Using a partial-wave method for sound–mean-flow scattering problems

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Using a partial-wave method for sound–mean-flow scattering problems

Remy Berthet, Christophe Coste

To cite this version:

Remy Berthet, Christophe Coste. Using a partial-wave method for sound–mean-flow scattering prob-

lems. Physical Review E , American Physical Society (APS), 2003. �hal-01406762�

Using a partial-wave method for sound– mean-flow scattering problems

R. Berthet

LPS, UMR CNRS 8550, E´ cole Normale Supe´rieure, 24 rue Lhomond, 75231 Paris Cedex 05, France C. Coste

GPS, Tour 23-13, 1^ere´tage, 2 place Jussieu, 75251 Paris, France 共Received 19 September 2002; published 14 March 2003兲

We present a semianalytical method, based on a partial-wave expansion and valid in the short wavelength limit for small Mach number flows, to analyze sound–vortical-flow interactions. It is more powerful than ray-tracing methods because it gives both amplitude and phase of the sound wave, and because it is less restrictive on the smallness of the wavelength. In contrast with the Born approximation approach, this method allows the computation of the sound field in the whole interaction domain 共including the near field兲, and preserves energy conservation. Vortical flows with finite circulation are amenable to our analysis, which gives a satisfactory description of wave front dislocation by vorticity, in good agreement with direct numerical simulations. We extend previous versions of this method to the case of smooth vorticity profiles which are observed in aeroacoustics experiments.

DOI: 10.1103/PhysRevE.67.036604 PACS number共s兲: 43.28.⫹h, 47.10.⫹g, 47.32.⫺y I. INTRODUCTION

Sound–mean-flow interactions play a prominent part in many physical phenomena, dwelling from atmospheric and oceanographic context关1兴to laboratory flow instabilities关2兴. It is still an open question to know how sound propagates in turbulent media, and how its characteristics are influenced by the mean flow. We will focus our attention in this paper on the short wavelength limit, when the sound wavelength␭ is smaller than the typical mean-flow scale L. In atmospheric and oceanographic context关1,3,4兴, this approximation is of- ten well verified. This limit is also useful to analyze some aspects of sound propagation in turbulent media in labora- tory experiments 关5–7兴.

In the very small wavelength limit, the acoustical interac- tion can be interpreted using the ray-tracing method 关4,8兴. Starting with a power law expansion of the wave amplitude in the wavelength, the zeroth order共geometric limit兲leads to two coupled differential equations involving the mean-flow specifications, the wave vector关储kជ(␶⁾储⫽2␲^/␭(␶⁾兴, and the ray path rជ(␶) in terms of␶, the acoustical ray parameter关9兴. The solution of these sets of equations gives a qualitative description of the mean-flow effects on the geometry of the rays 关10–13兴. A more quantitative approach requires the acoustical field values. The first-order terms of the wave- length power law expansion lead to the conservation of the acoustical energy along a ray 关9,14兴. Thus, one can easily compute the field properties along a ray, and then estimate the spatial field repartition关15兴. However, this method is no more valid in presence of caustics, when two or more rays cross: the physical requirement of a finite acoustical energy leads to a more complicated way of computation关14兴. Nev- ertheless, the geometric limit remains a qualitative but rel- evant method of analysis 关16,17兴 in many problems involv- ing sound-flow interactions.

An alternative way of solving this problem is to start with the linearized equations in the sound variables. One can then

write a wave equation for the sound wave 关18兴 with source terms expressing the coupling of sound with mean flow and solve the problem in the far-field limit in the first Born ap- proximation 关19,20兴. This treatment, valid for small Mach numberMⰆ␭/L, is useful in the large wavelength limit. As the wavelength decreases, ML/␭ must remain small to en- sure the validity of the Born approximation. Thus, the short wavelength limit cannot be easily analyzed with this method.

