## Abstract

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.

## 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:

The addition of equivalent source terms to allow for boundaries (either real or virtual) in the flow field (Curle 1955; Ffowcs Williams & Hawkings 1969).

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*p*+*ρu*^{2}/3 (Ffowcs Williams 1969; Kambe 1984), stagnation enthalpy*h*+*u*^{2}/2 (Howe 1975), (*P*/*P*_{0})^{1/γ}−1 (Goldstein 2001), etc.The use of a different base flow to 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 3*a*); 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 (§3*b*). Two possible choices for the new variable, both related to the local pressure, are introduced (§3*c*); the source terms in the resulting acoustic analogy formulations are analysed and compared with FWH, with the help of the energy equation (§§3*c*–*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 §4*b*. The thermoacoustic volume sources neglected in §4*b* 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 be a moving closed surface in three dimensions that separates region from an adjacent region , as illustrated in figure 1. The idea is that may contain solid boundaries; alternatively information on the flow in may be inaccessible. In either case, the aim of the acoustic analogy formulation is to describe the fluctuating pressure or density field in ; the field in is ignored for this purpose, although data on will be used to provide boundary conditions. Any acoustic influence of will be accounted for by equivalent sources on , and no use will be made of the equations of fluid motion within . Likewise, no use will be made of information for *t*<0; the acoustic influence of events prior to *t*=0 will be accounted for by impulsive sources at *t*=0, distributed throughout region . Let *f*(** x**,

*t*) be a continuous indicator function such that

*f*<0 in ,

*f*>0 in , and let |∇

*f*|=1 on . Smoothness of is assumed, so that |∇

*f*| is single valued.1 Let

*n*be a local normal coordinate, defined for points near by

*n*=

*f*; then evaluated on is the gradient operator normal to , in the direction from to . Define the spatial and temporal Heaviside functions(2.1)(2.2)which henceforth will be written without their arguments. From the definition of the Heaviside function, it follows that(2.3)where

*δ*(.) is the Dirac delta function and , the unit normal to . The time derivative of

*H*is found by noting that

*H*is constant in a reference frame moving with the surface, so that(2.4)Here , with

*v*the normal velocity of the surface directed into . The material derivative of

*H*is given by(2.5)where .

In what follows, a line over any variable or quantity means that it is multiplied by *HΘ*, 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 *ξ*(2.6)The identity(2.7)also holds wherever *u*_{i} is defined; here 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 . Conservation of mass and momentum are expressed by(3.1)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), *ρ* denotes fluid density; *u*_{i} is the fluid velocity in the *x*_{i} direction; *p*_{ij}=*P*_{ij}−*P*_{0}*δ*_{ij}, where *P* is absolute pressure and *P*_{ij} is the compressive stress in the fluid; *δ*_{ij} is the Kronecker delta and *G*_{i} is an applied body force per unit volume. The quantities (*ρ*−*ρ*_{0}), *ρu*_{i}, (*ρu*_{i}*u*_{j}+*p*_{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,(3.2)and(3.3)By eliminating from (3.2) and (3.3), an expression for the second time derivative of , the windowed density perturbation, that is valid for all (*x*_{i}, *t*) is obtained,(3.4)

### (a) Density form of the acoustic analogy

Subtracting from (3.4) leads directly to(3.5)Symbols *T*_{ij}, *J*_{i} and *L*_{ij} on the right of (3.5) stand for the Lighthill stress tensor(3.6)the surface mass flux vector(3.7)and the surface momentum flux tensor(3.8)

The sources on the right-hand side of (3.5) can be interpreted as follows:

the first two terms represent the impulsive addition of mass and momentum needed to start the flow from its initial reference state,

the second line contains the usual FWH surface monopoles and dipoles, windowed by

*Θ*, andvolume 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*HΘ*.

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 from the simulation domain to the acoustic far field (§4*a*). 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, and , 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 . 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 *ρ*(** x**)=

*ρ*

_{0}everywhere except in a limited region . Applying the acoustic analogy equation (3.5) in this frame gives

*u*

_{i}=0, while the control surface translates uniformly. As cuts through , the surface monopole distribution on varies with time, being given by −(

*ρ*−

*ρ*

_{0})

*v*

_{n}; but the radiated sound field

*ρ*−

*ρ*

_{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 is not justified under such conditions, since the physically unrealistic surface sources described above are cancelled by the term in the quadrupole distribution . For wave extrapolation purposes, therefore, there is a strong incentive to find alternative formulations that cope better with advected density disturbances crossing .

