application to computational aeroacoustics Extensions of Lighthill's acoustic analogy with

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September 2007
, published 8
, doi: 10.1098/rspa.2007.1864
463 2007 Proc. R. Soc. A
 
C.L Morfey and M.C.M Wright
 
application to computational aeroacoustics
Extensions of Lighthill's acoustic analogy with
 
 
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Extensions of Lighthill’s acoustic analogy with
application to computational aeroacoustics
B
Y
C.L.M
ORFEY
*
AND
M.C.M.W
RIGHT
Institute of Sound and Vibration Research,University of Southampton,
Southampton SO17 1BJ,UK
Lighthill’s aeroacoustic analogy is formulated for bounded domains in a general way that
allows pressure-based alternatives to the fluid density as wave variable.The advantage
relative to the standard version (Ffowcs Williams & Hawkings 1969 Phil.Trans.R.Soc.
A 264,321–342) is that the equivalent surface source terms needed for boundary value
problems do not involve the local density.Difficulties encountered in computational
aeroacoustics with standard wave extrapolation procedures,due to advection of density
inhomogeneities across the control surface,are thereby avoided.Likewise,in initial-value
problems,the equivalent volume source terms that represent initial conditions do not
involve the density either.The paper ends with an extension to parallel shear flows,in
which a modified aeroacoustic analogy due to Goldstein (Goldstein 2001 J.Fluid Mech.
443,231–236) is formulated for bounded domains using a similar windowed-variable
approach.The results provide a basis for acoustic wave extrapolation from jets and
boundary layers,where the control surface cuts through a sheared mean flow.
Keywords:aeroacoustics;acoustic analogy;initial-value problem;
wave extrapolation;thermoacoustics;jet noise
1.Introduction
The idea of replacing a region of unsteady fluid flow by a distribution of
equivalent sources that drive linear perturbations to a base flow has been
extremely useful in the field of acoustics.Rayleigh (1894) used equivalent sources
to describe scattering of sound in a non-uniform unbounded medium.Lighthill
(1952) used the same idea to develop his acoustic analogy in which the equations
of fluid motion,expressing conservation of mass and momentum,are combined to
yield a linear wave equation with nonlinear forcing terms.In both cases,the ‘base
flow’ is a uniform fluid at rest.Provided that the forcing terms can be estimated
independently of the far-field radiation,Lighthill’s equation can be said to
describe the nonlinear generation of sound by unsteady flows.
Subsequent extensions and variations of the acoustic analogy include:
(i) The addition of equivalent source terms to allow for boundaries
(either real or virtual) in the flow field (Curle 1955;Ffowcs Williams &
Hawkings 1969).
Proc.R.Soc.A (2007) 463,2101–2127
doi:10.1098/rspa.2007.1864
Published online 19 June 2007
*Author for correspondence (clm@isvr.soton.ac.uk).
Received 14 February 2007
Accepted 10 May 2007
2101
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(ii) Various rearrangements of the source terms to highlight physical
processes,often accompanied by a change of wave variable,such as
unsteady pressure p (Morfey 1973;Lilley 1974,1996),the quantity
pCru
2
/3 (Ffowcs Williams 1969;Kambe 1984),stagnation enthalpy
hCu
2
/2 (Howe 1975),(P/P
0
)
1/g
K1 (Goldstein 2001),etc.
(iii) The use of a different base flowto match the characteristics of a particular
situation,as in the parallel shear flow analogy of Lilley (1974) and its
modification by Goldstein (2001).Howe (1998) and Goldstein (2003,
2005) discuss a number of such extensions to the original concept of
Lighthill (1952).
The present paper aims to extend the usefulness of the Ffowcs Williams &
Hawkings formulation (1969,hereafter referred to as FWH) in three directions,
corresponding to the categories above.The new formulations found below offer
particular advantages in computational aeroacoustics (CAA),where they
provide the basis for improved wave extrapolation techniques.
The structure of the paper is as follows.First,the spatial window applied in
FWH is generalized to a spatiotemporal window (§§2 and 3a);this allows initial-
value problems to be treated in a consistent way,with initial conditions
represented by equivalent volume sources.Next,the windowed density
perturbation used as a wave variable in FWH is substituted by a new density-
type variable (§3b).Two possible choices for the new variable,both related to the
local pressure,are introduced (§3c);the source terms in the resulting acoustic
analogy formulations are analysed and compared with FWH,with the help of the
energy equation (§§3c–e).Implications for CAA are discussed in §4,with
emphasis on surface source distributions for wave field extrapolation;the
contribution of unsteady heat fluxes across the control surface is explicitly
identified in §4b.The thermoacoustic volume sources neglected in §4b are
reviewed in detail in appendix C;they are generally unimportant,but a possible
exception occurs in turbulent mixing of hot and cold streams when the fluid is not
a perfect gas.Finally,in §5,Goldstein’s formulation (2001) of the Lilley analogy
for parallel sheared base flow is extended to bounded domains,with boundary-
value and initial-value equivalent source distributions that are analogous to those
found in §3 for zero base flow.The resulting boundary-value surface sources allow
wave extrapolation from a control surface placed in a parallel shear flow,via
solution of the Lilley equation with a linearized wave operator.
2.Notation and definitions
Let S be a moving closed surface in three dimensions that separates region V
0
from an adjacent region V,as illustrated in figure 1.The idea is that V
0
may
contain solid boundaries;alternatively information on the flow in V
0
may be
inaccessible.In either case,the aim of the acoustic analogy formulation is to
describe the fluctuating pressure or density field in V;the field in V
0
is ignored for
this purpose,although data on S will be used to provide boundary conditions.
Any acoustic influence of V
0
will be accounted for by equivalent sources on S,and
no use will be made of the equations of fluid motion within V
0
.Likewise,no use
will be made of information for t!0;the acoustic influence of events prior to tZ0
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will be accounted for by impulsive sources at tZ0,distributed throughout
region V.Let f(x,t) be a continuous indicator function such that f!0 in V
0
,f O0
in V,and let jVf jZ1 on S.Smoothness of S is assumed,so that jVf j is single
valued.
1
Let n be a local normal coordinate,defined for points near S by nZf;
then v=vn evaluated on S is the gradient operator normal to S,in the direction
from V
0
to V.Define the spatial and temporal Heaviside functions
HðnÞ Z
1 in V and on S
0 in excluded region V
0
;
(
ð2:1Þ
QðtÞ Z
1 for tR0
0 for t!0;
(
ð2:2Þ
(a) (b)
(c) (d)
v
v
v
u
u u
v
1
Â
v
'
v
'
v
'
v
'
v
'
v
'
v
'
( f < 0)
( f < 0)( f > 0)
( f > 0)
( f > 0)
( f > 0)
u
2
1
3
2
Figure 1.Schematic showing the complementary regions V
0
(about which no knowledge is
available) and V,and the interface S between them.In region V,the equations of fluid motion
apply for tO0.Region V may be (a) exterior or (b) interior to S.More generally,(c) the excluded
region V
0
may be multiply connected.In all these cases,VgV
0
fills the entire space.(d) A further
option is to have VgV
0
surrounded by a closed surface S,that lies in a region of linear acoustic
disturbances to the reference state (r
0
,c
0
),and represents an acoustically absorbing or scattering
boundary.Case (a) can be regarded as a limiting case of (d) in which S becomes a sphere of infinite
radius and the Sommerfeld radiation condition is applied.
1
An extension of this description to cusped surfaces,such as a sharp-edged aerofoil,has been
presented by Farassat & Myers (1990).
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which henceforth will be written without their arguments.From the definition of
the Heaviside function,it follows that
vQ
vt
ZdðtÞ;
vQ
vx
i
Z0;
vH
vn
ZdðnÞ;
vH
vx
i
Z^n
i
dðnÞ;ð2:3Þ
where d($) is the Dirac delta function and ^n
i
Zvn=vx
i
,the unit normal to S.The
time derivative of H is found by noting that H is constant in a reference frame
moving with the surface,so that
vH
vt
ZKv
i
vH
vx
i
ZKv
i
^n
i
dðnÞ ZKvdðnÞ:ð2:4Þ
Here v
i
Zv^n
i
,with v the normal velocity of the surface S directed into V.The
material derivative of H is given by
DH
Dt
Zðu
i
Kv
i
Þ^n
i
dðnÞ;ð2:5Þ
where D=DtZðv=vtCu
i
v=vx
i
Þ.
In what follows,a line over any variable or quantity means that it is multiplied
by HQ,thus windowing it in space and time.For consistency,generalized
functions are usually written at the end of a product,with spatial generalized
functions preceding temporal ones.The relations given above can be used to find
the result of commuting the windowing operation with differentiation,with
respect to space and time respectively,for any field variable x
vx
vx
i
K
v