We will use in this paper another way of investigation: we linearize the basic equations and take the short wavelength limit. In this limit, if ␺0 (␺s) is any quantity related to the mean flow共the sound wave兲, we have

␺s储ⵜជ␺0储⬃␺s␺0

L Ⰶ␺0储ⵜជ␺s储⬃k␺s␺0. 共1兲 Then, at first order beyond the geometric limit, the sound propagation can be expressed by the modified wave equation

冉^{⳵⳵t⫹vជ0•ⵜជ}冊^{2ps⫺c2⌬ps⫽0, 共2兲}

wherevជ₀ is the mean-flow velocity, c is the speed of sound and p_s is the sound pressure perturbation. Then, an angular Fourier transform is applied to Eq.共2兲and allows the com- plete computation of the sound field. Each Fourier mode is called a partial wave, and the complete acoustic field is the superposition共interference pattern兲of all those partial waves.

This method was already used for sound–mean-flow interac- tion problems, to study sound scattering 共introducing, in analogy with scattering problems in quantum mechanics, the concept of phase shifts兲 关21,22兴, spiral waves phenomena 关23兴, and sound absorption by a vortex关24兴.

The partial-wave method that we propose here is an alter- native way to ray-tracing methods for sound–mean-flow scattering problem in the short wavelength limit: it allows the full computation of the acoustic fields 共amplitude and phase兲in the whole domain of interaction共both the far field

and the near field兲. It will be applied to the scattering of a sound wave 共wavelength ␭) by a vortical stationary mean flow 共typical length scale L) in the small wavelength limit

␭ⰆL, but in our calculations this limit is much less restric- tive than in geometrical acoustics approaches.

The paper is organized as follows: in the first part, we derive a modified wave equation for the interaction of a sound wave with a mean stationary flow. In the second part, we solve this equation in the bidimensional case using the partial-wave method and we formally compute the full acoustical field. In the last part, we present examples to char- acterize and illustrate the method.

II. THE WAVE EQUATION IN THE SHORT WAVELENGTH LIMIT

Let us consider an acoustic wave 共density variation ␳s, fluid velocity vជ_s, and wave celerity c⫽冑^ps/␳s) moving in an isentropic mean flow (␳0,vជ₀). The mean flow is assumed to be incompressible, which means a small Mach number MⰆ1, and the acoustic wave is assumed to be a small per- turbation of the mean flow. Moreover, the sound frequency is assumed to be larger than the typical frequency of the mean flow, which physically means that the flow is frozen during its interaction with the sound wave.

Starting with the decomposition of the physical fields ␳

⫽␳0⫹␳s, vជ_⫽vជ_0⫹vជ_s, and p⫽p₀⫹p_s, we can write the mass and momentum conservation equations for the mean flow. To the leading order in the small quantities ␳s/␳0, vs/v0, and p_s/ p₀, the sound wave fields are governed by

D␳s

Dt ⫹div共␳0vជ_s兲⫽0, 共3兲

␳0

Dvជ_s

Dt ⫹␳0共vជ_s•gradជ兲vជ_0⫹␳s共vជ_0•gradជ兲vជ_0⫽⫺c²gradជ␳s. 共4兲 We have introduced the particle derivative with respect to vជ₀:

D Dt⬅ ⳵

⳵^t⫹vជ_0•gradជ. 共5兲 From Eqs. 共3兲and共4兲, we can write a wave equation with a source term. This source term is at least of orderM/␤^{2, with}

␤⫽k L⫽2␲^L/␭. In the short wavelength limit␭ⰆL, which means ␤Ⰷ1, and to the first order in MⰆ1, this source term is negligible and the wave equation becomes关23兴

冋^{DtD22⫺c02⌬}册^{␳s⫽0 共6兲}

with c₀ the sound celerity in the medium at rest 关c⯝c₀

⫹O(M²)兴. Then, one gets easily Eq.共2兲for the sound pres- sure. In the following, we will focus on the density solutions

of Eq.共6兲. Equation共6兲was also derived by Obukhov for the sound velocity potential in his studies of sound scattering by turbulent flows 关25兴.

It should be mentioned here that the wave equation must be derived first and only then the short wavelength limit be taken. Taking the short wavelength limit of Eqs.共3兲and共4兲 and writing a wave equation can lead to a misleading physi- cal analysis. When deriving Eq. 共6兲, one must take the tem- poral derivative of Eq. 共3兲and the spatial derivative of Eq.