### (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 from the 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 expression (3.4), and use the kinematic relation (A 5) for the second time derivative of an arbitrary windowed variable, , to replace *ρ* by a new variable *ρ*^{+} related to the local pressure. By defining(3.9)and subtracting from , an expression for is obtained that exhibits the properties mentioned above. A generic acoustic analogy can then be written as(3.10)Here, , and are defined in the same way as *J*_{i}, *L*_{ij} and *T*_{ij} with *ρ* replaced by *ρ*^{+}, and *Q*^{+} is defined as(3.11)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(3.12)which is valid throughout .

Like (3.5), equation (3.10) is exact; it applies to bounded domains (*f*>0, *t*>0); 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*^{+} and *ρ*−*ρ*^{+}. 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 *ρ*^{+}.

### (c) Determination of Q^{+} from the energy equation

If the acoustic density approximation *ρ*^{☆}, defined by(3.13)where *K* is the isentropic compressibility 1/(*ρc*^{2}), is chosen as the substituted density variable *ρ*^{+}, then the corresponding value of *Q*^{+} is given by (3.11) as(3.14)From the energy equation for a single-component2 viscous heat-conducting fluid, with heat input rate per unit volume, D*ρ*/D*t* and D*p*/D*t*, are related by(3.15)where *Φ* is the viscous dissipation function(3.16)and *q*_{i} is the heat flux vector; *α* is the volumetric thermal expansivity and *c*_{p} is the constant-pressure specific heat. The quantity is the difference between the actual dilatation rate and that due to isentropic compression (Morfey 1976); is therefore referred to as the entropic dilatation rate. Alternative expressions for *Q*^{☆} in terms of follow from combining (3.14) and (3.15) with the continuity equation,(3.17)(3.18)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 §3*e*). 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 *ρ*^{+}, 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, *K*=(*γP*)^{−1}, where *γ* is the (constant) specific heat ratio. The substituted density variable, denoted in this case by , is defined by(3.19)(3.20)It follows from (3.19) that . The corresponding value of *Q*^{+}, denoted here by , is given by combining (3.11) and (3.15) with the continuity equation,(3.21)The advantage in simplicity relative to (3.17) or (3.18) is clear, while 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 as a substituted variable is possible, however, only for fluids with *K*=*K*(*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 *ρ* in the acoustic analogy is *ρ*^{☆} 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 *ρ*^{+}=*ρ*^{☆} (general fluid) or (fluid with *K*=*K*(*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.

#### (i) *ρ*^{+}=*ρ*^{☆}, general fluid

(3.22)Here, is an equivalent body force per unit volume, defined by(3.23)and , are defined in the same way as *J*_{i}, *L*_{ij} and *T*_{ij} with *ρ* replaced by *ρ*^{☆}, so(3.24)where *τ*_{ij} is the viscous stress such that *p*_{ij}=*pδ*_{ij}−*τ*_{ij}.

The presence of convected density inhomogeneities in the flow will make *ρ* differ from *ρ*^{☆}, 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}/*ρ*, 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) , fluid with *K*=*K*(*P*)

(3.25)Here(3.26)and are defined in the same way as *J*_{i}, *L*_{ij} and *T*_{ij} with *ρ* replaced by . Equation (3.21) has been used to substitute 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 form to date, with initial-value and boundary-value equivalent source terms that do not involve the local density. 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 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)(3.27)where *γ* is the specific heat ratio. For a perfect gas (*γ*=const.), the second term vanishes; but in this situation, it is simpler to use formulation (3.25) based on Goldstein's variable, since .

To interpret *Q*^{☆} for the general case of an arbitrary fluid, define the excess compressibility *K*_{e} as(3.28)the partial derivative is evaluated holding the specific entropy *s* constant and *β* is the fundamental derivative (Thompson 1988). Then (3.17) gives(3.29)

The three terms in the first bracket each have a physical interpretation.