x
vx
i
ZKx
^
n
i
dðnÞQ;
vx
vt
K
v

x
vt
Zx½v
i
^
n
i
dðnÞQKHdðtÞ:ð2:6Þ
The identity
vx
vt
h
Dx
Dt
K
v
vx
i
ðxu
i
Þ CxD;ð2:7Þ
also holds wherever u
i
is defined;here DZvu
i
=vx
i
is the dilatation rate.An
important,but lengthy,derivation of the second time derivative of an arbitrary
windowed variable is given in appendix A.
3.Initial–boundary value formulations for aeroacoustics
As a starting point for deriving a generalized statement of Lighthill’s acoustic
analogy that incorporates both initial and boundary conditions,consider the
windowed equations of motion for a fluid occupying region V.Conservation of
mass and momentum are expressed by
v
vt
ðrKr
0
Þ C
v
vx
i
ðru
i
Þ Z0;
v
vt
ðru
i
Þ C
v
vx
j
ðru
i
u
j
Cp
ij
Þ Z
G
i
:ð3:1Þ
Here and throughout,subscript ‘0’ denotes the properties of a uniform reference
medium,chosen to coincide with the actual flow in the acoustic far field.Without
loss of generality,a frame of reference is chosen that makes the fluid velocity zero
at infinity.In (3.1),r denotes fluid density;u
i
is the fluid velocity in the x
i
direction;p
ij
ZP
ij
KP
0
d
ij
,where P is absolute pressure and P
ij
is the compressive
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stress in the fluid;d
ij
is the Kronecker delta and G
i
is an applied body force per
unit volume.The quantities (rKr
0
),ru
i
,(ru
i
u
j
Cp
ij
),G
i
in (3.1) all vanish in the
far-field region.
An acoustic analogy in terms of windowed variables is now sought.Applying
equations (2.6) to the conservation equations (3.1) produces additional terms on
the right-hand side,
v
vt
rKr
0
ð Þ C
v
vx
i
ru
i
ð Þ ZðrKr
0
ÞHdðtÞ C½ru
i
KðrKr
0
Þv
i
 ^n
i
dðnÞQ ð3:2Þ
and
v
vt
ru
i
ð Þ C
v
vx
j
ru
i
u
j
Cp
ij
 
Z
G
i
Cru
i
HdðtÞ C ru
i
ðu
j
Kv
j
Þ Cp
ij
 
^n
j
dðnÞQ:
ð3:3Þ
By eliminating
ru
i
from (3.2) and (3.3),an expression for the second time
derivative of
rKr
0
,the windowed density perturbation,that is valid for all
(x
i
,t) is obtained,
v
2
vt
2
rKr
0
ð Þ Z
v
vt
ðrKr
0
ÞHdðtÞ½ K
v
vx
i
ru
i
HdðtÞ½ 
C
v
vt
ru
i
KðrKr
0
Þv
i
½  ^n
i
dðnÞQf g
K
v
vx
i
ru
i
ðu
j
Kv
j
Þ Cp
ij
 
^n
j
dðnÞQ
 
K
v
G
i
vx
i
C
v
2
vx
i
vx
j
ru
i
u
j
Cp
ij
 
:
ð3:4Þ
(a) Density form of the acoustic analogy
Subtracting c
2
0
V
2
rKr
0
ð Þ from (3.4) leads directly to
1
c
2
0
v
2
vt
2
KV
2
 
c
2
0
rKr
0
ð Þ Z
v
vt
ðrKr
0
ÞHdðtÞ½ K
v
vx
i
ru
i
HdðtÞ½ 
C
v
vt
J
i
^n
i
dðnÞQ½ K
v
vx
i
L
ij
^n
j
dðnÞQ
 
K
v
G
i
vx
i
C
v
2
T
ij
vx
i
vx
j
:
ð3:5Þ
Symbols T
ij
,J
i
and L
ij
on the right of (3.5) stand for the Lighthill stress tensor
T
ij
Zru
i
u
j
Cp
ij
Kc
2
0
ðrKr
0
Þd
ij
;ð3:6Þ
the surface mass flux vector
J
i
Zru
i
KðrKr
0
Þv
i
Zrðu
i
Kv
i
Þ Cr
0
v
i
;ð3:7Þ
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and the surface momentum flux tensor
L
ij
Zru
i
ðu
j
Kv
j
Þ Cp
ij
:ð3:8Þ
The sources on the right-hand side of (3.5) can be interpreted as follows:
(i) the first two terms represent the impulsive addition of mass and
momentum needed to start the flow from its initial reference state,
(ii) the second line contains the usual FWH surface monopoles and dipoles,
windowed by Q,and
(iii) volume source terms appear in the third line,with the body force G
i
and
the Lighthill stress tensor T
ij
windowed spatially and temporally by HQ.
Equation (3.5) without the initial-value source terms is the standard FWH
equation and has been widely used in CAA,where it provides a means of
extrapolation fromthe simulation domain to the acoustic far field (§4a).However,
in that context,(3.5) is not well suited to applications involving heated flows,or
flows in which mixing occurs between different fluids (Shur et al.2005).The
reason is that the surface monopole and dipole distributions,J
i
^n
i
and L
ij
^n
j
,
depend on the local density;so fluctuations in these quantities occur when local
hot spots,or regions of different fluid compositions,are advected across the fixed
control surface S.Such fluctuations are present even when the flow is entirely
silent,as can be seen by considering the advection of density inhomogeneities by a
uniform steady flow.Suppose that the density field is steady in a frame of
reference moving with the flow,with r(x)Zr
0
everywhere except in a limited
region R.Applying the acoustic analogy equation (3.5) in this frame gives u
i
Z0,
while the control surface S translates uniformly.As S cuts through R,the surface
monopole distribution J
i
^n
i
on S varies with time,being given by K(rKr
0
)v
n
;but
the radiated sound field rKr
0
is zero.
It is important to recognize that (3.5) nevertheless remains valid for heated
and inhomogeneous flows.The point is that neglect of the volume quadrupoles
T
ij
is not justified under such conditions,since the physically unrealistic surface
sources described above are cancelled by the c
2
0
ðrKr
0
Þ term in the quadrupole
distribution
T
ij
.For wave extrapolation purposes,therefore,there is a strong
incentive to find alternative formulations that cope better with advected density
disturbances crossing S.
(b ) Density-substituted forms of the acoustic analogy
Two formulations of the extended Lighthill analogy are presented below in
which the local density is absent,both from the surface monopole and dipole
distributions,and fromthe initial-value source terms.The first version applies to
an arbitrary fluid,and the second version applies to a particular class of fluids
that includes perfect gases.
Both versions begin from the v
2
rKr
0
ð Þ=vt
2
expression (3.4),and use the
kinematic relation (A5) for the second time derivative of an arbitrary windowed
variable,
x,to replace r by a new variable r
C
related to the local pressure.
By defining
x ZrKr
C
;ð3:9Þ
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and subtracting v
2
x=vt
2
from v
2
rKr
0
ð Þ=vt
2
,an expression for v
2
r
C
Kr
0


=vt
2
is obtained that exhibits the properties mentioned above.A generic acoustic
analogy can then be written as
1
c
2
0
v
2
vt
2
KV
2
 
c
2
0
r
C
Kr
0


Z
v
vt
ðr
C
Kr
0
ÞHdðtÞ
 
K
v
vx
i
r
C
u
i
HdðtÞ
 
C
v
vt
½J
C
i
^n
i
dðnÞQK
v
vx
i
L
C
ij
^n
j
dðnÞQ
 
C
v
Q
C
vt
K
v
vx
i
Q
C
u
i
C
r
C
r
G
i
C
rKr
C
r
vp
ij
vx
j
!
C
v
2
T
C
ij
vx
i
vx
j
:
ð3:10Þ
Here,J
C
i
,L
C
ij
and T
C
ij
are defined in the same way as J
i
,L
ij
and T
ij
with r
replaced by r
C
,and Q
C
is defined as
Q
C
ZK
DðrKr
C
Þ
Dt
CðrKr
C
ÞD

Z
Dr
C
Dt
Cr
C
D;ð3:11Þ
where the second version follows from mass conservation.The penultimate term
of (3.10) has been obtained by writing the equation of conservation of momentum
in the form
Du
i
Dt
Z
G
i
r
K
1
r
vp
ij
vx
j
;ð3:12Þ
which is valid throughout V.
Like (3.5),equation (3.10) is exact;it applies to bounded domains ( f O0,tO0);
and no assumption has been made about the fluid equation of state.The fifth
and sixth terms on the right contains additional volume terms not present in
(3.5),related to Q
C
and rKr
C
.The usefulness of (3.10),as the basis of an
acoustic analogy,depends on these terms being sufficiently small that their
contribution from any region of purely acoustic linear disturbances can
be neglected;this issue is examined next,for two particular choices of the
variable r
C
.
(c ) Determination of Q
C
from the energy equation
If the acoustic density approximation r
-
,defined by
r
-
Zr
0
Cc
K2
0
p Zr
0
ð1CK
0
pÞ;ð3:13Þ
where K is the isentropic compressibility 1/(rc
2
),is chosen as the substituted
density variable r
C
,then the corresponding value of Q
C
is given by (3.11) as
Q
-
Zr
0
DCK
0
pDC
Dp
Dt
 
:ð3:14Þ
From the energy equation for a single-component
2
viscous heat-conducting fluid,
with heat input rate _q per unit volume,Dr/Dt and Dp/Dt,are related by
K
1
r
Dr
Dt
CK
Dp
Dt
Z
a
rc
p
FK
vq
i
vx
i
C_q
 