共4兲. Then, both the wave vector k and the sound frequency appear and can modify the order of magnitude of the source terms.

Equation共6兲, valid in the short wavelength limit for small Mach number flows, shows that the interaction between sound and flows is twofold. First, the sound wave is locally advected by the mean flow, which is a purely kinematic ef- fect: for a uniform mean flow velocity, Eq.共6兲corresponds to a Galilean transformation of the wave equation from the mean-flow reference to the laboratory reference. Second, Eq.

共6兲is the analogous, in a compact form due to the geometri- cal approximations, of the wave equation derived in the sound–mean-flow interaction studies 关18,26兴 as mentioned above. Then, the scattering by the mean-flow velocity gradi- ents is also considered in Eq.共6兲.

It should be noticed that acoustical problems and surface wave problems in the shallow water approximation are simi- lar关27兴because both type of waves are nondispersive. Thus, it is not surprising that Eq. 共6兲 is stricly analogous to the wave equation 共2.7兲derived by Coste et al.关28兴to describe the interaction between a short wavelength surface wave and a vortical flow in the shallow water approximation.

III. PARTIAL-WAVE METHOD IN THE SOUND SCATTERING PROBLEM

The partial-wave共hereafter PW兲method is generally used for scattering problems both in electromagnetism 关29兴 and quantum mechanics关30兴when the scatterers have a spherical or a cylindrical symmetry. In a three-dimensional problem, introducing the spherical harmonic functions allows to com- pute the scattering field 共or the wave function兲 in terms of PW, depending only on the radius r. In the two-dimensional case, using the polar coordinates, the PW method consists in an angular Fourier transform and also leads to PW that only depends on r.

In the context of sound scattering by a vortical flow, we will consider an axisymmetric bidimensional single vortex of spatial extension L_m: its vorticity vanishes for r⭓L_m. Its velocity only depends on the radius r:vជ₀(rជ)⫽U(r)␪^{ˆ , where}

␪ˆ is the unit orthoradial vector of the polar coordinates. As- suming a small Mach number M⫽v₀/c₀Ⰶ1 and a short wavelength with respect to the flow length L_m, ␤⫽kL_m

⫽2␲^Lm/␭Ⰷ1, Eq.共6兲describes the interaction of the vor- tical flow with the sound wave. Then, we use the PW method to compute the sound fields. In this case, the PW method simply consists in an angular Fourier transform, as pointed out before.

A. Partial-wave equation Since the flow is axisymmetric, Eq.共5兲reads

D Dt⫽ ⳵

⳵^t⫹ U共r兲

r

⳵

⳵␪ ^共7兲

and Eq.共6兲becomes

冋^{⳵⳵r22⫹1r ⳵⳵r⫹r12 ⳵␪⳵22⫺c102冉⳵⳵t⫹U共rr兲 ⳵␪⳵ 冊2}册^{␳s⫽0. 共8兲}

Introducing a partial-wave development for a monochro- matic sound wave of pulsation 2␲␯^,

␳s共rជ,t兲

␳s0

⫽Re冋冉^{n⫽⫺⬁兺⫹⬁ ␳n共r兲ein␪}冊^{e⫺i2␲␯t}册

⫽Re冋^{n⫽⫺⬁兺⫹⬁ ␳n共r兲ei(n␪⫺2␲␯t)}册^{, 共9兲}

(Re关z兴 is the real part of the complex number z), Eq. 共8兲 leads, at order M, to an ordinary differential equation for each partial wave ␳n(r),

␳n

⬙

⬘

r ⫹冋^{k2⫺2n k Uc0r共r兲⫺nr22}册^{␳n⫽0 共10兲}

with k⫽2␲␯^/c0, the prime denoting a derivation with re- spect to r.

The complete solution is obtained when Eq.共10兲is solved for each partial wave, and the full sound wave density varia- tion ␳s(rជ,t) is constructed using Eq. 共9兲.

B. Partial-wave solution

We consider mean flows of typical vortical size L_m. We describe such a flow with a piecewise vorticity profile:

⍀⫽⍀in共r兲, r⭐L_i,

⍀⫽⍀inter共r兲, L_i⭐r⭐L_m, 共11兲

⍀⫽0, r⭓L_m.

