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

P

r

o

g

r

e

s

s

i

n

C

o

m

p

u

t

a

t

i

o

n

a

l

A

e

r

o

a

c

o

u

s

t

i

c

s

i

n

P

r

e

d

i

c

t

i

n

g

t

h

e

N

o

i

s

e

R

a

d

i

a

t

e

d

f

r

o

m

T

u

r

b

u

l

e

n

t

F

l

o

w

s

G

e

o

f

f

r

e

y

M

.

L

i

l

l

e

y

,

X

i

n

Z

h

a

n

g

a

n

d

A

l

d

o

R

o

n

a

*

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)

˜

,

v˜

,

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=

/,

=-2/3I+(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

2v: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

2v

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)

2v

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

dlnH

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

tx

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

tx

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.

REFERENCES

1

Colonius, T., Lele, S.K., and Moin, P. (1995) "The sound gener-

ated by a two dimensional shear layer: A comparison of direct

computations and acoustic analogies." Proceedings CEAS/AIAA

Aeroacoustics Conference Munich, Germany.

2

Mitchell, B.E., Lele, S.K., and Moin, P. (1995) "Direct compu-

tation of the sound generated by an axisymmetric jet." AIAA

Paper 95-0504.

3

Laufer, J., and Yen, T.C. (1983) "Noise generation in a low

speed jet." J.Fluid.Mech.134, pp. 1-31.

4

Lighthill, M.J. (1952) "On sound generated aerodynamically: 1.

General theory." Proc. Roy. Soc.A211, pp. 564-587.

5

Goldstein, M. (1976) "Aeroacoustics" New York: McGraw-Hill.

6

Sarkar, S., and Hussaini, M.Y. (1993) "Computation of the

sound generated by isotropic turbulence." ICASE Report 93-74.

7

Lilley, G.M. (1994) "The radiated noise from isotropic turbu-

lence."Theoretical and Computational Fluid Dynamics.6, pp.

281-301.

8

Lilley, G.M. (1996) "The radiated noise from isotropic turbu-

lence with applications to the theory of jet noise." J.Sound.Vib.

190 (3), pp. 463-476.

9

Zhang, X., Rona, A., and Lilley, G.M. (1995) "Far field noise

radiation from an unsteady supersonic cavity flow." Proceed-

ings CEAS/AIAA Aeroacoustics Conference Munich, Germany.

G. M. Lilley, X. Zhang and A. Rona: PROGRESS IN COMPUTATIONAL AEROACOUSTICS

International Journal of Acoustics and Vibration, Vol. 2, No. 1, 1997

9

10

Morris, P.J., Giridharan, M.G., and Lilley, G.M. (1990) "On the

turbulent mixing of compressible shear layers." Proc.Roy.Soc.

A431, pp. 219-243.

11

Brown, G.L., and Roshko, A. (1974) "On the density effects and

large structures in turbulent mixing layers." J.Fluid.Mech.64,

pp. 775-781.

12

Wilcox, D.C. (1993) "Turbulence modeling for CFD." DCW

Industries Inc, La Canada, California.

13

Zhang, X., and Edwards, J.A. (1990) "An investigation of super-

sonic oscillatory cavity flow driven by a thick shear layer." The

Aeronautical Journal.94, pp. 355-364.

14

Rona, A. (1996) "Aerodynamic and aeroacoustic estimations of

oscillatory supersonic flows." unpublished Ph.D. thesis, Depart-

ment of Aeronautics and Astronautics, University of Southamp-

ton, U.K.

15

Phillips, O.M. (1960) "On the generation of sound by super-

sonic turbulent shear flows." J.Fluid.Mech.9, pp. 1-28.

16

Pridmore-Brown, D.C. (1958) "Sound propagation in a fluid

flowing in an attenuating duct." J.Fluid.Mech.4, pp. 393-406.

17

Goldstein, M.E. (1984) "Aeroacoustics of turbulent shear

flows."Annual Reviews of Fluid Mechanics.16, pp. 263-285.

18

Tam, C.K.W. (1971) "Directional acoustic radiation from a

supersonic jet generated by shear layer instability."

J.Mech.Fluid.46, pp. 757-768.

19

Tam, C.K.W. (1972) "On the noise of a nearly ideally expanded

supersonic jet" J.Fluid.Mech.51, pp. 69-96.

20

Tam, C.K.W., and Burton, D.E. (1984a) "Sound generated by

instability waves of supersonic flows. Part 1. Two-dimensional

mixing layers." J.Fluid.Mech.138, pp. 249-271.

21

Tam, C.K.W., and Burton, D.E. (1984b) "Sound generated by

instability waves of supersonic flows. Part 2. Axisymmetric

jets."J.Fluid.Mech.138, pp. 273-295.

22

Tam, C.K.W. (1995) "Supersonic jet noise." Annual Reviews of

Fluid Mechanics. 27, pp. 17-43.

23

Lin, S.Y., Chen, Y.F., Shih, S.C. "Numerical study of MUSCL

schemes for computational aeroacoustics." AIAA Paper

97-0023.

24

Lilley, G.M. (1973) "On the noise from air jets." AGARD CP

131, pp. 13.1-13.12.

25

Musafir, R.E. (1993) "Some notes on the description of jet noise

sources."Proc. J.O.A 15, pp. 901-908.

26

Tester, B.L., and Morfey, C.L. (1976) "Developments in jet

noise modelling-theoretical predictions and comparisons with

measured data." J.Sound.Vib.46, pp. 79-103.

27

Howe, M.S. (1975) "Contributions to the theory of aerodynamic

sound with applications to excess jet noise and the theory of the

flute."J.Fluid.Mech.71, pp. 625-673.

G. M. Lilley, X. Zhang and A. Rona: PROGRESS IN COMPUTATIONAL AEROACOUSTICS

International Journal of Acoustics and Vibration, Vol. 2, No. 1, 1997

10

## Comments 0

Log in to post a comment