when a turbulent velocity field impinges upon a
rotors’ blades and, due to the condition of zero normal velocity on the blades’ surfaces,
induces a pressure jump across those blades. In this section we use Smith’s cascade theory
to determine the lift response of the blades to the particular incoming distorted velocity
ui given in equation (2.62). This is not the only possible approach, alternatives include
neglecting the cascade effect and using an Amiet model (Blandeau et al., 2009, Blandeau
and Joseph, 2010) or Sears function (Wright, 2000, Brouwer, 2010) to consider the response
of individual blades, or the use of alternative linear cascade models (Atassi and Logue,
2009). Recent work by Roger and Carazo (2010) has begun to take into account a more
realistic representation of the open rotor blade shape within a blade response function,
including effects of sweep, continuous variation of chord length in the spanwise direction
56 2. Axisymmetric Rotor Systems
(for correct tone prediction), and the rectangular blade tips typical of open rotors. In
their work, incorporating sweep was found to have the most significant effect, and this is
possible within our model by adjusting the axial position of the leading edge, x0(r).
2.3.1 LINSUB
Outline of Smith’s theory
Smith (1973) analysed the response of a cascade of idealised flat plate blades to an input
velocity perturbation. The aim of the paper was to treat the discrete frequency noise gen-
erated, transmitted and reflected by a blade row. Five particular forms of upwash velocity
are treated by Smith: bending vibration; torsional vibration; a convected sinusoidal wake;
an upstream or downstream going acoustic wave. In each case there are then five different
physical quantities which can be calculated directly as output: total lift; total moment;
the vortex sheet shed from the blades (when the lift force varies in time new vortices are
shed); and, if conditions are not cut-off, the upstream and downstream travelling waves.
Certain modelling assumptions apply. Firstly, radial gradients are neglected, and thus
we treat each radial station separately - i.e. ‘unwrapping’ the blades and considering a
two-dimensional set-up. (Within analytic models it is usual for the blade normal to be
assumed to lie in the eˆx and eˆφ directions only, and hence radial gradients will not come
into the ∇ · f forcing term in the wave equation, (2.107).) This approximation allows the
interaction between adjacent blades to be taken into account, rather than treating each
blade individually. Radial variation is still taken into account via the leading edge velocity.
Secondly, at each radial station the blades are modelled as a continuous distribution of
line vortices (lying along the third dimension) along the chord, see Figure 2.5. In addition,
only subsonic flows are considered and the blades are assumed to operate at zero incidence
to the mean flow at their leading edge.
This model further extends the vortex theory of a propeller employed in the actuator
disk flow model; we now represent the rotor blades by two-dimensional flat plates rather
than one-dimensional lines. A vortex at zˆ0 gives rise to upstream and downstream going
pressure perturbations (leading to contributions to the upwash velocity at all zˆ 6= zˆ0),
and a downstream going vorticity wave (leading to contributions to the upwash velocity
at zˆ > zˆ0). There is a singularity in the lift at the leading edge where, in this inviscid
model, the fluid is forced to turn at a sharp angle.
Our input velocity takes the form of a convected sinusoidal wake and we are interested
2.3 Blade pressure jump 57
zˆ
zˆ0
Γ(zˆ0)
leading edge
trailing edge
β
s
c
Figure 2.5: The vortices provide a representation of the circulation generated by the
motion of the aerofoils (a viscous effect), while using the inviscid equations of motion in
the surrounding fluid.
in the lift output from the model. We do not directly use the acoustic waves as calculated
by Smith, because this two-dimensional cascade theory model is only valid very near the
rotor. If we simply considered the output sound waves as they propagate, this would not
give a good prediction for the far-field noise. Instead the lift distribution is input as the
forcing term in the cylindrical far-field wave equation in §2.4.
Smith’s strip theory analysis gives a computationally efficient method for calculating
the strengths of the vortices, Γ(zˆ), and equivalently the force distribution induced along
the blade due to the condition of zero normal velocity on the blades’ surfaces. Whitehead
(1987) then created the LINearized SUBsubsonic (LINSUB) code to implement Smith’s
theory where a discrete number of points, nc, are considered along the blade in order to
calculate Γ.
LINSUB theory shows particularly good agreement with experiment under conditions
of low Mach no., zero mean blade loading, and a high hub-to-tip ratio, and is extremely
widely used in analytic aeroacoustics (e.g. Cheong et al. (2006), Evers and Peake (2002)).
There are three potential limitations of strip theory in the present context of our work.
Firstly, the key assumption of neglecting radial gradients becomes more acceptable as
the hub-to-tip ratio increases, but for open rotors the hub-to-tip ratio tends to lie in
the fairly low range of 0.3-0.4. Incorporating radial effects into LINSUB can lead to
significant changes at low frequencies (Lloyd, 2009), and we give a method for including
the contribution of the radial wavenumber to∇2 in §2.3.3. Secondly, varying the camber of
open rotor blades is an important tool used when designing blades where variable tip speed
is used to achieve optimum performance at different operating conditions. In contrast,
58 2. Axisymmetric Rotor Systems
LINSUB employs a flat blade approximation. However, Evers and Peake (2002) showed
that, whereas for an individual frequency component the camber can have a significant
effect, when averaging over many frequencies, the effects of camber cancel out to a large
degree. Finally, LINSUB seems to underpredict the output wave amplitudes in the case of
non-zero blade loading, but we do not use these acoustic waves. In all the cases considered
in this thesis we assume that locally the blades operate at zero incidence.
As the number of blades on an open rotor is typically small, cascade effects will be
less significant than for high blade number fans. As mentioned at the start of this section,
an alternative approach would be to use an isolated blade model for the blade response.
This could be a useful area for future model development, as it would allow other effects
(for example, a more detailed blade geometry) to be taken into account.
Application of LINSUB
In using this theory, we assume that as soon as a velocity perturbation wave reaches the
rotor, it is simply convected by the mean flow there (which is in the chordwise direction)
i.e. it becomes ‘frozen in the flow’ at that point and there are no further distortion effects.
We decompose the normal velocity at the leading edge into azimuthal and temporal
harmonics of the form e−iωΓt+imφ, and treat each harmonic separately; different azimuthal
harmonics will have different inter-blade phase angles. Note that Smith used the opposite
sign convention (eiωΓt) in his analysis. We input the mean flow speed in the chordwise
direction, W , and the assumption of pure convection within the LINSUB model then sets
the upwash velocities at each chordal station.
In the notation of Smith, at a particular radial station, the lift across the blade aerofoil
is given by
−ρ0WwW
∫ 1
0
ΓWW (zˆ) dzˆ, (2.66)
where the circulation ΓWW and chordwise coordinate zˆ have been non-dimensionalised
with respect to the amplitude of the velocity perturbation times chord length, wW c, and
chord length, c, respectively. The upwash velocity along the chord, w(zˆ), is given by
w(zˆ)
wW
=
∫ 1
0
K(zˆ − zˆ0)ΓWW (zˆ0)dzˆ0, (2.67)
where K contains all the information regarding the contribution of the vortex at zˆ0 to
the normal velocity at zˆ (see equation (31) in Smith (1973) for the full expression for K).
2.3 Blade pressure jump 59
This integral equation is converted to a matrix equation in order to solve numerically for
ΓWW . The kernel, K, and the upwash velocity, w, are calculated at a discrete number,
nc, of collocation points. The upwash velocity at those points is then given in a vector U .
We determine ΓWW by solving the matrix equation
U = KΓ, (2.68)
(equation (50) in Smith (1973)). The lth entry of the matrix Γ then gives
pi
2nc
sin
(
pil
nc
)
ΓWW (zˆl), (2.69)
where zˆl =
1
2
(
1− cos pi(2l + 1)
2nc
)
. (2.70)
A MATLAB program5 was used to solve equation (2.68). Inputs are
stagger angle β(r) = tan−1 (Ωr/Uf) ,
Mach no. in chordwise direction MW = W/c0 =
(
Ω2r2 + U2f
) 1
2 /c0,
inter-blade phase angle χ = −2pim/B,
reduced frequency λ = −ωΓc/W,
space-chord ratio s/c,
number of collocation points nc.
The inter-blade phase angle was arrived at as follows. The difference in incident velocity
between blades as φ varies is given by e−iχ (by definition). For a velocity with eimφ
dependence, if φ → φ + 2piB , the velocity is multiplied by a factor e
2mpii
B , and thus we set
χ = −2pimB . The optimum nc is dependent upon the other parameters. For our purposes
nc ≥ 6 is sufficient, and we use nc = 10 in our numerical calculations, since this does not
significantly slow down the calculations.
Once ΓWW has been calculated, the pressure jump at zˆ is given by
∆p =ρ0WwWΓWW (zˆ, r;ωΓ, χ) e−iωΓteimφ. (2.71)
5Based on one developed by Dr Vincent Jurdic at the Institute of Sound and Vibration Research,
University of Southampton.
60 2. Axisymmetric Rotor Systems
2.3.2 Calculating the blade pressures
Since our information about the form of the upstream turbulence comes via its spectrum
tensor in wavevector space we treat each upstream uˆ∞j (k) gust separately. We can then
take the correlation of the blade pressure jump due to gusts of this form to obtain an
expression for the total blade pressure spectrum in terms of S∞ij . Decomposing ui (as
given in equation (2.62)) with respect to k gives
ui (x, t) =
1
(2pi)3
∫
ℜ3
Aij (x,k) uˆ∞j (k) exp {i [kx (X (x)− U∞t) + kyY (x) + kzZ (x)]} d3k
=
1
(2pi)3
∫
ℜ3
ugusti (x, t;k) d
3k. (2.72)
The quantity we require in order to use LINSUB is the normal velocity (N · u) at the
leading edge of a blade, in a reference frame which rotates with the blades, decomposed
into eimφ harmonics. Majumdar also took the approach of treating each uˆ∞j (k) gust
separately, but neglected the φ dependence in the amplitude Aij which arises when con-
sidering a specific k value, treating only the φ dependence in the phase term of the velocity,
exp {i [kyY + kzZ]}.
The unit normal to the blade is given by
N(r, φ) = sin β(r)eˆx + cosβ(r)eˆφ. (2.73)
As Majumdar did, we use the relation
exp {i (ky cosφ+ kz sinφ)R} =
∞∑
n=−∞
Jn
(√
k2y + k2zR
)
exp
{
in
[
φ− tan−1
(
kz
ky
)]}
,
(2.74)
where the Jn denote Bessel functions, thus finding
[N ·A]j exp {i [kxX + (ky cosφ+ kz sinφ)R]} =
exp (ikxX) [sin βAxj + cos βAφj]
∞∑
n=−∞
Jn (rkR) exp {in (φ− φk)} , (2.75)
where rk =
√
k2y + k2z , φk = tan
−1 (kz/ky) as in Figure 2.1.
In order to bring out the φ dependence of Aij in exponential form we define a new
2.3 Blade pressure jump 61
tensor Cpij as follows
Aij (x, r, φ;k) =
∞∑
p=−∞
Cpij (x, r;k) e
ipφ, (2.76)
where Cpij (x, r;k) =
1
2pi
∫ 2pi
0
Aij (x, r, φ;k) e−ipφdφ. (2.77)
Note that in the asymmetric case the integrand of Cpij will need to include the phase terms
ei(kxX+kyY+kzZ) as X, Y and Z then have φ dependence which cannot be written out in
explicit form so readily as in equation (2.74).
Since φk only appears within Aij in the form (φk − φ) (via kr = rk cos (φk − φ) and
kφ = rk cos (φk − φ)), we note that
Aij (x, r, φ; kx, rk, φk) = Aij (x, r, φ− φk; kx, rk, 0) , (2.78)
and thus
Cpij (x, r; kx, rk, φk) =
1
2pi
∫ 2pi−φk
−φk
Aij
(
x, r, φ˜; kx, rk, 0
)
e−ip(φ˜+φk)dφ˜
= e−ipφkCpij (x, r; kx, rk, 0) , (2.79)
making the substitution φ˜ = φ − φk, and using the fact that all terms in the integrand
are 2pi periodic.
Going back to the expression (2.72), it will be easiest to use the cartesian components
of uˆ∞j (k), as we wish to use the cartesian definition of S
∞
ij . A polar representation is
trickier, as quantities such as eˆr become ambiguous - depending on whether they relate
to x (when calculating Aij) or k (when inputting S∞ij ). However we would also like to
express Aij as a function of x, r, φ,X,R, ∂X/∂x, ∂X/∂r, ∂R/∂x, ∂R/∂r, R/r (rather than
the cartesian equivalents), as these are simpler to calculate. We therefore calculate the
‘polar’ coefficients Cpxx, C
p
xr, C
p
xφ, C
p
φx, C
p
φr, C
p
φφ, and then translate these to ‘mixed suffix’
quantities Axx, Axy, Axz, Aφx, Aφy, Aφz for use in (2.75). This process is shown in the
following example
eˆx ·A · eˆy = [eˆx ·A · eˆr] cosφ− [eˆx ·A · eˆφ] sinφ
=
1
2
∞∑
p=−∞
{
eiφ
(
Cpxr + iC
p
xφ
)
+ e−iφ
(
Cpxr − iCpxφ
)}
eipφ. (2.80)
62 2. Axisymmetric Rotor Systems
Similarly, we substitute the appropriate Cpij terms into the following
eˆx ·A · eˆz = [eˆx ·A · eˆr] sin φ+ [eˆx ·A · eˆφ] cos φ,
eˆφ ·A · eˆx,
eˆφ ·A · eˆy = [eˆφ ·A · eˆr] cos φ− [eˆx ·A · eˆφ] sinφ,
eˆφ ·A · eˆz = [eˆφ ·A · eˆr] sin φ+ [eˆx ·A · eˆφ] cos φ. (2.81)
Defining Dpj
We next define new quantities Dpj , for use in the expression given in equation (2.75), as
follows
sin βAxx + cosβAφx =
∞∑
p=−∞
[
sin βCpxx + cosβC
p
φx
]
eipφ
≡
∞∑
p=−∞
Dpx (x, r; kx, rk, φk) e
ipφ
=
∞∑
p=−∞
Dpx (x, r; kx, rk, 0) e
ip(φ−φk),
sin βAxy + cosβAφy =
∞∑
p=−∞
[
sin β
1
2
{(
Cp−1xr + iC
p−1
xφ
)
+
(
Cp+1xr − iCp+1xφ
)}
+ cosβ
1
2
{(
Cp−1φr + iC
p−1
φφ
)
+
(
Cp+1φr − iC
p+1
φφ
)}]
eipφ
≡
∞∑
p=−∞
Dpy (x, r; kx, rk, φk) e
ipφ,
sin βAxz + cosβAφz =
∞∑
p=−∞
[
sin β
1
2
{(
−iCp−1xr + Cp−1xφ
)
+
(
iCp+1xr + C
p+1
xφ
)}
+ cosβ
1
2
{(
−iCp−1φr + C
p−1
φφ
)
+
(
iCp+1φr + C
p+1
φφ
)}]
eipφ
≡
∞∑
p=−∞
Dpz (x, r; kx, rk, φk) e
ipφ. (2.82)
From our previous relation for Cpij given in equation (2.79), we have used
Dpj (x, r; kx, rk, φk) = e
−ipφkDpj (x, r; kx, rk, 0) + E. (2.83)
2.3 Blade pressure jump 63
The extra terms within E include additional e±iφk factors. However, when the full integral
over d3k is taken these terms integrate to zero, and thus we do not include E in subsequent
expressions.
Investigation of dominant p modes
Looking back at the definition of Aij (equation (2.37)) we see that the denominator and
numerator of each term (except the unity terms) contain pairs of li components. As φ
only appears within l through cos (φk − φ) and sin (φk − φ) (see equation (2.45)) we see
that the numerators and denominators contain only terms with eipˆφ dependence where pˆ
varies between −2 and 2. We find in fact that the coefficients Dpj die off fairly rapidly,
and for most values of k seven or fewer values of p are significant, as seen in Figure 2.6.
0 1 2 3 4 5 6 7
−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
p
Re(D
x
)
Im(D
x
)
Re(Dy)
Im(Dy)
Re(D
z
)
Im(D
z
)
Figure 2.6: Here we see the rapid decay of all Dpj values as p increases. Since D
−p
j =
(
Dpj
)∗
we do not need to calculate Dpj for negative values of p to assess their decay. (Due to
other terms which appear in the sum over p in the final expression for radiated sound the
+p and −p terms do not directly cancel each other.)
Further investigation has shown that as rk →∞ or kx → ∞ or kx → 0, fewer and fewer
terms are significant, (i.e. we only require p = −2,−1, ..., 2 for convergence), whereas for
small rk and kx mid-range a few more terms are needed. Including p = −3,−2, ...3 is
sufficient for our results.
64 2. Axisymmetric Rotor Systems
Moving to rotating frame
Substituting equations (2.82) into equation (2.72) via equation (2.74) we find
N · u = 1
2pi3
∫
ℜ3
uˆ∞j (k) e
−ikxU∞t [N ·A]j exp {i [kxX + (ky cos φ+ kz sinφ)R]} d3k
=
1
2pi3
∫
ℜ3
uˆ∞j (k) e
−ikxU∞t exp (ikxX)
∞∑
m=−∞
∞∑
p=−∞
Jm−p (rkR)D
p
j (x, r; kx, rk, 0) e
im(φ−φk)d3k. (2.84)
We move into the rotating frame by substituting φ = φ′ +Ωt, where φ′ is the angular
coordinate which remains constant when rotating with the rotor, creating urot.i . The
amplitude of the normal velocity at the leading edge of a blade, in the rotating frame, to
be substituted into equation (2.71), is given by
∞∑
m=−∞
wW e−iωΓt+imφ
′
= Niurot.i leading edge
= Ni (x0(r), r, φ′)urot.i (x0(r), r, φ
′, t;k)
= eikxX
∞∑
m=−∞
{ ∞∑
p=−∞
Jm−p (rkR)D
p
j (x0(r), r; kx, rk, 0)
}
uˆ∞j (k) exp [i (mΩ− kxU∞) t] exp [im (φ′ − φk)] , (2.85)
where x0(r) is the axial position of the leading edge of the blade. The effect of moving to
the rotating frame has determined the correct frequency for use in the response function,
ωΓ = kxU∞ −mΩ; we can now replace φ′ by φ − Ωt again. Substituting into equation
(2.71), the pressure jump across a blade which passes through the position x, r, φ at time
t, due to an upstream gust of the form uˆ∞(k)ei(k·x−kxU∞t), is
∆p (x, r, φ, t) =ρ0W
∞∑
m=−∞
ΓmWW
(
zˆ =
x− x0 (r)
cosβ(r)
, r;ωΓ = kxU∞ −mΩ, χ = −
2mpi
B
)
∞∑
p=−∞
Jm−p (rkR)D
p
j (x0(r), r; kx, rk, 0) uˆ
∞
j (k) e
ikxXe−ikxU∞teim(φ−φk). (2.86)
The arguments of ΓWW have moved the leading edge of the blade to the correct position,
and defined the frequency ωΓ and inter-blade phase angle χ. The full pressure jump is
2.3 Blade pressure jump 65
found by integrating equation (2.86) over k
(
1
(2pi)3
∫
...d3k
)
. The pressure jump across
blade b which cuts through the position x, r at time t, denoted by ∆pb (x, r, φ, t), is
found by substituting the azimuthal position of the leading edge of the bth blade, φb =
φ0(r) + 2pib/B + Ωt into the above expression.
Definition of Pˆblade
We are now in a position to calculate the spectrum of blade pressures, defined as
Pˆblade (x, r, ω) =
∫ ∞
−∞
Pblade (x, r, τ) eiωτdτ
=
∫ ∞
−∞
〈
(∆pb (x, r, t))∗ ,∆pb (x, r, t+ τ)
〉
eiωτdτ. (2.87)
This expression denotes taking a large number of different realisations of the upstream
turbulence, i.e. a large number of different u∞k (x
′), averaging over these and then averag-
ing over time as well. We denote the ensemble average by 〈, 〉 and the temporal average by
the overbar. In this way we obtain a statistical average over many rotations of the blade
pressure differences across a particular point on a blade - a potentially useful quantity for
calibration against experimental observation. By averaging over time we should remove
the dependence upon blade number b.
Substituting in the expression from equation (2.86) we find
Pblade (x, r, τ) =
〈
ρ0W
∫
ℜ3
∞∑
m=−∞
[ΓmWW (x, r, kx)]
∗ exp (ikxU∞t) exp (−ikxX)
{ ∞∑
p=−∞
Jm−p (rkR) e−imφkD
p
j (x0(r), r; kx, rk, 0)
}∗
exp
[
−imφb(r, t)
] [ 1
(2pi)3
∫
ℜ3
u∞j (y) exp (ik · y) d3y
]
d3k,
ρ0W
∫
ℜ3
∞∑
m′=−∞
Γm
′
WW (x, r, k
′
x) exp (−ik′xU∞(t+ τ)) exp (ik′xX)
{ ∞∑
p′=−∞
Jm′−p′ (r′kR) e
−im′φ′kDp
′
k (x0(r), r; k
′
x, r
′
k, 0)
}
exp
[
im′φb(r, t+ τ)
] [ 1
(2pi)3
∫
ℜ3
u∞k (y
′) exp (−ik′ · y′) d3y′
]
d3k′
〉
66 2. Axisymmetric Rotor Systems
=ρ20W
2
∫
ℜ3
∫
ℜ3
∞∑
m=−∞
[ΓmWW (x, r, kx)]
∗ e−im(φ0(r)+
2pib
B )
{ ∞∑
p=−∞
Jm−p (rkR) e−imφkD
p
j (x0(r), r; kx, rk, 0)
}∗
∞∑
m′=−∞
Γm
′
WW (x, r, k
′
x) e
im′(φ0(r)+ 2pibB )
{ ∞∑
p′=−∞
Jm′−p′ (r′kR) e
−im′φ′kDp
′
k (x0(r), r; k
′
x, r
′
k, 0)
}
1
(2pi)6
[∫
ℜ3
∫
ℜ3
〈
u∞j (y
′) , u∞k (y)
〉
∞ e
ik·ye−ik
′·y′d3yd3y′
]
〈 ei(m′−m)Ωtei(kx−k′x)U∞t 〉t ei(k
′
x−kx)Xe−i(k
′
xU∞−m′Ω)τd3k′d3k. (2.88)
As before (when calculating the velocity correlation Rij , equation (2.63)) the double
integral over y and y′ reduces to (2pi)6δ(k−k′)S∞kl (k) via a change of variables, y′ = y+η,
and setting k′ = k when performing the k′ integral removes several terms. The long time
average then acts on ei(m
′−m)Ωt only, which reduces to δmm′ using
δmm′ = lim
T→∞
1
2T
∫ T
−T
ei(m−m
′)t′dt′. (2.89)
Note that blade number dependence in the expression given in equation (2.88) has indeed
vanished.
We take the Fourier transform in τ to obtain Pˆblade; a factor 2piU∞ δ (kx − ω/U∞ −mΩ/U∞)
is brought in and we can perform the kx integral. Thus only one value of kx contributes
for a particular value of ω and m. We can integrate over polar k instead of the cartesian
representation by writing ky = rk cos φk, kz = rk sinφk
Pˆblade (x, r, ω) =
2piρ20W
2
U∞
∞∑
m=−∞
|ΓmWW (x, r, ω)|2
∫ ∞
0
{ ∞∑
p=−∞
Jm−p (rkR)D
p
j (x0(r), r; k
m
x , rk, 0)
}∗
{ ∞∑
p′=−∞
Jm−p′ (rkR)D
p′
k (x0(r), r; k
m
x , rk, 0)
}[∫ 2pi
0
S∞jk (k
m
x , ky, kz) dφk
]
rkdrk.
(2.90)
2.3 Blade pressure jump 67
where in the above expression kmx = (ω +mΩ) /U∞, that is the time dependence has been
set to e−iωτ , and we sum over cartesian suffices j = x, y, z.
The φk integral above can be carried out analytically
∫ 2pi
0
S∞jk (kx, ky, kz) dφk = 0 if j 6= k, (2.91)
as these terms involve only the integral of sinφk, cosφk or sinφk cosφk over a 2pi period,
and
∫ 2pi
0
S∞xx (kx, ky, kz) dφk = 2pir
2
kG, (2.92)
∫ 2pi
0
S∞yy (kx, ky, kz) dφk =
∫ 2pi
0
S∞zz (kx, ky, kz) dφk =
(
2pik2 − pir2k
)
G, (2.93)
where G = 55g1u2∞,1/36piL
2
3 (g2/L2 + k2x + r
2
k)
17/6 for the von Ka´rma´n spectrum. We thus
have an integral expression for the blade pressures which involves an infinite summation
over m, and an infinite integral over rk. Evaluation of Pˆblade numerically requires further
analysis to determine which indices and regions of the integral give the dominant contri-
butions; we discuss the latter in the next subsection. The sum over p is less problematic
as the coefficients die off rapidly as outlined above.
2.3.3 Limiting the rk integral
As highlighted previously, LINSUB neglects radial gradients. If we consider 3D input
gusts which take the form eikxx+imφ+iktr, LINSUB will treat the eiktr factor as part of
the amplitude at a particular r station6. Eventually, for large kt, the approximation
of neglecting the contribution of the wavenumber in the radial direction to derivatives
will break down. The output we use from LINSUB, the blade pressure jumps, do not
need to satisfy a dispersion relation. However, the sound waves which we will ultimately
obtain from solving the wave equation will satisfy a dispersion relation obtained via the
(∇2 − ω2/c20) operator. Thus here we limit our attention to kt values which eventually
contribute to the radiated sound, and in this way obtain a convergent integral expression.
6Note that LINSUB assumes that once a disturbance reaches the leading edge it is simply convected.
This means the wavenumbers within the LINSUB analysis are not the same as for the incident velocity.
The axial wavenumber within LINSUB, denoted by Smith as α, is not equal to kx as we define it in
the expression for N · u; m is input via the inter-blade phase angle but then several different tangential
wavenumbers, β, are found, some of which will correspond to imaginary α.
68 2. Axisymmetric Rotor Systems
Determining the effective radial wavenumber
The expression for the velocity field input into LINSUB, as it stands in equation (2.84),
does not have r dependence written explicitly in the form eiktr, where we can simply pick
out the appropriate kt upon which to impose limits. We therefore need to determine an
expression for the ‘effective’ radial wavenumber given a general r dependence within the
velocity input. The situation is not completely straightforward as we wish to translate
between a full cylindrical gust and the rectilinear system of LINSUB in a consistent
manner and this requires some approximations.
Posson et al. (2010) and Glegg and Jochault (1998) have looked at similar problems of
translating between these two coordinate systems. Posson noted that, in addition to the
wave equation being satisfied in each of the two frames, there are two conditions which
we might like to satisfy when moving from the annular to the rectilinear frame. Firstly,
keeping the tangential wavevector the same in the two frames and secondly keeping the
structure of the radial function the same, e.g. by decomposing Bessel functions into
cosines, and then into exponentials, and taking kt from these waves. However, she showed
it was not possible to satisfy both of these conditions at once, in general.
