Progress in Computational Aeroacoustics in Predicting the Noise Radiated from Turbulent Flows

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1. INTRODUCTION
In recent years computational aeroacoustics has been
shown to be a growing powerful tool in the prediction of the
radiated noise from unsteady flows. At low Reynolds
numbers Colonius et al.
1
and Mitchell et al.
2
have extended
the method of Direct Numerical Simulation (DNS) for calcu-
lations of the radiated sound from shear layers. In these
calculations it was found that one of the dominant noise
sources involved the interaction of vortices and in particular
their pairing. This was one of the noise sources Laufer had
argued was a prime noise source in a jet and pioneered
research on this subject throughout the 1970's and early
1980's (see Laufer and Yen
3
). The calculated unsteady flow
results of Colonius et al. and Mitchell et al. were also used to
predict the far field radiated noise using Lighthill's acoustic
analogy
4
and by using Kirchhoff`s theorem. Good agreement
was obtained except at shallow angles to the jet where the
use of Lighthill's equation significantly overpredicted the
far-field sound possibly due to the neglect of refraction of the
sound by the mean flow. Even for these flows of relatively
simple geometry the computations are expensive since,
unlike standard CFD calculations, they must be time and
space accurate. They are only possible when the number of
modes and frequencies are strictly limited. Although such
pioneering work can not handle fully developed turbulence it
clearly is important in examining the methodology to be
followed in the numerical study of aeroacoustics.
Further extensive calculations have been undertaken by
Colonius et al.
1
on the complex interaction between the
sound generated and the flow field using both DNS and also
making comparisons with the extension to Lighthill's
equation to include flow-acoustic interaction as proposed by
Lilley and Goldstein
5
. The DNS results were found to be in
good agreement with solutions to the Goldstein-Lilley
equation, and displayed the importance of refraction in flow
fields having extensive volumes in the streamwise direction.
For turbulent flows the main attack has been made by
Sarkar and Hussaini
6
and Lilley
7
on studying isotropic turbu-
lent flows at low Mach numbers and low Reynolds numbers
using the space-time flow data base to predict the radiated
noise using Lighthill's acoustic analogy. As reported above,
such low Reynolds number data is unrepresentative of fully
developed turbulent flows. Although databanks exist for
compressible isotropic turbulent flows, no results have so far
been reported of the retarded-time space correlations required
for the prediction of the far field radiated sound. An approxi-
mate extension of the incompressible space-time data has
been reported by Lilley
8
for the case of temperature fluctua-
tions and the use of the isotropic turbulence model as an
approximate predictor for the noise radiated from a jet over a
wide range of Mach numbers and temperatures. What is
urgently needed is the numerical calculation of the character-
istics of the fourth-order space-time covariance as a function
of typical homogeneous and non-homogeneous turbulent
shear flows to validate the above results and establish if it is
possible to use them as the building blocks for the prediction
of noise in more complex flows. Such computations are,
however, expensive since they require an accurate resolution
in space and time. These statements in respect to DNS apply
also to methods of Large Eddy Simulation (LES).
In a number of flows of practical importance it has been
found that the flow is dominated by large scale structures and
that the flow, including the large scale structures, can be
calculated using the Reynolds Averaged Time Dependent
Navier-Stokes equations. Once the unsteady flow has been
solved to the necessary accuracy the radiated noise can then
be calculated using Lighthill's acoustic analogy or if needs
be, one of the more complex acoustic analogies when impor-
International Journal of Acoustics and Vibration, Vol. 2, No. 1, 1997
3
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*
Department of Aeronautics and Astronautics, University of Southampton, SO17 1BJ, United Kingdom
In recent years a number of simple unsteady flows involving the interaction between vortices have been studied
using computational fluid dynamics. These have been extended to include the sound radiated to the far field
either by Direct Numerical Simulation, by the use of acoustic analogies, or by the use of Kirchoff methods. For
more complex flows results have been obtained using methods based on solving the time dependent large scale
flow structures using the unsteady Reynolds Averaged Navier-Stokes equations and then using acoustic analo-
gies to derive the noise in the radiation field. Some success has been made with the latter methods in the predic-
tions of the noise radiated from the flow over cavities at supersonic speeds, where the noise characteristics are
dominated by large scale events associated with self-excited flow oscillations. Similar methods are being applied
to other self-excited flows, and ultimately to turbulent flows such as jets. The paper describes these methods and
results together with some limited preliminary comparisons with experimental data. In an Appendix an extension
of Lighthill's equation for aerodynamic noise is presented covering the effects of flow-acoustic interaction.
*
Fourth International Congress on Sound and Vibration, St. Petersburg, Russia, June 24-27, 1996
tant flow-acoustic interactions are present. A typical class of
flows for which this method is applicable is that of self-
excited flows, such as the cavity exposed to a supersonic
freestream. Recent results for the flow have been reported by
Zhang et al.
9
The success in predicting the far-field noise
depends on the accuracy of the flow solver in providing a
numerical data base for comparison with experimental flow
data. This paper discusses the theoretical background used in
this work and finally discusses some early preliminary
comparisons with experiment.