ZD

;ð3:15Þ
2
For a mixture of two fluids,a generalization of eqn (16) is given in appendix II of Morfey (1976).
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where F is the viscous dissipation function
FZt
ij
vu
i
vx
j
;ð3:16Þ
and q
i
is the heat flux vector;a is the volumetric thermal expansivity and c
p
is the
constant-pressure specific heat.The quantityD

is the difference betweenthe actual
dilatationrate andthat due toisentropic compression(Morfey1976);D

is therefore
referredtoas the entropic dilatationrate.Alternative expressions for Q
-
interms of
D

follow fromcombining (3.14) and (3.15) with the continuity equation,
Q
-
Zr
0
ð1 CK
0
pÞD

Kr
0
ðKKK
0
CK
0
KpÞ
Dp
Dt
;ð3:17Þ
Zr
0
ðK
0
=KÞD

Cr
0
ð1KK
0
=K CK
0
pÞD:ð3:18Þ
It is clear from(3.17) that in a region where the only disturbances are sound waves,
Q
-
is indeed small (this is explained in more detail in §3e).Its presence as a
monopole source term in the acoustic analogy (equation (3.22)) accounts for
thermal attenuation of sound and nonlinear acoustic phenomena in such a region.
An alternative choice for r
C
,in the context of perfect-gas flows,was
heuristically proposed by Shur et al.(2005) following Goldstein (2001).In its
most general form,this alternative definition is valid for any fluid whose
isentropic compressibility is a function K(P) of the pressure alone;note that for a
perfect gas,KZ(gP)
K1
,where g is the (constant) specific heat ratio.The
substituted density variable,denoted in this case by r

,is defined by
r

Zr
0
exp
ð
P
P
0
KðP
0
ÞdP
0

;ð3:19Þ
Zr
0
P
P
0
 
1=g
ðfor a perfect gasÞ:ð3:20Þ
It follows from (3.19) that dr

=r

ZK dP.The corresponding value of Q
C
,
denoted here by Q

,is given by combining (3.11) and (3.15) with the continuity
equation,
Q

Zr

D

:ð3:21Þ
The advantage in simplicity relative to (3.17) or (3.18) is clear,while Q

shares
with Q
-
the property that in a small-amplitude sound field its sole effect as a
monopole source is to account for thermal attenuation.Use of r

as a substituted
variable is possible,however,only for fluids with KZK(P).This is a reasonable
model for most gas flows encountered in aircraft turbomachinery;on the other
hand for liquid flows,including bubbly liquids,the appropriate substitute for r in
the acoustic analogy is r
-
defined in (3.13).
(d) Pressure-related forms of the acoustic analogy
Two alternatives to the standard acoustic analogy of Lighthill (1952),
expressed for bounded domains as in (3.5),are now presented.They result
from choosing r
C
Zr
-
(general fluid) or r
C
Zr

(fluid with KZK(P)) in (3.10).
They both offer the advantage that the local density does not appear in either the
surface or the initial-value source terms.
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(i) r
C
Zr
-
,general fluid
1
c
2
0
v
2
vt
2
KV
2
 
p Z
1
c
2
0
v
vt
pHdðtÞ½ K
v
vx
i
r
-
u
i
HdðtÞ½ 
C
v
vt
J
-
i
^n
i
dðnÞQ
 
K
v
vx
i
L
-
ij
^n
i
dðnÞQ
 
C
v
Q
-
vt
K
v
vx
i
F
-
i
C
rKr
-
r
vp
ij
vx
j
!
C
v
2
T
-
ij
vx
i
vx
j
:
ð3:22Þ
Here,F
-
i
is an equivalent body force per unit volume,defined by
F
-
i
Z
r
-
r
G
i
CQ
-
u
i
;ð3:23Þ
and J
-
i
;L
-
ij
and T
-
ij
,are defined in the same way as J
i
,L
ij
and T
ij
with r replaced
by r
-
,so
T
-
ij
Zr
-
u
i
u
j
Cp
ij
Kc
2
0
ðr
-
Kr
0
Þd
ij
Zr
-
u
i
u
j
Kt
ij
;ð3:24Þ
where t
ij
is the viscous stress such that p
ij
Zpd
ij
Kt
ij
.
The presence of convected density inhomogeneities in the flow will make r
differ from r
-
,even in a non-conducting fluid.The dipole body force term in
(3.23) then depends on fluctuations in the body force per unit mass,G
i
/r,rather
than G
i
,and an extra dipole term appears (the term in p
ij
on the last line of
(3.22)).The p
ij
term acts like an additional body force applied to the reference
medium;it is the generalization of the dipole source identified by Morfey (1973)
and Howe (1998) for inviscid flows,and by Lilley (1974,eqn (23)) for viscous
perfect gas flows.
(ii) r
C
Zr

,fluid with KZK(P)
1
c
2
0
v
2
vt
2
KV
2
 
c
2
0
r

Kr
0
 
Z
v
vt
ðr

Kr
0
ÞHdðtÞ½ K
v
vx
i
r

u
i
HdðtÞ½ 
C
v
vt
J

i
^n
i
dðnÞQ½ K
v
vx
i
L

ij
^n
j
dðnÞQ
 
C
v
vt
r

D

 
K
v
vx
i
F

i
C
rKr

r
vp
ij
vx
j
!
C
v
2
T

ij
vx
i
vx
j
:
ð3:25Þ
Here
F

i
Z
r

r
G
i
Cr

D

u
i
;ð3:26Þ
and J

i
;L

ij
and T

ij
are defined in the same way as J
i
,L
ij
and T
ij
with r replaced
by r

.Equation (3.21) has been used to substitute Q

in the first source term on
the third line.
Equations (3.22) and (3.25) are key results.They represent the Lighthill–FWH
acoustic analogy in its most general and useful formto date,with initial-value and
boundary-value equivalent source terms that do not involve the local density.
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For that reason,they are well suited to wave extrapolation in computational
acoustics,as discussed in §4.
(e ) Interpretation of the monopole source term
The monopole density
Q
-
in (3.22) is non-zero in general.However,in an ideal
fluid,its effect is limited to the scattering of sound by sound (nonlinear
acoustics),or to scattering in an inhomogeneous medium by variations of
compressibility (for example in a bubbly liquid);whereas in real turbulent flows,
fluctuations in Q
-
also arise from unsteady viscous or thermal dissipation.
An exact expression for Q
-
in ideal gas flows that is convenient for
computational studies follows from (3.15) and (3.18)
Q
-
Z
gK1
c
2
0
FKpDK
vq
i
vx
i
C_q
 
Kr
0
g
g
0
K1
 
D;ð3:27Þ
where g is the specific heat ratio.For a perfect gas (gZconst.),the second term
vanishes;but in this situation,it is simpler to use formulation (3.25) based on
Goldstein’s r

variable,since Q

Zr

D

.
To interpret Q
-
for the general case of an arbitrary fluid,define the excess
compressibility K
e
as
K
e
ZKKK
0
K
vK
vP
 
s;0
p ZKKK
0
Cð2b
0
K1ÞK
2
0
p;ð3:28Þ
the partial derivative ðvK=vPÞ
s
is evaluated holding the specific entropy s
constant and b is the fundamental derivative c
K1
vðrcÞ=vrð Þ
s
(Thompson 1988).
Then (3.17) gives
Q
-
Zr
0
ð1 CK
0
pÞ D

KK
e
Dp
Dt
Cðb
0
K1ÞK
2
0
Dp
2
Dt

Cr
0
K
3
0
p
2
Dp
Dt
Zr
0
D

KK
e
Dp
Dt
Cðb
0
K1ÞK
2
0
Dp
2
Dt

1 COðK
0
pÞ½ :ð3:29Þ
The three terms in the first bracket each have a physical interpretation.
(i) The entropic dilatation rate D

is given by the energy equation (3.15).It
contains contributions
D

m
Z
a
rc
p
F due to viscous dissipation;ð3:30Þ
D

k
ZK
a
rc
p
vq
i
vx
i
Z
a
rc
p
v
vx
i
k
vT
vx
i
 
due to heat conductivity k;ð3:31Þ
D

q
Z
a
rc
p
_q due to external heat sources:ð3:32Þ
In an Euler equation model,only D