We identify r in the annular frame with z in the rectilinear one. Given a disturbance
of the form f(r)eikpx+i
m
r rφ−iωt we wish to equate this to a wave in the rectilinear set-up
of the form eikxx+ikyy+iktz−iωt (where x remains the axial direction and y is the transverse
direction), and thus pick out kt. For a disturbance which satisfies the wave equation, and
has the same time dependence, this comes down to equating the output of ∇2 in the two
frames. If we also wish to keep the wavenumber vector in a plane of constant radius the
same in the unwrapped configuration then
k2p +
(m
r
)2
= k2x + k
2
y, (2.94)
tan θˆ =
m
kpr
=
ky
kx
, (2.95)
(Posson et al., 2010). Thus we identify kp with kx and m/r with ky. Equating the result
of applying the ∇2 operator we find
k2x + k
2
y + k
2
t = −
1
r
∂
∂r
(
r ∂f∂r
)
f(r)
+
m2
r2
+ k2p, (2.96)
⇒ k2t = −
1
r
∂
∂r
(
r ∂f∂r
)
f(r)
. (2.97)
2.3 Blade pressure jump 69
This gives a general formula for picking out the effective radial wavenumber from an
expression with r dependence. The r dependence within our velocity expression is given
by Jn (rkR). Thus we find
k2t = r
2
k
(
∂R
∂r
)2(
1− n
2
r2kR2
)
−
(
rk
r
∂R
∂r
+
rk
r
∂2R
∂r2
− rk
R
(
∂R
∂r
)2) J ′n(rkR)
Jn(rkR)
, (2.98)
and we can rewrite J ′n(rkR) =
1
2 (Jn−1(rkR)− Jn+1(rkR)). The second set of terms here
comes to zero if R is a linear function of r. In Figure 2.7 we see that R is indeed close
to a linear function of r for the actuator disk model we employ, except near the blade
tip. In his work on UDN, Cargill (1993) took R = αr, and thus used a simplified form
for the effective radial wavenumber with ∂R/∂r = α. Note that, as Cargill did, we have
neglected the r dependence within eikxX(x,r) in determining kt.
0.2 0.4 0.6 0.8 1 1.2 1.4
0
2
4
6
8
10
12
r/rd
R
/r d
High distortion
Low distortion
Figure 2.7: The upstream r value of a streamline which originates at the rotor face,
denoted by R, is shown to be linear in the case of low distortion, and near to linear along
the majority of the blade span in the high distortion case. Very near the blade tip in
the high distortion case linearity does break down, but in our numerical calculation of
radiated sound we will consider discrete radial positions, the outermost of which can be
chosen to lie within the linear range.
70 2. Axisymmetric Rotor Systems
Limiting condition
We invert expression (2.98) in order to impose finite limits upon the rk integral in (2.90).
We therefore find
r2k =
k2t(
∂R
∂r
)2 +
(m− p)2
R2
. (2.99)
A first approximation limit upon k2t , taking into account its contribution to ∇2, is given
by k2t ≤ ω2/c20. This is essentially the limit that would be imposed if we were considering
a stationary source on an infinite span airfoil. An area for future work, outlined in
Appendix D, is to implement more precise limits for r2k based on a rigorous stationary
phase argument. For now, in our numerical calculation of expression (2.90), we impose
√
(m− p)2
R2
≤ rk ≤
√(
∂R
∂r
)−2 ω2
c20
+
(m− p)2
R2
. (2.100)
Since (m− p) is the order of the Bessel function which governs radiation, and rkR is
its argument, the lower limit here is a statement that it is when the argument is greater
than the order that we get radiation, as noted by Parry (1988). Within the upper limit
the quantity ∂R/∂r appears, which gives the magnitude of the streamtube contraction.
Far upstream we have ∂R/∂r = 1, and it increases as we travel towards the rotor. This
contraction is analogous to that experienced in a contracting duct. For a duct of radius
a the Helmholtz number, ωa/c0, gives the non-dimensional frequency and controls the
number of propagating modes. In our case, we therefore interpret the quantity
(
∂R
∂r
)−1 ω
c0
, (2.101)
as proportional to an effective Helmholtz number.
2.3 Blade pressure jump 71
2.3.4 Blade pressure plots
In Figures 2.8 and 2.9 we plot the blade pressure spectra as calculated from equation
(2.90) for a range of distortion levels and integral lengthscales. Peaks are found at all
integer multiples of Ω.
100 150 200 250 300 350 400
−5
0
5
10
15
20
ω
High distortion
Medium distortion
Low distortion
lo
g
(Pˆ
b
la
d
e
)
Figure 2.8: The blade pressure spectrum at the leading edge of the blades, near the hub
(r/rd = 0.4). High distortion corresponds to Uf/U∞ = 100; medium to Uf/U∞ = 10;
low to Uf/U∞ = 1.18. As distortion is increased, the blade pressure peaks increasingly
sharply at integer multiples of the rotor angular velocity.
100 150 200 250 300 350
−5
0
5
10
15
20
ω
L = 0.1
L = 1
L = 10
lo
g
(Pˆ
b
la
d
e
)
Figure 2.9: High distortion case (Uf/U∞ = 100), comparing L values. As L is increased
above the streamtube radius (L ∼ 1) the peaks remain but the overall level is lowered.
72 2. Axisymmetric Rotor Systems
2.3.5 Unsteady force generated by a single rotor
Single wavevector component
The total force per unit volume, f , generated by the rotor is found by summing the
pressure jumps across each blade, as given in equation (2.86). The argument which
follows is similar to Majumdar’s analysis (where we have expanded details of the use of
the delta function), the difference being our treatment of the φ dependence within Aij
terms. Blade number is denoted by b and runs from 1, ..., B and we first consider a single
upstream wavevector component k, with amplitude uˆ∞j (k), for which the force upon the
fluid is given by
fki (x, r, φ, t;k) =
B∑
b=1
ρ0WNi(r) exp (−ikxU∞t)
∞∑
m=−∞
ΓmWW (x, r; kx, χ
m)
eikxX
∞∑
p=−∞
Jm−p (rkR) exp (−imφk)Dpj (x0(r), r; kx, rk, 0)
uˆ∞j (k) exp
(
im
[
φ0(r) +
2bpi
B
+ Ωt
])
∞∑
s=−∞
δ
[
r
(
φ− Ωt− φ0(r)−
2bpi
B
)
− (x− x0(r)) tan β(r)− 2pis
]
.
(2.102)
Here Ni gives the direction of the force, normal to the blade. The sum over p has come
from eipφ dependence within Aij. The delta function7 defines the blade surface, noting
that ΓmWW = 0 for x values outside the chord.
We wish to separate out the φ and t dependence within f explicitly which will allow
us to solve the wave equation in the next section. Firstly we re-express the delta function
as
∞∑
s=−∞
δ
[
r
(
φ− Ωt− φ0(r)−
2bpi
B
)
− (x− x0(r)) tanβ(r)− 2spi
]
=
1
2pir
∞∑
s=−∞
exp
(
is
[(
φ− Ωt− φ0(r)−
2bpi
B
)
− (x− x0(r)) tan β(r)
r
])
,
(2.103)
7The dimension of the delta function’s argument is of length, and thus the delta function’s output has
dimensions 1/length, giving the correct dimensions for the full expression of a force per unit volume.
2.3 Blade pressure jump 73
where we have used the relation δ(rx) = δ (x) /r (for r 6= 0), and the Poisson summation
formula 2pi
∑∞
−∞ δ(x− 2pis) =
∑∞
−∞ exp(−isx). For particular values of m and l we can
now collect together all those terms in equation (2.102) involving blade number b, which
are
B∑
b=1
exp
(
2bmpii
B
)
1
2pir
∞∑
s=−∞
exp
(
is
[(
φ− Ωt− φ0(r)−
2bpi
B
)
− (x− x0(r)) tanβ(r)
r
])
=
B
2pir
∞∑
l=−∞
exp
(
−i (m+ lB)
[
(φ− Ωt− φ0(r))−
(x− x0(r)) tanβ(r)
r
])
. (2.104)
Here we have used
B∑
b=1
exp
(
2piib(m+ s)
B
)
= 0 if
m+ s
B
non-integer,
= B if
m+ s
B
integer, (2.105)
and thus we have set s = −lB − m and summed over l instead of s. For a partic-
ular l and m value, collecting together all the t dependent terms in (2.102), we find
exp (−i [kxU∞ + lBΩ] t) dependence. The φ dependence is given by exp (i [m+ lB]φ).
2.3.6 Total force
Thus, if we now consider the full incident velocity over all wavenumbers in place of a
single harmonic, uˆ∞j (k), we find the force term is given by
fi (x, r, φ, t) =ρ0WNi(r)
B
2pir
∞∑
m=−∞
1
(2pi)3
∫
ℜ3
ΓmWW (x, r; kx, χ
m) exp (−ikxU∞t)
eikxX
∞∑
p=−∞
Jm−p (rkR) exp {−imφk}Dpj (x0(r), r; kx, rk, 0)
[∫
ℜ3
u∞j (x
′) exp (−ik · x′) d3x′
]
d3k
∞∑
l=−∞
exp
(
i (m+ lB)
[
(x− x0(r)) tanβ(r)
r
])
exp (−ilBφ0(r))
exp (−ilBΩt) exp (i [m+ lB]φ) . (2.106)
74 2. Axisymmetric Rotor Systems
2.4 Far-field solution of the wave equation
The final part of our calculation is to substitute the forcing term (2.106) into the wave
equation, in order to calculate the far-field pressure spectrum, and hence the Unsteady
Distortion Noise heard by someone beneath the flight-path of an Advanced Open Rotor
driven aircraft.
r =
√
y2 + z2
−x Mσ0
σ0
θ
observer
U∞
rotor at
reception
time
rotor at
emission
time
Figure 2.10: Far-field coordinate system.
We work in two sets of coordinates (as shown in Figure 2.10): reception coordinates,
(x, y, z) or (x, r, φ), where the observer’s position is given at the time the sound is heard;
and emission (‘wind-tunnel’) coordinates, (σ0, θ, φ), where the observer’s position is given
at the time the sound was emitted. In addition, the source coordinates (xs, rs, φs) give
the position of a blade source with respect to the rotor’s origin. We work in the reference
frame in which the rotor is stationary in the axial direction, and assume a uniform flight
speed U∞, in which case the wave equation is given by
∇2p− 1
c20
(
∂
∂t
+ U∞
∂
∂x
)2
p =∇ · f (x, t) . (2.107)
In the previous section we saw that the φ dependence within each term of the triple
infinite series which makes up f takes the form exp (iqφ), where q = m + lB. Similarly,
we saw that each t harmonic is of the form exp (−iωt) where ω = kxU∞ + lBΩ, and all
terms are subsequently integrated over kx. It is the extra kxU∞ term within the frequency
ω which contains information about the turbulent aspect of the problem, these are not
purely periodic source terms. We can solve for each value of l, m, p and kx separately
and then integrate over kx and sum over all indices.
2.4 Far-field solution of the wave equation 75
Thus we seek the solution, pl,m,p (x, t, kx), to the problem
∇2pl,m,p − 1
c20
(
∂
∂t
+ U∞
∂
∂x
)2
pl,m,p =∇ ·
{
N(r)F l,m,p (x, r, kx) exp (iqφ)
}
exp (−iωt) ,
(2.108)
where
F l,m,p (x, r, kx) =ρ0W
B
2pir
ΓmWW (x, r; kx, χ
m) eikxX
1
(2pi)3
∫
ℜ2
Jm−p (rkR) exp {−imφk}Dpj (x0(r), r, kx, rk, 0)
[∫
ℜ3
u∞j (x
′) exp (−ik · x′) d3x′
]
dkydkz
exp
(
−i (m+ lB)
[
(x− x0(r)) tanβ(r)
r
])
exp (−ilBφ0(r))]. (2.109)
Note that, as u∞j is a statistical quantity, we will only be able to calculate explicitly the
auto-correlation or some other average of p.
Since there is no diffraction in our problem, p must have the same azimuthal and time
dependence as F . Writing
pl,m,p (x, t, kx) = pˆl,m,p (x, r, kx) exp (iqφ) exp (−iωt) , (2.110)
we find pˆl,m,p obeys the equation
1
r
∂
∂r
(
r
∂pˆl,m,p
∂r
)
− q
2
r2
pˆl,m,p +
∂2pˆl,m,p
∂x2
− 1
c20
(
−iω + U∞
∂
∂x
)2
pˆl,m,p =
sin β(r)
∂F l,m,p
∂x
+
iq
r
cosβ(r)F l,m,p. (2.111)
Here we have substituted N = (sin β(r), 0, cosβ(r)) for the blade normal and thus we
neglect any radial component of N, i.e. we do not consider lean of the blade. We can
find the solution to (2.111) in integral form by seeking the appropriate Green’s function,
H l,m, in x and r. The solution will then be given by (noting that (σ0, θ) can be directly
76 2. Axisymmetric Rotor Systems
related to (x, r))
pˆl,m,p (σ0, θ, kx) =
∫
rs
∫
xs
H l,m (σ0, θ; xs, rs)
[
sin β(rs)
∂F l,m,p (xs, rs, kx)
∂xs
+
iq
rs
cosβ(rs)F l,m,p (xs, rs, kx)
]
rsdrsdxs.
(2.112)
We now derive the Green’s function using the same framework as Hanson (1995),
which will allow us to extend to the non-zero incidence case in a straightforward manner
in Chapter 5. In Hanson8, the force term is re-expressed as
∇ · f = −
∑
ω
Qω (x, y, z) e−iωt, (2.113)
where ω is given by mBΩ. Thus he treats periodic sources with frequencies which are
multiples of BPF only. The exact solution is then given by
p (x, y, z, t) =
∑
ω
Pω (x, y, z) e−iωt, (2.114)
where
Pω (x, y, z) =
∫
ℜ3
Qω (xs, ys, zs) e
i ωc0
σ
4piS
dxsdysdzs, (2.115)
S =
√
(x− xs)2 + (1−M2)
[
(y − ys)2 + (z − zs)2
]
, (2.116)
σ =
[−M (x− xs) + S]
(1−M2) . (2.117)
Here M = U∞/c0.
Our sources Qω take the form
Qω (xs, rs, φs) = −∇ ·
{
N(rs)F l,m,p (xs, rs; kx) exp (iqφs)
}
= −F˜ (xs, rs) eiqφs . (2.118)
In the far-field we can make some simplifying approximations to (2.116) and (2.117) and
then find the appropriate Green’s function by substituting (2.118), which has simple φ
8Note that the positive x direction in Hanson (1995) is the reverse of ours.
2.4 Far-field solution of the wave equation 77
dependence, into (2.115). The first simplifying approximation is to substitute the leading
order term of S into the denominator of (2.115). This leading order approximation is
given by
S ≈ S0 =
√
x2 + (1−M2) (y2 + z2) = σ0 (1−M cos θ) , (2.119)
using −x = σ0 (cos θ −M) and y2 + z2 = σ20 (1− cos2 θ) (see Figure 2.10). The second
simplifying approximation is to calculate terms up to the first order of S and substitute
this into the phase. We find
S =
√
x2 + (1−M2) (y2 + z2)− 2xxs + (1−M2) (−2yys − 2zzs) +O(xs2)
= S0
√
1− 2xxs
S20
+
(1−M2) (−2yys − 2zzs)
S20
+O(xs2), (2.120)
⇒ S ≈ S0 + S1 = S0 −
xxs
S0
−
(
1−M2
) (yys + zzs)
S0
= S0 −
xxs
S0
−
(
1−M2
) (r cos φrs cosφs + r sinφrs sin φs)
S0
. (2.121)
The phase approximation is therefore
σ ≈ −M (x− xs) + S0 + S1
1−M2
=
−M (x− xs) + S0 − xxsS0
1−M2 −
rrs (cos φ cosφs + sinφ sinφs)
S0
= σ0 +
cos θ
1−M cos θxs −
sin θ
1−M cos θrs cos (φ− φs) , (2.122)
using S0 = σ0 (1−M cos θ), x = −σ0 (cos θ −M) and r = σ0 sin θ. Substituting (2.118),
(2.119) and (2.122) into (2.115), we find
Pω (x, y, z) =
∫
ℜ3
−F˜ (xs, rs) eiqφs
4piσ0 (1−M cos θ)
ei
ω
c0
[σ0+ cos θ1−M cos θxs− sin θ1−M cos θ rs cos(φ−φs)]rsdxsdrsdφs
=
ei
ω
c0
σ0
4piσ0 (1−M cos θ)
∫
ℜ2
−F˜ (xs, rs) ei
ω cos θ
c0(1−M cos θ)
xs
[∫ pi
−pi
e−i
ω sin θ
c0(1−M cos θ)
rs cos(φ−φs)eiqφsdφs
]
rsdxsdrs. (2.123)
78 2. Axisymmetric Rotor Systems
The φs integral can be found analytically, yielding a Bessel function, as follows. Defining
γ0 = ω sin θ/c0 (1−M cos θ), and τ = −pi/2 + (φ− φs) we have
∫ pi
−pi
e−i
ω sin θ
c0(1−M cos θ)
rs cos(φ−φs)eiqφsdφs = −
∫ pi
−pi
e−iγ0rs sin(−τ)eiq(−τ+φ−
pi
2 )dτ
= −2pieiqφe−iq pi2 Jq (γ0rs) , (2.124)
using the definition of the Bessel function
Jq (γ0rs) =
1
2pi
∫ pi
−pi
e−i(qτ−γ0rs sin τ). (2.125)
We thus obtain the following Green’s function
H l,m (σ0, θ; xs, rs) =
ei
ω
c0
σ0
4piσ0 (1−M cos θ)
(−2pi) Jm+lB (γ0rs) ei
ω cos θ
c0(1−M cos θ)
xse−
i(m+lB)pi
2 .
(2.126)
A final trick (now that we know the xs dependence of H l,m) is to integrate the ∂F l,m,p/∂xs
term within (2.112) by parts, yielding
pˆl,m,p (σ0, θ, kx) =
∫
rs
∫
xs
H l,m (σ0, θ; xs, rs)F l,m,p (xs, rs, kx)
[−iω cos θ sin β(rs)
c0 (1−M cos θ)
+
i (m+ lB) cosβ(rs)
rs
]
rsdrsdxs. (2.127)
The full acoustic pressure in the far-field will then be given by
p (σ0, θ, φ, t) =
∞∑
l=−∞
∞∑
m=−∞
∞∑
p=−∞
∫ ∞
−∞
pˆl,m,p (σ0, θ, kx) exp (−iωt) dkx exp (iqφ) . (2.128)
In this triple summation, the p index arose from splitting the distortion amplitude Aij
into circumferential harmonics (see equation (2.76)) and the m index corresponds to the
total azimuthal order of the incident turbulence field upon the blades. Thus m includes
the contribution to azimuthal order from the phase of the distorted turbulence field, as
expanded in equation (2.74). Therefore if Aij was perfectly axisymmetric we would still
have a summation over m. The l index arose from the 2pi/b periodicity of the blades, as
shown in §2.3.5.
2.4 Far-field solution of the wave equation 79
2.4.1 Auto-correlation of pressure
We take the auto-correlation of p to form P (x, τ), and then take the Fourier transform
to find the Power Spectral Density (PSD), Pˆ (x, ω), as follows
Pˆ (x, ω) =
∫ ∞
−∞
〈(p (σ0, θ, φ, t))∗ p (σ0, θ, φ, t+ τ)〉 eiωτdτ
=
∫ ∞
−∞
P (σ0, θ, φ, τ) eiωτdτ. (2.129)
We see that Pˆ has dimensions of [pressure2 · time] =
[
ρ2 L
4
T 3
]
. Majumdar (1996) then
integrated Pˆ over a shell of radius σ0, removing φ and σ0 dependence, to find
P (ω) =
∫ 2pi
φ=0
∫ pi
θ=0
Pˆ (x, ω)σ20 sin θdθdφ, (2.130)
which has dimensions of
[
ρ2 L
6
T 3
]
; power has dimensions of
[
ρL
5
T 3
]
. Using the plane wave
result, acoustic intensity = p′2/ρ0c0, we can therefore obtain a measure of power per
frequency as follows
Pf (ω) =
P (ω)
ρ0c0
, (2.131)
where Pf has dimensions
[
ρL
5
T 2
]
. To obtain a Power Level (PWL) quantity we could
integrate Pf over ω. However, in this thesis, we instead plot a ‘spectral power’ quantity
SP (ω) = 10 log10
(
P (ω)
10−12
)
dB. (2.132)
This quantity gives the correct trends of location and relative heights of tones which
are the key results of this work, since the absolute levels are ultimately controlled by
the magnitude of the turbulence intensity. In Appendix B we discuss further the use of
different sound metrics.
The first action of the statistical average of equation (2.129) is upon the Fourier
transforms of u∞ (contained within F l,m,p), which become (2pi)6δ(k − k′)S∞kl (k). The
time average then acts upon exp (ilBΩt) and exp (−il′BΩt), leading to a δll′ term, and
reducing the remaining infinite series from four (l, l′, m and m′) to three. The Fourier
80 2. Axisymmetric Rotor Systems
transform integral over τ brings out a δ(kxU∞ + lBΩ− ω) factor. The expression9 for Pˆ
is therefore given by
Pˆ (σ0, θ, φ, ω) =ρ20
B2
4pi2U∞
1
4σ20 (1−M cos θ)2
∑
l,m,m′
∫
rs,xs
[Jm+lB (γ0rs)]
∗ [W (rs)ΓmWW (xs, rs;ω)]
∗ e−ikxX
[
iω cos θ sin β(rs)
c0 (1−M cos θ)
− i(m+ lB) cosβ(rs)
rs
]
exp
{ −iω cos θxs
c0 (1−M cos θ)
+ i(m+ lB)
[xs − x0(rs)] tan β(rs)
rs
+ ilBφ0(rs)
}
∫
r′s,x′s
Jm′+lB (γ0r′s)W (r
′
s)Γ
m′
WW (x
′
s, r
′
s;ω) e
ikxX′
[
−iω cos θ sin β(r
′
s)
c0 (1−M cos θ)
+
i(m′ + lB) cosβ(r′s)
r′s
]
exp
{
iω cos θx′s
c0 (1−M cos θ)
− i(m′ + lB) [x
′
s − x0(r′s)] tan β(r′s)
r′s
− ilBφ0(r′s)
}
∑
p,p′
∫
ℜ
J∗m−p (rkR)D
p∗
k (x0(rs), rs; kx, rk, 0)Jm′−p′ (rkR
′)Dp
′
l (x0(r
′
s), r
′
s; kx, rk, 0)
[∫ 2pi
0
ei(m−m
′)φkS∞kl (k) dφk
]
rkdrkdr′sdx
′
sdrsdxse
i(m′−m)(φ−pi2 ). (2.133)
Here R′ = R(x0(r′s), r
′
s), and X
′ = X(x0(r′s), r
′
s). Note that, unlike for the spectrum of
blade pressures, the e−ikxX and eikxX
′
terms do not cancel each other10. Averaging over
φ, when calculating P , leads to m = m′. The input parameters to this expression are:
c0, ρ0 (physical parameters); U∞, S∞ij (far upstream conditions); Aij (x) , Uf (dependent on
mean flow model); B,Ω, rd, φ0(r), x0(r), c(r) (rotor parameters). In addition, the following
quantities depend upon those parameters: kx = (ω − lBΩ)/U∞, M = U∞/c0, β(r) =
tan−1(Ωr/Uf ), W =
√
U2f + (Ωr)2 and ωΓ = ω − (m+ lB).
9Checking dimensions of the above expression, it should have dimensions of pressure2.time ≡
[
ρ2 L
4
T 3
]
.
We find
[
Pˆ
]
=
[
ρ20
1
U∞σ20
W 2 1r2s S
∞dkydkzdrsdxsdr′sdx
′
s
]
≡ ρ2 L4T 3 , using [S∞] ≡ L
5
T 2 .
10Majumdar and Peake (1998) did not include these terms in their expression for radiated sound.
2.4 Far-field solution of the wave equation 81
2.4.2 Modal analysis
Here we detail the approximations employed when implementing (2.133) numerically for
a given frequency ω.
Summation over the l and m indices
We wish to restrict the infinite sums over the l,m indices to the dominant terms only in
order to reduce the computational intensity of calculations. To achieve this, we consider
first the l values which result in kx = ω− lBΩ/U∞ being as close to zero as possible, due
to the sharp decay of S∞kl as k increases. Majumdar included l = 0, 1, 2 only for all ω
from 0 to 2 BPF. However, for some values of ω within this range it is more appropriate
to take values of l centred on 0 or 2. The difference between using a total of 3 or 5 l
values is found to be negligible and so we take nl (the total number of l values included
in the sum) to be 3 in all our calculations. Next, we include an odd number of m values
centred on −lB, due to the decay of the Jm+lB term as the order increases. This sum is
more sensitive to the total number of m values, nm, as the decay of the Bessel functions
is not as rapid as that of S∞kl . A noticeable difference is found in SP between the nm = 3
(the number included by Majumdar) and nm = 5 cases for the low distortion conditions
- an increase in m values leads to an increase in broadband level. In the high distortion
case, the height of the tonal peaks is unaffected by nm, although the broadband level does
change slightly. We set nm = 5 in our calculations.
Sum over p, p′, and rk integral limits
When calculating the SP (and thus the φ-averaged version of (2.133)) numerically a
discrete set of rs, r′s values are chosen (typically around 6 values). The integrand is then
evaluated for all pairs of rs, r′s and summed using the trapezium rule, first over r
′
s then
over rs. The integral
∫
ℜ
J∗m−p (rkR)D
p∗
j (x0(rs), rs; kx, rk, 0)Jm−p (rkR
′)Dpk (x0(r
′
s), r
′
s; kx, rk, 0)
[∫ 2pi
0
S∞kl (k) dφk
]
rkdrk, (2.134)
82 2. Axisymmetric Rotor Systems
for a particular pair of rs, r′s does not converge with an infinite upper limit for rk. We use
a limited range for this integral, as discussed earlier in §2.3.3, of
√
(m− p)2
R2
≤ rk ≤
√(
∂R
∂r
)−2 ω2
c20
+
(m− p)2
R2
. (2.135)
Since R and ∂R/∂r differ for rs and r′s we have a choice as to whether to take the inner
or outer of each limit. We use the outer limits, thus using the larger value of R and the
smaller value of ∂R/∂r, although in fact there is little difference if the inner limits are
used. We also find that setting p = p′ gives sufficiently accurate results which are very
close to the p 6= p′ case, as well as significantly speeding up the numerics.
Justifying |k| ≫ 1/ |X|
Finally, we remember that the expression for Aij required |k| ≫ 1/ |X|. We can now
check this assumption. Note that, for the rk limits outlined above
|k| =
√
k2x + r
2
k ≥ rk ≥
|m− p|
R
, (2.136)
and for m 6= p then
|m− p|
R
≥ 1
R
≥ 1√
X2 +R2
=
1
|X| . (2.137)
Since m ∼ lB and |p| ≤ 3 we see that |m− p| will be of the same order as B except in the
case of l = 0. Thus for l 6= 0, since B is typically greater than 10, |k| ≫ 1/ |X| holds. For
l = 0 we have kx = ω/U∞, and thus require ω ≫ U∞/
√
X2 +R2 to satisfy |k| ≫ 1/ |X|.