2. FLOW FIELD
We consider a turbulent compressible flow field governed
by the conservation equations of mass, momentum, and
energy for a perfect gas satisfying given initial and boundary
conditions. We assume the turbulent flow possesses a well-
defined and deterministic large scale structure which coexists
with an uncorrelated random small scale structure which
provides the mechanism for the dissipation of kinetic energy
into heat. Our flow solver has a resolution in space and time
sufficient to capture the large scale structure and the compu-
tation is run for a sufficiently long time for the flow to
exhibit its self-excited oscillation characteristics when these
exist. The flows studied include large regions of separated
flow and hence present a challenge to find a flow-solver
which is sufficiently robust to predict accurately the time-
dependent large scale features of such an unsteady complex
flow.
An earlier attempt at studying the properties of the large
scale deterministic structure in turbulent fluids was made by
Morris et al.
10
In that study of the two-dimensional mixing
region the large scale eddy structure was determined by a
linearised modal approach where it was found the initially
unstable modes eventually reached a saturation amplitude as
they were convected downstream into a region of greater
total thickness where the mode became damped. One impor-
tant feature of this methodology was the requirement to
include a dissipation model and it was found that an adequate
model was simply to allow the large eddy structure to disap-
pear once it passed through neutral equilibrium. The detailed
calculations showed results in good agreement with the
experiments of Brown and Roshko
11
. In this case non-linear
interactions were not found to be significant in the determi-
nation of the primary structure but secondary structures
involving the presence of longitudinal vortices require more
complex three-dimensional and finite amplitude treatment.
The present approach is numerical rather than analytic
and is fully non-linear and three-dimensional, although in the
example discussed here it has been restricted to a two-
dimensional flow only.
3. LARGE SCALE COMPRESSIBLE TURBULENT FLOW
TIME DEPENDENT EQUATIONS
3.1. Notation
The flow variables
￿￿
p,h,v are respectively the instantane-
ous density, pressure, enthalpy and vector velocity. They are
decomposed into a time mean, (bar), a large scale structure,
(prime), and an uncorrelated small scale structure, (double
prime). Thus
, (1)
￿￿x,t￿ ￿ ￿
￿x￿ ￿￿
￿
￿￿
￿￿
and similarly for all other flow variables. It is convenient
to combine the mean flow and large scale fluctuations by
introducing the tilde variable such that
, (2)
￿
˜
￿x,t￿ ￿ ￿
￿x￿ ￿￿
˜
￿
and similarly for all other flow variables. The tilde opera-
tion on results in no change to the argument. The tilde
￿
￿x￿
variable may be thought of as a time average which is long
compared with the characteristic time of the small scale
fluctuations, but small compared with the characteristic time
for the development of the large scale structures. Thus
, (3)
￿
￿￿
￿ 0
and similarly for all other flow variables. With this
decomposition the large fluctuations in the flow variables are
generated by the large eddies. However short-time averaged
flow equations contain contributions from products such as
and . These terms involving products with
￿
￿￿
v
￿￿
￿
￿￿
v
￿￿
v
￿￿
￿
￿￿
can either be modelled using Favre averaging or can be
neglected whenever . It is the latter case which
￿
￿￿
v
￿￿
￿
￿˜v
˜
we regard as appropriate in the work described below. Thus
although density fluctuations are present in the tilde variables
involving the density, it is assumed their small scale fluctua-
tions make little added contribution when the added variables
involve , , and similarly with other products involving
v
￿￿
v
￿￿
v
￿￿
small scale fluctuations. A consequence is that the instantane-
ous equation of continuity reduces to
. (4)
￿￿ ￿￿˜v
￿￿
￿ ￿ 0
The equations of conservation of mass, momentum and
energy are obtained by performing a short-time average as
discussed above.