The entropic dilatation rate Δ

^{•}is given by the energy equation (3.15). It contains contributions(3.30)(3.31)(3.32)In an Euler equation model, only survives.The −

*K*_{e}D*p*/D*t*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.The nonlinear D

*p*^{2}/D*t*term combines with the quadrupole term in the last line of (3.22) to produce the Westervelt source term of 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 here , 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 , which is chosen to extend as far into the surrounding region of smaller-amplitude unsteadiness as computational costs allow. Numerical boundary conditions on are chosen so as to minimize the reflection of outgoing acoustic waves. The resulting simulation in 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 FWH method based on *ρ*−*ρ*_{0} as acoustic variable, referred to as FWH(*ρ*) in what follows, and the related Kirchhoff method based on *p*, referred to as Kirchhoff(*p*). Both these rely on the flow outside 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(*ρ*) and Kirchhoff(*p*) methods as applied to transonic rotor noise. By calculating the far-field radiation with taken progressively further from the rotor, they were able to show that FWH(*ρ*) converged more rapidly with increasing distance. A similar conclusion was reached by Singer *et al*. (2000) who studied the sound field of a long rigid cylinder in subsonic cross-flow (*M*=0.2) with a turbulent wake. Since the FWH and 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 (*ρ*−*ρ*^{☆})/*ρ*_{0}∼*M*^{2}, they are due to the FWH volume term being of quadrupole order, i.e. , which has zero monopole and dipole moments; whereas the corresponding volume term in the Kirchhoff formulation is a spatially windowed quadrupole distribution, , 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 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 contribution to *T*_{ij} cannot be neglected and decays slowly in the 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 can be neglected if *ρ* is replaced in the FWH surface terms either by , or by 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

(4.1)with(4.2)

#### (ii) Fluid with *K*=*K*(*P*)

(4.3)where 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 , lies in the restriction of the equivalent sources to a control surface . Contributions from volume-distributed sources are rendered arbitrarily small by moving 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 of (3.5) reduces in this case to , while the surface monopole distribution vanishes. Curle showed by dimensional reasoning that for homentropic flows where , the volume quadrupole distribution, *T*_{ij}≈*ρu*_{i}*u*_{j}−*τ*_{ij}, and the surface dipole distribution make contributions *I*_{Q}, *I*_{D} to the far-field intensity, such that(4.4)Here, *M*=*U*/*c*_{0} is the Mach number of the incident flow, and *Re*=*UL*/ν 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.

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 (*Θ*=1). The volume monopole distribution *Q*^{☆} is given by (3.29), noting that *K*_{0}*p*∼*M*^{2}; thus, in the limit *M*^{2}→0,(4.5)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 . Then in flows with Δ*T*/*T*=*O*(1), the dominant term in is due to heat conduction,(4.6)giving (for *Θ*=1) the following expression for in (4.5),(4.7)These three terms are referred to below as .

When boundaries are present, and is substituted in (4.5), the normal heat flux at the boundary, (positive into the fluid), leads to a surface monopole distribution of strength per unit area. This result holds for either fixed or moving boundaries. An oscillating heat flux *q*_{n} on is thus acoustically equivalent to vibrating an impermeable boundary with a normal velocity of , if terms in 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), , contribute volume monopole and volume dipole sources of sound, respectively, when substituted in (4.5). The 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 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 boundary-layer thickness is much less than an acoustic wavelength; thus, like , 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:(4.8)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 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 . 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 () 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(5.1)are introduced, where is the pressure-related density variable defined in (3.19). Then (B 6) and (B 7) give(5.2)(5.3)where is the material derivative following the base flow. On the left, (5.2) and (5.3) are linear in *π*, *m*_{i}; the source terms on the right account for dissipative and nonlinear effects, with(5.4)(5.5)(5.6)In (5.3), is the base-flow sound speed, related to the base-flow density by , and is defined by(5.7)

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 *HΘ*=*Ω*, as in §3, leads to windowed versions of these equations. As before, windowed variables are written with a line over. Rearranging derivatives of *π*, *m*_{i} and *σ*_{ij}, so that the derivatives act on , gives(5.8)(5.9)Next (5.8) – (5.9) is taken, in order to partially eliminate , noting that vanishes for prescribed base flow,(5.10)Final elimination of from the left of (5.10) follows on taking of that equation, and using the base-flow description(5.11)from (B 3). In the second term on the left of (5.10), *j*=1 does not contribute and, for *j*≠1, the momentum equation (5.9) gives(5.12)The final result is an inhomogeneous Lilley–Goldstein equation in the windowed variable . It may be written in the compact form(5.13)where the operators and *L*_{j} are defined by(5.14)Equivalently, since ,(5.15)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,(5.16)Note that the base-flow velocity, *U*(*x*_{2}, *x*_{3}), is implicit in the operators , *L*_{j} and(5.17)but nowhere appears explicitly in the source terms of (5.13).