We are primarily interested in frequencies at and above the BPF, for which ω is an order
of magnitude larger than typical values of U∞/rh (where rh is the hub radius) and since√
X2 +R2 > R > rh for a streamtube contraction, the condition on ω will be satisfied for
the parameter ranges of interest.
2.4.3 SP plots
In Figures 2.11 and 2.12 we have plotted SP, as given in equation (2.132), for a range of
distortion levels and integral lengthscales. The main points to note are that the broadband
level in the low distortion case (which corresponds approximately to take-off at Mach no.
2.4 Far-field solution of the wave equation 83
0.25) is 30-40 dB lower than the tonal peaks which occur in the high distortion case
(which corresponds approximately to static testing), and that the integral lengthscales
which produce the highest tonal peaks are of the order of the rotor radius.
1000 1500 2000 2500
50
60
70
80
90
100
110
120
130
140
ω
SP
High distortion
Medium distortion
Low distortion
Figure 2.11: Radiated sound results, varying levels of distortion. L = 1 in all cases. We
see the strikingly tonal nature for high distortion levels. Peaks are at BPF and 2 BPF.
1000 1500 2000 2500
70
80
90
100
110
120
130
ω
SP
L = 0.05 rd
L = 0.5 rd
L = 5 rd
Figure 2.12: Varying the turbulent integral lengthscale, L, whilst keeping the distortion
level constant, Uf/U∞ = 10. As L increases we see sharper tonal peaks, but once L has
passed a critical value the whole spectrum shifts down almost uniformly.
84 2. Axisymmetric Rotor Systems
In Figure 2.13 we have reproduced the two sets of parameter values which were calcu-
lated in Majumdar and Peake (1998), as shown in Figure 2.14. We see excellent qualitative
agreement between the prediction schemes, however the effect of full inclusion of the non-
zero kφ and Aφφ terms and consideration of φ dependence within the distorted amplitude
as well as the phase is seen to lower the broadband level in flight with respect to the static
testing tonal peaks.
2000 4000 6000 8000 10000 12000 14000
ω
SP
20 dB
Flight
Static
Figure 2.13: Results from our prediction scheme, run for the same parameter values as in
Figure 2.14. We note that this meant nm was taken to be 3, and thus a lower number of
azimuthal orders were included than in Figures 2.11 and 2.12.
Figure 2.14: Figure reproduced from Majumdar and Peake (1998), comparing the radiated
sound in the flight and static testing cases.
Chapter 3
Alternative inputs: mean flow and
turbulence models
3.1 Chapter outline
In this chapter we firstly revisit the mean flow model employed in Chapter 2, introducing
extra features to more realistically simulate the flow into an AOR, see schematic below.
Note that it is possible to input any axisymmetric flow (which is irrotational and tends
to a uniform velocity far upstream) into the analysis of Chapter 2, if we can track along
the streamlines sufficiently far back to find convergent values for X and R. For example a
computationally generated flow could be used. However, analytic models allow for more
rapid calculation of the distortion quantities. Secondly, we explore the use of different
input turbulence models and integral lengthscales to more accurately capture the key
features of atmospheric turbulence. As part of this work, two approximations for the
distorted turbulence 3D energy spectrum, E1(k) and E2(k), are proposed.
Inputs Outputs
Mean flow model, U:
1) variable circulation
actuator disk
2) diversion of flow
around the bullet
3) co-axial, contra-
rotating actuator disks
Upstream turbulence
model, u∞:
1) Liepmann spectrum
2) Gaussian spectrum
3) role of L investigated
Turbulence energy
spectra at rotor face:
E1(k), E2(k)
Far-field pressure: Pˆ
leading to spectral power
Distortion of upstream
turbulence, u∞, by
mean flow, U, using
Rapid Distortion Theory.
Analysis as for Chapter
2, with appropriate
expressions derived for
∂Xi/∂xj for each new
mean flow.
85
86 3. Alternative inputs: mean flow and turbulence models
3.2 Mean flow model
We consider three new axisymmetric mean flows in this chapter, see Figure 3.1.
−20 −15 −10 −5 0
10
5
0
5
10
x, axial direction
r,
r
a
di
al
d
ire
ct
io
n
High Distortion, Variable Circulation
−20 −15 −10 −5 0
3
2
1
0
1
2
3
x, axial direction
r,
r
a
di
al
d
ire
ct
io
n
Low Distortion, Variable Circulation
−20 −15 −10 −5 0
10
5
0
5
10
x, axial direction
r,
r
a
di
al
d
ire
ct
io
n
High Distortion with Bullet
−20 −15 −10 −5 0
3
2
1
0
1
2
3
x, axial direction
r,
r
a
di
al
d
ire
ct
io
n
Low Distortion with Bullet
−20 −15 −10 −5 0
10
5
0
5
10
x, axial direction
r,
r
a
di
al
d
ire
ct
io
n
High Distortion, Co−axial
−20 −15 −10 −5 0
3
2
1
0
1
2
3
x, axial direction
r,
r
a
di
al
d
ire
ct
io
n
Low Distortion, Co−axial
Figure 3.1: Illustration of the three new mean flows we consider in this chapter. Top: a
variable circulation actuator disk model. Middle: a model which simulates diversion of the
flow streamlines around the engine hub (the bullet), using a point source (whose position
is shown in red). Bottom: two co-axial actuator disks, modelling the presence of the rear
blade row. Shown in red for comparison on the high distortion plots are streamlines which
run through the root and tip of the blade in a constant circulation actuator disk model.
Note that the flow for x > 0 includes a non-zero azimuthal component, not shown here.
The actuator disk remains the basis for these improved mean flow models. However,
we note briefly here a couple of alternative flow models found during the literature review
3.2 Mean flow model 87
undertaken in Chapter 1. Simonich et al. (1990) used a set of discrete vortex rings to
represent the wake of the rotor, rather than our continuous vortex line shedding model.
One disadvantage of the vortex ring model is that it cannot predict velocities very close to
the blade tip. Mish and Devenport (2006b), when modelling the distortion of an isotropic
turbulence field incident upon an airfoil, applied Rapid Distortion Theory with the mean
flow given by the flow around a cylinder, to simulate the leading edge of the blade. In
Paterson and Amiet (1982), first the noise spectrum due to each point along the blade is
calculated and then blade-to-blade correlations are introduced. This means a non-axial,
azimuthally varying flow can be used, allowing consideration of a non-zero shaft angle;
this is discussed in more detail in Chapter 5. Finally, we note that whilst the effects of
swirl upstream of the rotor are neglected in our work, this extension was made by Wright
(2000), as discussed in §2.2.4.
3.2.1 Vortex theory of a propeller
In Chapter 2 we simply stated, in equations (2.47) and (2.48), the mean flow velocities
found within Hough and Ordway’s vortex theory model for the constant circulation actu-
ator disk. We now outline the derivation in more detail, before introducing the variable
circulation actuator disk model.
In Hough and Ordway’s vortex model for rotor-induced flow each blade is modelled as
a vortex line. Thus information about the blade shape is used only as far as it influences
circulation, denoted at each radial location by Γ (r). Trailing helical vortex sheets are shed
behind the blades as the propeller translates forward, with circulation given by −Γ′ (r),
as shown in Figure 3.2. In the limit of infinite blade number this model reproduces the
actuator disk jump in flow properties, but also provides us with expressions for the velocity
field everywhere.
The Biot-Savart Law gives the velocity induced by a line vortex of unit length and
strength lying in direction eˆ from (xv, rv, φv) as follows
u (x, r, θ) =
eˆ× r
4pi |r|3
, (3.1)
where eˆ is a unit vector, and r is the vector (x−xv, r−rv, φ−φv). This can be integrated
along each vortex line, and summed to give the full induced velocity fields. We find that
the vortices along the blades contribute solely to the tangential velocity, Uφ. In fact it
88 3. Alternative inputs: mean flow and turbulence models
Figure 3.2: Illustration reproduced from Hough and Ordway (1965) showing the vortex
line system we are considering for a single blade, together with one of the trailing helical
vortex lines which are shed from each point along the blade. Here U is the axial velocity
at upstream infinity, denoted by U∞ in this thesis. Thus the shed vortices lie along the
vector U∞eˆx + Ωreˆφ.
is found that upstream of the propeller (i.e. in the region x < 0) and also in the region
r > rd this contribution is exactly balanced by the contribution to Uφ from the trailing
vortex sheets. Thus there is zero swirl upstream of the propeller and in r > rd, but this
jumps to a non-zero value immediately downstream.
In Chapter 2 we used a simple constant circulation actuator disk model, i.e. with
circulation along each blade given by Γ(r) = Γ. The assumption of constant blade load-
ing along the length of each blade is essentially a statement that each part of the blade
contributes equally to the thrust generated. Although certain properties of open rotor
blades, such as their chord length, vary little over the majority of the blade, this approx-
imation will certainly break down near the tip. It is therefore of interest to examine the
difference that a more realistic, variable circulation model makes to our predictions, and
this is the first new model we consider.
3.2.2 Variable circulation actuator disk
The constant circulation actuator disk model gives a constant velocity profile at the rotor
face, whereas in reality it will vary along the span. For a real rotor the axial velocity
increases with r as we move out from the hub to reach a maximum within the region of
greatest loading, and then decreases towards the tip. This qualitative behaviour can be
3.2 Mean flow model 89
seen in Figure 3.3, where we have plotted output from a typical numerical strip theory
calculation, as used by industry.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
0.4
0.5
0.6
0.7
0.8
0.9
1
U
x
/U
∞
r/r
d
take−off
cutback
approach
Figure 3.3: Representative profiles of the axial velocity at the rotor face as output from
a typical industry numerical strip theory calculation. This data was provided by Dr. M.
Kingan, Dr P. J. G. Schwaller and Dr A. B. Parry.
We can therefore model the velocity profile more realistically using Hough and Ord-
way’s variable blade circulation model, where the circulation is given by
Γ (r) =
105
16
piU∞Ud
BΩ
r
rd
√
1− r
rd
, (3.2)
Γ′ (r) =
105
16
piU∞Ud
BΩ
(
1− 3r2rd
)
rd
√
1− rrd
. (3.3)
This profile was developed by Hough and Ordway as it compares well to the Goldstein
optimum distribution of blade loading, which minimises the energy lost in the slip stream
for a given propeller thrust (Goldstein, 1929). For this form of Γ(r), the induced axial and
radial steady velocities upstream of the rotor (equivalent to equations (2.47) and (2.48))
90 3. Alternative inputs: mean flow and turbulence models
are
Uvar.x (x, r) =
105
32
Udx
2pir
3
2
∫ rd
0
√
r′
rd
√
1− r
′
rd
Q′− 12
(ω′)dr′; for x < 0, (3.4)
Uvar.r (x, r) =
105
32
Ud
2pir
1
2
∫ rd
0
(
1− 3r′2rd
)
rd
√
1− r′rd
√
r′Q 1
2
(ω′)dr′. (3.5)
As before, Qn are Legendre functions, see Appendix A for definitions. The variable
and constant circulation axial velocity profiles differ significantly from each other at the
disk face. However, moving upstream they quickly produce very similar profiles, see
Figure 3.4. From this figure we also note that the majority of the axial distortion occurs
within five disk radii, and this leads us to conclude that coherent flow structures shed
from airframe features which lie several radii upstream of an open rotor could undergo
significant distortion before they hit the rotor blades.
0 50 100
0
0.2
0.4
0.6
0.8
1
U
x
/U
∞
r/r
d,
ra
di
al
d
ire
ct
io
n
x =−5rd
0 50 100
0
0.2
0.4
0.6
0.8
1
U
x
/U
∞
r/r
d,
ra
di
al
d
ire
ct
io
n
x =−0.5rd
0 50 100
0
0.2
0.4
0.6
0.8
1
U
x
/U
∞
r/r
d,
ra
di
al
d
ire
ct
io
n
x =−0.25rd
0 50 100
0
0.2
0.4
0.6
0.8
1
U
x
/U
∞
r/r
d,
ra
di
al
d
ire
ct
io
n
x =−0.05rd
Variable Circ
Constant Circ
Figure 3.4: Here we show the axial velocities at x = −5rd, x = −0.5rd, x = −0.25rd,
x = −0.05rd. The total thrust generated is held constant between the constant and
variable circulation actuator disk models and thus the captured streamtube is the same
in both cases. Within a distance of half of the radius of the disk the variable and constant
circulation profiles are virtually the same.
3.2 Mean flow model 91
In Figure 3.5a we compare the streamlines of the constant and variable circulation
cases. The streamlines behave quite differently close to the disk. Instead of r decreasing
monotonically along each streamline as we travel toward the disk, in the variable circula-
tion case the radius increases as the streamline gets very close to the disk, as seen in the
zoomed in Figure 3.5b. This leads to negative values of ∂X/∂x and ∂R/∂x at the disk
face for lower values of r, as seen in Figure 3.6.
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0
10
8
6
4
2
0
2
4
6
8
10
x, axial direction
r,
r
a
di
al
d
ire
ct
io
n
Constant circ.
Variable circ.
(a)
−0.4 −0.2
0.5
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
0.5
x
r
(b)
Figure 3.5: a) The variable circulation model leads to slightly decreased R values at the
disk face over most of the blade span. b) Zoomed-in figure near to the disk.
−5 0 5 10 15 20
0.4
0.6
0.8
1
dX/dx
r/r
d
High Distortion
Constant circ.
Variable circ.
−5 0 5 10 15
0.4
0.6
0.8
1
dR/dx
r/r
d
0.8 0.85 0.9 0.95 1
0.4
0.6
0.8
1
dX/dx
r/r
d
Low Distortion
−0.1 0 0.1 0.2 0.3
0.4
0.6
0.8
1
dR/dx
r/r
d
Figure 3.6: Plot of ∂X/∂x and ∂R/∂x at the disk face, comparing the constant and
variable circulation cases.
92 3. Alternative inputs: mean flow and turbulence models
These altered values of the derivatives of X then affect the distorted turbulence spec-
trum at the rotor face, plotted in Figure 3.7 (we recall that Sxx was defined in equation
(2.60)). In Figure 3.6 we see that for the high distortion case the values of the deriva-
tives, ∂X/∂x and ∂R/∂x, are higher in the variable circulation case towards the blade
tip, and reduced near the hub when compared to the constant circulation case. There is
a corresponding increase in distorted turbulence level towards the tip of the blades and a
reduction near the hub seen in Figure 3.7. For low levels of distortion we see very little
change to the turbulence spectrum between the two flows, as might be expected. Even
though the values of ∂Xi/∂xj differ in that case too, they are much smaller in absolute
size and are dominated by the δij term of Aij (given in equation (2.30)). The largest
difference in turbulence level between the constant and variable cases is found near the
hub, for medium levels of distortion.
−2 −1 0 1 2 3 4
−12
−10
−8
−6
−4
−2
0
2
4
6
log(k
x
)
lo
g(S
xx
)
High distortion, near tip
High distortion, near hub
Medium distortion, near tip
Medium distortion, near hub
Low distortion, all r values
Figure 3.7: Distorted turbulence spectrum (axial component) for the variable circulation
model, at different positions along the rotor blade and for varying distortion levels. The
constant circulation case with the same input parameters is shown in red (as was shown
in Figure 2.4 in blue). L = 1 in all cases.
3.2 Mean flow model 93
Validation against numerical data
In Figure 3.8 we compare the axial velocity as output from a typical strip theory calcu-
lation with that given by the variable circulation actuator disk model. It is found that
a significantly closer fit to the strip theory data can be obtained by scaling the velocity
field as follows
Ux(x, r) = Uvar.x
(
x,
(r − rh)
(rd − rh)
rd
)
, (3.6)
where rh is the radius of the hub and the ‘var’ superscript indicates Hough and Ordway’s
original variable circulation model, equations (3.4) and (3.5). That is, we take account
of the radial limits of the blade, rh ≤ r ≤ rd, and scale the profile to lie between these
limits.
In §3.4, we examine the effect of these different mean flows on the radiated sound,
including an investigation of the use of a scaled velocity profile as given above.
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
U
x
/U
∞
r/r
d
strip theory numerical output, take−off conditions
const. circ. a.d. model, thrust as for numerics
const. circ. a.d. model, (63% thrust)
variable circ. a.d. model, (70% thrust)
Figure 3.8: Here we have used the adjusted variable circulation model (given in equation
(3.6)) to fit to the output from the numerical strip theory calculation. Note also the use
of a lower thrust than that output from the numerics gives an improved fit.
94 3. Alternative inputs: mean flow and turbulence models
3.2.3 Modelling the bullet
The engine core extends upstream of an AOR in rig testing, as shown in Figure 3.9, and
affects the mean flow onto the rotors. Previous work by Zachariadis and Hall (2009) has
indicated that the bullet can make a significant difference to the thrust generated by the
rotors, altering the axial velocity onto the blades as well as the flow angle. To simulate
the mean flow distortion around the bullet we have developed a new and simple analytical
model whereby a point source is added to the flow upstream of an actuator disk. This
is similar to a Rankine half-body model, except the point source is superposed upon an
actuator disk flow rather than a uniform flow.
Figure 3.9: The large ‘bullet’ protruding upstream of the rotors in Rig 140 testing in the
1980s. Reproduced with permission from Rolls-Royce.
The streamfunction for a general source of strength m, placed at x = x0, along the
axis r = 0, is given in cylindrical polar coordinates (x, r, φ) by
Ψs = − m (x− x0)√
r2 + (x− x0)2
. (3.7)
Superposing this upon an actuator disk flow
(
U∞ + Ua.d.x (x, r)
)
eˆx +Ua.d.r (x, r)eˆr will lead
to a stagnation point upstream of x0, on the r = 0 axis, and we denote its position by
x = x1. The two source parameters, m and x0, can be chosen to specify x1 and the
downstream radius rh of the centremost streamline, see Figure 3.10.
The axial velocity due to the point source is
U sx =
1
r
∂Ψs
∂r
=
m (x− x0)
(
r2 + (x− x0)2
) 3
2
. (3.8)
3.2 Mean flow model 95
x0
x1
actuator disk
r = rh
r = 0
Figure 3.10: Actuator disk streamlines diverted around a point source at x0. The stream-
line which runs along r = 0 as x → −∞ intersects the disk at r = rh. By adjusting the
strength and position of the point source, x1 and rh can be specified.
As the axial velocity, U∞ + U sx + U
a.d.
x , vanishes at the stagnation point x1 we find
x0 = x1 +
√
m
U∞ + Ua.d.x (x1, 0)
. (3.9)
The positive square root has been chosen since (x1 − x0) < 0. Here
(
U∞ + Ua.d.x (x1, 0)
)
is the axial velocity which would be obtained with an actuator disk only. This is the first
condition needed to specify x0 and m. The second condition is found by equating the
value of the streamfunction at two points on the centremost streamline (which starts on
r = 0 as x → −∞). The actuator disk streamfunction, Ψa.d., tends to zero as x → −∞
(see Appendix A for the full definition of Ψa.d.). Thus, if the total streamfunction for this
flow is given by Ψ, we have
Ψ (x, 0)|x→−∞ =
1
2
U∞r2
∣∣
r=0
+ Ψs (x, 0)|x→−∞ + Ψa.d. (x, 0)
∣∣
x→−∞ = m. (3.10)
At the rotor face, the value of the streamfunction is
Ψ (0, rh) =
mx0√
r2h + x
2
0
+
1
2
(
U∞ +
Ud
2
)
r2h. (3.11)
Setting these two values equal, we find the desired source strength is given by
m =
r2h
(
U∞ + Ud2
)
2
(
1− x0√
r2h+x
2
0
) . (3.12)
We can solve for x0 and m by specifying the length of the bullet as x1, and the hub
96 3. Alternative inputs: mean flow and turbulence models
width as rh, then substituting (3.12) into (3.9) and seeking x0 numerically. For example,
if the length of the bullet is chosen as −x1 = 2rd and the hub radius is rh = 0.35rd, then
for a low distortion case, (an actuator disk which induces a velocity at the rotor face of
Uf = 1.18U∞) we find x0 = −1.8rd and m = 2.9r2dU∞. If instead we consider a medium
distortion case, Uf = 10U∞, the source position moves toward the disk, x0 = −1.5rd, and
the source strength decreases proportionally to the change in U∞, m = 0.29r2dU∞.
In Figure 3.11 we have plotted the axial component of the distorted turbulence spec-
trum, comparing the cases with and without the bullet. We see that the addition of the
point source simulating the bullet tends to raise the energy level slightly, but not in all
cases (for example the medium distortion case near the hub). There will be a level of
distortion between the high and medium cases shown where the contraction induced by
the actuator disk balances the expansion induced by the bullet, and the two spectra will
coincide. However, the value of U∞ (which determines the level of distortion) at which
this occurs will vary for different points along the blade. There will be no level of distor-
tion for which the spectra exactly coincide at all radial positions. This low level of change
indicates that although the values of X change within this new model, the derivatives
∂Xi/∂xj do not vary significantly from the no-bullet case.
−2 0 2 4
0
20
40
60
80
100
log(k
x
)
S x
x
High distortion, near tip
With bullet
Without bullet
−2 0 2 4
0
0.2
0.4
0.6
0.8
log(k
x
)
S x
x
Medium distortion, near tip
−2 0 2 4
0
2
4
6
log(k
x
)
S x
x
High distortion, near hub
−2 0 2 4
0
0.01
0.02
0.03
0.04
0.05
log(k
x
)
S x
x
Medium distortion, near hub
Figure 3.11: Distorted turbulence spectrum at the rotor face with and without the bullet,
L = 1 in all cases. The differences observed are not large enough to show up on a log-log
plot.
3.2 Mean flow model 97
3.2.4 Co-axial propellers
Finally, a simple adjustment to the model can be made by including the effect of the rear
blade row upon the distortion. This was in fact a first step towards more realistically
capturing the flow through an AOR undertaken at the start of this Ph.D. We can model
the mean flow induced by a co-axial, contra-rotating, propeller system by superposing two
actuator disks, separated by a mid-chord to mid-chord gap of length g. We note that both
the radius, rd, and the ‘strength’, Ud, may vary between the two rotors, as they do not in
general contribute equally to the thrust generated. Essentially this involves replacing x
by (x− g) in the velocities induced by the second rotor, as given in equations (2.47) and
(2.48), remembering that the arguments of the Legendre functions also involve x
Uco-ax. (x, r) = Ua.d. (x, r;Ud1, rd1) +Ua.d. (x− g, r;Ud2, rd2) . (3.13)
Figure 3.12 shows the streamlines induced by such a system. We see that a smaller
upstream streamtube radius intersects with the front blade row, compared to the single
disk case, also shown.
−30 −25 −20 −15 −10 −5 0 5
10
8
6
4
2
0
2
4
6
8
10
x/g, axial direction
r/r
d,
1,
ra
di
al
d
ire
ct
io
n
High distortion, single a.d.
High distortion, co−axial a.d.s
Figure 3.12: Comparing the streamlines of the single actuator disk to the co-axial disk
case. The total thrust generated is kept constant between the two cases, but in the twin
disk case the thrust is split front:rear in the ratio 60:40. Indicated by asterisks are two
streamlines which intersect the front disk at the same position.
We compare ∂Xi/∂xj between the single actuator disk and co-axial disk systems in
Figure 3.13. The opposite effect from the variable circulation case is found, with values
of the derivatives being increased near the tip when compared to the single actuator disk.
The tip is also where most change is seen in Sxx, as plotted in Figure 3.14. This is due
98 3. Alternative inputs: mean flow and turbulence models
to the effect of the second blade row on the captured streamtube.
0 5 10 15
0.4
0.6
0.8
1
dX/dx
r/r
d
High Distortion
Single disk
Contra disks
0 5 10 15
0.4
0.6
0.8
1
dR/dx
r/r
d
0.84 0.86 0.88 0.9 0.92
0.4
0.6
0.8
1
dX/dx
r/r
d
Low Distortion
0 0.05 0.1 0.15 0.2
0.4
0.6
0.8
1
dR/dx
r/r
d
Figure 3.13: Plot of ∂X/∂x and ∂R/∂x, comparing the single and co-axial disk cases. R
also changes in order to preserve det(∂Xi/∂xj) = 1. All parameters are as in Figure 3.6.
−2 −1 0 1 2 3 4
−12
−10
−8
−6
−4
−2
0
2
4
6
log(k
x
)
lo
g(S
xx
)
High distortion, near tip
High distortion, near hub
Medium distortion, near tip
Medium distortion, near hub
Low distortion, all posns
Figure 3.14: Distorted turbulence for the co-axial case (shown in blue). The level is the
same as the single actuator disk case (shown in red) near the hub, but lowered near the
tips.
These results raise the question of the level of UDN produced by the rear row, especially
towards the tip of the rear blades. Modelling the flow there would require taking account
of the tip vortices shed by the front row, and we do not pursue this analysis here.
3.2 Mean flow model 99
3.2.5 Calculating ∂Xi/∂xj for new flows
In order to calculate the distortion amplitude Aij in each of the new mean flow cases,
we require the derivatives of X. By substituting R =
√
2Ψ/U∞, we find the following
relations for any axisymmetric flow
∂R
∂x
= − rUr
RU∞
,
∂R
∂r
=
r (U∞ + Ux)
RU∞
,
∂X
∂x
=
∂X
∂x
∣∣∣
R
+
∂X
∂R
∣∣∣
x
∂R
∂x
∣∣∣
r
=
U∞
(U∞ + Ux)
+
∂X
∂R
∣∣∣
x
(
− rUr
RU∞
)
,
∂X
∂r
=
∂X
∂R
∣∣∣
x
∂R
∂x
∣∣∣
x
=
∂X
∂R
∣∣∣
x
(
r [U∞ + Ux]
RU∞
)
. (3.14)
Thus, when altering the mean flow model most of the changes can be made by simply
inserting the appropriate expression for Ux and Ur. The only term which requires re-
derivation is
∂X
∂R
∣∣∣
x
= U∞
∂∆ˆ
∂R
∣∣∣
x
, (3.15)
where ∆ˆ is the drift function re-expressed as a function of R and x, rather than r and x.
Now, referring to the definition of drift given in equation (2.22), we have
∂∆ˆ(x,R(x, r))
∂R
∣∣∣
x
= −
∫ x
−∞
∂Ux
∂rs
∣∣∣
xs
∂rs
∂R
∣∣∣
xs
(U∞ + Ux)
2 dxs,
= −U∞R
∫ x
−∞
∂Ux
∂rs
∣∣∣
xs
rs (U∞ + Ux)
3dxs, (3.16)
(Majumdar (1996)). Here (xs, rs) are coordinates along the streamline which runs through
(x, r). For the case of a single constant circulation actuator disk we had
∂Ua.d.x
∂rs
= r
− 32
s
∂f a.d.
∂rs
− 3
2
r
− 52
s f a.d., (3.17)
100 3. Alternative inputs: mean flow and turbulence models
where
f a.d. =
Udx
2pi
∫ rd
0
1√
rv
Q′− 12
(ω1) drv, (3.18)
∂f a.d.