q
survives.
(ii) The KK
e
Dp/Dt term is Rayleigh’s monopole scattering term (Rayleigh
1894).It accounts for sound attenuation and scattering by bubble clouds
in liquids (e.g.Commander & Prosperetti 1989;Leighton et al.2004),or
by any variation in the compressibility of the medium.
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(iii) The nonlinear Dp
2
/Dt termcombines with the quadrupole termin the last
line of (3.22) to produce the Westervelt source termof nonlinear acoustics
(Hamilton & Morfey 1997).
The relative error term O(K
0
p) in (3.29) may also be written as O(M
2
),in
aeroacoustic applications where p scales on r
0
u
2
ref
;here MZu
ref
=c
0
,and u
ref
is a
typical flow velocity.
4.Implications for computational aeroacoustics
(a) Wave extrapolation procedures
In CAA,a two-stage procedure—called direct noise computation in the reviews by
Bailly & Bogey (2004) and Colonius & Lele (2004)—is used to calculate the far-
field sound radiated by a region of turbulent or unsteady flow.An accurate
numerical simulation is first performed to capture the unsteady flow in a limited
domain D,which is chosen to extend as far into the surrounding region of smaller-
amplitude unsteadiness as computational costs allow.Numerical boundary
conditions on D are chosen so as to minimize the reflection of outgoing acoustic
waves.The resulting simulation in D is then extended to the far field by one of
several methods that typically involve linearized approximations to the flow
equations and are less demanding computationally (Colonius & Lele 2004).
Since the late 1980s,two popular choices for far-field extension of accurate near-
field simulations have been the standard FWHmethod based on rKr
0
as acoustic
variable,referred to as FWH(r) in what follows,and the related Kirchhoff method
based on p,referred to as Kirchhoff( p).Both these rely on the flow outside D
approximating a uniform acoustic medium with small-amplitude disturbances
governed by the wave equation.Brentner & Farassat (1998) have carried out a
detailed comparison of the FWH(r) and Kirchhoff( p) methods as applied to
transonic rotor noise.By calculating the far-field radiation with S taken
progressively further from the rotor,they were able to show that FWH(r)
converged more rapidly with increasing distance.Asimilar conclusion was reached
by Singer et al.(2000) who studied the sound field of a long rigid cylinder in
subsonic cross-flow(MZ0.2) with a turbulent wake.Since the FWHand Kirchhoff
formulations are both exact if all the terms are retained,these differences must be
due to the neglected volume terms being different.Specifically,since both studies
were for unheated,homogeneous fluid flows with (rKr
-
)/r
0
wM
2
,they are due to
the FWHvolume termbeing of quadrupole order,i.e.v
2
T
ij
=vx
i
vx
j
,which has zero
monopole and dipole moments;whereas the corresponding volume term in the
Kirchhoff formulation is a spatially windowed quadrupole distribution,
v
2
T
ij
=vx
i
vx
j
,and lacks this property.Since the far-field solution was obtained
with the free-field Green’s function in both cases and the radiating surface S was
compact with respect to the lower radiated frequencies,weaker radiation is
expected from the volume terms in the FWH formulation.
For CAA calculations of jet noise,different problems arise with the standard
FWH and Kirchhoff techniques for far-field extrapolation,because jets of
practical interest are typically heated (as in aircraft gas turbine exhausts).The
c
2
0
ðrKr
0
Þd
ij
contribution to T
ij
cannot be neglected and decays slowly in the
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downstream direction.A review of CAA results for turbulent jets by Shur et al.
(2005) drew attention to this problem,and offered a pragmatic solution:the
authors suggest that the FWH volume quadrupole distribution
T
ij
can be
neglected if r is replaced in the FWH surface terms either by r
-
Zr
0
Cc
K2
0
p,or
by r

from (3.20).These changes occur naturally in the density-substituted
acoustic analogy formulations (3.22) and (3.25).
Since initial values are not usually involved in wave extrapolation,and by
definition the volume sources are ignored,the appropriate equations are as follows.
(i) General fluid
1
c
2
0
v
2
vt
2
KV
2
 
pz
v
vt
J
-
i
^n
i
dðnÞ
 
K
v
vx
i
L
-
ij
^n
j
dðnÞ
 
;ð4:1Þ
with
J
-
i
Zr
0
u
i
Cc
K2
0
pðu
i
Kv
i
Þ and L
-
ij
Zðr
0
Cc
K2
0
pÞu
i
ðu
j
Kv
j
Þ Cp
ij
:ð4:2Þ
(ii) Fluid with KZK(P)
1
c
2
0
v
2
vt
2
KV
2
 
c
2
0
r

Kr
0
 
z
v
vt
J

i
^n
i
dðnÞ½ K
v
vx
i
L

ij
^n
j
dðnÞ
 
;ð4:3Þ
where J

i
and L

ij
are defined as in (3.25).Spalart & Shur (P.R.Spalart & M.L.
Shur 2007,unpublished work) have used both versions to calculate the sound
radiated from an LES-simulated hot jet and found the difference to be negligible.
Their numerical study also provided detailed evidence that these formulations
are better suited to such problems than standard FWH.
(b ) Sound radiation from low Mach number flows past solid boundaries
The computational convenience of (4.1) and (4.3),for purposes of representing
the sound field in V,lies in the restriction of the equivalent sources to a control
surface S.Contributions from volume-distributed sources are rendered
arbitrarily small by moving S further from the source region.In the case of
low Mach number flows past solid boundaries,however,a control surface placed
on the boundary may already yield the dominant contribution to the far-field
sound,without enclosing any other part of the flow.
This possibility was first recognized by Curle (1955),who reformulated
Lighthill’s analogy for flows past rigid obstacles.The surface dipole distribution
L
ij
^n
j
of (3.5) reduces in this case to p
ij
^n
j
,while the surface monopole distribution
J
i
^n
i
vanishes.Curle showed by dimensional reasoning that for homentropic flows
where c
2
0
ðrKr
0
Þzp,the volume quadrupole distribution,T
ij
zru
i
u
j
Kt
ij
,and
the surface dipole distribution p
ij
^n
j
make contributions I
Q
,I
D
to the far-field
intensity,such that
I
Q
=I
D
zM
2
f ðReÞ/0 as M/0:ð4:4Þ
Here,MZU/c
0
is the Mach number of the incident flow,and ReZUL/n is the
Reynolds number based on a typical dimension L.Physically,the asymptotic
dependence (4.4) applies when the wavelength of the radiated sound is large
compared with L.The obstacle is then described as acoustically compact.
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Curle’s surface-source description is generalized below to flows with unsteady
heat transfer at impermeable moving boundaries.The aim is to represent the
boundary by an equivalent surface distribution of monopoles and dipoles,which
will account for almost the entire sound field when M
2
is small.The starting
point is (3.22),with the initial-value terms removed (QZ1).The volume
monopole distribution Q
-
is given by (3.29),noting that K
0
pwM
2
;thus,in the
limit M
2
/0,
v
Q
-
vt
Zr
0
v
D

vt
1 COðM
2
Þ
 
:ð4:5Þ
Here it is assumed that K
e
/K
0
is O(M
2
);in other words,any variations of fluid
compressibility due to gradients of entropy or composition are of the same order
of magnitude as those due to pressure variations.
3
It is further assumed that
external heat sources are absent,so that _qZ0.Then in flows with DT/TZO(1),
the dominant term in D

is due to heat conduction,
D

ZK
a
rc
p
vq
i
vx
i
1 COðM
2
Þ
 
;ð4:6Þ
giving (for QZ1) the following expression for
D

in (4.5),
D

HzK
a
rc
p
H
vq
i
vx
i
Z
a
rc
p
q
i
^n
i
dðnÞ Cq
i
H
v
vx
i
a
rc
p
 
K
v
vx
i
a
rc
p
q
i
H
 
:ð4:7Þ
These three terms are referred to below as D

1
;D

2
and D

3
.
When boundaries are present,and D

1
is substituted in (4.5),the normal heat
flux at the boundary,q
i
^n
i
Zq
n
(positive into the fluid),leads to a surface
monopole distribution of strength r
0
ða=rc
p
Þq
n
per unit area.This result holds for
either fixed or moving boundaries.An oscillating heat flux q
n
on S is thus
acoustically equivalent to vibrating an impermeable boundary with a normal
velocity of ða=rc
p
Þq
n
,if terms in q
2
n
are neglected.
In some situations—when the solid surface is acoustically compact,for
example,or (in turbulent flows) when the heat flux q
n
is coherent on a scale much
less than the acoustic wavelength—the equivalent surface monopole distribution
identified above is the dominant source of far-field sound associated with Q
-
.It
has previously been identified by Landau & Lifshitz (1987) using matched
expansions,Howe (1975,§8) using volume sources in an acoustic analogy and
Kempton (1976,§2) who compared both these methods with a surface heat flux
formulation.The examples discussed by these authors relate to the small-
amplitude case with the solid boundary,either an infinite plane surface,or an
acoustically compact body.The results from all three methods are equivalent to
the more general result stated here.
The remaining terms in (4.7),D

2
and D

3
,contribute volume monopole and
volume dipole sources of sound,respectively,when substituted in (4.5).The D

2
monopole contribution is nonlinear in the temperature gradient,and so is not
relevant to the linearized examples mentioned above;its potential as a source of
sound in turbulent flows is discussed in appendix C.The D

3
dipole contribution,
to the extent that it represents a layer of dipoles close to a solid wall and oriented
normal to the wall,will be a relatively weak radiator,provided that the thermal
3
In Howe (1998) §2.3.2,such variations are set equal to zero.An extreme case where this
assumption fails is a bubbly liquid.For ideal gas flows,eqn (26) follows directly from eqn (17).
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boundary-layer thickness is much less than an acoustic wavelength;thus,like D