3.2. Conservation of Mass
(5)
￿
￿t
￿˜ ￿ ￿ ￿ ￿￿˜v
˜
￿ ￿ 0
3.3. Conservation of Momentum
(6)
￿
￿t
￿￿˜v
˜
￿ ￿ ￿ ￿ ￿￿˜v
˜
v
˜
￿ ￿ ￿￿p˜ ￿ ￿ ￿ ￿￿
˜
￿ ￿˜
v
￿￿
v
￿￿
￿
3.4. Conservation of Kinetic Energy of Short-time
Averaged Motion
￿
￿t
￿￿˜v˜ ￿ v˜/2￿ ￿ ￿ ￿ ￿￿˜v˜v˜ ￿ v˜/2￿ ￿
(7)￿ ￿v
˜
￿ ￿p˜ ￿ ￿ ￿
￿￿
˜
￿ ￿
˜
v
￿￿
v
￿￿
￿ ￿ v
˜
￿ ￿￿
˜
￿ ￿
˜
v
￿￿
v
￿￿
￿:￿
v
˜
G. M. Lilley, X. Zhang and A. Rona: PROGRESS IN COMPUTATIONAL AEROACOUSTICS
International Journal of Acoustics and Vibration, Vol. 2, No. 1, 1997
4
3.5. Conservation of Kinetic Energy of Small
Scale Motion
This is obtained from the short-time average of the scalar
product of the instantaneous equation of motion with
v
￿￿
￿
￿t
￿￿˜
v
￿￿
￿ v
￿￿
/2￿ ￿ ￿ ￿ ￿￿˜v˜
v
￿￿
￿ v
￿￿
/2￿ ￿ ￿
v
￿￿
￿ ￿p˜
￿￿
￿
￿￿ ￿ ￿￿
˜
v
￿￿
v
￿￿
￿ v
￿￿
/2￿ ￿ ￿ ￿ ￿
￿
￿￿
￿ v
￿￿
￿ ￿
￿
￿￿
:￿v
￿￿
￿ ￿￿
˜
v
￿￿
v
￿￿
￿:￿v
˜
(8)
3.6. Conservation of Heat Energy
￿
￿t
￿￿˜h
˜
￿ ￿ ￿ ￿ ￿￿˜v
˜
h
˜
￿ ￿
￿p˜
￿t
￿ v
˜
￿ ￿p˜ ￿ ￿ ￿ q˜ ￿ ￿
˜
:￿v
˜
￿
(9)￿
￿
￿￿
:￿v
￿￿
￿
v
￿￿
￿ ￿p
￿￿
￿ ￿ ￿ ￿￿˜
v
￿￿
h
￿￿
￿
3.7. Conservation of Total Enthalpy
We define the short-time averaged total or stagnation
enthalpy as , which includes the
h
˜
s
￿ h
˜
￿ v˜ ￿ v˜/2 ￿
v
￿￿
￿ v
￿￿
/2
kinetic energy of the small scale turbulence. Thus adding
Eqs. (7), (8) and (9), and noting ,
h
s
￿￿
￿ h
￿￿
￿ v
￿￿
￿ v
￿￿
/2 ￿ v˜ ￿ v
￿￿
￿
￿t
￿￿˜h
˜
s
￿ ￿ ￿ ￿ ￿￿˜v˜h
˜
s
￿ ￿
￿p˜
￿t
￿
(10)￿ ￿ ￿ ￿q˜ ￿ ￿˜
v
￿￿
h
s
￿￿
￿ ￿
˜ ￿ v˜ ￿
￿
￿￿
￿ v
￿￿
￿
3.8. Equation of State
, (11)p˜ ￿
￿￿1
￿
￿˜h
˜
where
￿
is the ratio of the specific heats. This set of
equations is similar to the standard Reynolds averaged
equations of compressible turbulent flow except that all the
flow equations and the flow variables are time dependent.
They can be solved provided similar modelling assumptions
are made for the Reynolds stresses, the turbulent dissipation
and the turbulent diffusion terms. In this approach we use a
2-equation k-
￿
turbulence model with a local kinematic eddy
viscosity equal to k/
￿
. Our 8-equations for the 8-variables
,
￿
are eqs. (5), (6), (8), (10), (11)
￿
˜
,

,
p
˜
,
h
˜
,
k
￿
￿
v
￿￿
￿ v
￿￿
/2
together with the
￿
equation. Either Wilcox's
12
k-
￿
or the
standard k-
￿
turbulence model can be used, provided care is
taken in noting that here k and
￿
are related to the contribu-
tions from the small scale eddies only, which equal the
properties of the turbulent scales unresolved in the calcula-
tions. These are much smaller quantities than used in the
conventional Reyn-olds-averaged methodology. Naturally
the new time-depend-ent Reynolds averaged scheme cannot
use the same constants as formulated in conventional
Reynolds averaged schemes and require calibration against
good, reliable and repeatable experimental data. The method
is used for free shear flows as well as wall bounded flows
with both attached and separated boundary layers. However
it is to be noted that in its present form it is mostly applicable
to flows which are self-excited where the large scale fluctua-
tions dominate the flow-field.
These equations can be solved for any flow Reynolds
number. The only restriction on the computational grid is that
it must be sufficient to capture the large eddies in the flow.
The success of the method depends on the flow solver used
for these time dependent Reynolds averaged equations.
Zhang and Edwards
13
have used a Roe flux difference split
approximate Riemann solver to calculate the inviscid fluxes
and to resolve the shock waves outside the boundary layer.
Full details of the numerical method are given in Zhang et
al.
9
In Fig. 1 we show just one of a family of interferograms
taken by Zhang and Edwards
13
in experiments on a cavity
flow at supersonic speeds, whereas Fig. 2 shows the compari-
son with one of the family of numerical interferograms
obtained by Zhang et al.
9
It is shown that most of the large
scale features have been obtained in the numerical calcula-
tions. This is confirmed by Rona
14
in a detailed quantitative
comparison between the calculated mean flow field and
experimental data. It is only when such agreement can be
obtained with a flow solver that it is possible to obtain a
reasonable accuracy for the noise radiated to the far field by
using Lighthill's acoustic analogy. This involves finding the
double divergence of through-
T
ij
￿
￿v
i
v
j
￿ ￿
ij
￿ ￿p ￿ c
￿
2
￿￿￿
ij
out the flow domain. Our attempts at finding the total acous-
tic power output have met limited success.