∂rs
=
Udx
2pi
∫ rd
0
1√
rv
Q′′− 12
(ω1)
(r2s − r2v − x′2)
2rvr2s
drv. (3.19)
This expression for ∂Ux/∂rs is thus inserted into (3.16). For the case of variable circula-
tion, f is adjusted as follows
fvar. =
105
32
Udx
2pi
∫ rd
0
√
rv
rd
√
1− rv
rd
Q′− 12
(ω1) drv, (3.20)
∂fvar.
∂rs
=
105Udx
64pi
∫ rd
0
√
rv
rd
√
1− rv
rd
Q′′− 12
(ω1)
(r2s − r2v − x′2)
2rvr2s
drv. (3.21)
For the case of flow around the bullet
Ubul.x = U
a.d.
x + U
s
x, (3.22)
U sx =
m (x′ − x0)
(
r2s + (x′ − x0)2
) 3
2
, (3.23)
∂U sx
∂rs
= − 3m (x
′ − x0) rs
(
r2s + (x′ − x0)2
) 5
2
. (3.24)
The extra term, given in equation (3.24), can be directly substituted into the numerator
of (3.16). For the case of twin, co-axial rotors, f is straightforward to adjust
f co-ax. =
Ud1x
2pi
∫ rd1
0
1√
rv
Q′− 12
(ω1) drv +
Ud2 (x− g)
2pi
∫ rd2
0
1√
rv
Q′− 12
(ω2) drv, (3.25)
∂f co-ax.
∂rs
=
Ud1x
2pi
∫ rd1
0
1√
rv
Q′′− 12
(ω1)
(r2s − r2v − x′2)
2rvr2s
drv
+
Ud2 (x− g)
2pi
∫ rd2
0
1√
rv
Q′′− 12
(ω2)
(
r2s − r2v − (x′ − g)2
)
2rvr2s
drv, (3.26)
where ω2 = 1+ [(x′− g)2 +(rs− rv)2]/2rsrv. When calculating ∂∆ˆ/∂R or ∆ numerically,
it is most efficient to cut off the integral at some large lower limit, and then add on the
‘tail’ of the integral as calculated analytically using the leading order behaviour for Ux
and ∂Ux/∂rs as x→ −∞.
3.3 Atmospheric turbulence models 101
In the second section of the chapter we consider in more detail the input turbulence
model, which in Chapter 2 was taken to be the von Ka´rma´n spectrum. In the third and
final section of the chapter we will look at the effect that the various different mean flow
and turbulence input models have on the radiated sound spectrum.
3.3 Atmospheric turbulence models
A fundamental challenge of modelling turbulence is that no closed set of equations exists
for determining statistical properties, such as 〈u2〉, the ensemble average of velocity. Much
work in turbulence theory is therefore concerned with obtaining closed, solvable sets of
equations for statistical quantities by introducing further laws, for example based on ex-
perimental observation. The need for modelling assumptions to close the set of turbulence
equations, and also because there are often significant differences between turbulence gen-
erated in seemingly similar situations, means a wide range of turbulence models exists.
It also means that numerical simulations are heavily relied upon in this field. Here we
undertake a review of different analytical descriptions of turbulence generated in the at-
mosphere. Any model will of course be an approximation to this complex phenomenon.
Simonich et al. (1990) previously reviewed several studies in order to choose an appro-
priate turbulence spectrum for their UDN model, and in particular they highlighted the
work of Snyder (1981) which includes effects of heat transfer.
3.3.1 The three-dimensional energy spectrum
As described in Chapter 1, the two primary methods of turbulence representation are
either through modelling the statistical distribution of key eddy characteristics or via a
spectrum representation. In Ganz’s UDN model (1980), the former method was employed.
The von Ka´rma´n spectrum was represented by a statistical distribution of elements and
then RDT was used to simulate the distortion of these elements. A key conclusion of
Ganz’s work was that tonal noise may be dominated by a relatively limited range of
transverse eddy lengthscales, of the order of 25% of the blade spacing at the rotor tip.
An ‘instantaneous’ spectrum is a further type of approximation, as used in Paterson
and Amiet (1982) and Simonich et al. (1990). Here, results for a flat-plate airfoil in
rectilinear motion are averaged over time, weighted by the appropriate Doppler factor.
This approximation will be valid if the rotor directivity does not change significantly
102 3. Alternative inputs: mean flow and turbulence models
during each blade-eddy interaction, i.e. for eddies which are sufficiently small.
We use the full spectrum representation of turbulence, as defined in Chapter 2 equa-
tions (2.54) - (2.57) and (2.60) - (2.61). Turbulence spectra give an approximation of the
turbulent kinetic energy contained within a range of eddy frequencies. Given a turbulent
signal, for example experimental measurements of velocity correlations, we might wish to
reconstruct a picture of the distribution of eddy sizes to get an idea of what the turbu-
lence ‘looks like’. The three-dimensional energy spectrum, E(k), gives one (imperfect)
measure for this distribution of eddy sizes1. In equation (2.55), we introduced the general
expression for an isotropic turbulence spectrum, in terms of E(k). If all eddies present are
of the same size, L, then E(k) will peak at k ∼ pi/L, however it is important to remember
that eddies of a fixed size contribute to E(k) for all k. In addition, the Fourier Transform
involved in obtaining S∞ij means information about the phase of the velocities is lost, and
this is why E(k) is imperfect: an infinite number of different distributions of eddy size
can give rise to the same spectrum. However, changes in E(k) do give an indication of
changes in turbulent behaviour.
Our upstream spectrum, which is assumed to be isotropic, obeys the following relations
E(k) = 2pik2S∞ii (k) (3.27)
= k3
d
dk
[
1
k
dΘ∞xx
dk
]
, (3.28)
(summation convention assumed). The quantity E(k)dk gives the contribution of wavenum-
ber k to 12〈u2〉, and is therefore seen as representing energy.
We can construct two corresponding expressions, E1(k) and E2(k), for our distorted
spectrum Sij , as follows
E1(x, ω) = 2piSii(x, ω), (3.29)
E2(x, ω) = k3x
d
dkx
[
1
kx
dΘxx
dkx
]
, (3.30)
where Θxx =
∫
ℜ2
Axk(x;k)Axl(x;k)S∞kl (k)dkydkz. (3.31)
These quantities will not represent energy spectra, as Sij is a temporal (rather than
spatial) Fourier Transform and is no longer isotropic. However, they allow comparison of
the change in total spectral level, rather than a single component, such as Sxx.
1An alternative measure is given by the structure function, (Davidson, 2004).
3.3 Atmospheric turbulence models 103
In Figures 3.15 - 3.17 we have plotted Sxx, E1(k) and E2(k) for the full range of mean
flows defined in this chapter, both near the hub and the blade tip. In Figure 3.15 the
biggest differences in Sxx between the different flows are seen for the co-axial case near
the tip, and for the variable circulation case near the hub. By summing all diagonal com-
ponents and plotting Sii (Figure 3.16) the hub and tip results are brought closer together,
although the variable circulation case near the hub is still seen to differ significantly from
all other cases. These results indicate that, within the variable circulation model, the
turbulent energy is shifted somewhat away from the hub, towards the mid-blade. In the
co-axial case, a lower level of turbulent energy is seen near the tips of the front row of
blades, as some of the upstream turbulence has been diverted to the rear row.
The plot of E2 (Figure 3.17) compares closely to Sxx, as it is based on the axial com-
ponent Θxx. However, the spectra now display a clear peak at the dominant lengthscale,
which is
Ld ∼
pi
kx
=
piU∞
ωd
≈ 1.7. (3.32)
having increased from the upstream integral lengthscale of 1.
−2 −1 0 1 2 3 4
−8
−6
−4
−2
0
2
4
6
log(ω)
lo
g(S
xx
)
Const. circ. a.d.
Bullet model
Contra−rot. a.d.s
Var. circ. a.d.
No distortion
Figure 3.15: This figure shows the Sxx component of the distorted turbulence spectrum,
the quantity which was investigated by Majumdar and Peake (1998), for all the flows
considered in this chapter. The dashed lines indicate calculations at points near the
blade tip (r/rd = 0.96), the solid lines are near the hub (r/rd = 0.34 for the const. and
contra cases and 0.35 for the bullet and var. cases). Uf/U∞ = 100 in all cases, i.e. high
distortion.
104 3. Alternative inputs: mean flow and turbulence models
−2 −1 0 1 2 3 4
−6
−4
−2
0
2
4
6
8
log(ω)
lo
g(S
ii)
Const. circ. a.d.
Bullet model
Contra−rot. a.d.s
Var. circ. a.d.
No distortion
Figure 3.16: Sum of all diagonal elements Sii = E1/2pi. The dashed lines indicate points
near the blade tip, the solid lines are near the hub.
0 0.5 1 1.5 2 2.5 3
−5
−4
−3
−2
−1
0
1
2
3
4
5
log(ω)
lo
g(E
2)
Const. circ. a.d.
Bullet model
Contra−rot. a.d.s
Var. circ. a.d.
No distortion
Figure 3.17: Plot of E2. The dashed lines indicate points near the blade tip, the solid
lines are near the hub. We see that the spectra peak at log(ωd) ≈ 0.6.
3.3 Atmospheric turbulence models 105
3.3.2 Turbulence shed by installation features
Hanson’s comparison in 1974 of experimental results of the blade pressure spectrum due
to UDN with that due to a cylinder placed in the inlet flow led him to conclude that atmo-
spheric turbulence may in fact be similar to that caused by a narrow cylinder. Sharland
(1964) had previously undertaken an experimental study of noise caused by a circular rod
bent into a ring placed upstream of a model fan rotor, and found this increased broadband
levels. The lack of ducting around an open rotor leads to interaction with flow structures
shed from installation features, such as the large pylon which joins the engine to the
airframe. In this section we explore the possibilities for modelling such interactions.
Figure 3.18: Turbulence behind a chimney. Image reproduced from Davidson (2004).
Image copyright Henri Werle´ of ONERA.
By considering different upstream turbulence models we are able to assess the impact
of different forms of S∞ij , to see how critical this choice may be. We look at two new
models here: the Liepmann and Gaussian spectra.
106 3. Alternative inputs: mean flow and turbulence models
Liepmann spectrum
The Liepmann spectrum is given by
ΘLMxx (kx) =
u2∞,1L
2pi (1 + k2xL2)
, (3.33)
⇒ ELM(k) =
8L5u2∞,1
pi
k4
(1 + L2k2)3
, (3.34)
⇒ SLMij (k) =
2L5u2∞,1
pi2
[k2δij − kikj]
(1 + L2k2)3
, (3.35)
and is widely used (Cheong et al., 2006, Lloyd, 2009). For kL ≪ 1, we have ELM ∼ k4,
as for the von Ka´rma´n spectrum. This power law fits that observed for grid-generated
turbulence (Davidson, 2004). For kL ≫ 1 we see that ELM decays like k−2, and this
differs from the von Ka´rma´n spectrum for which E(k) ∼ k− 53 as k →∞.
Gaussian spectrum
The Gaussian spectrum is given by
ΘGxx (kx) =
u2∞,1
2
√
pi
lG exp
(
−k
2
xl
2
G
4
)
, (3.36)
where lG =
2Γ
(
5
6
)
Γ
(
1
3
) L. (3.37)
Thus
EG(k) =
k4l5Gu
2
∞,1
8
√
pi
exp
(
−k
2l2G
4
)
, (3.38)
⇒ SGij (k) =
l5Gu
2
∞,1
32pi
3
2
(k2δij − kikj)
exp
(
k2l2G
4
) . (3.39)
Using the Gaussian spectrum to represent atmospheric turbulence has been shown to
underpredict ingestion noise due to isotropic turbulence at low and very high frequencies,
and overpredict for high frequencies (Atassi and Logue, 2008). However, as we are also
interested in representing the turbulence shed by features like the pylon we include it for
comparison.
3.3 Atmospheric turbulence models 107
Figure 3.19 compares the von Ka´rma´n, Liepmann and Gaussian energy spectra. A
particular point of difference is the rate of fall off. The very rapid fall off of the Gaussian
spectrum is unrealistic for atmospheric turbulence, and this model is most useful in a re-
stricted region of smaller kL values, as might be found for wakes due to coherent shedding
from bluff bodies.
−3 −2 −1 0 1 2 3
−15
−10
−5
0
5
log (kL)
E(
k)/
(4
pi
u
2
L3
k2
)
von Karman
Liepmann
Gaussian
Figure 3.19: Here we compare possible candidates for the upstream spectrum model:
von Ka´rma´n, Liepmann and Gaussian. The main difference between the three is their
behaviour as kL increases. Note the separation between the energy containing and inertial
subranges (located at the peak of E(k)) is found at k = L−1 for the von Ka´rma´n and
Liepmann spectra, and is slightly shifted to k > L−1 for the Gaussian spectrum.
Effect of turbulence model on distorted spectrum
In Figures 3.20 and 3.21 we have plotted Sxx when the upstream turbulence field is given
by the Liepmann and Gaussian models respectively, compared to the von Ka´rma´n case.
We see in the Liepmann case a slight change in decay rate. The very rapid drop off in
the Gaussian case means that under that model very little energy is transferred to eddies
with higher values of kx (that is, eddies with a smaller lengthscale than the integral
lengthscale).
108 3. Alternative inputs: mean flow and turbulence models
−2 −1 0 1 2 3 4
−14
−12
−10
−8
−6
−4
−2
0
2
4
6
log(k
x
)
lo
g(S
xx
)
High distortion, near tip
High distortion, near hub
Medium distortion, near tip
Medium distortion, near hub
Low distortion, all r vals
Figure 3.20: Axial component of the distorted turbulence spectrum, with the mean flow
induced by a single actuator disk. The Liepmann spectrum case is shown in blue, with
von Ka´rma´n in red for comparison. L = 1 in all cases.
−2 −1 0 1 2 3 4
−14
−12
−10
−8
−6
−4
−2
0
2
4
6
log(k
x
)
lo
g(S
xx
)
High distortion, near tip
High distortion, near hub
Medium distortion, near tip
Medium distortion, near hub
Low distortion, all r vals
Figure 3.21: Axial component of the distorted turbulence spectrum, mean flow induced
by a single actuator disk. The Gaussian spectrum case is shown in blue, with von Ka´rma´n
in red for comparison. L = 1 in all cases.
3.3 Atmospheric turbulence models 109
Anisotropic models
For completeness we note here certain UDN models which directly model the turbulence
at the rotor face, instead of the process of distortion.
Kerschen and Gliebe (1981) developed an axisymmetric, but anisotropic, model for
turbulence at the rotor face. The axial and transverse lengthscales, denoted by La and Lt,
are independent within this model (as well as the axial and transverse mean square speeds
u2∞,a and u
2
∞,t). An interesting area for future work would be to investigate mapping our
distorted spectrum onto their expression, by choosing appropriate values for La, Lt, u2∞,a
and u2∞,t.
In the case of uniform distortion it is possible to write down an analytic expression for
the distorted, anisotropic turbulence, using the Batchelor and Proudman (1954) model.
The distorted wavevector, given in our model by equation (2.34), is then given by
li =
ki
ei
, (3.40)
(no summation convention). The constant factors ei give the distortion in each direction
and so must multiply to give 1 to satisfy incompressibility. This model has been recently
used by Atassi and Logue (2009) and Devenport et al. (2010).
3.3.3 Integral lengthscale, L
Along with the form of E(k), which defines S∞ij , our model requires an input value for
the integral lengthscale parameter, L. The integral lengthscale can be defined via Rij
(referring back to equation (2.61)) as follows
L =
∫ ∞
0
Rxx(reˆx)
u2∞,1
dr, (3.41)
(Davidson, 2004). As before, u2∞,1 is the mean square speed of the axial component of
turbulent velocity. The dependence of the eventual spectrum on this second parameter,
u2∞,1, is purely multiplicative i.e. doubling u
2
∞,1 will lead to a 10 log10 2 dB rise in PWL.
Turbulence always contains a wide range of scales; the value of L shifts the dominant
eddy size but there will still be a range of eddy sizes present. We input L for the up-
stream turbulence, and the particular form of distortion will then determine the dominant
110 3. Alternative inputs: mean flow and turbulence models
lengthscale of the distorted turbulence. In the case of distortion due to a streamtube con-
traction, this distortion lengthscale will be related to the rotor radius (Hanson and Horan,
1998). In view of this, UDN studies often restrict themselves to turbulent lengthscales
less than the scale of the streamtube contraction, claiming that larger eddies appear only
as slowly varying mean flow changes (Simonich et al., 1990). The experimental study of
Mish and Devenport (2006b) did indeed confirm that for integral scales which are large
relative to the chord, the distortion of inflow is not as critical. However, we note that
experimental studies do not always measure turbulence lengthscales, and a study by Wil-
son and Thomson (1994) found that large-scale turbulence can cause acoustic fluctuations
which are often neglected.
Hanson and Horan (1998), when considering turbulent flow onto a stator, observed
that the 1/3 Octave Band Power was highly sensitive to L, exhibiting distinct behaviour
in the low or high frequency regimes. They used the Liepmann spectrum as the basis for
their inhomogeneous model (equation (3.35)) and thus for high frequencies, with large
k, the 1 in the denominator is dominated by L2k2, and the spectrum behaves as 1/L.
The noise was thus found to decrease like 1/L at high frequencies. Conversely, for low
frequencies when both terms in the denominator play a role, the noise was found to
increase with L.
Measurements of L
Choosing an appropriate value for L is not straightforward, as it will vary with atmospheric
conditions from place to place and from day to day. In addition, it is important to take
account of the effects of scaling when extrapolating from rig testing to flight data. Wilson
and Thomson (1994) reviewed the large range of experimental measurements which had
been made up to that date, and in Table 3.1 we have summarised their results. We see
that L is found to vary with height off the ground, z, but different regimes are observed
depending on whether buoyancy (due to temperature differences) or inertia (due to wind)
is the dominant turbulence generation mechanism. The ratio of buoyancy to inertia forces
can be quantified by the ‘normalized height’
ζ = z/Lmo, (3.42)
where Lmo is the Monin-Obukhov length (see Wilson and Thomson (1994) equation (3)
for full definition). When |ζ | ≪ 1, shear instabilities (i.e. inertia forces) dominate, and
3.3 Atmospheric turbulence models 111
when |ζ | ≫ 1, buoyancy dominates.
Note that in several instances in Table 3.1 the scales found are significantly larger
than 1 metre, the order of L commonly used by acousticians. Wilson observed that
the short timescale of typical acoustic measurements, of a few minutes, contrasted with
atmospheric measurements which are often taken over several days, and this may lead
to large scale structures being missed in acoustic tests. In addition, the lengthscales in
the two transverse directions, Lu and Lv, are often found to differ, an indication of the
distortion and elongation of eddies which can take place in atmospheric conditions even
before a rotor is introduced.
Correlation lengthscales Height off ground
Study (for velocities separated in measurement taken
the wind direction) (in metres)
dimensional quantities
given in metres
Kader et al, 1989 Lu ≈ 10.3z, Lv ≈ 7.5z (not given)
Johnson et al, 1987 Lu ≈ 0.3z + exp(−0.3z) 1m < z < 33m
Zubkovskii and Fedorov, 1986 Lu ≈ 18m− 26ζ (not given)
Lenschow and Stankov, 1986 Lu = Lv ≈ 0.45z z > 20m (from aircraft)
Daigle et al, 1983 0.5m < Lu < 1.2m 0.15m < z < 1.2m
Mizuno, 1982 20m < Lu, Lv < 40m z = 10m
Kaimal et al, 1972 Lu ≈ 22z, mainly shear (not given)
Lu ≈ 1.3z, mainly buoyancy
Panofsky, 1962 Lu = 34m, Lv = 55m z = 2m
Table 3.1: Lu denotes the lengthscale measure based on velocities in the axial (wind)
direction, Lv is based on velocities in transverse horizontal direction. All measurements
are taken from observation towers except Lenschow and Stankov.
Simonich et al. (1990) also obtained expressions for the turbulent integral lengthscale
and mean square velocity at a variety of heights (50, 120 and 150m), see Table 3.2. They
found L ≈ 0.4z, and u2∞,1 decreases slightly with height. Simonich’s values give us an
indication of the range of appropriate values for u2∞,1 to use when making predictions.
112 3. Alternative inputs: mean flow and turbulence models
Height, z Lu
√
u2∞,1 u
2
∞,1
50m 20m 0.296 m/s 0.0876 m2/s2
122m 48.8m 0.261 m/s 0.0681 m2/s2
152m 61m 0.248 m/s 0.0615 m2/s2
Table 3.2: Atmospheric turbulence parameters given by Simonich et al. (1990) at a range
of heights.
It is possible that if the sub-range of lengthscales which have most significant effect on
the sound are modelled closely then results will be accurate even if the model spectrum
diverges significantly elsewhere from that observed. Thus for UDN modelling, use of an
integral lengthscale of the order of the rotor radius can be justified as we expect these
eddies to contribute most to the noise (Ganz, 1980).
In conclusion, Wilson’s summary of L values indicates it would be informative to
consider some higher values of L. We note that in Figure 2.12 tonal peaks were observed
for large values of L, although the highest noise case was confirmed to be when L ∼ rd.
3.4 Radiated noise results
In this final section of the chapter we plot the radiated sound, as predicted by our model,
for a wide range of inputs. Firstly we consider the different mean flows defined in this
chapter, namely the variable circulation model, the ‘bullet’ simulation model, and the
co-axial twin rotor model. Each of these is compared back to the single actuator disk
model employed in Chapter 2. Secondly we consider different upstream turbulence mod-
els, namely the Liepmann and Gaussian spectra, and compare these to the results ob-
tained with the von Ka´rma´n model. Thirdly we consider a wider range of L values than
previously, including those used by Simonich et al. (1990).
3.4.1 Adjusting the mean flow
Variable circulation
In Figures 3.22 and 3.23 we investigate the use of different profile scalings, as first outlined
in equation (3.6). The first adjustment is made to the values of ∂Xi/∂xj used within Aij
3.4 Radiated noise results 113
and to the quantity R which appears in the phase term of the velocity, as follows
∂Xi
∂xj
∣∣∣∣
(xˆ,rˆ)
=
∂Xvar.i
∂xj
∣∣∣∣
x=xˆ,r= (rˆ−rh)(rd−rh)
rd
, (3.43)
R(x, r) = Rvar.
(
x,
(r − rh)
(rd − rh)
rd
)
. (3.44)
Here Xvar. and Rvar. are the values which would be obtained using the original variable
circulation model, and thus we have scaled the X and R values from the region (0, rd) at
the disk face to lie between rh and rd instead. A secondary adjustment can be made to
R as follows
R(x, r) = rh +Rvar.
(
x,
(r − rh)
(rd − rh)
rd
)
. (3.45)
Here the addition of rh ensures that R is always greater than r, as occurs in a streamtube
contraction. In order to compare like with like, in Figure 3.23 we plot the PWL found in
the constant circulation actuator disk case using exactly the same adjustments.
1200 1400 1600 1800 2000 2200 2400
70
80
90
100
110
120
130
ω
SP
Variable circ.
Variable circ. dXi/dxj and R adjusted 1
Variable circ. dXi/dxj and R adjusted 2
Figure 3.22: Medium distortion level, L = 1, comparing the original variable circulation
mean flow model with two different types of profile scaling. ‘Adjusted 1’ corresponds to
∂Xi/∂xj and R being evaluated at a new radial position as given in equations (3.43) and
(3.44), and ‘adjusted 2’ corresponds to the same values for ∂Xi/∂xj , but R being given
by the expression in equation (3.45).
In Figure 3.22 we see that the change in ∂Xi/∂xj values leads to a lowering of PWL
114 3. Alternative inputs: mean flow and turbulence models
1200 1400 1600 1800 2000 2200 2400
70
80
90
100
110
120
130
ω
SP
Variable circ. dXi/dxj and R adjusted
Constant circ.
Constant circ. dXi/dxj and R adjusted
Figure 3.23: Medium distortion level, L = 1, comparing the variable and constant circu-
lation mean flow models, where profile scaling has been implemented identically.
at all frequencies. Interestingly, when the value of R is further adjusted the broadband
level is unaffected, but the tonal peaks are. We conclude that the broadband level is most
affected by the detailed nature of the distortion, as parametrised by ∂Xi/∂xj , whereas
the peaks are primarily affected by the overall magnitude of the streamtube contraction,
as given by R. In Figure 3.23 we see that the difference between the constant and variable
circulation cases is in fact increased when both models use the ‘scaled’ profiles, with a
difference of more than 5 dB seen in the peak tonal level.
Figure 3.24 compares the (scaled) variable and (non-scaled) constant circulation cases
at several levels of distortion. We see a shift in broadband level in each case when the
variable model is used, but the direction of the shift depends on the level of distortion.
For high distortion, the broadband level is lowered in the variable case whereas for low
distortion, it is raised. As the variable flow model is more realistic, UDN models which use
a constant circulation actuator disk to represent the streamtube contraction may under-
or over-estimate broadband levels due to this.
In Figure 3.25 we again compare the variable and constant circulation cases, but now
varying the integral lengthscale parameter. The qualitative change in all three cases is
the same as L is increased, with the variable circulation model resulting in a lowered
broadband PWL at all frequencies, but a raised tonal peak around 1 BPF, and a lowered
peak around 2 BPF.
3.4 Radiated noise results 115
1200 1400 1600 1800 2000 2200 2400
40
50
60
70
80
90
100
110
120
130
140
ω
SP
Variable circ. high distort.
Variable circ. medium distort.
Variable circ. low distort.
Constant circ.
Figure 3.24: Variable vs. constant circulation models, varying the distortion level. The
actuator disk strength, Ud, is kept constant between the two models.
1200 1400 1600 1800 2000 2200 2400
60
70
80
90
100
110
120
130
ω
SP
Variable circ. L = 0.05 rd
Variable circ. L = 0.5 rd
Variable circ. L = 5 rd
Constant circ.
Figure 3.25: Variable vs. constant circulation, varying the integral lengthscale L. The
actuator disk strength, Ud, is kept constant between the two models.
116 3. Alternative inputs: mean flow and turbulence models
Bullet and co-axial cases
Figure 3.26 shows the effect on the radiated sound of adding a point source to the upstream
flow, simulating the bullet. The broadband level is consistently reduced for all distortion
levels by 1-3 dB, however the peak tonal level remains unchanged. We note that it is
not completely straightforward to predict the final sound spectrum from the distorted
turbulence spectrum at the rotor face, as plotted for example in Figure 3.11. This is
because the changes in Sxx tend to differ at different radial positions along the blade,
each of which contributes a component to the far-field pressure.
In Figure 3.27 we have plotted the radiated sound comparing the co-axial and single
disk cases at different distortion levels. The difference seen is minimal, as is the case when
different L values are compared.
1200 1400 1600 1800 2000 2200 2400
40
50
60
70
80
90
100
110
120
130
140
150
ω
SP
Bullet, high distortion
Bullet, medium distortion
Bullet, low distortion
Figure 3.26: Bullet (blue) vs. no bullet (red) case, for a variety of distortion levels.