Solutions of the linear equation(5.18)for arbitrary axisymmetric base-flow profiles, , , *U* are discussed in Tester & Morfey (1976), for the generic source distribution(5.19)here *μ*=0, 1, 2, … is the spatial order of the source distribution and *ν*=0, 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 , and *ψ* is identified with . 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. In either case, Tester & Morfey (1976) show how one can use a Green's function for the (*μ*=0, *ν*=3) monopole source, together with its first radial derivative, to obtain any of the *μ*=(0, 1, 2) Green's functions for arbitrary *ν*.

### (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’ *σ*_{ij} in (5.6) with an additional stress Δ*σ*_{ij}, and subtracting from the right of (5.13) to compensate. The square-bracketed ‘surface stress’ in the third line of (5.13) gains an extra term Δ*σ*_{ij}, and the ‘applied force’ *σ*_{i}—originally defined in (5.5)—gains an extra term .

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(5.20)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(5.21)the new ‘surface stress’ (in square brackets) is analogous to in the modified Lighthill-analogy equation (3.25), the only differences being that the Lighthill base-flow density *ρ*_{0} is replaced by in (5.21), and *u*_{j} is replaced by .

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):(5.22)(5.23)The ‘applied body force’ *σ*_{j} in (5.22) may be compared with the dipole distribution on line three of (3.25), and *σ*_{ij} may be compared with 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 for *t*>0, become(5.24)after substituting , these are identical in form to the governing equations of Lighthill's original 1952 paper, namely(5.25)apart from the extra source terms *σ*, *σ*_{j}. One can therefore write down a wave equation for the windowed variable immediately, by following the steps used to obtain (3.5),(5.26)The same result is obtained from (5.13) by noting that when the base flow is a uniform fluid at rest,(5.27)removal of the common operator then leads directly to (5.26).

Despite starting from a common basis and using the same wave variable (apart from a *ρ*_{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 distributions, 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

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 form more suitable for heated and inhomogeneous fluid flows. Two formulations are given, both exact: (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 FWH boundary-value source terms involve the local density, the corresponding terms in (i) and (ii) do not.

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

*ρ*. The corresponding sources obtained by extending the standard FWH formulation, as in (3.5) where*ρ*−*ρ*_{0}is the wave variable, inevitably involve the local density at*t*=0.In all the three cases—standard FWH and modified versions (i) and (ii)—the acoustic analogy equation for bounded domains may be written as(6.1)The volume distributions

*Q*^{+}, and are defined by(6.2)(6.3)where*G*_{i}is an applied body force per unit volume. Comparative expressions for*ρ*^{+},*Q*^{+}, , and are listed in table 1 for each of the three formulations.View this table:For purposes of wave extrapolation in CAA, the absence of

*ρ*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 . The modified acoustic analogy formulations of §3 are more tolerant to truncation of volume sources under these conditions.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 acoustically compact; (4.8) then generalizes the results of Curle (1955) to flows where heat transfer is important.

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 .In some applications of the wave extrapolation method (WEM) to aeroacoustics, the control surface 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 §5*b*, provides the appropriate boundary-source formulations needed for WEM in parallel shear flows.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.

## Acknowledgments

The authors thank Dr Philippe Spalart for stimulating discussions, and in particular for suggesting the 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.

## Footnotes

↵An extension of this description to cusped surfaces, such as a sharp-edged aerofoil, has been presented by Farassat & Myers (1990).

↵For a mixture of two fluids, a generalization of eqn (16) is given in appendix II of Morfey (1976).

↵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).

↵The nonlinear term may be written as where

*ζ*is the dimensionless pressure and*β*is the fundamental derivative (Thompson 1988).↵Overbars were used in Goldstein (2001) but could here be confused with windowed quantities.

↵There is a very small entropy perturbation associated with the acoustic mode, given by

*s*^{(p)}≈(*κ*/*ρc*_{p})(*α*/*ρc*^{2})*∂p*/*∂t*, that is neglected in this appendix. Likewise, the entropy mode is accompanied by first-order pressure and velocity perturbations that are also neglected.↵In eqn (20), T should be replaced by 1/

*ρ*.- Received February 14, 2007.
- Accepted May 10, 2007.

- © 2007 The Royal Society