Figure 1. Experimental Interferogram showing shear layer
impingement M =1.5 Zhang and Edwards
13
.
￿
G. M. Lilley, X. Zhang and A. Rona: PROGRESS IN COMPUTATIONAL AEROACOUSTICS
International Journal of Acoustics and Vibration, Vol. 2, No. 1, 1997
5
Figure 2. Numerical Interferogram M =1.5 showing shear
￿
layer impingement Zhang et al.
9
4. THE LIGHTHILL ACOUSTIC ANALOGY FOR A
TURBULENT FLOW FIELD DOMINATED BY LARGE
SCALE STRUCTURES
The flow equations derived in Section 3 for the time-
dependent characteristics of a turbulent flow can be used to
determine the external acoustic field. Following Lighthill's
procedure we find, where I is the idem factor,
￿
￿
2
￿t
2
￿c
￿
2
￿
2
￿￿￿˜ ￿￿
￿
￿ ￿
, (12)￿ ￿￿ ￿￿ ￿￿˜v˜v˜ ￿￿p˜ ￿c
￿
2
￿˜￿I ￿￿
˜ ￿￿˜
v
￿￿
v
￿￿
￿
where the source terms involve the time dependent large
scale vortical motion superimposed on the mean flow. In
most aeroacoustic calculations it is found that the dominant
noise sources arise from the energy containing structures in
the flow. The viscous stress terms at sufficiently high
Reynolds numbers normally make only a very small contri-
bution to the total acoustic power. However in Eq. (12)
above the effective stress tensor involving the contributions
of the small scale (unresolved) turbulence makes a small but
non-neglible contribution to the acoustic power when based
on our eddy viscosity model for the small scale turbulence as
described in Section 3, which is perhaps not surprising in a
wall-bounded flow such as a cavity. The details of solving
the Lighthill acoustic analogy equation, with a simple correc-
tion to allow for refraction, follow closely the method
described in Lilley
8
. In order to investigate flow-acoustic
interaction we need to solve the basic anisentropic flow
equations for the time-dependent large scale flow, including
the modelling of the small scale structure, as described in
Section 3 above. The perturbations, which are assumed to be
isentropic, about this basic anisentropic flow, lead to a third-
order convected wave equation, similar to that proposed by
Goldstein and Lilley, which reduces to the acoustic wave
equation external to the flow. The source terms, which are
quadratic in the disturbances about the mean flow, involve
fluctuations in the velocity and enthalpy only, and do not
involve the perturbation densities within the flow. In our
treatment, so far, we have not considered this more complex
approach since the simpler Lighthill's acoustic analogy gave
an adequate approximation for the total acoustic power.
Details of the extended method, involving flow-acoustic
interaction, is given in the Appendix, where a comparison is
given between the acoustic sources and those obtained in
Lighthill's acoustic analogy.
5. CONCLUSIONS
The study of the noise radiated from complex turbulent
flows at high Reynolds numbers has been advanced by
numerically solving the time dependent Reynolds averaged
flow equations for the large scale structure, in which the
unresolved small scale structures are modelled using standard
two-equation turbulence models. The flow field databank
obtained is then interrogated to enable the acoustic source
terms to be evaluated using Lighthill's acoustic analogy. By
this procedure the radiated total acoustic power is found to be
of the correct order of magnitude. Further work is required to
extend the method to include flow-acoustic interaction
effects.
The method has been shown to be adaptable to complex
flows such as the supersonic cavity flow where self-excited
oscillations dominate the flow field. The success of the
method clearly depends on the accuracy of the flow solver in
comparison with experimental data. In the case of the super-
sonic cavity flow good agreement has been obtained with
most of the large scale features of the flow which dominate
the far-field sound radiation.
APPENDIX: THE CONVECTED WAVE EQUATION
FOR AERODYNAMIC NOISE IN SHEAR FLOW
TURBULENCE
Geoffrey M. Lilley
This Appendix describes the derivation of the equations
of aeroacoustics when in a complex turbulent flow the mean
flow field is given. The focus is on the description of the
acoustic sources in terms of the large scale fluctuations about
the mean flow.
In Lighthill's derivation of the equation for aerodynamic
noise the conservation equations of mass and momentum are
combined and give the inhomogeneous acoustic wave
equation
(A.1)
￿
2
￿
￿t
2
￿ c
￿
2
￿
2
￿ ￿
￿
2
T
ij
￿x
i
￿x
j
where is Lighthill's stress
T
ij
￿
￿v
i
v
j
￿
￿
ij
￿
￿
p ￿ c
￿
2
￿￿
tensor which is assumed known throughout the flow field.
Equation (A.1) is based on the existance of a finite fluctuat-
ing dilatation in the flow, which though in general a small
quantity, is responsible for the generation and radiation of
sound from the flow. Its value cannot be assumed or
modelled. Equation (A.1) is exact and involves, as shown by
Lighthill, a rearrangement of the Navier-Stokes equations.