3.4.2 Adjusting the turbulence spectrum
In order to implement a different upstream turbulence model in the code for calculating
the radiated sound expression (as given in equation 2.133) we must adjust the integral of
S∞jk over φk, which was given in equations (2.91) - (2.93) for the von Ka´rma´n spectrum.
3.4 Radiated noise results 117
1200 1400 1600 1800 2000 2200 2400
40
50
60
70
80
90
100
110
120
130
140
150
ω
SP
Co−axial, high distortion
Co−axial, medium distortion
Co−axial, low distortion
Figure 3.27: Comparing the co-axial (blue) and single disk (red) cases, for a variety of
distortion levels.
We find
∫ 2pi
0
S∞xx (k
m
x , ky, kz) dφk = 2pir
2
kG, (3.46)
∫ 2pi
0
S∞yy (k
m
x , ky, kz) dφk =
∫ 2pi
0
S∞zz (k
m
x , ky, kz) dφk =
(
2pik2 − pir2k
)
G, (3.47)
where, for the Liepmann spectrum, G =
2L5u2∞,1
pi2 (1 + L2k2)3
, (3.48)
and, for the Gaussian spectrum, G =
l5Gu
2
∞,1
32pi
3
2
exp
(
−k
2l2G
4
)
. (3.49)
In both cases we still have
∫ 2pi
0
S∞jk (k
m
x , ky, kz) dφk = 0 if j 6= k, (3.50)
as these terms involve only the integral of sin φk, cosφk or sinφk cosφk over a 2pi period.
We compare the Liepmann to von Ka´rma´n models in Figure 3.28. Batchelor and
Proudman (1954) found that, in the uniform distortion case, the turbulence distortion
becomes independent of the precise form of the homogeneous upstream turbulence as the
distortion level is increased. However, in this non-uniform case, we see that in fact the
broadband PWL is lowered by 5-10 dB at all levels of distortion when the Liepmann
118 3. Alternative inputs: mean flow and turbulence models
spectrum is used in place of the von Ka´rma´n, although the tonal peak heights remaining
unchanged.
1200 1400 1600 1800 2000 2200 2400
40
50
60
70
80
90
100
110
120
130
140
ω
SP
Liepmann
von Karman
Figure 3.28: Comparing the von Ka´rma´n and Liepmann models for the upstream turbu-
lence, for varying distortion level. Highly peaked = high distortion; somewhat peaked =
medium distortion; broadband = low distortion.
In Figure 3.29 we compare the Gaussian and von Ka´rma´n models. As predicted, the
Gaussian model produces a less realistic spectrum, which is very highly peaked around
multiples of BPF for L = 0.5rd. For larger values of L the dB level found is extremely
low (off the scale plotted here), due to the rapid decay of the Gaussian spectrum as L
increases that was seen in Figure 3.19.
Varying the integral lengthscale
Lastly, we have input a wider range of L values to include the higher values which are
observed in the real atmosphere. In Figure 3.30 we can confirm that the highest tonal
levels are seen when L = rd, however even when L≫ rd the tonal peaks remain, whereas
for L≪ rd we find a broadband spectrum with no tones. In Figure 3.31 we see a decrease
of 4-5 dB in UDN level as the open rotor travels from 50m to 122m height, primarily due
to the change in atmospheric turbulence intensity. The difference between 122m and 152m
height is less pronounced. The high distortion cases are given for comparison, as would
be seen if realistic atmospheric conditions were successfully simulated in static testing.
3.4 Radiated noise results 119
1200 1400 1600 1800 2000 2200 2400
50
60
70
80
90
100
110
120
130
140
ω
SP
Gaussian, L = 0.05 rd
Gaussian, L = 0.5 rd
von Karman, L = 0.05 rd
von Karman, L = 0.5 rd
von Karman, L = 5 rd
Figure 3.29: Here we compare the von Ka´rma´n and Gaussian models for the upstream
turbulence for a variety of integral lengthscales, all at the same level of distortion
1200 1400 1600 1800 2000 2200 2400
60
70
80
90
100
110
120
130
ω
SP
L = 0.005 rd
L = rd
L = 2.5 rd
L = 50 rd
Figure 3.30: Comparing several L values for the medium distortion case (Uf/U∞ = 10).
A single actuator disk model is used for the mean flow.
120 3. Alternative inputs: mean flow and turbulence models
1200 1400 1600 1800 2000 2200 2400
20
40
60
80
100
120
140
ω
SP
L = 20m, u2 = 0.0876 m2/s2
L = 48.8m, u2 = 0.0681 m2/s2
L = 61m, u2 = 0.0615 m2/s2
Figure 3.31: Implementing the values of L and u2∞,1 as used by Simonich to simulate
atmospheric conditions at 50m (L = 20m), 122m (L = 48.8m) and 152m (L = 61m).
Chapter 4
Generalisation to asymmetric rotor
systems
4.1 Chapter outline
In this chapter, we rework the analysis of Chapter 2 for non-axisymmetric mean flows.
This involves changes to the Fourier Transform of the distortion amplitude, denoted in
Chapter 2 by Cpij. We test the model by inputting a simple mean flow which is asymmetric
at the rotor face, given by two actuator disks side by side. The turbulence spectrum at
the rotor face now varies with azimuthal angle, φ, as values of ∂Xi/∂xj depend upon φ.
This then leads to changes in the φ dependence of the force term in the wave equation.
Both the sound from one rotor with an asymmetric inflow, and that which arises from
the two rotor system are calculated.
The extensions made in this chapter allow any irrotational mean flow, which tends to
a uniform velocity far upstream, to be incorporated within our UDN model.
Inputs Outputs
Mean flow model, U:
two actuator disks side
by side (equivalent to a
rotor next to an∞ wall)
Upstream turbulence
model, u∞:
von Ka´rma´n spectrum
Turbulence energy spectra
at rotor face: now plotted
at several φ positions
Far-field pressure: Pˆ
leading to spectral power
Directivity patterns from
two adjacent rotors
Distortion of upstream
turbulence, u∞, by
mean flow, U, using
Rapid Distortion Theory.
Analysis reworked for
asymmetric mean flow,
as ∂Xi/∂xj now depends
upon φ.
121
122 4. Generalisation to asymmetric rotor systems
4.2 Effects of asymmetry
The mean flow of air into a fully installed open rotor on an aircraft will vary with φ
due to the presence of the fuselage and airframe features, such as the pylon each engine
is mounted on. Such ‘installation effects’ are a current priority in AOR noise testing
programmes (Peake and Parry, in press, 2012) and are receiving much attention from
engine manufacturers, for example within the current Airbus/Rolls-Royce programme of
wind tunnel testing discussed below, and the CFD programmes at ONERA and Airbus
mentioned in §1.3.4. With the positioning of the engines still an open question both
the pusher and puller configurations are being investigated; each of these presents its
own challenges. In the pusher configuration, the rotor is mounted behind the pylon and
therefore interacts with the (asymmetric) pylon wake. The puller configuration entails
higher blade angle of attack (due to the flow induced by the nearby wing) and we discuss
non-zero incidence in detail in Chapter 5.
In Chapter 3 we discussed the modelling of turbulent structures shed by installation
features, and found that for large-scale turbulent structures (i.e. higher L values), the
broadband UDN level was significantly affected by the precise form of the upstream tur-
bulence. However, as in many of the studies reviewed in Chapter 1, we dealt solely with
the axisymmetric situation. Such analyses neglect a key aspect of real installation effects,
namely asymmetry. Cargill (1993) noted the need to consider whether an axisymmetric
inflow is good enough to model real intakes, but also highlighted the increased complexity
of the analysis in the asymmetric case.
4.2.1 Previous work on asymmetry
Recent joint Airbus/Rolls-Royce wind tunnel testing of Rig 145 has included both rotor
alone testing, and testing with mock-ups of the pylon in place, as shown in Figure 4.1 (see
also the discussion of pylon blowing in §1.2.2, Figure 1.7). This testing has confirmed the
asymmetry of pylon-rotor interaction, with azimuthal directivity being affected (Ricouard
et al., 2010). Pylon-rotor interaction noise was found to be less sensitive to operating
conditions than rotor alone tones, and dominated the isolated rotor noise. The exact
design of the pylon did not affect results significantly. The presence of the pylon was also
found to affect the front rotor tones more than the rear rotor or interaction tones, and in
this chapter we consider the noise from the front rotor only.
Further experimental observations of asymmetry include work by Carolus and Stremel
4.2 Effects of asymmetry 123
Figure 4.1: Rig 145 testing between 2008-2010 included measurements with the pylon in
place. Image reproduced with permission from Rolls-Royce. The pylon of an open rotor
must be substantial in order to withstand blade-off failure testing.
(2002), who conducted experiments measuring the pressure fluctuations on rotating fan
blades due to ingested grid generated turbulence. Interestingly, they noted unexpected
peaks at the Rotational Shaft Frequency (RSF) which they believed to be due to a slight
asymmetry of the mean flow, which they could not remove. More recently, Koch (2009)
measured the SPL generated by a rotor placed downstream of an asymmetrical array of
cylindrical rods, as a model for fan inflow distortion noise. Extra peaks were observed
between BPF tones, which were somewhat ‘haystacked’. The BPF tonal level was also
increased in some cases, see Figure 4.2.
As described in §1.3.3, the extension of analytic models to include three-dimensional
effects is of interest generally, to more realistically capture true flow characteristics. Si-
monich et al. (1990) and Amiet et al. (1990) looked at the effect of the direction of
distortion (not just its strength) on UDN levels, and noted that the eddy-blade intersec-
tion area would be affected by asymmetries in the mean flow. In their model, turbulence
stretching along the rotor axis was found to decrease noise, compared to stretching which
occurred at an angle to the axis. The work of both Simonich et al. and Paterson and
Amiet (1982) primarily dealt with helicopter rotors, hence their interest in asymmetric
inflows. Paterson and Amiet noted that non-axial mean flows tend to smooth harmonic
peaks and shift them away from BPFs. They highlighted three different asymmetric ef-
fects. Firstly, radial wandering of an ingested eddy with respect to the blade leads to
the amplitude of tones varying in time, as the relative velocity between eddy and blade
will vary with r. Secondly, random azimuthal wandering may lead to varying frequency
124 4. Generalisation to asymmetric rotor systems
Figure 4.2: Image reproduced from Koch (2009). Plotted here is the spectrum radiated
from a fan placed in an inflow which has been distorted by an upstream array of cylindrical
rods. The top figure is for the case of an asymmetric array, the bottom is the baseline
plot with no rods present.
of tones, away from BPF. Thirdly, steady asymmetry of the mean flow will lead to non-
random azimuthal and radial wandering, potentially shifting the frequency of tones, but
this effect will be steady in time.
4.3 Asymmetric turbulence distortion
In this chapter we aim to investigate the general effects of asymmetry upon UDN. We
do not try to recreate all aspects of a realistic AOR inflow-field, for example we do not
try to approximate the flow field around the pylon. However, any mean flow (including
a numerically defined flow) could now be substituted into our framework. This is a
substantial extension of Majumdar’s work where only flows which had a streamfunction
were considered, thus limiting attention to two-dimensional or axisymmetric flows.
4.3.1 A simple asymmetric mean flow
We consider in this chapter a simple asymmetric system where the mean flow is readily
calculated. The set-up is shown in Figure 4.3, and consists of two actuator disks which
lie in the y − z plane, separated by a distance d in the direction perpendicular to their
axes.
The mean flow we use is therefore given by the sum of two actuator disk flows, centred
4.3 Asymmetric turbulence distortion 125
y
x
z
y = d
φ
φˆrˆ
r Actuator disk 1
Ud1, rd1
Actuator disk 2
Ud2, rd2
Figure 4.3: Illustration of our coordinate system. These two actuator disks side by side
give rise to an asymmetric flow.
at (0, 0, 0) and (0, d, 0). Note that the sign of Ud = BΩΓ/2piU∞ = T/pir2dρ0U∞, the
‘strength’ of each actuator disk, does not indicate the direction of rotation but rather the
direction of thrust generation1.
The above set-up can be used to model two adjacent rotors of different strengths
and/or sizes, and below we give the expression for the mean flow induced by two actuator
disks with general Ud1, Ud2, rd1, rd2. Examples of aircraft with four engines, two beneath
each wing in close proximity to each other, include the Airbus A340 and A400, and the
military Avro Shackleton, see Figure 4.4. Pusher configurations of AOR, with fuselage-
mounted rather than wing-mounted rotors also have two adjacent rotors.
If the properties of both actuator disks are the same (Ud1 = Ud2 and rd1 = rd2) then
by symmetry we have zero velocity in the eˆy direction across the surface y = d/2. This
set-up is thus equivalent to a system in which a single rotor at the origin is placed next
to an infinite wall running through the plane y = d/2, on which a condition of zero
normal velocity is satisfied. This flow can therefore be used to approximate the presence
of the fuselage alongside a single open rotor. Note that we will not be considering the
effect of sound scattering off the fuselage. This has been treated recently by Kingan and
McAlpine (2010), who approximated the fuselage by an infinite cylinder, leading to an
adjusted Green’s function.
1The direction of rotation has no bearing on the velocities given in an actuator disk model, which is
steady in time and can be thought of as the limit of an infinitely bladed propeller.
126 4. Generalisation to asymmetric rotor systems
(a) (b) (c)
Figure 4.4: a) The Airbus A400m. b) The counter-rotating propellers of the Avro Shack-
leton. Photo by NJR ZA, licenced under Creative Commons Licence. c) Example of the
rear mounted AOR concept being developed by Airbus.
For the axisymmetric, single actuator disk system we considered in Chapter 2, cylindri-
cal polar coordinates were most appropriate, with mean flow velocities given in equations
(2.47) and (2.48). When considering an asymmetric set-up it is more convenient to switch
to cartesian coordinates. For the first disk, centred at (0, 0, 0), we re-write r dependence
within the velocity expressions in terms of y and z (using r2 = y2 + z2), and there is no φ
dependence. For the second rotor, we replace r with rˆ, where rˆ2 = (y − d)2 + z2. We also
decompose the radial velocities, Ur, induced by the two disks into velocity components in
the y and z directions. The azimuthal velocity, Uφ, is zero upstream of the disks. Thus
U · eˆy ≡ Uy = Ur1 cosφ+ Ur2 cos φˆ =
Ur1y
(y2 + z2)
1
2
+
Ur2 (y − d)
[
(y − d)2 + z2
] 1
2
, (4.1)
U · eˆz ≡ Uz = Ur1 sin φ+ Ur2 sin φˆ =
Ur1z
(y2 + z2)
1
2
+
Ur2z
[
(y − d)2 + z2
] 1
2
, (4.2)
where tanφ = z/y and tan φˆ = z/(y − d). We therefore find the velocities in the region
x < 0 are given by
Ux =
Ud1x
2pi (y2 + z2)
3
4
∫ rd1
0
1√
r′
Q′− 12
[ω (y, r′)] dr′
+
Ud2x
2pi
[
(y − d)2 + z2
] 3
4
∫ rd2
0
1√
r′
Q′− 12
[ω (y − d, r′)] dr′, (4.3)
4.3 Asymmetric turbulence distortion 127
Uy =−
Ud1
√
rd1yQ 1
2
[ω (y, rd1)]
2pi (y2 + z2)
3
4
−
Ud2
√
rd2 (y − d)Q 1
2
[ω (y − d, rd2)]
2pi
(
[y − d]2 + z2
) 3
4
, (4.4)
Uz =−
Ud1
√
rd1zQ 1
2
[ω (y, rd1)]
2pi (y2 + z2)
3
4
−
Ud2
√
rd2zQ 1
2
[ω (y − d, rd2)]
2pi
(
[y − d]2 + z2
) 3
4
. (4.5)
The argument, ω, of the Legendre functions Qn and their derivatives is defined as
ω (y′, r′) = 1 +
x2 +
[
(y′2 + z2)
1
2 − r′
]2
2 (y′2 + z2)
1
2 r′
. (4.6)
Note that ω is even in its y′ argument. Thus by inspection of equation (4.4) we see that
for y = d/2 we have Uy = 0, i.e. the no penetration condition on the line of symmetry is
satisfied.
By following a particular streamline far upstream, we can use these velocities to find
the unique y and z position at which it originated. See Figure 4.5 for an example of
the streamlines generated by this velocity field. Figure 4.6 compares the asymmetric
streamlines to the axisymmetric case. We can compute quantities of interest for different
values of d, the distance between the disks, and thus quantify the asymmetric effect.
−20 −18 −16 −14 −12 −10 −8 −6 −4 −2 0 2
−20
−15
−10
−5
0
5
10
15
20
x, axial direction
y = d/2
y
−20
−10
0
−10
−5 0 5 10
−10
−5
0
5
10
x, axial direction
y
z
Figure 4.5: Streamlines for the asymmetric, two-disk system (with Ud1 = Ud2 = 198U∞
and rd1 = rd2), side-on and head-on views. The disk positions are indicated in black. The
surface of zero normal velocity is indicated by a dashed line.
128 4. Generalisation to asymmetric rotor systems
−20 −18 −16 −14 −12 −10 −8 −6 −4 −2 0
−10
−8
−6
−4
−2
0
2
4
6
8
10
x, axial direction
y
Asymmetric
Axisymmetric
Figure 4.6: Comparing the asymmetric streamlines to the axisymmetric case. The values
of Ud and rd are kept constant between the two. The infinite wall is located at y = 2.5,
above the asymmetric streamlines shown.
4.3.2 Turbulence at the rotor face
In Chapter 2 we found the distorted turbulence spectrum was given by the simple expres-
sion
Sij (x, ω) =
2pi
U∞
∫
ℜ2
Aik (x,k)Ajl (x,k)S∞kl (k) dkydkz, (4.7)
where, in the above expression, kx = ω/U∞, i.e. only one kx component contributes for
each particular frequency ω. When calculating this numerically, it is more successful to
make the substitutions
ky = rk cos φk, kz = rk sinφk, (4.8)
reducing the two infinite integrals to one infinite and one finite integral as follows
Sij (x, ω) =
2pi
U∞
∫ ∞
0
∫ 2pi
0
Aik (x,k)Ajl (x,k)S∞kl (k) rkdφkdrk. (4.9)
The main point to note is that for an asymmetric mean flow the φ dependence within
Aij is not removed when the k integrals are performed, as it was for the axisymmetric
situation where φ dependence only came in through kr and kφ (as seen in equations (2.42)
and (2.43)). Thus, when we consider the various Sij spectra for an asymmetric mean flow,
their levels will be different at different azimuthal positions around the rotor.
In Figures 4.7 and 4.8 we have plotted the axial component of the distorted turbulence
4.3 Asymmetric turbulence distortion 129
spectrum, Sxx, at a range of azimuthal positions around the disk at the origin. We see, in
Figures 4.7 and 4.8, that the spectrum increases as we move round from φ = 0, nearest
the second rotor where less streamtube contraction occurs, to φ = pi, where the greatest
level of contraction occurs, see Figure 4.6. In Figures 4.9 and 4.10 we show the spectra for
a range of distortion levels and radial positions, holding the azimuthal position constant.
The spectrum level near the blade tip is found to be higher than that near the hub
in all cases. However, the dependence of spectrum level on distortion level is not so
straightforward, as the low distortion case lies between the high and medium distortion
cases, in contrast to the axisymmetric case. Finally, in Figure 4.11, we have varied the
separation between rotors, d. As expected, as d increases, and the mean flow approaches
the axisymmetric mean flow, the spectra at different φ positions get closer together.
−2 −1 0 1 2 3 4
−8
−6
−4
−2
0
2
4
6
log(k
x
) = log(ω/U
∞
)
lo
g(S
xx
)
φ = 0
φ = pi/6
φ = pi/3
φ = pi/2
φ = 2pi/3
φ = 5pi/6
φ = pi
undistorted
(a)
plane of symmetry
z
y
(b)
Figure 4.7: a) Here we have calculated Sxx in the high distortion case (Uf/U∞ = 100)
near the rotor tip (r/rd = 0.9), at a range of azimuthal positions. We see the energy level
is higher than the undistorted spectrum in all cases. As we move from φ = 0 (nearest
the other rotor) to φ = pi in equal increments of φ, the levels get closer together. b) This
subfigure shows the positions at which Sxx has been calculated.
130 4. Generalisation to asymmetric rotor systems
−2 −1 0 1 2 3 4
−12
−10
−8
−6
−4
−2
0
log(k
x
) = log(ω/U
∞
)
lo
g(S
xx
)
φ = 0
φ = pi/6
φ = pi/3
φ = pi/2
φ = 2pi/3
φ = 5pi/6
φ = pi
undistorted
Figure 4.8: Here we show a medium distortion case (Uf/U∞ = 10) near the rotor hub
(r/rd = 0.4). At this position the energy level is below that of the undistorted spectrum
in all cases. We also see a peak in the φ = 0 spectrum not present in the other spectra.
−2 −1 0 1 2 3 4
−10
−8
−6
−4
−2
0
2
4
log(k
x
) = log(ω/U
∞
)
lo
g(U
∞
S x
x)
High distortion, near tip
High distortion, near hub
Medium distortion, near tip
Medium distortion, near hub
Low Distortion, all r values
Undistorted
Figure 4.9: Here we show the spectrum at φ = 0 for a range of distortion levels and radial
positions. We have collapsed the data by multiplying Sxx by U∞. By normalising in this
way the undistorted case (which corresponds to ∂Xi/∂xj = δij) is the same for each of
the three cases (high, medium and low distortion levels). The level near the tip is higher
than that near the hub in all cases, but dependence on distortion level is more complex,
with the low distortion case in between the high and medium distortion cases.
4.3 Asymmetric turbulence distortion 131
−2 −1 0 1 2 3 4
−8
−6
−4
−2
0
2
4
6
log(k
x
) = log(ω/U
∞
)
lo
g(S
xx
)
High distortion, near tip
High distortion, near hub
Medium distortion, near tip
Medium distortion, near hub
Low Distortion, all r values
Undistorted
Figure 4.10: Plot as in Figure 4.9 above, but at φ = pi. There is less variation in spectrum
level between the hub and tip than at φ = 0.
−2 −1 0 1 2 3 4
−12
−11
−10
−9
−8
−7
−6
−5
−4
−3
−2
log(k
x
) = log(ω/U
∞
)
lo
g(S
xx
)
d = 2.5 rd, φ = 0
d = 2.5 rd, φ = pi/3
d = 2.5 rd, φ = 2pi/3
d = 2.5 rd, φ = pi
d = 5 rd, φ = 0
d = 5 rd, φ = pi/3
d = 5 rd, φ = 2pi/3
d = 5 rd, φ = pi
Figure 4.11: Here we have adjusted d, the level of asymmetry in the mean flow. As
expected, the more highly asymmetric flow (with the smaller value of d) leads to larger
differences between the spectrum at different azimuthal positions. As d increases the
φ = 0 and φ = pi spectra move towards each other.
132 4. Generalisation to asymmetric rotor systems
4.4 Blade pressures
In order to calculate NiAij (to substitute into the expression for the normal velocity at the
leading edge, for use in LINSUB) we require expressions for Axx, Axy, Axz and the ‘mixed
suffices’ quantities Aφx, Aφy, Aφz. For this we would like to calculate the most general
form for the components of Aij in the polar directions, in terms of the polar expressions lr,
lφ, ∂/∂r, ∂/∂φ, R, Φ,and φ. In the next subsection we derive these quantities, before going
on to calculate the normal velocity at the leading edge. We can then obtain expressions for
the blade pressure jumps, spectrum of blade pressures, and the forcing term to substitute
into the wave equation, all within the asymmetric framework.
4.4.1 The general form of Aij
In Chapter 2, an approximation to the distorted amplitude was given
Aij =
(
δim −
lilm
|l|2
)
∂Xj
∂xm
, (4.10)
(summation convention assumed). This quantity is straightforwardly interpreted in the
cartesian reference frame, but care must be taken when considering the components with
respect to polar coordinates. The most general form for l is
l = ki∇Xi
=
{
eˆx
∂
∂x
+ eˆy
∂
∂y
+ eˆz
∂
∂z
}
[kxX + kyY + kzZ] , (4.11)
where k is a constant wavevector. Thus lr and lφ are given in terms of the polar quantities
X,R,Φ as follows
lr = l · eˆr = kx
∂X
∂r
+ kφ
(
− sin φ∂(R cosΦ)
∂r
+ cosφ
∂(R sin Φ)
∂r
)
+ kr
(
cosφ
∂(R cosΦ)
∂r
+ sin φ
∂(R sinΦ)
∂r
)
, (4.12)
lφ = l · eˆφ = kx
1
r
∂X
∂φ
+ kφ
1
r
(
− sin φ∂(R cosΦ)
∂φ
+ cos φ
∂(R sinΦ)
∂φ
)
+ kr
1
r
(
cosφ
∂(R cos Φ)
∂φ
+ sinφ
∂(R sinΦ)
∂φ
)
. (4.13)
4.4 Blade pressures 133
To calculate the mixed suffices components of Aij we first write out the ‘polar’ com-
ponents in terms of the cartesian components
eˆx ·A · eˆx = Axx,
eˆx ·A · eˆr = cosφAxy + sinφAxz,
eˆx ·A · eˆφ = − sin φAxy + cosφAxz,
eˆr ·A · eˆx = cosφAyx + sinφAzx,
eˆr ·A · eˆr = cos2 φAyy + cosφ sinφ (Ayz + Azy) + sin2 φAzz,
eˆr ·A · eˆφ = − cos φ sinφAyy + cos2 φAyz − sin2 φAzy + sin φ cosφAzz,
eˆφ ·A · eˆx = − sin φAyx + cosφAzx,
eˆφ ·A · eˆr = − sin φ cosφAyy − sin2 φAyz + cos2 φAzy + sin φ cosφAzz,
eˆφ ·A · eˆφ = sin2 φAyy − sin φ cosφ (Ayz + Azy) + cos2 φAzz. (4.14)
We then rewrite each cartesian component of Aij using the following relations
ly = cos φlr − sin φlφ, (4.15)
lz = sin φlr + cosφlφ, (4.16)
∂
∂y
= −sin φ
r
∂
∂φ
+ cosφ
∂
∂r
, (4.17)
∂
∂z
=
cos φ
r
∂
∂φ
+ sinφ
∂
∂r
, (4.18)
Y = R cosΦ, (4.19)
Z = R sin Φ. (4.20)
For example
Axy =
(
1− lxlx|l|2
)
∂Y
∂x
− lxly|l|2
∂Y
∂y
− lxlz|l|2
∂Y
∂z
=
(
1− lxlx|l|2
)
∂ (R cosΦ)
∂x
− lxlr|l|2
∂ (R cos Φ)
∂r
− lxlφ|l|2
1
r
∂ (R cosΦ)
∂φ
. (4.21)
134 4. Generalisation to asymmetric rotor systems
Finally, substituting the expressions for the cartesian Aij components, in terms of the
polar quantities lr, lφ, R,Φ, into the expressions of (4.14), we find
Aix =
(
δix −
lilx
|l|2
)
∂X
∂x
+
(
δir −
lilr
|l|2
)
∂X
∂r
+
(
δiφ −
lilφ
|l|2
)
1
r
∂X
∂φ
,
Air =
(
δix −
lilx
|l|2
)
∂Xr
∂x
+
(
δir −
lilr
|l|2
)
∂Xr
∂r
+
(
δiφ −
lilφ
|l|2
)[
cosφ
r
∂ (R cosΦ)
∂φ
+
sin φ
r
∂ (R sinΦ)
∂φ
]
,
Aiφ =
(
δix −
lilx
|l|2
)
∂Xφ
∂x
+
(
δir −
lilr
|l|2
)
∂Xφ
∂r
+
(
δiφ −
lilφ
|l|2
)[
−sin φ
r
∂ (R cos Φ)
∂φ
+
cosφ
r
∂ (R sinΦ)
∂φ
]
, (4.22)
where Xr = R cos(Φ− φ), and Xφ = R sin(Φ− φ). In these expressions the suffix i runs
over x, r and φ, and δxx = δrr = δφφ = 1, with all other values of δij being zero. We
note that, for example, Aφr 6= (δφm − lφlm/|l|2) (∂Xr/∂xm), as might have been expected.