2
,
it is not significant in the examples mentioned above.In an unbounded fluid,it
represents sound generation by entropy diffusion and is shown to be a weak effect
in appendix C.
Retaining the surface monopole distribution identified above,and neglecting
all other volume source terms on the right of (3.22),leads to the following result
for the pressure field radiated by acoustically compact impermeable solid
boundaries:
1
c
2
0
v
2
vt
2
KV
2
 
pzr
0
v
vt
v
i
C
a
rc
p
q
i
 
^n
i
dðnÞ

Kr
0
v
vx
i
p
ij
^n
j
dðnÞ
 
:ð4:8Þ
Note that the explicit inclusion of the q
n
surface monopole in the acoustic
analogy removes one of Tam’s objections to the latter as a description of
aeroacoustic sources (Tam 2002,example 2).
5.Extension to parallel shear flow
A limitation of Lighthill-type acoustic analogies is that the base flow must be
taken as uniform.Thus,extrapolation of far-field sound from data on a control
surface S can be accomplished using (3.22) or (3.25) only if the exterior fluid
can be modelled as an ideal acoustic medium at rest,since only under
these conditions will the right-hand side of either equation vanish everywhere
outside S.The ability to extend (3.22) or (3.25) to more general base flows would
offer greater freedom in the choice of control surface for wave extrapolation.One
of the most crucial issues in applying surface extrapolation methods to the jet
noise problem is the downstream closure of the FWH surface,which must pass
through the non-uniform flow in the jet.
In this section,an equation similar to (3.25) is derived,but with a steady
parallel shear flow as base flow.The starting point is a set of equations obtained
by Goldstein (2001) that describe mass and momentum conservation in terms of
base-flow (~) and perturbation ($
0
) variables.These are summarized in appendix
B,in a form appropriate to fluids with viscosity and heat conduction (see (B 6)
and (B 7)).Following Goldstein (2001),the modified perturbation variables
p Z
r

r
0
K1;m
i
Z
r

r
0
u
0
i
Zð1 CpÞu
0
i
;ð5:1Þ
are introduced,where r

is the pressure-related density variable defined in (3.19).
Then (B 6) and (B 7) give
e
Dp
Dt
C
vm
i
vx
i
Zs;ð5:2Þ
e
Dm
j
Dt
Cm
i
v~u
j
vx
i
C~c
2
vp
vx
j
Zs
j
K
vs
ij
vx
i
;ð5:3Þ
where
e
D=DtZv=vtC~u
i
v=vx
i
is the material derivative following the base flow.
On the left,(5.2) and (5.3) are linear in p,m
i
;the source terms on the right
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account for dissipative and nonlinear effects,with
s Zð1 CpÞD

;ð5:4Þ
s
j
ZKðc
2
Þ
0
vp
vx
j
Cð1CpÞg
j
Cð1CpÞ u
0
j
D

C
1
r
vt
ij
vx
i
 
;ð5:5Þ
s
ij
Zð1CpÞu
0
i
u
0
j
:ð5:6Þ
In (5.3),~cðx
2
;x
3
Þ is the base-flow sound speed,related to the base-flow density
~rðx
2
;x
3
Þ by ~r~c
2
Zconst:,and ðc
2
Þ
0
is defined by
ðc
2
Þ
0
Zc
2
K~c
2
:ð5:7Þ
Equations (5.2) and (5.3) are exact;they are derived for a fluid whose
isentropic compressibility depends only on pressure (e.g.a perfect gas),and with
the base flow defined by (B 3).They represent a generalization of equations (3.8)
and (3.9) in Goldstein (2001),there derived for a perfect gas with dissipative
effects omitted.
(a) Lilley–Goldstein equation for bounded domains
Multiplying equations (5.2) and (5.3) by HQZU,as in §3,leads to windowed
versions of these equations.As before,windowed variables are written with a line
over.Rearranging derivatives of p,m
i
and s
ij
,so that the derivatives act on
p;
m
i
and
s
ij
,gives
e
D
p
Dt
C
v
m
i
vx
i
Zp
e
DU
Dt
Cm
i
vU
vx
i
C
s;ð5:8Þ
e
D
m
j
Dt
C
m
i
v~u
j
vx
i
C~c
2
v
p
vx
j
Z~c
2
p
vU
vx
j
Cm
j
e
DU
Dt
C
s
j
K
v
s
ij
vx
i
Cs
ij
vU
vx
i
:ð5:9Þ
Next
e
D=Dt (5.8) – v=vx
j
(5.9) is taken,in order to partially eliminate
m
i
,noting
that v~u
i
=vx
i
vanishes for prescribed base flow,
e
D
2
p
Dt
2
K2
v
~
u
i
vx
j
v
m
j
vx
i
K
v
vx
j
~c
2
v
p
vx
j
 
Z
e
D
Dt
p
e
DU
Dt
Cm
i
vU
vx
i
C
s
!
K
v
vx
j
~c
2
p
vU
vx
j
Cs
ij
vU
vx
i
Cm
j
e
DU
Dt
C
s
j
!
C
v
2
s
ij
vx
i
vx
j
:
ð5:10Þ
Final elimination of
m
j
from the left of (5.10) follows on taking
e
D=Dt of that
equation,and using the base-flow description
f~u
i
g ZfUðx
2
;x
3
Þ;0;0g;ð5:11Þ
from(B 3).In the second term on the left of (5.10),jZ1 does not contribute and,
for js1,the momentum equation (5.9) gives
e
D
m
j
Dt
ZK~c
2
v
p
vx
j
Cðs
ij
C~c
2
pd
ij
Þ
vU
vx
i
Cm
j
e
DU
Dt
C
s
j
K
v
s
ij
vx
i
:ð5:12Þ
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The final result is an inhomogeneous Lilley–Goldstein equation in the windowed
variable
p.It may be written in the compact form

pÞ Z
e
D
2
Dt
2
½pHdðtÞKL
j
ð1CpÞu
0
j
HdðtÞ
 
C
e
D
2
Dt
2
u
0
j
Cpðu
i
Kv
i
Þ
 
^n
i
dðnÞQ
 
KL
j
~c
2
pd
ij
Cð1 CpÞðu
i
Kv
i
Þu
0
j
 
^n
i
dðnÞQ
 
C
e
D
2
s
Dt
2
CL
j
v
s
ij
vx
i
K
s
j
 
;
ð5:13Þ
where the operators L and L
j
are defined by
LðxÞ Z
e
D
3
x
Dt
3
KL
j
~c
2
vx
vx
j
 
;L
j
ðhÞ Z
e
D
Dt
vh
vx
j
 
K2
v
vx
1
h
vU
vx
j
 
:ð5:14Þ
Equivalently,since ~r~c
2
Zconst:,
1
~c
2
LðxÞ Z
e
D
Dt
1
~c
2
e
D
2
x
Dt
K~r
v
vx
j
1
~r
vx
vx
j
 
"#
C2
v
vx
1
vU
vx
i
vx
vx
i
 
;ð5:15Þ
this form of the wave operator,applied to p,describes small-amplitude pressure
waves on a parallel sheared mean flow in a general fluid (Pridmore-Brown 1958;
Tester & Morfey 1976).Derivatives of the window function have been written in
(5.13) as delta functions using the relations,
e
DU
Dt
ZHdðtÞ Cð~u
i
Kv
i
Þ^n
i
dðnÞQ;
vU
vx
j
Z^n
j
dðnÞQ:ð5:16Þ
Note that the base-flow velocity,U(x
2
,x
3
),is implicit in the operators L,
L
j
and
e
D
Dt
Z
v
vt
CU
v
vx
1
;ð5:17Þ
but nowhere appears explicitly in the source terms of (5.13).
Solutions of the linear equation
1
~c
2
LðjÞ Z~rG;ð5:18Þ
for arbitrary axisymmetric base-flow profiles,~r,~c,U are discussed in Tester &
Morfey (1976),for the generic source distribution
G Z
e
D
Dt
!
v
ðKvÞ
m
S
ðm;nÞ
ij/
vx
i
vx
j
/
;ð5:19Þ
here mZ0,1,2,.is the spatial order of the source distribution and nZ0,1,2,.
is the temporal order.It follows from (5.15) that the Lilley–Goldstein equation
(5.13) can be put in the general form (5.18) and (5.19),if both sides are multiplied
by ~r,and j is identified with ~r~c
2
p.Analytical expressions are available for far-field
radiation in the low- and high-frequency limits,for the special case of
axisymmetric base flows;alternatively (5.18) and (5.19) can be solved numerically.
C.L.Morfey and M.C.M.Wright
2116
Proc.R.Soc.A (2007)
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In either case,Tester &Morfey (1976) showhowone can use a Green’s function for
the (mZ0,nZ3) monopole source,together with its first radial derivative,to
obtain any of the mZ(0,1,2) Green’s functions for arbitrary n.
(b ) Alternative arrangement of the Lilley–Goldstein source terms
The arrangement of source terms on the right of (5.13) is not unique.It can be
altered by augmenting the ‘applied stress’ s
ij
in (5.6) with an additional stress
Ds
ij
,and subtracting L
j
v
Ds
ij
 