Only a very small fraction of the flow kinetic energy escapes
as noise whereas is itself the same order of magnitude as
T
i
j
the local flow kinetic energy. In Lighthill's acoustic analogy
the flow field is replaced by an equivalent distribution of
acoustic sources of strength per unit volume, acting in a
T
i
j
medium at rest, in which the acoustic sources may move but
not the medium. The small fraction of the flow kinetic energy
that escapes as noise and is radiated to the far-field is derived
from the space-time properties of the double divergence of
the stress tensor, , allowing for the retarded time from
T
i
j
source to far-field observer. Equation (A.1) involves the
instantaneous values of the flow variables and thus contains
contributions from the mean and turbulent flow together with
the near source sound field generated by the flow and its
resulting refraction, diffraction and absorbtion from interac-
tion with the flow. These latter effects may often be of small
amplitude and may then be neglected in finding the total
acoustic power generated by the unsteady flow. Nevertheless
unless they are included in the directivity of the acoustic
T
i
j
radiation and its spectrum cannot be predicted satisfactorily.
Equation (A.1) is the equation for the density and pressure
fluctuations inside the flow and all its terms are, in general,
of equal magnitude. The exception is when the flow is of
very low Mach number and is nearly incompressible. Exter-
G. M. Lilley, X. Zhang and A. Rona: PROGRESS IN COMPUTATIONAL AEROACOUSTICS
International Journal of Acoustics and Vibration, Vol. 2, No. 1, 1997
6
nal to the flow Eq. (A.1) reduces to the ordinary acoustic
wave equation having the constant speed of sound, , since
c
￿
there is assumed to be zero, and the medium is disturbed
T
i
j
by sound waves alone.
When the mean flow convection velocity is finite an alter-
native approach is to describe the flow equations using
logarithmic thermodynamic variables in place of the
pressure,p, density,
￿
, and enthalpy, h. We define
. The square of the speed of
￿ ￿ ln
p
1/￿
:
￿
￿ ln
￿
:￿ ￿ lnh
sound, . The dilatation, . The basic flow
c
2
￿
￿
￿
￿1
￿
h
￿ ￿ ￿￿ v
equations of continuity, motion, state and energy for a perfect
gas in this notation, are respectively
, (A.2)
D
￿
D
t
￿ ￿￿
+ + , (A.3)
D
v
D
t
￿ ￿c
2
￿￿
￿￿
t
t
￿ ￿
￿
, (A.4)￿
D￿
d
t
￿
D
￿
D
t
￿
D￿
D
t
,
D￿
Dt
￿ ￿￿ ￿ 1￿
D￿
Dt
￿
D
s
/
C
p
Dt
￿ ￿￿ ￿ q ￿ q ￿ ￿￿ ￿ t:￿v￿/h
(A.5)
where t=￿
/￿,￿￿
=-2/3￿￿I+￿(￿v+v￿), and q=(￿/Pr)￿h.
On eliminating D￿/Dt between Eqs. (A.2) and (A.3) we
obtain Phillips equation
15
.
D
2
￿
Dt
2
￿ ￿ ￿ ￿c
2
￿￿￿ ￿ ￿v:v￿ ￿
D
2
s
/
C
p
Dt
2
￿ ￿ ￿ ￿￿ ￿ t ￿ t ￿ ￿￿￿
(A.6)
The final two terms, which arise from fluctuations in the
entropy and viscous stress gradients, can normally be
neglected at high Reynolds numbers. Thus at high Reynolds
number the dominant source term in the convected wave
equation, Eq. (A.6), is ￿v:v￿ so that
, (A.6a)
D
2
￿
Dt
2
￿ ￿ ￿ ￿c
2
￿￿￿ ￿ ￿v:v￿
or
. (A.6b)
D
2
￿
Dt
2
￿ ￿c
2
￿
2
￿￿ ￿ ￿v:v￿ ￿ c
2
￿￿ ￿ ￿￿
The physical process governed by the acoustic source
￿v:v￿ arises from the rate of deformation of the fluid
element, noting the three invariants of the rate of deformation
tensor include ￿,D￿/Dt, as well as the divergence of
D
2
￿/
D
t
2
the flow acceleration, , which is given by
￿￿
a
. (A.7)￿ ￿ a ￿ ￿ ￿
D
v
D
t
￿
D
￿
D
t
￿ ￿v:v￿
Thus the volume integration of ￿v:v￿ is the major contri-
bution to the flow acceleration, a, arising from rate of defor-
mation of the fluid element at a given point in the fluid. ￿v:v￿
is also a contribution to the rate of dissipation, ￿, since
, but its contribution to the
￿￿
￿
￿
￿2/3￿
2
￿ ￿
2
￿ 2￿v:v￿
￿
dissipation is small compared with the dominant term
, except near the boundaries of wall-bounded
￿
￿
￿2/3￿
2
￿ ￿
2
￿
flows.