Essentially, the derivatives within Aij do not act upon φ within the expressions for Xr
and Xφ.
In the axisymmetric case, we have Φ = φ, Xr = R, Xφ = 0 and ∂X/∂φ = ∂R/∂φ = 0,
and thus we regain equations (2.37).
4.4.2 Input into LINSUB
To implement LINSUB we require the normal velocity at the leading edge of a blade.
That is, we require
1
2pi3
∫
ℜ3
uˆ∞j (k) e
−ikxU∞t [N ·A]j exp {i [kxX + kyY + kzZ]} d3k, (4.23)
re-written in a reference frame which rotates with the blades and decomposed into eimφ
′
harmonics, where φ′ = φ− Ωt. Thus we wish to rewrite the above in the form
∞∑
m=−∞
wW (x, r) eiωΓt+imφ
′
, (4.24)
which will allow us to pick out the appropriate frequency ωΓ.
4.4 Blade pressures 135
Using the fact that Y = R cosΦ, Z = R sin Φ, and the relation
exp {i (ky cosΦ + kz sinΦ)R} =
∞∑
n=−∞
Jn
(√
k2y + k2zR
)
exp
{
in
[
Φ− tan−1
(
kz
ky
)]}
,
(4.25)
we have
[N ·A]j exp {i [kxX + kyY + kzZ]} = exp (ikxX (x, r, φ)) [sin βAxj + cos βAφj]
∞∑
n=−∞
Jn (rkR (x, r, φ)) exp {in (Φ (x, r, φ)− φk)} ,
(4.26)
where rk =
√
k2y + k2z and φk = tan
−1 (kz/ky), as in Figure 2.1. Note that X,R, and Φ
depend upon φ in this asymmetric case.
Defining new quantities C˜m,nij and D˜
m,n
j
Next we need to bring out the φ dependence of equation (4.26) explicitly in exponential
form. Previously we did this by expanding Aij as a Fourier Series, but now that we
have extra φ dependence within the eikxX , Jn(rkR) and einΦ terms we must define a new
quantity C˜m,nij (note the extra superscript in comparison to the corresponding quantities
in Chapter 2) as follows
Aij (x, r;k) eikxXJn(rkR)einΦ =
∞∑
m=−∞
C˜m,nij (x, r;k) e
imφ, (4.27)
where C˜m,nij (x, r;k) =
1
2pi
∫ 2pi
0
Aij (x, r;k) eikxXJn(rkR)einΦe−imφdφ.
The relation
C˜m,nij (x, r; kx, rk, φk) = e
−imφkC˜m,nij (x, r; kx, rk, 0) , (4.28)
still holds, since φk does not appear in the extra terms within the C˜
m,n
ij integrand.
136 4. Generalisation to asymmetric rotor systems
As in §2.3.2, we next define D˜m,nij as follows
D˜m,nx ≡
[
sin βC˜m,nxx + cosβC˜
m,n
φx
]
,
D˜m,ny ≡
[
sin β
1
2
{(
C˜m−1,nxr + iC˜
m−1,n
xφ
)
+
(
C˜m+1,nxr − iC˜m+1,nxφ
)}
+ cosβ
1
2
{(
C˜m−1,nφr + iC˜
m−1,n
φφ
)
+
(
C˜m+1,nφr − iC˜
m+1,n
φφ
)}]
,
D˜m,nz ≡
[
sin β
1
2
{(
−iC˜m−1,nxr + C˜m−1,nxφ
)
+
(
iC˜m+1,nxr + C˜
m+1,n
xφ
)}
+ cosβ
1
2
{(
−iC˜m−1,nφr + C˜
m−1,n
φφ
)
+
(
iC˜m+1,nφr + C˜
m+1,n
φφ
)}]
. (4.29)
Here we have used the relation
D˜m,nj (x, r; kx, rk, φk) = e
−imφkD˜m,nj (x, r; kx, rk, 0) . (4.30)
Normal velocity at the blade leading edge
We can now obtain our leading edge velocity. For each wavevector k we have
∞∑
m=−∞
wWeiωΓt+imφ
′
=
∞∑
m=−∞
{ ∞∑
n=−∞
D˜m,nj (x0(r), r; kx, rk, 0) exp [−i(m+ n)φk]
}
uˆ∞j (k) exp [i (mΩ− kxU∞) t] exp [imφ′] . (4.31)
We note the changes to this expression from the axisymmetric case, as given in equation
(2.85). Firstly, for fixed frequency and azimuthal order (given by (mΩ− kxU∞) and m
respectively), the φk dependence now contains an extra factor n, which previously exactly
balanced the exp(inΦ) term. Secondly, we pick out (via D˜m,nj ) the e
imφ component of
AijJneinΦeikxX , rather than the eipφ component of Aij alone.
4.4.3 Output from LINSUB
As in Chapter 2, we use Smith’s LINSUB method to calculate the blade pressure jumps
induced by a sinusoidal disturbance impinging upon a cascade of blades. As we wish
to substitute the forcing term obtained into the RHS of the wave equation (2.107), we
calculate both the pressure jump across a single blade and the force found after summing
over all blades.
4.4 Blade pressures 137
Pressure jump
Substituting (4.31) into equation (2.71), the pressure jump across a blade which passes
through the position x, r, φ at time t, due to an upstream gust of the form uˆ∞(k)ei(k·x−kxU∞t),
is now given by
∆p (x, r, φ, t) =ρ0W
∞∑
m=−∞
ΓmWW
(
zˆ =
x− x0 (r)
cosβ(r)
, r;ωΓ = kxU∞ −mΩ, χ = −
2mpi
B
)
{ ∞∑
n=−∞
D˜m,nj (x0(r), r; kx, rk, 0) e
−i(m+n)φk
}
uˆ∞j (k) e
−ikxU∞teimφ. (4.32)
Again, we note the differences from the axisymmetric expression of equation (2.86): the
Bessel function has been absorbed into the quantity D˜m,nj , and the order of the φk exponent
has been adjusted by n.
Spectrum of blade pressures
For completeness we give the expression for the blade pressure spectrum for this asym-
metric system. The equivalent to equation (2.90) is
Pˆblade(x, r, ω) =
2piρ20W
2
U∞
∞∑
m=−∞
|ΓmWW (x, r, ω)|2
∫ ∞
0
{ ∞∑
n=∞
D˜m,nj (x0(r), r; kx, rk, 0)
}∗{ ∞∑
n′=∞
D˜m,n
′
k (x0(r), r; kx, rk, 0)
}
[∫ 2pi
0
S∞jk (k
m
x , ky, kz) e
i(n−n′)φkdφk
]
rkdrk, (4.33)
where kmx = (ω +mΩ)/U∞. The main difference here is the appearance of a new expo-
nential term within the integral over φk.
138 4. Generalisation to asymmetric rotor systems
Forcing term
To obtain the force exerted by the rotor on the fluid, f , we must sum the blade pressure
jumps from equation (4.32) over all blades, exactly as in §2.3.5. This involves defining
the blades’ surfaces through use of a delta function, and then re-expressing that delta
function as a sum of exponentials. The sum over blade number b, from 1 to B, then
means certain combinations of terms sum to zero, and we can sum instead over the new
index l, with φ dependence given by m+ lB. We find
fi (x, r, φ, t) =ρ0WNi(r)
B
2pir
∞∑
m=−∞
1
(2pi)3
∫
ℜ3
ΓmWW (x, r; kx, χ
m) exp (−ikxU∞t)
∞∑
n=−∞
D˜m,nj (x0(r), r; kx, rk, 0) exp {−i(m+ n)φk}
[∫
ℜ3
u∞j (x
′) exp (−ik · x′) d3x′
]
d3k
∞∑
l=−∞
exp
(
i (m+ lB)
[
(x− x0(r)) tanβ(r)
r
])
exp (−ilBφ0(r))
exp (−ilBΩt) exp (i [m+ lB]φ) . (4.34)
4.5 Far-field noise 139
4.5 Far-field noise
4.5.1 Spectrum of radiated sound
In order to calculate the SP level (defined in (2.132)) we take the correlation of the sound
pressure, using the same procedure and the same Green’s function as in Chapter 2. The
equivalent to equation (2.133) is found to be
Pˆ (σ, θ, φ, ω) =ρ20
B2
4pi2U∞
1
4σ20 (1−M cos θ)2
∑
l,m,m′
∫
rs,xs
[Jm+lB (γ0rs)]
∗ [W (rs)ΓmWW (xs, rs;ω)]
∗
[
iω cos θ sin β(rs)
c0 (1−M cos θ)
− i(m+ lB) cosβ(rs)
rs
]
exp
{ −iω cos θxs
c0 (1−M cos θ)
+ i(m+ lB)
[xs − x0(rs)] tanβ(rs)
rs
+ ilBφ0(rs)
}
∫
r′s,x′s
Jm′+lB (γ0r′s)W (r
′
s)Γ
m′
WW (x
′
s, r
′
s;ω)
[
−iω cos θ sin β(r
′
s)
c0 (1−M cos θ)
+
i(m′ + lB) cosβ(r′s)
r′s
]
exp
{
iω cos θx′s
c0 (1−M cos θ)
− i(m′ + lB) [x
′
s − x0(r′s)] tanβ(r′s)
r′s
− ilBφ0(r′s)
}
∑
p,p′
∫
ℜ
[
D˜m,pj (x0(rs), rs; kx, rk, 0)
]∗
Dm
′,p′
k (x0(r
′
s), r
′
s; kx, rk, 0)
[∫ 2pi
0
ei(m+p−m
′−p′)φkS∞kl (k) dφk
]
rkdrkdr′sdx
′
sdrsdxse
i(m′−m)(φ−pi2 ). (4.35)
4.5.2 SP plots
In Figures 4.12 and 4.13 we have plotted the radiated sound from the rotor at the origin,
with an asymmetrical inflow as described in this chapter. Also shown are the axisymmetric
SP spectra for comparison. We see that in the asymmetric case, the broadband level is
generally lowered, although the heights of the tonal peaks remain at the same level. The
exception is the broadband level around 1BPF in the low distortion (flight) case, which
is raised for the asymmetric inflow. This effect on the broadband level is consistent with
our earlier observation that the precise form of distortion primarily affects the broadband
level rather than tones.
140 4. Generalisation to asymmetric rotor systems
1000 1500 2000 2500
20
40
60
80
100
120
140
ω
SP
Asymm. high distortion
Asymm. medium distortion
Asymm. low distortion
Axisymmetric
Figure 4.12: In blue are plotted the SP as radiated from a single rotor which experiences
an asymmetric inflow of varying distortion levels. Shown in red are the equivalent ax-
isymmetric levels. We note that the number of azimuthal harmonics summed over, nm,
was taken to be 3.
1000 1500 2000 2500
50
60
70
80
90
100
110
120
130
ω
SP
Asymm. L = 0.05 rd
Asymm. L = 0.5 rd
Asymm. L = 5 rd
Axisymmetric
Figure 4.13: As in the figure above we have plotted, in blue, the SP as radiated from a
single rotor which experiences an asymmetric inflow, this time as L is varied. Shown in
red are the equivalent axisymmetric levels.
4.5 Far-field noise 141
In Figure 4.14 we have compared two different values of d, thus varying the level of
asymmetry. In the case of increased d, both the high and low distortion cases are similar,
whereas for a medium level of distortion we see a lowering of the tonal peak at 1 BPF
and an increase in broadband level.
1000 1500 2000 2500
20
40
60
80
100
120
140
ω
SP
d = 5 rd high distortion
d = 5 rd, medium distortion
d = 5 rd low distortion
d = 2.5 rd
Figure 4.14: In blue is shown the radiated sound from a single rotor with an asymmetric
inflow due to two actuator disks separated by a distance d = 5rd. The case d = 2.5rd,
which is more highly asymmetric and was also plotted in blue in Figure 4.12, is shown in
red for comparison. We see that the level is most affected by asymmetry in the medium
distortion case.
4.5.3 Sound from two adjacent rotors
Thus far in this chapter we have calculated the sound as radiated from one rotor only,
that at the origin, where the inflow to that rotor was asymmetric. We can also obtain an
expression for the far-field pressure generated by the two rotors side by side by superposing
two single rotor solutions. To leading order in the far-field, within the expression for
pressure, we only need adjust the observer position in the frequency phase term, e
iωσ
c0 , in
the Green’s function, which we recall is given by
H l,m (σ0, θ; xs, rs) ∼
ei
ω
c0
σ0
4piσ0 (1−M cos θ)
(−2pi) Jm+lB (γ0rs) ei
ω cos θ
c0(1−M cos θ)
xse−
i(m+lB)pi
2 .
(4.36)
142 4. Generalisation to asymmetric rotor systems
Terms in the amplitude which depend upon observer position decay much more rapidly
in the far field, and the same far-field coordinates can be used for both rotors.
σ1
θ2
observer
U∞
rotor at
origin
θ1
σ2
d
a) side-on view b) head-on view
z
y
rotor at
origin
observer
φ2
φ1
Figure 4.15: The far-field observer coordinate system, for two adjacent rotors.
The observer’s position with respect to the rotor centred at the origin is given in
spherical polars by (σ1, θ1, φ1), as shown in Figure 4.15, and we require an expression
for the distance of the observer from the second rotor, σ2, in terms of (σ1, θ1, φ1). The
cartesian distances from the observer to the second rotor are given by
x2 = σ1 cos θ1, (4.37)
y2 = σ1 sin θ1 cos φ1 + d, (4.38)
z2 = σ1 sin θ1 sin φ1. (4.39)
Combining to give σ22 we find
σ22 = x
2
2 + y
2
2 + z
2
2
= σ21 + 2dσ1 sin θ1 cos φ1 + d
2. (4.40)
Thus, to first order in d/σ1, the distance of the observer from the second rotor is
σ2 ≈ σ1 + d sin θ1 cosφ1, (4.41)
4.5 Far-field noise 143
and we substitute this into the e
iωσ2
c0 factor of the sound pressure due to the second rotor.
As σ1 → ∞, we have θ2 → θ1 and φ2 → φ1, and thus no changes need be made to the θ
and φ terms.
The properties of each of the two rotors (radius, strength, angular velocity) may of
course vary - U∞ will necessarily be the same. We consider here the case of identical
rotors, and for completeness give the full general solution in Appendix C.
Identical rotors - co-rotation
If both rotors have the same properties, including Ω1 = Ω2, then only the observer’s
distance from the rotor will need to be adjusted. Thus the pressure field due to two rotors
is
p2r(σ1, θ1, φ1) = p(σ1, θ1, φ1; Ω1, rd1, Ud1) + p(σ1 − d sin θ1 cosφ1, θ1, φ1; Ω1, rd1, Ud1)
= p1(σ1, θ1, φ1) + exp
[
iωd sin θ1 cosφ1
c0
]
p2(σ1, θ1, φ1). (4.42)
Here we have denoted the pressure fields due to each rotor by p1 and p2. This is because,
within the expression for p, the quantity u∞j (x
′) occurs in the force term, as was seen in
equation (4.34). This upstream field, u∞j , will be different for p1 and p2, as each rotor
experiences a different inflow of turbulence. We can recognise this by using the notation
u∞1,j (x
′) and u∞2,j (x
′) within p1 and p2 respectively. However, when the cross-correlation
is taken between u∞1,j (x
′) and u∞2,j (x
′) (or the cross-correlation of either field with itself)
we regain the same turbulence spectrum, S∞ij , as both fields originate in the same region
far upstream.
When calculating Pˆ in the single rotor case, complex conjugation meant the exponen-
tial terms e
iωσ
c0 cancelled out, but now we obtain two cross terms where it does not cancel
and thus
Pˆ2r (x, ω) =
∫ ∞
−∞
〈p∗2r, p2r〉 eiωτdτ
=
(
2 + 2 cos
(
ωd sin θ1 cosφ1
c0
))
Pˆ (x, ω) . (4.43)
To plot the Sound Pressure Level (SPL) directivity we calculate the expression given in
equation (4.35) with m 6= m′ for one particular frequency, such as 1 BPF. (See Appendix
144 4. Generalisation to asymmetric rotor systems
B for the definition of SPL.) In Figure 4.16 we have plotted the (single rotor) directivity
at a range of distortion levels for two different values of separation distance d. We see
that the highest noise level is straight ahead of the rotor, due to the presence of zeroth
order Bessel functions in our radiated sound expression.
0 0.5 1 1.5 2 2.5 3
observer angle, θ
SP
L
10 dB
d = 5 rd high distortion
d = 5 rd medium distortion
d = 5 rd low distortion
d = 2.5 rd
Figure 4.16: Directivity patterns for a single rotor for the frequency ω = 1 BPF, at a
variety of distortion levels. In blue are shown the patterns when d = 5rd, in red are the
patterns for a more asymmetric mean flow when d = 2.5rd. Absolute levels have not been
given in this case, as they will depend upon the observer’s distance from the rotor, σ0.
Figure 4.17 then compares the UDN radiated from both rotors together, as given in
(4.42), against the UDN radiated from a single rotor. Directly beneath the rotors, at
φ = pi/2, the sound simply adds (the argument of the cosine function becomes zero in
equation (4.43)), but out to the sides at φ = 0 (or pi) we find a significantly altered
directivity pattern due to interference when cosφ1 = ±1. The sound level directly ahead
and behind the rotors is raised and the sound level for a section underneath the flight
path is reduced.
4.5 Far-field noise 145
0 0.5 1 1.5 2 2.5 3
observer angle, θ
SP
L
10 dB
φ = 0, d = 5 rd
φ = pi/4, d = 5 rd
φ = pi/2, d = 5 rd
d = 2.5 rd
Figure 4.17: Directivity patterns for the twin rotor system, for the frequency ω = 1 BPF,
at three different azimuthal positions. Again both the d = 5rd and d = 2.5rd cases are
plotted. Shown in black is the single rotor case.
146 4. Generalisation to asymmetric rotor systems
Chapter 5
Rotor at incidence: an important
asymmetric case
5.1 Chapter outline
At the start of this Ph.D. one of the questions we asked was: what might the effect of
asymmetry due to incidence be on both the distortion of turbulence and the subsequent
radiation of UDN to the far-field? As a first step towards investigating this, in Chapter 4
we considered a more straightforward system in which the distortion is asymmetric (due
to an asymmetric mean flow) but the radiation is calculated for a rotor travelling in the
direction of its axis, as in Chapters 2 and 3. In this chapter, we consider the full case of
UDN for a rotor at incidence.
Our model must be altered in a number of ways, in particular the mean flow stream-
lines, and the appropriate Green’s function.
Inputs Outputs
Mean flow model, U:
actuator disk at inci-
dence, superposed upon
a uniform upstream flow
Upstream turbulence
model, u∞:
von Ka´rma´n spectrum
Turbulence energy spectra
at rotor face plotted at
several φ positions
Far-field pressure: Pˆ
leading to spectral power
and directivity
Distortion of upstream
turbulence, u∞, by
mean flow, U, using
Rapid Distortion Theory.
Analysis as for asymmet-
ric mean flow in Chapter
4, but with an adjusted
Green’s function.
147
148 5. Rotor at incidence: an important asymmetric case
5.2 Effects of incidence
In reality, most aeroplane engines operate at ‘incidence’, that is with their shaft at some
non-zero angle to the flight direction, for some periods during flight; incidence typically
being highest during take-off. For AOR engines, incidence is significantly increased in the
puller configuration (as compared to the pusher) due to the proximity of the wing which
induces an altered mean flow upstream of the rotors. In addition, AOR driven aircraft can
climb and descend more steeply than turbofan driven aircraft (high blade stresses being
a limiting factor for the latter) which also increases incidence. The effect of incidence
is often neglected in theoretical work as the assumption of axisymmetry simplifies the
analysis. It is standard procedure to assume sound radiates into a uniform flow parallel
to the rotor’s axis, and this is the approach we employed in previous chapters.
Incidence could potentially affect UDN levels in several ways. Firstly, the mean flow
is altered, and thus the distortion of turbulence will differ from the axisymmetric case.
Secondly, the location where each turbulent eddy hits the blades will vary. Thirdly, the
blade response will change. Fourthly, the sound now radiates into an angular inflow. In
the model presented here we do not consider the effect of incidence upon blade response.
We continue to assume zero mean blade loading, as this is one of the modelling assump-
tions within the LINSUB theory. Authors who have considered angle of attack effects
on blade response include Myers and Kerschen (1995) and Peake and Kerschen (1997).
Incorporating a different blade response function into our framework could be an area for
future model development.
Incidence mainly affects the sound radiated from the front rotor, and we continue to
assess the UDN produced by the front rotor only, as in previous chapters.
5.2.1 Previous work on non-zero incidence
Little work has been done previously on the effect of incidence upon the process of tur-
bulence distortion, with most authors concentrating on the effect of angle of attack upon
a blade’s response to isotropic turbulence.
In a recent example, Devenport et al. (2010) undertook a series of experiments upon
a single blade in isotropic turbulence, and found that angle of attack effects upon the
resulting sound spectrum were small. This was due to significant cancellation between
components with varying k2, the wavenumber in the direction of lift. This finding is
5.2 Effects of incidence 149
consistent with that of Paterson and Amiet (1976), who showed that for incompressible
flows (i.e. for low frequencies) it is the component of an incident gust in the direction
of the blade normal which induces the unsteady response, and this will be proportional
to angle of attack, which is typically small. For anisotropic turbulence however, blade
response is expected to vary more significantly with angle of attack, due to a reduced level
of cancellation, although this may not necessarily result in a significantly altered noise
spectrum.
Mish and Devenport (2006a) had previously examined angle of attack effects, again for
a blade immersed in grid-generated (and therefore close to isotropic) turbulence, but with
particular consideration of the eddy stretching which takes place at the leading edge of
the blade. Their measurements of surface pressure showed a lack of dependence on angle
of attack, and they concluded that the additional eddy stretching due to incidence in
the leading edge region does not significantly affect the intensity of pressure fluctuations.
They suggested that incidence effects would only be significant for turbulence of a small
scale compared to the airfoil chord. Within our model we can now include the altered
eddy stretching due to incidence not just in the leading edge region, but throughout the
full upstream flow.
Paterson and Amiet (1976, 1982) noted that non-zero incidence can shift the dominant
frequency of turbulence ingestion noise away from multiples of BPF, due to the change
in azimuthal position at which each eddy passes through the rotor. The effect of non-
convected turbulence, such as random wanderings due to a non-uniform inflow, was found
to be a reduction in tonal peaks and an increase in their widths.
The effect of incidence upon radiation to the far-field was considered in detail by
Hanson (1995), and we follow his framework for our coordinate system in order to use his
form of Green’s function. Hanson noted that authors considering the problem of a rotor
whose shaft is at some non-zero angle of attack previously had assumed that the resultant
unsteady loading would be the dominant effect, rather than radiation into a non-axial
flow field. In fact, as his paper confirmed, flow angularity can be more significant than
unsteady loading in certain cases.
150 5. Rotor at incidence: an important asymmetric case
5.3 Distortion of turbulence by rotor at incidence
5.3.1 Adapting the mean flow model
We wish to model the streamtube contraction induced by a rotor placed at a non-zero
angle, α, to the flight direction. The set-up is shown in Figure 5.1.
θ
−x
−x′
z z′
α
rotor axis
observer
tilted rotor
flight direction
σ0
(a) Side on view. The angle between the flight direction and the
vector between the rotor origin and the observer is denoted by θ.
We denote the angle between the rotor axis and vector between
rotor origin and the observer by θ′. Note that θ′ 6= θ + α, in
general.
y
z
observer
φ
observer
y′ = y
z′
coordinate system tilted around the y axis
φ′
(b) Head-on view.
Figure 5.1: Coordinate system of interest, after Hanson (1995), except the directions of
both the positive x and x′ axes have been reversed. Note that in this chapter, φ′ denotes
the azimuthal coordinate in the tilted system, not in the rotating frame, as it did in
Chapter 2.
The tilted coordinates (denoted by primes) are related to the flight-direction coordi-
5.3 Distortion of turbulence by rotor at incidence 151
nates via
x′ = x cosα− z sinα,
y′ = y,
z′ = x sinα + z cosα,
r′2 = y′2 + z′2,
tanφ′ =
z′
y′
. (5.1)
We construct the new flow by superposing an actuator disk flow in the x′ coordinate
system, on top of the flight component U∞eˆx, which tends to U∞ cosαeˆx′ as x′ → −∞. We
neglect the interaction between the cross-wind component of the flight speed, U∞ sinαeˆz′,
and the actuator disk flow. This approximation will be valid when α is sufficiently small,
or when the strength of the actuator disk is sufficiently high. As given in Chapter 1, for
the pusher configuration of AOR, |α| is typically less than 3◦ (negative values of α can
occur during approach) whereas in the puller configuration the maximum value of |α| is
of the order of 15◦. Therefore the cross-wind component will be dominated by the axial
component in the parameter regimes of interest to us.
The mean flow field is therefore given by
U · eˆx = U∞ + U inc.x (x, y, z) = U∞ + Ux (x′, y′, z′) cosα + Ur (x′, y′, z′) sinφ′ sinα, (5.2)
U · eˆy = U inc.y (x, y, z) = Ur (x′, y′, z′) cosφ′, (5.3)
U · eˆz = U inc.z (x, y, z) = −Ux (x′, y′, z′) sinα + Ur (x′, y′, z′) sin φ′ cosα, (5.4)
where (Ux, Ur) is an axisymmetric actuator disk flow as given in equations (2.47) and
(2.48), with strength
U ′d =
T
pir2dρ0 (U∞ cosα)
. (5.5)
Figure 5.2 shows some representative plots of the streamlines for this tilted actuator disk
set-up.
5.3.2 Distorted turbulence spectrum
In Figures 5.3 - 5.7 the axial component of the distorted turbulence spectrum has been
plotted for a range of positions at the tilted rotor face, and at a number of distortion
152 5. Rotor at incidence: an important asymmetric case
−20 −18 −16 −14 −12 −10 −8 −6 −4 −2 0
−6
−4
−2
0
2
4
6
8
10
12
x, axial direction
z
(a)
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0
−3
−2
−1
0
1
2
3
4
5
6
x, axial direction
z
(b)
Figure 5.2: a) In blue are shown the streamlines induced by the tilted actuator disk system.