=vx
i
 
fromthe right of (5.13) to compensate.The
square-bracketed ‘surface stress’ in the third line of (5.13) gains an extra term
Ds
ij
,and the ‘applied force’ s
i
—originally defined in (5.5)—gains an extra term
vðDs
ij
Þ=vx
i
.
In particular,(5.13) can be brought into closer correspondence with (3.25),
which describes sound generation in a uniform medium at rest,by defining
Ds
ij
Z
1
~r
ðpK~r~c
2
pÞd
ij
Kt
ij
 
:ð5:20Þ
The quantity in square brackets equals the viscous stress,plus a term nonlinear
in the pressure perturbation p.
4
The surface dipole term in line three of (5.13)
then becomes
KL
j
p
ij
~r
Cð1 CpÞðu
i
Kv
i
Þu
0
j

^
n
i
dðnÞQ

;ð5:21Þ
the new ‘surface stress’ (in square brackets) is analogous to r
K1
0
L

ij
in the modified
Lighthill-analogy equation (3.25),the only differences being that the Lighthill
base-flow density r
0
is replaced by ~r in (5.21),and u
j
is replaced by u
0
j
.
The last line of (5.13),representing volume-distributed sources of sound,
retains the same form but with the following revised definitions in place of (5.5)
and (5.6):
s
j
Z
1
~r
K
1 Cp
r
 
vp
ij
vx
j
Cð1 CpÞ g
j
Cu
0
j
D

 
C p
ij
K~r~c
2
pd
ij
 
v~r
K1
vx
i
;ð5:22Þ
s
ij
Zð1CpÞu
0
i
u
0
j
C
1
~r
p
ij
K~r~c
2
pd
ij
 
:ð5:23Þ
The ‘applied body force’ s
j
in (5.22) may be compared with the dipole
distribution on line three of (3.25),and s
ij
may be compared with T

ij
in (3.25).
Each of the terms in this version of the Lilley–Goldstein analogy has its parallel
in the modified Lighthill analogy,except for the final term of (5.22),which
involves the base-flow density gradient.
(c ) Reduction to the Lighthill analogy for uniform base flows
For the special case in which the base flow is a uniform fluid at rest,a
Lighthill-type acoustic analogy is recovered from(5.13).The governing equations
(5.2) and (5.3),valid in V for tO0,become
vp
vt
C
vm
i
vx
i
Zs;
vm
j
vt
Cc
2
0
vp
vx
j
Zs
j
K
vs
ij
vx
i
;ð5:24Þ
4
The nonlinear termmay be written as ~r~c
2
½ð
~
bK1Þz
2
COðz
3
Þ where z is the dimensionless pressure
p=~r~c
2
and b is the fundamental derivative c
K1
vðrcÞ=vrj
s
(Thompson 1988).
2117
Extensions of Lighthill’s acoustic analogy
Proc.R.Soc.A (2007)
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after substituting ~u
i
Z0;~cZc
0
,these are identical in form to the governing
equations of Lighthill’s original 1952 paper,namely
vr
vt
C
v
vx
i
ðru
i
Þ Z0;
v
vt
ðru
j
Þ Cc
2
0
vr
vx
j
ZK
vT
ij
vx
j
;ð5:25Þ
apart from the extra source terms s,s
j
.One can therefore write down a wave
equation for the windowed variable
p immediately,by following the steps used to
obtain (3.5),
1
c
2
0
v
2
vt
2
KV
2
 
c
2
0
p
 
Z
v
vt
pHdðtÞ½ K
v
vx
i
ð1 CpÞu
i
HdðtÞ½ 
C
v
vt
½ð1CpÞu
i
Kpv
i
 ^n
i
dðnÞQf g
K
v
vx
i
½ð1 CpÞu
i
ðu
j
Kv
j
Þ Cc
2
0
pd
ij
 ^n
j
dðnÞQ
 
C
v
s
vt
K
v
s
i
vx
i
C
v
2
s
ij
vx
i
vx
j
:ð5:26Þ
The same result is obtained from (5.13) by noting that when the base flow is a
uniform fluid at rest,
L/
v
vt
v
2
vt
2
Kc
2
0
V
2
 
;L
j
/
v
vt
v
vx
j
;ð5:27Þ
removal of the common v=vt operator then leads directly to (5.26).
Despite starting froma common basis and using the same wave variable (apart
from a r
0
factor),equations (5.26) and (3.25) are not identical.Differences appear
in the surface dipole distribution,and in the volume dipole and quadrupole distri-
butions,that represent a rearrangement between the various terms similar to that
discussed in §5b.Equation (5.26) appears to offer no obvious advantage over (3.25).
6.Conclusions
(i) The acoustic analogy formulation of aerodynamic sound,due to Lighthill
(1952) and extended to spatially bounded domains by Ffowcs Williams &
Hawkings (1969),is presented in a modified formmore suitable for heated
andinhomogeneous fluid flows.Two formulations are given,bothexact:(i)
for general fluids,with pressure as the wave variable,and (ii) for a
restricted class of fluids that includes perfect gases.In the latter case,the
wave variable is the pressure-related density introduced by Goldstein
(2001).Whereas the standard FWHboundary-value source terms involve
the local density,the corresponding terms in (i) and (ii) do not.
(ii) The modified FWH formulations presented in §3 also include equivalent
source terms appropriate for initial-value problems.As with the
boundary-value source terms,the initial-condition sources do not involve
r.The corresponding sources obtained by extending the standard FWH
formulation,as in (3.5) where rKr
0
is the wave variable,inevitably
involve the local density at tZ0.
(iii) In all the three cases—standard FWH and modified versions (i) and
(ii)—the acoustic analogy equation for bounded domains may be
C.L.Morfey and M.C.M.Wright
2118
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written as
1
c
2
0
v
2
vt
2
KV
2
 
c
2
0
r
C
Kr
0


Z
v
vt
ðr
C
Kr
0
ÞHdðtÞ
 
K
v
vx
i
r
C
u
i
HdðtÞ
 
C
v
vt
½r
C
ðu
i
Kv
i
Þ Cr
0
v
i
 ^n
i
dðnÞQ
 
K
v
vx
i
½r
C
u
i
ðu
j
Kv
j
Þ ^n
j
dðnÞQ
 
C
v
Q
C
vt
K
v
vx
i
rKr
C
r
vp
ij
vx
j
CF
C
i
!
C
v
2
T
C
ij
vx
i
vx
j
:
ð6:1Þ
The volume distributions Q
C
,F
C
i
and T
C
ij
are defined by
Q
C
Z
Dr
C
Dt
Cr
C
D;F
C
i
Z
r
C
r
G
i
CQ
C
u
i
;ð6:2Þ
T
C
ij
Zr
C
u
i
u
j
Cp
ij
Cc
2
0
ðr
C
Kr
0
Þd
ij
;ð6:3Þ
where G
i
is an applied body force per unit volume.Comparative
expressions for r
C
,Q
C
,F
C
i
,T
C
ij
and c
2
0
ðr
C
Kr
0
Þ are listed in table 1 for
each of the three formulations.
(iv) For purposes of wave extrapolation in CAA,the absence of r from
boundary-value source terms offers significant advantages.In particular,
use of standard FWH boundary terms (with external volume sources
neglected) can lead to large errors in the extrapolated sound field when
the flow contains density inhomogeneities that are advected across the
control surface S.The modified acoustic analogy formulations of §3 are
more tolerant to truncation of volume sources under these conditions.
(v) For describing the radiation of sound from unsteady flows with
impermeable boundaries and heat transfer,equation (4.8) relates the
pressure to surface distributions of monopoles and dipoles on the flow
boundaries.The neglect of volume sources in this situation requires the
boundaries (or else the coherent length-scale of the surface sources) to be
Table 1.(Quantities appearing in the acoustic analogy equation (6.1),for three choices of r
C
.Here,
D

is the entropic dilatation rate,defined in equation (3.15).In the last column,
pKc
2
0
ðr

Kr
0
Þzðb
0
K1Þp
2
=r
2
0
c
2
0
,where b is the fundamental derivative c
K1
v(rc)/vrj
s
.)
variable FWH modified FWH (i) modified FWH (ii)
r
C
r
r
-
Zr
0
Cc
–2
0
p
r