In the ambient medium external to the flow the convected
wave equation, Eq. (A.6), reduces to the standard wave
equation with a constant speed of sound, . Similar
c
￿
equations to Eq. (A.6) can be found for the other dependent
variables, ￿ and ￿. It is also possible to find a convected
wave equation for the stagnation enthalpy, . The
h
s
￿ h
￿
v
2
/2
choice of dependent variable in the convected wave equation
appears to have no special significance and rests mainly on
how the flow field is defined. Here we consider the
convected wave equation in terms of the dependent variable
￿ only. The lefthand side of Eq. (A.6) is not linear in the flow
disturbance variable,
￿
, since the operator
D
/
D
t
￿ ￿/￿
t
￿
v ￿ ￿
is itself a function of the flow disturbance. We therefore
define a new convective operator based on the local mean
velocity, , so that is now independent
v
0
D
0
/
D
t
￿ ￿/￿
t
￿v
0
￿ ￿
of the disturbance field, and the difference terms between
D/Dt and are included on the righthand side of the
D
0
/
D
t
convected wave equation.
In order to examine the procedure in detail, let us now
consider the case of an anisentropic mixing region flow
where the mean flow is unbounded and has the following
properties. We assume the flow field is turbulent and the
dominant acoustic sources are based on the characteristics of
the large scale structures which contain the energy containing
eddies. In the work described above these large scale struc-
tures have been determined from the tilde variables covering
all the flow field in space and time. For convenience we will
write , where the mean value, , replaces and
￿˜
￿
￿
0
￿
￿
￿
￿
0
￿
the large scale fluctuations about the mean, , replaces .
￿
￿
￿˜
￿
We similarly express all other flow variables in terms of a
mean value and a large scale fluctuation about this mean. The
mean flow is therefore defined by , where
￿
v
0
,
￿
0
,
￿
0
,
￿
0
￿
, (A.8)
v
0
￿ U
i
, (A.9)
￿v
0
￿
j
￿U
￿x
2
i
, (A.10)
￿
￿￿
0
￿
￿￿
0
￿
dln
H
dx2
j
, (A.11)
￿
￿
0
￿
0
, (A.12)
￿￿ v
0
￿ ￿
0
￿ 0
where the mean velocity, U, and mean enthalpy, H, are
functions of the transverse distance, , only. The distur-
x
2
bance field in its larger scales is assumed isentropic so
, and .
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
1
￿
￿
￿
The isentropic disturbance flow equations are therefore
, (A.13)
D
0
￿
￿
Dt
￿ ￿￿
￿
￿ v
￿
￿ ￿￿
0
￿ ￿v
￿
￿ ￿￿
￿
￿￿ v
￿
￿ ￿￿
￿
￿￿
D
0
v
￿
Dt
￿ ￿c
0
2
￿￿￿
￿
￿
￿￿￿1￿
2
￿￿
￿
￿
2
￿ ￿
, (A.14)
￿ v
￿
￿ ￿v
0
￿￿v
￿
￿ ￿v
￿
￿￿ v
￿
￿ ￿v
￿
￿￿
where the linearised approximation to and the
h
￿
￿ c
0
2
￿
￿
convective operator, , are used as
D
0
/
D
t
￿ ￿/￿
t
￿v
0
￿ ￿
proposed above. The averaged value < > is the time average
taken over a time large compared with the large scale fluctua-
G. M. Lilley, X. Zhang and A. Rona: PROGRESS IN COMPUTATIONAL AEROACOUSTICS
International Journal of Acoustics and Vibration, Vol. 2, No. 1, 1997
7
tions. It is the contribution from the large scale fluctuations
to the mean flow equations.
The reduction of Eqs. (A.13) and (A.14) to form an
inhomogeneous convected wave equation follows the same
strategy introduced above whereby we eliminate
D
0
￿
￿
/
D
t
between Eqs. (A.13) and (A.14). The result is
D
0
2
Dt
2
￿ c
0
2
￿
2
￿
￿
￿ 2￿v
0
:v
￿
￿ ￿
D
0
Dt
(
v
￿
￿ ￿￿
￿
￿￿ v
￿
￿ ￿￿
￿
￿
)
￿
￿
D
0
Dt
￿v
￿
￿ ￿￿
0
￿ ￿￿￿
(
v
￿
￿ ￿v
￿
￿￿ v
￿
￿ ￿v
￿
￿
)
￿
￿
￿￿1
2
c
0
2
￿
2
(
￿￿
￿
￿
2
￿￿ ￿￿
￿
￿
2
￿
)
￿
. (A.15)￿c
0
2
￿￿
0
￿ ￿
￿
￿
￿
￿￿1
2
(
￿￿
￿
￿
2
￿￿ ￿￿
￿
￿
2
￿
)
The righthand side contains, in addition to quadratic
terms in and , terms linear in these perturbations.
v
￿
￿
￿
However all terms are of similar order to the linear terms on
the lefthand side since we must regard
(
v
￿
￿ ￿v
￿
￿￿ v
￿
￿ ￿v
￿
￿
)
as of similar order to the linear terms, noting that
(
￿ v
￿
￿ ￿v
￿
￿
)
is of zeroth order. All quadratic terms in Eq. (A.15)
involv-ing can be replaced by since .
￿
￿
￿
￿
￿￿
￿
￿ ￿￿ ￿1￿￿￿
￿
However in its present form Eq. (A.15) is unsatisfactory for
we need an equation which contains linear terms only in in￿
￿
its lefthand side and all terms nonlinear in and will then
v
￿
￿
￿
be contained in the righthand side. In this way we can derive
an inhomogeneous convected wave equation in , where in
￿
￿
the flow field all terms linear and non-linear are of similar
order, but whereas the linear lefthand-side represents the
effects of propagation of the near-field acoustic sources in
the flow, the righthand-side represents the generation.