Also shown for comparison are the axisymmetric streamlines (in red). The position of the
disk in each case is shown by a bold line. Total thrust, T , is kept constant between the
two cases. b) A zoomed-in view nearer the disk.
levels. In the first four figures we have taken α = pi/12 rad = 15◦. We see in Figures 5.3
and 5.4 that, as before, the spectrum level is greater at the blade tip than the hub in all
circumstances.
From Figure 5.2 (the plot of the streamlines at incidence) we see that the tilting of
the actuator disk increases the streamtube contraction in the region φ′ = pi/2 (where the
disk is tilted away from the negative x axis) and decreases it in the region φ′ = 3pi/2
(where the disk is tilted toward the negative x axis). Correspondingly, the spectrum level
is highest at φ′ = pi/2 in all cases, as seen in Figures 5.5 and 5.6 which show the variation
of Sxx with φ′ for two different sets of parameters.
Finally, in Figure 5.7 we vary the angle of incidence, α. Interestingly, as α increases the
variation of the turbulence spectrum with φ′ decreases. It is for small, but non-zero, values
of α that we have the greatest azimuthal variation. This is explained by considering that
as α is increased, and the radial position of interest r′ is kept constant, then the variation
in radial position, r, as we move around the disk decreases.
5.3 Distortion of turbulence by rotor at incidence 153
−2 −1 0 1 2 3 4
−8
−6
−4
−2
0
2
4
6
log(k
x
) = log(ω/U
∞
)
lo
g(U
∞
S x
x)
High distortion, near tip
High distortion, near hub
Medium distortion, near tip
Medium distortion, near hub
Low Distortion, all r values
Undistorted
Figure 5.3: Here we show the spectrum at φ′ = pi/2 for a range of distortion levels and
radial positions. The level near the tip is higher than that near the hub in all cases, and as
distortion is decreased the level also decreases. We have collapsed the data by multiplying
Sxx by U∞.
−2 −1 0 1 2 3 4
−10
−8
−6
−4
−2
0
2
4
log(k
x
) = log(ω/U
∞
)
lo
g(S
xx
)
High distortion, near tip
High distortion, near hub
Medium distortion, near tip
Medium distortion, near hub
Low Distortion, all r values
Undistorted
Figure 5.4: Plot as in Figure 5.3, but at φ′ = 3pi/2. The spectrum levels at the hub and
tip are now further apart than in the φ′ = pi/2 case. Interestingly, the dependence on
distortion level is now more complex, with the low distortion case lying between the high
and medium cases.
154 5. Rotor at incidence: an important asymmetric case
−2 −1 0 1 2 3 4
−8
−6
−4
−2
0
2
4
6
log(ω)
lo
g(S
xx
)
φ’ = pi/2
φ’ = 2pi/3
φ’ = 5pi/6
φ’ = pi
φ’ = 7pi/6
φ’ = 4pi/3
φ’ = 3pi/2
undistorted
Figure 5.5: Here we have calculated Sxx in the high distortion case near the rotor tip
(r′/rd = 0.9), at a range of azimuthal positions. We see that the energy level is higher
than the undistorted spectrum in all cases. As we move from φ′ = pi/2 (rotor tilted away
from -ve x axis) to φ′ = 3pi/2 in equal increments of φ′, the levels are fairly evenly spaced.
0 1 2 3 4 5 6
−14
−12
−10
−8
−6
−4
−2
0
log(ω)
lo
g(S
xx
)
φ’ = pi/2
φ’ = 2pi/3
φ’ = 5pi/6
φ’ = pi
φ’ = 7pi/6
φ’ = 4pi/3
φ’ = 3pi/2
undistorted
Figure 5.6: Here we show a medium distortion case near the rotor hub (r′/rd = 0.4). The
energy level is below that of the undistorted spectrum in most cases, but not all (as was
the case for the asymmetric flow in Chapter 4). We also see a peak in the φ′ = 3pi/2
spectrum not present in the other spectra.
5.4 Far-field noise calculation 155
0 1 2 3 4 5 6
−14
−12
−10
−8
−6
−4
−2
0
2
log(ω)
lo
g(S
xx
)
undistorted
α = 0
α = pi/24, φ = pi/2
α = pi/24, φ = 3pi/2
α = pi/12, φ = pi/2
α = pi/12, φ = 3pi/2
α = pi/4, φ = pi/2
α = pi/4, φ = 3pi/2
Figure 5.7: Here we have adjusted α, the angle of rotor incidence. We see that as α is
increased, the spectral levels increase. However, perhaps surprisingly, the variation with
φ′ decreases.
5.4 Far-field noise calculation
In re-deriving the expression for far-field sound pressure, we must first re-derive the forcing
term, before adjusting the appropriate Green’s function.
5.4.1 Adjusting the force term
The total force exerted by the rotor on the fluid, f , is simplest to define as a function of
the tilted (primed) coordinates. We wish to obtain expressions for the components of f
in the eˆx′ and eˆφ′ directions, in order to use both LINSUB and Hanson’s form of Green’s
function. Hanson’s analysis is given for a point source representation of blade loading and
we extend his theory here by integrating over the full blade planform, finding we are able
to perform the integral over the azimuthal source position, φ′s, explicitly.
An upstream gust which impinges upon the leading edge of the rotor blades, with
wavevector component k and constant amplitude uˆ∞ gives rise to a force on the fluid in
156 5. Rotor at incidence: an important asymmetric case
the eˆx′ direction as follows
fkx′ (x
′, r′, φ′, t) =
B∑
b=1
ρ0W sin β(r′) exp (−ikxU∞t)
∞∑
m=−∞
ΓmWW (x
′, r′; kx, χm)
eikxX
∞∑
p=−∞
Jm−p (rkR) exp (−imφk)D′pj (x0(r′), r′; kx, rk, 0)
uˆ∞j (k) exp
(
im
[
φ0(r′) +
2bpi
B
+ Ωt
])
∞∑
s=−∞
δ
[
r′
(
φ′ − Ωt− φ0(r′)−
2bpi
B
)
− (x′ − x0(r′)) tanβ(r′)− 2pis
]
.
(5.6)
In this expression, both the normal to the blade, N′, and the delta function which defines
the blades’ surfaces are given in the primed coordinate system. The fkφ′ component is
defined similarly, with sin β(r′) replaced by cosβ(r′). (We neglect the radial component
of the cross-wind term, U∞ sinαeˆz′, in our calculation of the blade normal for the unleant
blades assumed here.)
The distortion amplitude Aij is contained within the quantities D
′p
j , where D
′p
j is based
upon C ′pij in precisely the equivalent way as in Chapter 2 (equation (2.82)). This latter
quantity is defined as
C ′pij (x
′, r′;k) =
1
2pi
∫ 2pi
0
Aij (x′, r′;k) e−ipφ
′
dφ′. (5.7)
Note that X, Y and Z within the definition of Aij remain the upstream coordinates in
the non-primed system; X ′, Y ′ and Z ′ do not tend to constant values as x→ −∞.
The blade angle β is defined as follows
tan β (r′) =
Ωr′
U∞ cosα+
U ′d
2
=
Ωr′
U∞ cosα + Ud2 cosα
. (5.8)
We hold the total thrust induced by the actuator disk, T , and the flight speed, U∞,
constant when comparing the axisymmetric and incidence models, which leads to our
definition of U ′d in equation (5.5). This model differs from that of Wright (2000) who also
considered the case of an angular inflow into a propeller (with no turbulence present) but
held the magnitude of the cross-wind component at the rotor face constant instead, given
5.4 Far-field noise calculation 157
by U ′d/ cosα in our set-up. In Figure 5.8 we have plotted the blade angle β as a function
of rotor incidence angle α for a range of U∞ values, holding Uf = U∞ + Ud/2 constant.
We see that for the parameter space of interest to us, small α, the low and high distortion
cases behave differently. For the high distortion case β decreases with α, whereas for the
low distortion case β increases with α.
0 pi/24 pi/12
0.655
0.66
0.665
0.67
0.675
0.68
0.685
0.69
0.695
α (rad)
β (
α
)
Low distortion
Uf = 2 U∞
High distortion
Figure 5.8: Blade angle β plotted as a function of rotor incidence α, for r′/rd = 0.4. When
Uf is twice U∞ we have U∞ = Ud/2, and the two cosα factors nearly balance each other
for a large range of α values (up to around α = 0.7 rad).
To obtain the total axial force, fx′, we integrate fkx′ over all k. We separate out t and
φ′ dependence explicitly within fx′, following the same method as in Chapter 2, to find
fx′ (x′, r′, φ′, t) =ρ0W sin β(r′)
B
2pir
∞∑
m=−∞
1
(2pi)3
∫
ℜ3
ΓmWW (x
′, r′; kx, χm) exp (−ikxU∞t)
eikxX
∞∑
p=−∞
Jm−p (rkR) exp {−imφk}D′pj (x0(r′), r′; kx, rk, 0)
[∫
ℜ3
u∞k (y) exp (−ik · y) d3y
]
d3k
∞∑
l=−∞
exp
(
i (m+ lB)
[
(x′ − x0(r′)) tanβ(r′)
r′
])
exp (−ilBφ0(r′))
exp (−ilBΩt) exp (i [m+ lB]φ′) . (5.9)
158 5. Rotor at incidence: an important asymmetric case
5.4.2 Adjusted Green’s function
In Chapter 2, when solving the wave equation, we derived the form of Green’s function
used for a rotor translating in the direction of its axis. Here, we will instead derive the
Green’s function for a rotor travelling at some non-zero angle of attack to its axis. This
was calculated by Hanson (1995) for a point source representation of blade loading, and
we give an outline of his analysis here (noting once again that Hanson used the opposite
sign convention for the positive x axis) and proceed to cast the Green’s function into the
equivalent form to that used previously in our analysis.
We recall that in Chapter 2 the wave equation was given in the form
∇2p− 1
c20
(
∂
∂t
+ U∞
∂
∂x
)2
p =∇ · f (x, t) = −
∑
ω
Qω (x, y, z) e−iωt, (5.10)
with solution
p (x, y, z, t) =
∑
ω
Pω (x, y, z) e−iωt. (5.11)
In addition to t dependence, in equation (5.9), we have separated out φ′ dependence
explicitly within f . Thus we have
Qω (x, y, z) = −
∞∑
l,m,p=−∞
∇ ·
(
N′F l,m,peimφ
′
)
= −
∞∑
l,m,p=−∞
[
sin β(r′)
∂F l,m,p
∂x′
+
cosβ(r′)
r′
imF l,m,p
]
eimφ
′
, (5.12)
where F l,m,p is the appropriate (l,m, p) component of (5.9) and we have applied∇ directly
in (x′, y′, z′) coordinates.
In the tilted system, Hanson finds the following solution
Pω (σ0, θ′, φ′) =
e
iωσ0
c0
4piσ0 (1−M cos θ)
∫ rd
0
∫ ∞
−∞
∫ pi
−pi
Qω (x′s, r
′
s, φ
′
s) (5.13)
exp
[ −iω
c0 (1−M cos θ)
(x′s cos θ
′ + r′s sin θ
′ cos (φ′ − φ′s))
]
r′sdφ
′
sdx
′
sdr
′
s,
where we integrate over the source coordinates (x′s, r
′
s, φ
′
s). The changes here from the
standard representation of the Green’s function are only in the phase term; in the far-field,
5.4 Far-field noise calculation 159
the amplitude terms are the same to leading order. The form is identical to that found
in Chapter 2 with θ′ and φ′ replacing θ and φ, except that the non-primed θ appears in
the 1/ (1−M cos θ) Doppler factors. We can express θ in terms of primed coordinates as
follows
cos θ = cos θ′ cosα− sin θ′ sinφ′ sinα. (5.14)
We note again that θ′ 6= θ + α in general, but equality does hold when φ′ = pi/2. The
Green’s function, as compared to equation (2.126), is therefore given by
H ′l,m (σ0, θ′; x′s, r
′
s) ∼
ei
ω
c0
σ0
4piσ0 (1−M cos θ)
(−2pi) Jm+lB (γ′0r′s) e
i ω cos θ
′
c0(1−M cos θ)
x′se−
i(m+lB)pi
2 ,
(5.15)
where γ′0 = ω sin θ
′/c0 (1−M cos θ).
160 5. Rotor at incidence: an important asymmetric case
5.4.3 Correlation of far-field pressure
The Power Spectral Density is thus given by
Pˆ inc. (σ0, θ′, φ′, ω) =ρ20
B2
4pi2U∞
1
4σ20 (1−M cos θ)2
∑
l,m,m′
∫
r′s,x′s
[Jm+lB (γ′0r
′
s)]
∗ [W (r′s)Γ
m
WW (x
′
s, r
′
s;ω)]
∗
[
iω cos θ′ sin β(r′s)
c0 (1−M cos θ)
− i(m+ lB) cosβ(r
′
s)
r′s
]
exp
{ −iω cos θ′x′s
c0 (1−M cos θ)
+ i(m+ lB)
[x′s − x0(r′s)] tanβ(r′s)
r′s
+ ilBφ0(r′s)
}
∫
r′′s ,x′′s
Jm′+lB (γ′0r
′′
s )W (r
′′
s )Γ
m′
WW (x
′′
s , r
′′
s ;ω)
[
−iω cos θ
′ sin β(r′′s )
c0 (1−M cos θ)
+
i(m′ + lB) cosβ(r′′s )
r′′s
]
exp
{
iω cos θ′x′′s
c0 (1−M cos θ)
− i(m′ + lB) [x
′′
s − x0(r′′s )] tanβ(r′′s )
r′′s
− ilBφ0(r′′s )
}
∑
p,p′
∫
ℜ
D˜m,p∗j (x0(r
′
s), r
′
s; kx, rk, 0) D˜
m′,p′
k (x0(r
′′
s ), r
′′
s ; kx, rk, 0)
[∫ 2pi
0
ei(m+p−m
′−p′)φkS∞kl (k) dφk
]
rkdrkdr′′sdx
′′
sdr
′
sdx
′
se
i(m′−m)(φ′−pi2 ).
(5.16)
Since φ′ now appears within θ, the φ′ integral is no longer straightforward. This increases
the time it takes to numerically compute SP for each frequency ω.
5.4.4 SP and SPL plots
In Figure 5.9 we have plotted the SP for a rotor at angle of incidence α = pi/12, compared
to the axisymmetric case. We see similar effects as in the asymmetric case considered
in Chapter 4, namely a decreased broadband level in the high distortion case and an
increased broadband level around 1 BPF in the low distortion case. New effects due to
incidence include an increased width peak at 1 BPF in the high distortion case, and a
significant increase in peak tonal level in the medium distortion case. As α decreases
these effects are found to diminish.
In Figures 5.10, 5.11 and 5.12 we have plotted the directivity. As expected, the level
5.4 Far-field noise calculation 161
1200 1300 1400 1500 1600 1700
20
40
60
80
100
120
140
ω
SP
Incidence, high distortion
Incidence, medium distortion
Incidence, low distortion
Axisymmetric
Figure 5.9: In blue are plotted the SP as radiated from a rotor at incidence, for varying
distortion levels. Shown in red are the equivalent axisymmetric levels.
is significantly lower away from multiples of BPF in the high distortion case. In Figure
5.11 we see the variation of the directivity as φ′ changes. This is due to Hanson’s adjusted
Doppler factors, as well as the effect of non-zero incidence upon the distortion. In Figure
5.12 we see that the variation with α is marginally greater at φ′ = pi/2, that is on the side
of the rotor which is tilted away from the flight direction.
0 0.5 1 1.5 2 2.5 3
observer angle, θ
SP
L
20 dB
1 BPF
1.5 BPF
Figure 5.10: Directivity patterns for a rotor at incidence for the high distortion case, at
two different frequencies.
162 5. Rotor at incidence: an important asymmetric case
0 0.5 1 1.5 2 2.5 3
observer angle, θ
SP
L
5 dB
α = pi/12, φ’ = pi/2
α = pi/12, φ’ = 3 pi/2
α = pi/24, φ’ = pi/2
α = pi/24, φ’ = 3 pi/2
Figure 5.11: Directivity patterns for a rotor at two different angles of incidence, and at
two different azimuthal positions, for the frequency ω = 1 BPF.
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
observer angle, θ
SP
L
2 dB
α = pi/12, φ’ = pi/2
α = pi/12, φ’ = 3 pi/2
α = pi/24, φ’ = pi/2
α = pi/24, φ’ = 3 pi/2
Figure 5.12: Zoomed-in plot of Figure 5.11 above, showing the variation with α at the
two extremal azimuthal positions, φ′ = pi/2 and φ′ = 3pi/2. The spectra for φ′ = 0 and pi
will lie between these.
Chapter 6
Summary and Conclusions
6.1 Motivation
We first recall our original motivations for this work. The aim of this Ph.D. project,
supported in part through a Rolls-Royce Industrial CASE award, was to further our
understanding of Counter Rotating Open Rotor noise, and in particular the phenomenon
of Unsteady Distortion Noise. Rolls-Royce is currently running an Advanced Open Rotor
development programme, and this Ph.D. took place at the same time as a programme
of joint Rolls-Royce/Airbus AOR engine testing. It is believe that UDN will be more
significant for AORs than turbofans.
By building on the work of Majumdar and Peake in the mid-1990s we wished to develop
a model which could treat the problem of incidence, and other more realistic features of
the new generation of open rotors - such as the installation. As detailed in this thesis,
the analytic model has now been successfully extended to a non-axisymmetric mean flow
system, and applied to two different asymmetric set-ups, including that of incidence. In
addition, while undertaking this work certain terms which were neglected in Majumdar’s
original axisymmetric model were identified and included.
We have implemented several adjustments to the mean flow model to more closely
emulate the AOR system: diversion of the inflow around a ‘bullet’, use of a variable
circulation actuator disk model, and analysis of the most appropriate turbulence spectrum
and integral lengthscales to use when modelling atmospheric turbulence. At the end of
this chapter we list further areas of model development which have been identified but
not yet tackled.
163
164 6. Summary and Conclusions
6.2 Summary of key results
Here we summarise our findings as detailed in each chapter of this thesis.
6.2.1 Chapter 2 results
Correction to the polar representation of distorted wavevector, li, and the
distortion amplitude, Aij The quantity li = km∂Xm/∂xi was first introduced in
equation (2.28). We have shown that the lφ component is non-zero, even for axisymmetric
mean flows, in which case li is given by equation (2.34). This means several Aij terms
which were assumed by Majumdar to be zero are in fact non-zero, as can be seen from
examination of equations (2.37). These include
Aφφ =
(
1− lφlφ|l|2
)
R
r
.
In fact we see that even if lφ is zero, there is still an R/r component of Aφφ.
Expression derived for the distorted turbulence spectrum Sij which includes
all azimuthal orders. An expression for Sij , the Fourier Transform of the correlation
between velocities separated in time, was given in equation (2.65). In Majumdar and
Peake (1998), only specific azimuthal components were plotted, as given by
Smij =
2pi
U∞
∫
ℜ2
A∗ik (x,k) J
∗
m (rkR)Ajl (x,k)Jm (rkR)S
∞
kl (k) rkdrkdφk.
If the quantities Smij are summed over all m, the two Bessel function terms in fact cancel
each other. The motivation for separating Sij into azimuthal components was because Smxx
appeared directly in Majumdar’s expression for the spectrum of unsteady blade pressures.
However, as discussed in the next point, this is not true for our expression, as we have
fully included the azimuthal dependence within the amplitude Aij , not solely that within
the phase of the incident velocities.
Numerical results for Sxx were given throughout the thesis. The energy level was found
to be higher near the tip than the hub of the rotor in all cases considered, due to the
sharper change in velocities near the edge of the streamtube contraction. Dependence of
Sxx upon distortion level is more complex, and depends on the precise mean flow under
consideration.
6.2 Summary of key results 165
Inclusion of azimuthal dependence within distorted gust amplitude, Aij (x,k)
For each upstream gust of a particular wavevector k, Aij contains φ dependence which
should be taken into account when calculating the inter-blade phase angle, χ. This
has been implemented within our application of LINSUB to calculate the blade pressure
jumps. We have introduced the quantities
Cpij (x, r;k) =
1
2pi
∫ 2pi
0
Aij (x, r;k) e−ipφdφ,
in order to explicitly bring out the eipφ dependence within Aij (equation (2.76)). The quan-
tities Dpj , given in equation (2.82), were introduced to facilitate calculation of the normal
component of the velocity incident on the leading edge of the blades: [sin βAxi + cosβAφi].
In Figure 2.6 we showed that Dpj decay rapidly with increasing p, and thus we do not
need to include very many terms to achieve convergence.
Although their definition is straightforward, the use of these new quantities C and D
is one of the most significant aspects to our new model, as this framework allows ready
extension to the asymmetric case.
Expressions derived for the spectra of blade pressure and sound pressure. The
spectrum of blade pressures, for the axisymmetric case, was given in equation (2.90). In
Majumdar and Peake (1998) (equation (2.32)), the blade pressure spectrum was given by
Pˆblade (x;ω) =
∞∑
m=−∞
2pi
U∞
|ρ0WΓmWW |2NpNqSmpq. (6.1)
Only the Smxx component appeared in their final expression as all S
m
φi components were
assumed to be zero. We have found this not to be the case due to non-zero Aφi, as
discussed above.
Similarly, we have now included contributions from Aφi in our expression for the
spectrum of sound pressure (equation (2.133)), rather than solely those which arise from
the axial component of the normal vector Ni.
Limiting the rk integral. Both the spectra of blade pressures and of sound pressures
involve an infinite integral over the radial wavevector component, rk. This arises from
neglecting the contribution of the transverse wavenumber to derivatives, for example
within ∇2. When calculating these quantities numerically it is necessary to limit this
166 6. Summary and Conclusions
integral to achieve convergence. We presented a method of doing this in §2.3.3, which
involves calculating an ‘effective’ radial wavenumber kt and then placing an upper limit
upon k2t of ω
2/c20. We have implemented this in our numerical code. A further refinement,
using multi-variate stationary phase, is presented in Appendix D.
Numerical results for the blade pressure spectrum and radiated sound ob-
tained. In Figure 2.8 we saw that the blade pressure spectrum contains peaks at multi-
ples of the Rotational Shaft Frequency, and in Figure 2.9 we found that the amplitude of
these peaks is reduced when the integral length scale is less than the disk radius (L < rd)
but for all L > rd the heights of the peaks remained approximately constant.
Consistent with previous findings, in Figure 2.11 we found that higher distortion levels
lead to a more tonal noise spectrum. For Mach no. = 0.25, and induced velocity at the
rotor face 100 m s−1, we found a broadband spectrum of between 100 and 110 dB, with
no tonal peaks. Interestingly, as L increases above rd (Figure 2.12) the spectrum shifts
down fairly uniformly at all frequencies, and so we retain tonal peaks for large values of
L.
Comparison to Majumdar’s results In Figures 2.13 and 2.14 we directly compared
our prediction scheme to that of Majumdar. Qualitatively the agreement was excellent,
but the inclusion of full φ dependence within our model leads to a reduced broadband
level in the flight case relative to the tonal peaks of the static testing case.
6.2.2 Chapter 3 results
Implementation of the variable circulation actuator disk model within our
framework. The true velocity at the face of an open rotor is more closely modelled
by the variable circulation actuator disk than the constant circulation actuator disk.
However, in Figure 3.4 we showed that, for a high distortion case, within half a radius of
the rotor disk the axial velocity is virtually the same in both cases. Calibration against
output from a standard strip theory code showed that using a lower thrust (i.e. a lower
value of Ud) in either actuator disk model gives a closer fit. An adjusted velocity profile,
where the original Hough and Ordway profile was scaled to lie between rh and rd, was
also found to approximate the axial velocities better.
Re-derivation of ∂Xi/∂xj for the variable circulation flow was detailed in §3.2.5. In-
6.2 Summary of key results 167
creases in ∂X/∂x and ∂R/∂x near the tip and decreases near the hub were found when
compared to the constant circulation model (see Figure 3.6). Corresponding shifts in the
distorted turbulence spectrum, Sxx, were shown in Figure 3.7.
Use of the variable circulation model led to a lowered broadband level in the high
distortion case, but a raised level in the low distortion case, shown in Figure 3.24. In the
medium distortion case the broadband level was lowered but the height of the tonal peak
at 1 BPF was raised, shown in Figure 3.25.
Development of an analytic fluid dynamical model to approximate the pres-
ence of the ‘bullet’ upstream of the rotor. In §3.2.3 we introduced a fluid point
source upstream of the rotor, simulating the distortion of streamlines around a central
engine nacelle. In Figure 3.11 we saw that the addition of this point source raised Sxx
in the high distortion case at the majority of radial positions, but it was lowered in the
medium distortion case near the hub. Full calculation of the derivative of drift for the
bullet flow was then given in §3.2.5.
The presence of the bullet was found to lower the noise spectrum marginally, as shown
in Figure 3.26.
Inclusion of a secondary actuator disk behind the first, to simulate the pres-
ence of the rear blade row. In Figure 3.14 we saw that the addition of a rear actuator
disk lowered Sxx near the rotor tips in all cases. However, this did not translate to a change
in SP, as seen in Figure 3.27.
Investigation of the effect of the form of distortion upon broadband and tonal
levels. Through comparison of several different models using ‘scaled’ and ‘unscaled’
velocity profiles, we saw in Figures 3.22 and 3.23 that the broadband level is primarily
affected by the values of ∂Xi/∂xj , whereas the tonal peaks are primarily affected by the
values of X and R in the phase. Thus the precise form of the distortion seems to affect
the broadband UDN level, whereas the magnitude of the overall streamtube contraction
affects the UDN tones.
Investigation of the energy distribution of the distorted turbulent eddies. We
made an exploratory examination involving the newly defined quantities, E1 and E2. The
turbulent energy at the blade tip was found to be higher than at the hub in all cases, see
168 6. Summary and Conclusions
for example Figure 3.15. However, we also found that the sum of all diagonal elements of
the turbulence distortion tensor, Sii, brings the spectra at hub and tip closer together, as
in Figure 3.16.
Throughout Chapter 3 we investigated the role of the fundamental quantities ∂Xi/∂xj
in determining the distorted turbulence spectrum, for example in the pair of Figures 3.6
and 3.7, and the pair of Figures 3.13 and 3.14. The turbulence spectrum was seen to
change directly due to changes in ∂Xi/∂xj , except for terms where ∂Xi/∂xj ≪ 1, when
the δij terms of the diagonal elements of Aij dominate.
Investigation of appropriate turbulence models and lengthscales. In Chapter
2 we gave our justification for primarily considering the von Ka´rma´n model, as it was
found by Wilson et al. (1999) to most closely match true atmospheric spectra (except
for the drop-off rate for high wavenumber k), see Figure 2.3. However, to build a model
for turbulence shed from the pylon, for example, it was instructive to consider different
upstream turbulence models to assess how much S∞ij affected the eventual noise spectrum.
In Chapter 3, Figure 3.28, we compared the Liepmann to the von Ka´rma´n spectrum.
Across different distortion levels, the SP was lowered in the Liepmann case (when L was
of the same order as the rotor radius) although the tonal peaks remained similar. The
use of the Gaussian spectrum was seen to affect SP very significantly, in Figure 3.29, due
to the rapid spectral fall off.