(3.19) or (3.20)
Q
C
0 Q
-
(3.17) or (3.18) Q

Zr

D

F
C
i
G
i
(r
-
/r)G
i
CQ
-
u
i
ðr

=rÞG
i
Cr

D

u
i
T
C
ij
T
ij
r
-
u
i
u
j
Kt
ij
r

u
i
u
j
Kt
ij
C½pKc
2
0
ðr

Kr
0
Þ d
ij
c
2
0
ðr
C
Kr
0
Þ c
2
0
ðrKr
0
Þ
p
c
2
0
ðr

Kr
0
Þ (zp,in far field)
2119
Extensions of Lighthill’s acoustic analogy
Proc.R.Soc.A (2007)
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acoustically compact;(4.8) then generalizes the results of Curle (1955) to
flows where heat transfer is important.
(vi) Solving for the acoustic variable,in the acoustic analogy formulations
summarized by (6.1–6.3),is facilitated by the fact that (6.1) is valid for
all (x
i
,t).Thus,the free-field Green’s function may be used,or otherwise
any causal Green’s function that satisfies homogeneous boundary
conditions on S.
(vii) In some applications of the wave extrapolation method (WEM) to
aeroacoustics,the control surface S is required to cut through a sheared
high-speed flow;this can occur,for example,with numerical simulations
of turbulent jets,as discussed by Shur et al.(2005).A more appropriate
base flow for the acoustic analogy approach is then the one proposed by
Goldstein (2001),following Lilley (1974).Goldstein’s (2001) parallel
shear flow analogy,originally presented for an inviscid perfect gas,is
extended in §5 to viscous flows and bounded domains.The resulting
Lilley–Goldstein equation (5.13),or its alternative version in §5b,
provides the appropriate boundary-source formulations needed for WEM
in parallel shear flows.
(viii) An important limitation of the perfect-gas model commonly used in
aeroacoustics is discussed in appendix C;a certain thermodynamic
derivative,which vanishes for a perfect gas but not for real fluids,is
shown to be of crucial importance in determining sound radiation from
turbulent mixing of hot and cold streams at low Mach numbers.
The authors thank Dr Philippe Spalart for stimulating discussions,and in particular for suggesting
the r

form of the Lighthill analogy.They also thank Dr Gwenael Gabard and Dr Anurag Agarwal
for their helpful comments.
M.C.M.W.was supported by an EPSRC Advanced Research Fellowship.
Appendix A.Second time derivative of a windowed field variable
The first step is to note that
v
2
x
vt
2
Z
v
vt
xHdðtÞ½  C
v
vt
Q
v
vt
ðHxÞ

:ðA1Þ
The quantity ðv=vtÞðHxÞ can be rewritten in terms of the material derivative of x
using the identity (2.7).Applying this to Hx gives
v
vt
ðHxÞ Z
D
Dt
ðHxÞK
v
vx
i
ðxu
i
HÞ CHxD
Zx
DH
Dt
K
v
vx
i
ðxu
i
HÞ CH
Dx
Dt
CxD
 
:
ðA2Þ
When (A 2) is substituted in (A 1),a term containing ðv=vtÞðxu
i
HÞ appears.
This can be rewritten by again using (2.7),
v
vt
ðxu
i
HÞ Z
D
Dt
ðxu
i
HÞK
v
vx
j
ðxu
i
u
j
HÞ Cxu
i
HD
Zxu
i
DH
Dt
K
v
vx
j
ðxu
i
u
j
HÞ CH
Dx
Dt
u
i
Cx
Du
i
Dt
Cxu
i
D
 
:
ðA3Þ
C.L.Morfey and M.C.M.Wright
2120
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Therefore,
v
2
x
vt
2
Z
v
vt
xHdðtÞ½ K
v
vx
i
xu
i
HdðtÞ½ 
C
v
vt
x
DH
Dt
Q
 
K
v
vx
i
xu
i
DH
Dt
Q
 
C
v
vt
Dx
Dt
CxD
 
K
v
vx
i
x
Du
i
Dt
Cu
i
Dx
Dt
CxD
 
"#
C
v
2
vx
i
vx
j
xu
i
u
j
 
:
ðA4Þ
Using the expression (2.5) for DH/Dt gives the general kinematic relation,
v
2
x
vt
2
Z
v
vt
xHdðtÞ½ K
v
vx
i
xu
i
HdðtÞ½ 
C
v
vt
½xðu
i
Kv
i
Þ^n
i
dðnÞQK
v
vx
i
xu
i
ðu
j
Kv
j
Þ^n
j
dðnÞQ
 
K
v
Q
vt
K
v
vx
i
x
Du
i
Dt
KQu
i
 
C
v
2
vx
i
vx
j
xu
i
u
j
 
;
ðA5Þ
where KQZDx=DtCxDZvx=vtCðv=vx
i
Þðxu
i
Þ.If x is the density of a conserved
fluid property,Q vanishes.
Appendix B.Basic equations for disturbances to a parallel shear flow
The equations originally derived by Goldstein (2001) for an inviscid perfect gas
are here generalized to include:(i) effects of viscosity and heat conduction,and
(ii) external sources in the form of heat input and body forces.Rather than
restricting the fluid to a perfect gas,the isentropic compressibility is assumed to
be a function K(P) of pressure alone,which includes the perfect gas model as a
special case.This is equivalent to assuming the density to have the separable
form rZf(P)g(s),where s is the specific entropy (Musafir 2007).
With the pressure-dependent density variable r

defined as in (3.19),the
following differential relations apply:
KdP Z
1
rc
2
dP Z
1
r

dr

;
1
r
dP Z
c
2
r

dr

:ðB 1Þ
The continuity and momentum equations may therefore be written as
vu
i
vx
i
ZD

K
1
r

Dr

Dt
;
Du
j
Dt
Z
Kc
2
r

vr

vx
j
C
1
r
vt
ij
vx
j
Cg
j
:ðB 2Þ
Here D

,defined in (3.15),is the entropic dilatation rate and includes the effects
of external heat input,heat conduction and viscous dissipation;also t
ij
is the
viscous stress tensor,and g
j
is an externally applied body force per unit mass.
A steady,divergence-free velocity field is introduced as base flow,following
Goldstein (2001),with
~u
i
f g Z Uðx
2
;x
3
Þ;0;0f g;~r Z~rðx
2
;x
3
Þ;
~
P ZP
0
Zconst:ðB 3Þ
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Extensions of Lighthill’s acoustic analogy
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Here,the tilde denotes base-flow variables,
5
and x
1
is the streamwise direction.
Perturbations relative to the base flow will be denoted by primes,
u
0
i
Zu
i
K~u
i
;ðc
2
Þ
0
Zc
2
K~c
2
:ðB 4Þ
The base-flow material derivative is denoted by
e
D
Dt
Z
e
D
Dt
Z
v
vt
CU
v
vx
1
Z
D
Dt
Ku
0
i
v
vx
i
;ðB 5Þ
note
e
D
~
x=DtZ0,for any base-flow variable
~
x.Equations (B 2) are now multiplied
by r

and rearranged using the relations above to give
e
Dr

Dt
C
v
vx
i
r

u
0
i
 
Zr

D

;ðB 6Þ
e
D
Dt
ðr

u
0
j
Þ Cr

u
0
i
v~u
j
vx
i
Cc
2
vr

vx
j
ZK
v
vx
i
ðr

u
0
i
u
0
j
Þ C
r

r
vt
ij
vx
i
Cr

u
0
j
D

Cr

g
j
:ðB 7Þ
Equations (B 6) and (B 7) are exact,for any fluid with KZK(P),provided that
the base flow is a parallel shear flow.
Appendix C.Thermoacoustic volume sources in the absence
of boundaries
In a viscous heat-conducting compressible fluid,three perturbation modes—
acoustic,thermal and vorticity—propagate (or diffuse) independently at small
amplitudes and in the absence of boundaries (Pierce 1989).Solid boundaries
scatter incident disturbances in any mode into outgoing disturbances in all three
modes,but such scattering is not considered here.Attention is focused instead on
the nonlinear generation of sound away from boundaries,by the second-order
interaction of the thermal mode with itself (self-scattering).Since the thermal
mode is characterized by entropy perturbations (denoted here by s
0
),
6
the self-
scattering process is called s
0
Ks
0
interaction in what follows.The s
0
Ks
0
mechanism
of thermoacoustic soundgenerationis consideredinpart a for perfect gas flows,and
generalized in part b to arbitrary fluids.The case of turbulent mixing between hot
and cold fluid is considered separately in part c.
Chu & Kova
´
sznay (1958) studied all six possible bilinear interactions among
the three basic modes;the fluid was taken as a perfect gas with Prandtl number
3/4 and kZk(T).No explicit expression was given for the acoustic source density
associated with s
0
Ks
0
interaction.The perfect gas model is shown below to be a
special case:one of the two s
0
Ks
0
acoustic source terms associated with Q

Zr

D

in (3.25) is identically zero.However,both for a perfect gas (part a) and a
general fluid (part b),sound radiation from a hot spot diffusing in a stationary
fluid is shown to be extremely weak.Turbulent mixing (part c) appears to be the
only situation where s
0
Ks
0
interaction leads to significant sound radiation.
5
Overbars were used in Goldstein (2001) but could here be confused with windowed quantities.
6
There is a very small entropy perturbation associated with the acoustic mode,given by
s
(p)
z(k/rc
p
)(a/rc
2
)vp/vt,that is neglected in this appendix.Likewise,the entropy mode is
accompanied by first-order pressure and velocity perturbations that are also neglected.
C.L.Morfey and M.C.M.Wright
2122
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(a) Perfect gas flows
In this case,a=rc
p
ZðgK1Þ=gP with g constant,and KZ1/gP.Equation
(3.25) can therefore be used,with Q and H set equal to 1.Entropy–entropy mode
interactions arise from D

,which appears in both monopole and dipole source
terms of line three;here,we focus on the monopole term.Provided that external
heat sources are absent ( _qZ0),(3.30–3.32) give the monopole density as
r