For this given sample mean flow the linear terms in can
v
￿
be equated to linear terms in plus additional nonlinear
￿
￿
terms in and from
￿
￿
v
￿
, (A.16)
2￿v
0
:v
￿
￿ ￿ 2
dU
dx
2
￿v
2
￿
￿x
1
, (A.17)
v
￿
￿ ￿￿
0
￿
￿
dln
H
dx
2
v
2
￿
and from the equation of motion
x
2
D
0
Dt
￿
￿v
2
￿
￿x
1
￿ ￿ ￿
￿
￿x
1
(
v
￿
￿ ￿v
2
￿
￿￿ v
￿
￿ ￿v
2
￿
￿
)
￿
. (A.18)
￿ c
0
2
￿
2
￿x
1
￿x
2
￿
￿
￿
￿￿1
2
(
￿￿
￿
￿
2
￿￿ ￿￿
￿
￿
2
￿
)
Finally we obtain after finding of Eq. (A.15) and
D
0
/
D
t
using Eqs. (A.16) to (A.18)
, (A.19)
D
0
Dt
￿
D
0
2
Dt
2
￿ c
0
2
￿
2
￿ ￿ 2c
0
2
dU
dx
2
￿
2
￿x
1
￿x
2
￿
￿
￿ L￿￿
￿
￿
where
L￿￿
￿
￿ ￿
D0
Dt
￿
2
v
i
￿
v
j
￿
￿x
i
￿x
j
￿
￿￿￿1￿
2
c
0
2
￿
2
￿￿
￿
￿
2
￿ 2
dU
dx
2
￿
2
v
2
￿
v
i
￿
￿x
1
￿x
i
￿
￿￿￿1￿
2
￿
,￿ c
0
2
￿
2
￿￿
￿
￿
2
￿x
1
￿x
2
￿
D
0
Dt
D
0
Dt
￿ ￿ ￿v
￿
￿
￿
￿ ￿
d￿lnH￿
dx
2
￿ ￿ ￿v
￿
v
2
￿
￿
(A.20)
where we note . Hence the source function,
c
0
2
￿
￿
￿
h
￿
, is a function of only. In Eq. (A.20) all the
L￿￿
￿
￿
￿v
￿
,h
￿
￿
corresponding ensemble averaged quadratic terms have been
omitted for convenience only.
The homogeneous equation is the Pridmore-
L￿￿
￿
￿ ￿ 0
Brown
16
equation for the propagation of sound waves in a
uniform shear flow. It provides a major reason for justifying
that our procedure has provided a method which extracts the
mean effects of flow-acoustic interaction from those of sound
generation. Equation (A.19) also reduces to the standard
acoustic wave equation external to the flow. In the derivation
of Eq. (A.19) we have written the righthand side in a form
comparable with that of Lighthill's equation. To do this we
have strictly to include extra terms involving fluctuations in
the dilatation, which for sake of brevity we have excluded
from Eq. (A.16). Such approximations are unnecessary and
need not be used in numerical calculations. The important
observation is that all acoustic source terms involve quadratic
terms only in velocity and enthalpy fluctuations and include
the amplifying effects of mean shear and mean temperature
gradient. Apart from the statements made above, in respect of
the basic flow, Eq. (A.19) is exact (see Eq. (A.15)). Similarly
Eq. (A.19) can be presented in its exact form without
neglecting terms involving fluctuations in the flow dilatation.
It is this form of the convected wave equation used by
Colonius et al.
1
in comparisons with DNS in the low
Reynolds number two dimensional compressible shear layer.
In its derivation no assumptions have been made in respect of
the amplitude of the large scale fluctuations, , in the flow
￿￿￿
field relative to the basic long time averaged mean flow.
An alternative derivation of Eq. (A.19) has been
discussed by Goldstein
17
. In this derivation the flow variables
are expanded in a small parameter, ￿. The basic flow field is
found at order zero. To first order in ￿, which is the field of
linear fluctuations, Goldstein shows that the linearised form
of in Eq. (A.19) is zero, and thus the linear field
L￿￿
￿
￿
supports no sound generation at subsonic speeds. In fact the
first order equation is similar to the compressible linear
stability equation and possesses spatially growing eigen
solutions. However non-parallel effects restrict the rate of
growth of the initially exponentially growing waves and
these waves eventually become damped and generate the
large scale structures in the turbulent shear flow as shown by
Morris et al.
10
At supersonic speeds the linear instability
waves can support sound waves as shown by Tam
18,19
, Tam
and Burton
20,21
and others. When terms to second order in ￿
are included Goldstein shows that sound generation occurs at
subsonic speeds and its magnitude is the full non-linear form
of evaluated from the order ￿ flow field. Thus
L￿￿
￿
￿
Goldstein concludes that it is only the quadratic disturbance
G. M. Lilley, X. Zhang and A. Rona: PROGRESS IN COMPUTATIONAL AEROACOUSTICS
International Journal of Acoustics and Vibration, Vol. 2, No. 1, 1997
8
flow field which can support the generation of noise in a
subsonic turbulent flow.