Also in Chapter 3, we discussed the fact that the integral length scale L depends very
much on the particular atmospheric conditions, which will vary in space and time. A
key element is whether buoyancy or inertia forces are the dominant turbulence generating
mechanism. Often measured values of L are higher than those typically used by acousti-
cians, and in Figure 3.30 we calculated results for large values of L. We found that once
L was larger than rd the effect of increasing L simply shifts the spectrum down fairly
uniformly, but with the same tonal peaks as occur for L ∼ rd.
Finally, we input a range of realistic turbulence parameters (as specified by Simonich
et al. (1990)) into our model to produce Figure 3.31. A downward shift in the noise
spectrum of around 5 dB was found between 50m and 122m altitude, with a much smaller
downward shift seen between 122m and 152m.
6.2 Summary of key results 169
6.2.3 Chapter 4 results
Implementation of an asymmetric mean flow model, and calculation of dis-
torted turbulence. Consideration of asymmetric mean flows is of interest to more
realistically capture true features of inflow into an AOR and in Chapter 4 we considered a
mean flow induced by two actuator disks side by side, separated by distance d. In Figures
4.7 - 4.11 we plotted the distorted turbulence spectrum, and compared these results to the
undistorted case, as well as investigating the effect of reducing asymmetry by increasing
d.
As in the axisymmetric case, the energy level was found to be higher at the blade tip
than the hub in all cases, but dependence of energy level on distortion level was more
complex, with the low distortion case lying between the high and medium distortion cases,
seen in Figure 4.9. As Sij depends upon φ for the asymmetric case we plotted results for
a range of φ values. The largest difference in spectral level at different radial positions
was found to be in the region φ = 0, i.e. closest to the other rotor (see Figure 4.3), where
a reduced streamtube contraction is experienced (Figure 4.5). As d increases both the
φ = 0 level increases, and the φ = pi level decreases, and thus the axisymmetric case lies
between the two.
Extension of our UDN model to non-axisymmetric mean flows. In calculating
the distorted amplitude, Aij , in the asymmetric case we require both Y and Z (or equiva-
lently R,Φ) in place of R alone in the axisymmetric case. In Chapter 4, the general form
for Aij in polar coordinates was derived, see equation (4.22). We note the potential for
errors when translating between the cartesian and polar reference frames. The definition
of Aij in polars is not intuitively obvious and the derivatives were found to act only on
certain terms of the quantities Xr = R cos(Φ− φ), and Xφ = R sin(Φ− φ).
The key change to the model in the asymmetric case was then in the calculation of a
Fourier Series to represent Aij, as now the extra terms Jn(rkR), einΦ and eikxX also contain
φ dependence. New quantities C˜m,nij and D˜
m,n
j were defined. Through substituting these
into expressions for the pressure jump across blades, and the total force term, changes
were found in the sound pressure spectrum both to the indices which are summed over,
and the integral over φk.
Calculation of the effect of asymmetry upon noise radiated from one rotor.
In Figures 4.12, 4.13 and 4.14 we plotted the SP from a single rotor with asymmetric
170 6. Summary and Conclusions
inflow. Key findings are that asymmetry lowers the broadband level, except in the flight
(low distortion) case around 1 BPF, where the level is raised by more than 5 dB. The
tonal level in the high distortion case is lowered somewhat due to asymmetry, but not
as significantly. This agrees with our earlier observation that the precise form of the
distortion primarily affects the broadband level.
Calculation of the UDN radiated from two rotors, side by side. Directivity was
also calculated in Chapter 4, when we considered the far-field pressure due to two adjacent
open rotors. This extension involved substitution of the far-field observer distance from
the second rotor into the phase term of the Green’s function. Our results, plotted in
Figure 4.17, demonstrate that the sound directly beneath the rotors simply adds, whereas
out to the sides the sound level is decreased significantly at certain observer positions.
6.2.4 Chapter 5 results
Development of a mean flow model to capture the characteristics of a stream-
tube contraction into a rotor at incidence. By superposing a tilted actuator disk
upon a uniform flow this new velocity field was derived.
Calculation of the effect of incidence upon the distorted turbulence spectrum.
In Figures 5.3 - 5.7 several aspects of turbulence distortion at incidence were investigated.
The effect of the asymmetry due to incidence was seen to be qualitatively similar to
that of two adjacent rotors. Of particular interest is the observation that as the angle of
incidence, α, increases, the difference between the distorted spectra at varying azimuthal
position in fact diminishes.
Application of Hanson’s Green’s function to derive an expression for the sound
radiated from a rotor at incidence. As shown in Figure 5.9, the effect of incidence
upon the radiated sound is similar in certain respects to that of asymmetry due to two
adjacent rotors plotted in Figure 4.12. The broadband level in the high distortion case is
lowered, and in the low distortion case is raised around 1 BPF. In addition, in the medium
distortion case, there is a change to the tonal level in the case of non-zero incidence, which
increases and becomes narrower. When the distortion is of an intermediate level, and the
flow is neither dominated by the upstream axial flow or a strong actuator disk, the effects
of incidence are seen to be most significant. Finally, in Figure 5.11, the greatest differences
6.3 Overall conclusions 171
in directivity at different azimuthal positions were observed to be around θ = pi/4 and
θ = 2pi/3.
6.3 Overall conclusions
6.3.1 Findings of most relevance to industry
• Of the various new axisymmetric flow features we have considered, use of
the variable circulation actuator disk has the greatest effect. The bullet’s
effect appears to be fairly localised near the hub, where the blade travels more slowly,
and thus has a less significant effect on noise levels. Inclusion of a rear actuator disk
made very little difference to UDN within this model. Scaling the velocity profile to
lie between the hub and the tip, rather than using the original actuator disk model,
was also a useful technique.
• Tonal UDN peaks are primarily affected by the total streamtube con-
traction, whereas broadband level is primarily affected by the form of
the distortion. The form of the distortion, as parametrised by ∂Xi/∂xj will be
affected by upstream features and asymmetries.
• The form of the upstream turbulence can have a very significant effect on
UDN level. Further experimental investigation of the spectral form of turbulence
shed from installation features would be very useful in the construction of UDN
models.
• Atmospheric turbulence parameters vary widely. Appropriate values for use
within prediction models might be L ∼ 10 m and
√
u2∞,1 ∼ 0.25m s−1, however the
integral lengthscale varies proportionally with height off the ground in reality and
turbulence intensity tends to decrease with height. Within our model increasing
L above the radius of the rotor is found to shift the noise spectrum down fairly
uniformly, so the tonal peaks which are present for L ∼ rd remain.
• Asymmetries in the mean rotor inflow raise the broadband level for fre-
quencies around 1 BPF, but lower it everywhere else. In addition, the effect
of incidence is most significant when the level of distortion is neither very large nor
very small.
172 6. Summary and Conclusions
• Our model predicts UDN levels to be around 10 dB lower at approach
than at take-off. In Figure 6.1 we show the results of our model using real
industry parameters. As can be seen, the approach level is below that of take-off at
all frequencies. The equivalent static result has also been included, demonstrating
the formation of tones at high distortion.
1 BPF (t−o) 2 BPF (app) 2 BPF (t−o)
ω
SP
20 dB
Take−off, static
Take−off, flight
Approach, flight
Figure 6.1: Baseline plot, run for realistic open rotor parameters.
6.3.2 Findings of most relevance to theoreticians
• Overall, the precise form of the distortion does make a few dB difference
to peak levels, and can make a significant difference to broadband levels.
However, the magnitude of the streamtube contraction remains the overriding key
factor in determining UDN levels.
• It is essential to limit the integral over transverse wavenumber, rk, in
order to achieve convergence within this model. A method for doing so has
been detailed.
• Extension to the asymmetric case is possible within this framework, but
takes a relatively longer time computationally. Modal analysis to identify
which terms contribute most to expressions, and where to cut off infinite sums, is
of importance here.
6.4 Further work 173
• This work has highlighted the non-trivial nature of moving between carte-
sian and polar reference frames with tensor-like quantities. Inclusion of all
azimuthal terms was seen to affect UDN results.
• A simple fluid dynamical model for the streamlines induced around the
upstream ‘bullet’ has been presented.
6.4 Further work
Areas which could now be investigated further, which we did not have time to examine
in detail during this Ph.D., include
• Implementation of a full stationary phase argument when determining appropriate
limits for the integral over rk. Some progress has been made towards this, as outlined
in Appendix D.
• Use of a more precise isolated blade response model, in place of the LINSUB cascade
model.
• Inclusion of entropy fluctuations within the RDT analysis, to simulate the ‘hot roots’
configuration of AOR, where the engine exhaust interacts with blades.
• Investigation of the effect of different blade leading edge profiles upon UDN.
• Consideration of further turbulence models. For example anisotropic models such as
the Batchelor, Karman-Howarth and Birkhoff-Saffman spectra, and the turbulence
shed behind a cylinder. We note that it would be a straightforward extension to
adjust the Liepmann model, through the use of an exponential correction (Posson
and Roger, 2011), to reproduce a faster decay at the top of the inertial range.
• Analysis of how UDN varies during take-off, as flight speed and incidence angle vary
with time.
• Improved approximations for the distortion amplitude, Aij .
174 6. Summary and Conclusions
Appendix A
Legendre functions and actuator disk
streamfunction
Legendre functions
The form of Legendre Function used in this thesis is
Qn− 12 (ω) =
∫ pi
2
−pi2
cos 2nα
[
2 (ω − 1) + 4 sin2 α
] 1
2
dα. (A.1)
The derivatives with respect to ω, (Q′
n− 12
etc.) then follow straightforwardly.
Actuator disk streamfunction
The constant circulation actuator disk velocity streamfunction is given by
Ψa.d. =
Udx
2pi
(
−E(κ)κ
√
rdr +
κ(x2+2r2d+2r2)K(κ)
4
√
rdr
)
+ Ud4
(
r2 − |r
2
d−r2|Λ0(β,κ)
2
)
for r ≤ rd,
Udx
2pi
(
−E(κ)κ
√
rdr +
κ(x2+2r2d+2r2)K(κ)
4
√
rdr3
)
+ Ud4
(
r2d −
|r2d−r2|Λ0(β,κ)
2
)
for r > rd,
(A.2)
175
176 A. Legendre functions and actuator disk streamfunction
where
K(κ) =
1
κ
Q− 12 (ω) , (A.3)
E(κ) =
(
1− 1
2
κ2
)
K (κ)− κ
2
Q 1
2
(ω) , (A.4)
F (β, κ′) =
∫ β
0
dθ√
1− κ′2 sin2 θ
, (A.5)
E(β, κ′) =
∫ β
0
√
1− κ′2 sin2 θdθ, (A.6)
Λ0(β, κ) =
2
pi
[E(κ)F (β, κ′) +K(κ)E(β, κ′)−K(κ)F (β, κ′)] . (A.7)
Appendix B
Definition of sound metrics
There are many different metrics used to measure sound. In this appendix we give precise
definitions for the metrics used when plotting results for this thesis.
Firstly we note that any quantities we use must include a 〈, 〉-average, so that we can
substitute a closed form for the upstream turbulence spectrum (such as the von Ka´rma´n
spectrum).
Power Spectral Density (PSD)
The Power Spectral Density is defined (Dowling and Ffowcs Williams, 1983) as the Fourier
Transform of the autocorrelation of the acoustic pressure
Pˆ (x, ω) =
∫ ∞
−∞
〈(p (x, t))∗ p (x, t+ τ)〉 eiωτdτ. (B.1)
The acoustic pressure, p, is a real quantity of course (and thus the complex conjugate
is redundant), but Pˆ will in general be complex. This is one of the quantities for which
explicit expressions are given in the main text, see equation (2.133) for example. Pˆ has
dimensions of
[
ρ2 L
4
T 3
]
. Energy has dimensions
[
ρL
5
T 2
]
, and power has dimensions
[
ρL
5
T 3
]
(rate of energy).
177
178 B. Definition of sound metrics
Power and Power Level (PWL)
Majumdar (1996) then defined the total radiated ‘power’ in the far-field as
P (ω) =
∫
sphere of radius σ
Pˆ (x, ω)σ2 sin θdθdφ. (B.2)
This quantity in fact has dimensions of
[
ρ2 L
6
T 3
]
, not power. However, it does give the
correct trends of location and height of tones etc. From this, Majumdar defined Power
Level (PWL) as
PWL = 10 log10
(
P (ω)
10−12 watts
)
dB. (B.3)
Since dB are not an absolute unit but a relative one, it would be possible to choose
a lengthscale to use in order to give P the correct dimensions for this definition (i.e.
dimensions of power). We would simply be changing what the reference level in the
denominator refers to.
Sound Pressure Level (SPL)
Again, from Dowling and Ffowcs Williams we have
SPL = 20 log10
(
p′rms
2× 10−5watts/m3Hz
)
dB (B.4)
= 10 log10
(
(p′rms)
2
(2× 10−5watts/m3Hz)2
)
dB. (B.5)
The quantity SPL is used widely. However, it is typically plotted as a function of fre-
quency, rather than time, as would be the case if pressure was used directly, as above.
To convert to frequency space, we use the Fourier Transform of pressure, and change the
reference quantity in the denominator.
Thus we use the following definition for SPL in this thesis
SPL = 10 log10
(
(F.T. of p2)
(2× 10−5watts/m3Hz)2 /Hz
)
dB (B.6)
= 10 log10
(
Pˆ (x, ω)
(2× 10−5watts/m3Hz)2 /Hz
)
dB. (B.7)
179
An alternative approach is to ‘undo’ the Fourier Transform of Pˆ by integrating over
frequency, and this is what is done when 1/3 Octave bands are taken. However, 1/3
Octaves have the effect of smoothing the frequency profile, and we would thus lose the
tones which are a key characteristic of UDN.
Intensity Level (IL)
We can obtain an expression for Intensity Level per unit frequency using the plane wave
result, acoustic intensity = p′2/ρ0c0. Thus, in Robison and Peake (2010) we plot the
quantity
IL (x, ω) = 10 log10
(
Pˆ (x, ω)
ρ0c0 [10−12watts/m2Hz]
)
dB. (B.8)
Again, this is really an ‘intensity level per frequency’.
180 B. Definition of sound metrics
Appendix C
Sound from non-identical adjacent
rotors
Here we give the general expression for the correlation of far-field acoustic pressure due
to the UDN generated by two non-identical adjacent rotors.
Within the expression for Pˆ as given in equation (4.35) the affected quantities are: β,
W , ωΓ (an input to ΓWW ) and kx. In addition, rd and B appear in several places, and
we note that φ0(r) and x0(r) may change as the blade geometry is likely to be altered
if the rotation is in a different direction. (We have set φ0(r) = φ0 = constant and
x0(r) = x0 = constant throughout our numerical calculations.) Examining the process
which led to the form of Pˆ given in equation (4.35), we see firstly that the statistical
average which acts upon the Fourier Transforms of u∞ remains the same. The flow
upstream of both rotors is the same, and so u∞ is of the same form. Thus kx = k′x when
we perform the k′ integral. Next, the time average now acts on general exp (ilB1Ω1t)
and exp (−il′B2Ω2t) terms, whereby t dependence is removed from the expression, and
we require values of l and l′ which satisfy
lB1Ω1 = l′B2Ω2. (C.1)
If Ω1 = −Ω2 and B1 = B2, we find l = −l′. However, for completely general Ωi and Bi
we must have B1Ω1/B2Ω2 = q/s where q, s are integers in order to find any solutions.
Finally, the e
iωσ
c0 factors no longer cancel.
181
182 C. Sound from non-identical adjacent rotors
Thus we find the most general expression for the cross term of Pˆ due to the pressure
fields of two rotors is
Pˆcross (σ, θ, φ, ω) =ρ20
B1B2
4pi2U∞
e
−iω(σ1−σ2)
c0
4σ1σ2 (1−M cos θ)2
∑
l,l′,m,m′
∫
rs,xs
[Jm+lB (γ0rs)]
∗ [W1(rs)ΓmWW (xs, rs;ω,Ω1)]
∗
[
iω cos θ sin β1(rs)
c0 (1−M cos θ)
− i(m+ lB) cosβ1(rs)
rs
]
exp
{ −iω cos θxs
c0 (1−M cos θ)
+ i(m+ lB1)
[xs − x0,1(rs)] tanβ1(rs)
rs
+ ilB1φ0,1(rs)
}
∫
r′s,x′s
Jm′+lB (γ0r′s)W2(r
′
s)Γ
m′
WW (x
′
s, r
′
s;ω,Ω2)
[
−iω cos θ sin β2(r
′
s)
c0 (1−M cos θ)
+
i(m′ + lB) cosβ2(r′s)
r′s
]
exp
{
iω cos θx′s
c0 (1−M cos θ)
− i(m′ + l′B2)
[x′s − x0,2(r′s)] tanβ1(r′s)
r′s
− il′B2φ0,2(r′s)
}
∑
p,p′
∫
ℜ
D˜m,p∗j (x0,1(rs), rs; kx, rk, 0)D
m′,p′
k (x0,2(r
′
s), r
′
s; kx, rk, 0)
[∫ 2pi
0
ei(m+p−m
′−p′)φkS∞kl (k) dφk
]
rkdrkdr′sdx
′
sdrsdxs
ei(m
′+l′B2−m−lB1)(φ−pi2 )e−i(m
′−m)pi2 , (C.2)
where kx = (ω − lBΩ)/U∞. The total Power Spectral Density is then
Pˆ2r (x, ω) = Pˆ1 (x, ω) + 2 cos
(
ω sin θ1 cosφ1
c0
)
Pˆcross (x, ω) + Pˆ2 (x, ω) . (C.3)
Appendix D
Stationary phase argument
Finally we detail a more sophisticated argument which could be used when defining the
rk integral limits within
∫ 2pi
0 Pˆ dφ, but which we did not implement in the numerics.
Within the expression for the pressure spectrum (2.133) we see two integrals of the
form
I =
∫ rd
rh
f(rs)Jα (γ0rs) Jβ (rkR(rs)) drs. (D.1)
We can use a multivariate stationary phase argument (as detailed in §9.6 of Jones (1982))
to pick out the key rs values, given α, β and rk, which contribute most to this integral.
This relationship can then be inverted to allow us to consider a restricted range of rk
values for each pair of rs and r′s. Substituting in the exponential definition of the Bessel
function
Jα (γ0rs) =
∫ pi
−pi
e−i(ατ−γ0rs sin τ)dτ, (D.2)
we find
I =
∫ rd
rh
f(rs)
[∫ pi
−pi
e−i(ατ−γ0rs sin τ)dτ
] [∫ pi
−pi
ei(βτ
′−rkR(rs) sin τ ′)dτ ′
]
drs. (D.3)
We have used the complex conjugate of (D.2) when substituting for Jβ, which is a real
quantity. Now the phase term in three variables is
φ (rs, τ, τ ′) = −ατ + γ0rs sin τ + βτ ′ − rkR(rs) sin τ ′. (D.4)
183
184 D. Stationary phase argument
When extending the theory of stationary phase to expressions of more than one variable
we seek points of zero gradient of the phase, thus we require
∂φ
∂rs
= γ0 sin τ − rk
∂R
∂rs
sin τ ′ = 0, (D.5)
∂φ
∂τ
= −α + γ0rs cos τ = 0, (D.6)
∂φ
∂τ ′
= β − rkR cos τ ′ = 0. (D.7)
Using (D.6) and (D.7) to eliminate τ and τ ′ from (D.5), we find the following condition
at points of stationary phase, rs = r∗s
√
γ20r∗2s − α2
r∗s
=
∂R
∂r∗s
√
r2kR(r∗s)2 − β2
R(r∗s)
. (D.8)
To determine the stationary phase multiplicative factor we calculate the determinant
of the Hessian matrix at rs = r∗s . After some algebra, this is found to be
det (Hessian) =
1
R
∂2R
∂r2s
(
r2kR
2 − β2
) (
γ20r
2
s − α2
) 1
2
− α
2
r2s
(
r2kR
2 − β2
) 1
2 +
(
∂R
∂rs
)2 (
γ20r
2
s − α2
) 1
2 . (D.9)
The stationary phase result is that, as the phase term gets large,
I ∼
√
(2pi)3
|det (Hessian)|f(r
∗
s)e
−i(ατ∗−γ0r∗s sin τ∗)+i(βτ ′∗−rkR(r∗s ) sin τ ′∗)e−
ipi
4 sgn(det(Hessian)). (D.10)
Given m, p and p′, we can pick a discrete set of values of rk which give a range of values
of r∗s and r
′∗
s between rh and rd. Thus rk will lie between r
h
k and r
d
k, defined via
γ20 −
α2
r2h
=
(
∂R
∂rs
)2∣∣∣∣∣
rs=rh
(
(rhk)
2 − β
2
(R(rh))
2
)
, (D.11)
γ20 −
α2
r2d
=
(
∂R
∂rs
)2∣∣∣∣∣
rs=rd
(
(rdk)
2 − β
2
(R(rd))
2
)
. (D.12)
This will lead to an increased number of r stations at which ∂Xi/∂xi needs to be calculated
numerically, which is likely to be time consuming for asymmetric flows.
Nomenclature
Aij distortion amplitude tensor
α angle of rotor incidence
b label for a specific blade
B total number of blades
β blade angle
c blade chord length
c0 speed of sound
Cpij p
th Fourier coefficient of Aij
C˜m,nij Fourier coefficient of Aij in asymmetric system
d separation distance between rotors, in asymmetric flow
Dpj combination of C
p
ij components
D˜m,nj combination of C˜
m,n
ij components
D0
Dt material derivative in mean flow
D∞
Dt material derivative in uniform upstream flow
δ Dirac delta function
δij Kronecker delta
∆ drift function
185
186 Nomenclature
eˆi basis of unit vectors
E turbulent energy spectrum
EG Gaussian three-dimensional energy spectrum
ELM Liepmann three-dimensional energy spectrum
E1 a distorted spectrum energy measure
E2 a distorted spectrum energy measure
η spatial separation vector
f wave equation forcing term
fki forcing term due to single upstream wavevector component
F l,m,p component of forcing term
g axial gap between rotors in co-axial system
g1 constant used in the von Ka´rma´n spectrum
g2 constant used in the von Ka´rma´n spectrum
γ ratio of specific heats
Γ blade circulation
ΓWW non-dimensionalised blade circulation
ΓmWW non-dimensionalised blade circulation, for particular azimuthal order
H l,m Green’s function
Jn Bessel function of order n
k wavenumber vector
k modulus of wavevector k
kr component of wavevector in eˆr direction
kt effective radial wavenumber
Nomenclature 187
kφ component of wavevector in eˆφ direction
K LINSUB kernel
l distorted wavenumber vector
l summation index which arises from sum over blade number
L integral length scale of free-stream turbulence
λ reduced frequency
m azimuthal order
m strength of point source
M Mach no. of upstream uniform flow
MW Mach no. in chordwise direction
nˆ normal vector on solid body surface (RDT)
nc number of chordwise points considered in LINSUB
N unit blade normal
p total pressure (mean plus perturbation)
p far-field sound pressure
p (superscript) azimuthal order of Aij
p0 mean pressure
p′ perturbation pressure
pl,m,p component of far-field sound pressure
pˆl,m,p amplitude of component of far-field sound pressure
∆pb pressure jump across blade b
P correlation of far-field pressures
Pblade correlation of blade pressures
188 Nomenclature
Pˆblade spectrum of blade pressures
Pˆ Fourier transform of far-field pressure spectrum
Pˆ2r Pˆ due to two adjacent rotors
φ azimuthal angle
φ′ azimuthal angle in rotating frame
φ′ in Chapter 5, azimuthal angle in tilted coordinate system
φ0 azimuthal position of leading edge of blade
φk azimuthal component of wavevector
φs azimuthal source coordinate
Φ far upstream azimuthal position
ψ potential of irrotational component of distorted velocity
ψ′ component of ψ which satisfies the boundary condition on rotor blades
ψ0 component of ψ which satisfies the inhomogeneous RDT wave equation
Ψ streamfunction
Ψa.d actuator disk streamfunction
Ψs streamfunction due to point source
Qn Legendre functions
Q′n derivative of Legendre function, with respect to its argument
r radial coordinate
rd rotor radius
rh hub radius
rk radial component of wavevector
rs radial source coordinate
Nomenclature 189
R far upstream radial position
Rij distorted velocity correlation tensor
R∞ij upstream turbulence velocity correlation tensor
ρ total density (mean plus perturbation)
ρ′ perturbation density
ρ0 mean density
ρ∞ constant density of uniform upstream flow
s blade spacing
S amplitude radius
S0 leading order term of S
S1 first order term of S
Sij distorted turbulence spectrum
SGij Gaussian spectrum tensor
SLMij Liepmann spectrum tensor
S∞ij upstream turbulence spectrum tensor
σ phase radius
σ0 distance of observer from rotor, emission coordinate
t time
T total propeller thrust
τ temporal separation vector
θ polar angle between observer and rotor, emission coordinate
θ′ in Chapter 5, polar angle in tilted coordinate system
Θxx one-dimensional turbulent energy spectrum
190 Nomenclature
u perturbation velocity field (distorted turbulence)
u∞ turbulence velocity far upstream
ugusti distorted velocity due to single upstream gust
urot.i velocity in rotating reference frame
uvort vortical component of distorted velocity
〈u2〉 average of turbulent velocity
u2∞,1 mean square speed of the axial component of turbulent velocity
U mean flow induced by the rotor
U∞ uniform axial speed far upstream (flight speed)
Ua.d.i velocity induced by constant circulation actuator disk
Ubul.i velocity induced by bullet system
U co-ax.i velocity induced by co-axial actuator disks
Uvar.i velocity induced by variable circulation actuator disk
Ud ‘strength’ of the actuator disk
Uf total axial velocity at disk face
Ur radial velocity induced by rotor
Ux axial velocity induced by rotor
Usx axial velocity due to point source
Uφ azimuthal velocity induced by rotor
v total velocity (mean plus perturbation)
wW amplitude of velocity perturbation at leading edge of blade
W mean flow speed in chordwise direction
ω temporal frequency parameter
Nomenclature 191
ωΓ LINSUB temporal frequency
Ω angular velocity of the rotor
x position vector
x′ tilted coordinate system, for rotor at incidence
x0 position of point source
x0(r) axial position of blade leading edge
x1 position of stagnation point in point source (bullet) model
xs axial source coordinate
X axial component of distortion vector, U∞∆
Xi distortion vector
Xvar.i Xi in variable circulation actuator disk case
χ inter-blade phase angle
Y far upstream position along y axis
zˆ chordwise coordinate
Z far upstream position along z axis
AOR advanced open rotor
BPF blade passing frequency
CAA computational aeroacoustics
CFD computational fluid dynamics
dB decibel
DNS direct numerical simulation
EPNL effective perceived noise level
LES large eddy simulation
192 Nomenclature
LINSUB ‘linearized subsonic’ code
PSD power spectral density
PWL power level
RANS Reynolds averaged Navier-Stokes
RDT rapid distortion theory
RSF rotational shaft frequency
SP spectral power
SPL sound pressure level
UDF unducted fan
UDN unsteady distortion noise
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