D

Zr

D

k
Cðviscous dissipation termÞ;ðC 1Þ
with
r

D

k
Zr
0
ð1 CpÞ
gK1
g
 
q
i
v
vx
i
1
P
 
K
v
vx
i
q
i
P



;ðC 2Þ
from (4.7).The first term in square brackets corresponds to D

2
and the second
term to D

3
.Any second-order s
0
Ks
0
interaction present in (C 1) must come from
D

3
,since D

2
involves pressure gradients and the viscous term in (C 1) involves
products of velocity gradients.The D

3
contribution to (C 1) is
r

D

3
Zr
0
ð1 CpÞ
gK1
g
 
v
vx
i
k
P
vT
vx
i

:ðC 3Þ
To identify the s
0
Ks
0
component in (C 3),the quantity in square brackets is
expanded by splitting each variable into its unperturbed ð$Þ
0
and first-order
entropy mode ð$
0
Þ components,with self-scattering neglected,and assuming kZ
k(T) with derivative dk/dTZk
T
,
k
P
vT
vx
i
Z
k
P


0
vT
0
vx
i
C
k
T
P


0
T
0
vT
0
vx
i
Cðterms involving T
s
0
Ks
0
and p
s
0
Ks
0
Þ:ðC 4Þ
Here,T
0
ZðT=c
p
Þ
0
s
0
is the temperature perturbation associated with the entropy
mode,and T is the absolute temperature.The second term on the right therefore
represents the desired s
0
Ks
0
interaction for a perfect gas and gives
r

D

ð Þ
s
0
Ks
0
z
1
2
r
0
gK1
g
 
k
T
P


0
v
2
vx
2
i
½T
0
T
0
 Z
1
2
k
T
c
p
T
 
0
v
2
vx
2
i
½T
0
T
0
:ðC 5Þ
A simple scaling argument can be used to indicate the smallness of the sound
field radiated by the monopole distribution (C 5).Let k(T) be represented by a
power law,kfT
n
(for gases,n is of order 1).Consider the sound generated by
diffusion of a local hot spot in an otherwise undisturbed gas,where the hot spot
has temperature DT relative to its surroundings,and typical dimension L
ref
.The
acoustic analogy (3.25) reduces to
1
c
2
0
v
2
vt
2
KV
2
 
r
0
c
2
0
pzr
0
V
2
vS
vt
;ðC 6Þ
where S follows from (C 5) as
Sz
1
2
k
T
rc
p
T
 
0
T
02
wn
k
rc
p
 
0
;if
DT
T
0
w1:ðC 7Þ
The diffusion time-scale over which the hot spot equilibrates is L
2
ref
ðk=rc
p
Þ
K1
0
;
therefore,the far-field pressure,pzr
0
c
2
0
p,at distance R scales as
R
L
ref
 
p
r
0
c
2
0
wc
K4
0
k
rc
p
 
4
0
L
K4
ref
Z
L
L
ref
 
4
:ðC 8Þ
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Extensions of Lighthill’s acoustic analogy
Proc.R.Soc.A (2007)
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Here,LZk
0
=r
0
c
0
c
p
is a thermoacoustic length-scale for the gas,related to the
molecular mean free path.Equation (C 8) requires the hot spot to be acoustically
compact,which,in turn,requires L
ref
[L;for typical gases at sea-level pressure,
Lw0.1 mm.Thus,for hot spots with length-scale 1 mm or greater,the radiated
sound due to s
0
Ks
0
interaction in a perfect gas is very weak indeed.
In part b,the perfect-gas restriction is relaxed;s
0
Ks
0
interactions are then able
to generate sound via the D

2
contribution to (C 1).For a diffusing hot spot in a
fluid at rest,the D

2
and D

3
contributions to the far-field pressure are found to be
comparable.However,for turbulent mixing between hot and cold fluid—
discussed in part c—an asymptotic analysis for M/1 shows the D

2
contribution
to be dominant,and to represent a significant source of sound.
(b ) General fluid
The analysis of the s
0
Ks
0
contribution to D

3
given above for a perfect gas is
easily extended to a general fluid and leads to qualitatively similar conclusions.If
one defines c(T,P)Zka/rc
p
,with c
T
Zðvc=vTÞ
P
,the general-fluid version of
(C 7) has c
T
in place of k
T
/rc
p
T.
However,the s
0
Ks
0
contribution to D

2
no longer vanishes.It is useful to define
the quantity
eðT;PÞ Z
a
rc
p
Z
vV
vh
 
P
;ðC 9Þ
where h is the specific enthalpy and VZr
K1
.Then
D

2
Zq
i
v
vx
i
a
rc
p
 
zKke
T
ð Þ
0
vT
0
vx
i
 
2
;ðC 10Þ
where the last expression results from considering just the second-order s
0
Ks
0
interaction.Comparison of D

2
and D

3
,for the example of a diffusing hot spot in a
general fluid,gives
D

2
D

3








w
ke
T
c
T
 
0
;ðC 11Þ
for arbitrary DT/T
0
.Since ke/cZ1,it is tempting to estimate ke
T
/c
T
w1;but for
perfect gases,e
T
is zero and such an estimate is invalid.
(c ) Thermoacoustic sound generation in turbulent flow
The contribution of the r
-
D

monopole termin (3.22) to thermoacoustic sound
generation is finally examined below in the context of turbulent mixing between
hot and cold regions of a fluid.For low Mach number flows,r
-
zr
0
(with relative
error of order M
2
),and the component of D

due to heat conduction may be
written as
D

k
z
v
2
F
vx
2
i
Kke
T
vT
vx
i
 
2
;where FðT;P
0
Þ Z
ð
T
T
0
cðT
0
;P
0
ÞdT
0
:ðC 12Þ
Provided that the mixing region is acoustically compact,the far-field sound is
determined by
Ð
D

k
d
3
x evaluated over the entire turbulent flow.The integral
C.L.Morfey and M.C.M.Wright
2124
Proc.R.Soc.A (2007)
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follows from (C 12):
ð
D

k
d
3
x ZK
ð
ke
T
jVTj
2
d
3
x;ðC 13Þ
assuming that jVFj falls off faster than jxj
K3
as jxj/N.It is not immediately
obvious how one should scale the right-hand side of (C 13).However,a scaling
hypothesis suitable for low-Mmixing at high Reynolds numbers follows from(19)
and (20) of Morfey (1976)
7
ð
D

k
d
3
xw
ð
D

k
h id
3
xwu
ref
L
2
ref
f:ðC 14Þ
In (C 14),the turbulent flow is assumed statistically stationary,and angle
brackets denote time averaging.
The specific example studied by Morfey (1976) is a turbulent jet of
temperature T
1
mixing with ambient fluid at a different temperature T
0
.Symbol
f represents the time-average increase in total fluid volume due to the mixing
process,per unit volume of injected fluid,as the mixed fluid approaches ambient
temperature.For a general fluid,the value of f follows from appendix III of
Morfey (1976) as
fze
0
r
1
ðh
1
Kh
0
Þ C
r
1
r
0
K1;ðM
2
/1;Re/NÞ:ðC 15Þ
Here,states 1 and 0 are at the same pressure (constant-pressure mixing),but the
temperatures T
1
and T
0
differ by an arbitrary amount.For a perfect gas,f is
easily shown to vanish,and (C 14) predicts zero monopole strength;this agrees
with (C 13),since any perfect gas has e
T
Z0 as noted in part b.
Some insight into the dependence of f on T
1
–T
0
is obtained by expanding r
1
in
(C 15) in powers of h
1
Kh
0
.The final result is
fzK
e
T
rc
p
 
0
ðh
1
Kh
0
Þ
2
zK
e
T
c
p
r
 
0
ðT
1
KT
0
Þ
2
;ðC 16Þ
with error of order (T
1
KT
0
)
3
;the relations ðvr
K1
=vhÞ
P
ZeðT;PÞ and ðvh=vTÞ
P
Z
c
p
have been used.The fact that e
T
appears as a coefficient in (C 16),as it does in
(C 13) and (C 10),underlines the importance of this thermodynamic derivative
in thermoacoustics.
Finally,the far-field sound pressure radiated from the turbulent mixing region
scales as
R
L
ref
p
r
0
c
2
0
wfM
2
ðM
2
/1;Re/NÞ:ðC 17Þ
For comparison,in gas flows,the far-field pressure contributions due to turbulent
Reynolds stresses and pressure–entropy interactions in the same mixing region
scale as
R
L
ref
p
r
0
c
2
0
wM
4
;
DT
T
0
 
M
3
ðM
2
/1;Re/NÞ:ðC 18Þ
The first expression leads to Lighthill’s eighth-power law (Lighthill 1952) in
which density scattering is neglected,and the second represents the scattering of
near-field pressure by compact density inhomogeneities.The latter may be either
unsteady (as discussed by Morfey (1973)) or associated with the mean flow
7
In eqn (20),T should be replaced by 1/r.
2125
Extensions of Lighthill’s acoustic analogy
Proc.R.Soc.A (2007)
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(Tester &Morfey 1976).Provided f is non-zero (i.e.T
1
sT
0
and the fluid is not a
perfect gas),unsteady thermal dissipation becomes the dominant sound source in
turbulent mixing of hot and cold gases as M/0.
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