The source term in Eq. (A.20), , is evaluated in
L￿￿
￿
￿
terms of the known values of derived from the `tilde'
￿v
￿
,h
￿
￿
flow.In such a numerically calculated flow we need to
capture the periodic form of the solution at all relevant
frequencies. Thus for any finite difference algorithm it is
necessary to verify that the characteristics of the set of
algebraic equations are exactly equal to those of the
homogeneous part of the partial differential convected wave
equation and so ensure there are no additional numerical
waves emanating from the governing boundaries and cell
boundaries. This also requires a time and space accurate
solver having a sufficient number of points per wavelength
within the flow as discussed by Tam
22
and Lin et al.
23
Equation (A.20) is similar to the convected wave equation
found earlier by Lilley
24
and as described by Goldstein
5,17
.
Solutions for the limiting cases of low and high frequency
have been given by Goldstein and others. Far-field approxi-
mate solutions have been discussed recently by Musafir
25
.
Equation (A.20) does not predict the presence of non-linear
sound waves inside the flow or externally.
When considering non-linear acoustic distortion it is
necessary to consider the wave motion relative to the
combined mean and turbulent flow where the latter is frozen
during the passage of the sound wave packet. On the other
hand the generation of noise is dependent on fluctuations
relative to the mean flow, and as discussed earlier is mainly
dependent on the large scale eddies which contain most of
the turbulent energy. Thus the convected wave equation for
considering the generation of noise from turbulence needs to
involve , which is the operator following the mean
D
0
/
D
t
motion. However to consider non-linear propagation the
more relevant operator is D/Dt since it relates to the
combined effect of the mean and ‘frozen’ turbulent flow. For
the latter the convected wave equation becomes for isentropic
disturbances
, (A.21)
D
2
Dt
2
￿ c
0
2
￿
2
￿ ￿ ￿v:v￿ ￿
c
2
￿￿1
￿￿￿￿
2
confirming, as discussed earlier, that the source terms
relate to contributions from the square of both velocity and
temperature fluctuations.
In the special case where U=0 and H=constant and in the
source term we assume the dilatation of the turbulent fluctua-
tions is sufficiently small to have a negligible effect on the
noise production, we find Eq. (A.19) reduces to
, (A.22)
￿
2
￿
￿
￿t
2
￿ c
0
2
￿
2
￿
￿
￿
￿
2
￿x
i
￿x
j
￿v
i
￿
v
j
￿
￿ ￿
￿
2
￿t￿x
i
￿v
i
￿
h
￿
/c
0
2
￿
which is our equivalent to Lighthill's equation. In Eq.
(A.22) we see the quadrupole terms, involve quadratic veloc-
ity fluctuations, but which no longer contain the absolute
instantaneous density. (However the sound speed is that in
the flow and not the external ambient medium speed of
sound. In calculations matching boundary conditions must be
used between the flow field and the external medium.) In
addition we find the dipole terms arising from the correlation
between velocity and enthalpy fluctuations in agreement with
the results derived by Lilley
24
from Lighthill's acoustic
analogy, where
1
c
￿
2
￿
2
￿t
2
￿ c
￿
2
￿
2
p ￿
￿
2
￿x
i
￿x
j
￿￿v
i
v
j
￿ ￿
￿￿1
2c
￿
2
￿
2
￿t
2
￿￿v
2
￿ ￿
, (A.23)￿
￿
2
￿t￿x
i
(
￿v
i
￿h
s
￿ h
￿
￿/h
￿
)
when the viscous stresses are omitted. This reduces to Eq.
(A.22) when . However we note that Eq.
v
2
/2
￿
￿
￿
h
s
￿
h
￿
￿
(A.22) is only an approximation to Lighthill's equation since
in deriving the righthand side it is assumed the isentropic
disturbance dilatation is zero. The second term in Eq. (A.22),
which is of dipole type, bears similarities with that found by
Tester and Morfey
26
from the complete third-order convected
wave equation, and represents the effect of temperature
fluctuations in the shear flow.
As stated earlier, eq. (A.19) is not unique in respect of
deriving the noise radiation from a given turbulent flow. It
has been established by introducing some simplifications to
the given flow field as well as choosing as the appro-
￿v
￿
,h
￿
￿
priate flow variables to describe the equivalent acoustic
sources in the flow. In some flows this is clearly not the best
choice. When, for instance, the unsteady vorticity is known
throughout the flow then, as found by Howe
27
, equations in
the fluctuating total enthalpy, , can be used with acoustic
h
s
￿
source terms dominated by . However since the
￿￿ ￿v ￿￿
￿
database for the time-dependent flow will be the same,
irrespective of the choice made for the equations to resolve
the acoustic field, the sound radiation to the far-field must be
independent of the choice of flow variables and their
equations.
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G. M. Lilley, X. Zhang and A. Rona: PROGRESS IN COMPUTATIONAL AEROACOUSTICS
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