The corresponding velocity can then be expressed in the form

U共r兲⫽U_in共r兲, r⭐L_i,

U共r兲⫽U_inter共r兲, L_i⭐r⭐L_m, 共12兲 U共r兲⫽ ⌫

2␲^{r, r⭓Lm,}

⌫ being the circulation of the mean flow. We impose the continuity of the fluid velocity关and of the fluid pressure共or density兲兴 at the boundaries between the three domains:

U_in(L_i)⫽U_inter(L_i) and U_inter(L_m)⫽⌫/(2␲^Lm). This

structure of mean flows allows us to modelize a large number of physical situations involving axisymmetric mean flows 关31–33兴.

1. Outer solution

For r⭓L_m, injecting the velocity field 共12兲 in Eq. 共10兲, we get

␳n

⬙

⬘

r ⫹冋^k2⫺mr22册^{␳n⫽0, 共13兲}

where m⫽冑ⁿ²⫹2 n␣ ^and␣⫽k⌫/(2␲^c0). For a nonzero- circulation vortex (␣⫽0), the index m may be real or even complex if 0⬎n⬎⫺2␣. In either cases, the outer solution can be expressed in terms of Bessel and Hankel functions,

␳n

out共r兲⫽d_nJ_m共kr兲

J_m共␤兲 ⫹e_nH_m¹共kr兲

H_m¹共␤兲, 共14兲 where d_n and e_n are two numerical constants to be deter- mined by the boundary conditions.

In a seminal paper, Berry et al.关34兴have shown the close analogy between the crossing of a potential vector of finite circulation by a charged particle in quantum mechanics and the interaction between a surface wave and a vortical flow in an ordinary fluid. The vortex has the same effect on the sur- face waves as the vector potential on the wave function: the finite circulation implies a phase shift, characterized by the parameter␣, in either the wave function or the surface wave, and the wave fronts display a dislocation 共the so-called Aharonov-Bohm effect in quantum mechanics 关35兴兲. This dislocated wave is a non perturbative interaction term 共the dislocation amplitude is the same as that of the wave兲, and persists arbitrarily far from the interaction region, which is not the case for the scattered wave 共with amplitude decreas- ing as 1/冑r in two dimensions兲.

The wave front dislocation is a phase shift that cannot be directly observed in quantum mechanics关36兴, whereas in the acoustical or surface waves cases, both the phase and ampli- tude of the waves can be measured: the Berry’s analogy was experimentally confirmed by Vivanco et al. 关37兴 in the sur- face wave case and in the acoustical case by Roux et al.关31兴 using the time reversal mirror method and by Labbe´ and Pinton 关32兴with direct ultrasonic measurements.

The dislocation parameter ␣ is zero when the vortex has no circulation. It leads to Bessel functions of integer order in solution共14兲: there is no Aharonov-Bohm effect in this case.

Thus, in classical wave scattering experiments, a dislocated wave front can be interpreted as the signature of a vortical flow with a global circulation 关37,38兴.

2. Inner solution In the vortex region, we must solve

␳n

⬙

⬘

r ⫹冋^{k2⫺2n k Ucinter0r 共r兲⫺nr22}册^{␳n⫽0 共15兲}

for L_i⭐r⭐L_mand

␳n

⬙

⬘

r ⫹冋^{k2⫺2n k Uc0inr共r兲⫺nr22}册^{␳n⫽0 共16兲}

for r⭐L_i.

In the region r⭐L_i, U_in/r is an angular velocity: it must remain finite when r goes to zero. Thus, in this limit, U_in/rⰆn²/r² and Eq.共16兲 always becomes a usual Bessel equation: its solution is a linear combination of Bessel and Neumann functions J and N. When r goes to zero, the Neu- mann function diverges, and so does one of the solutions of Eq.共16兲. This latter must consequently be excluded, in order to get a finite sound field everywhere. The inner solution can thus be expressed in a formal way in both regions, using a set of only three numerical constants a_n, b_n, and c_n:

␳n

in共r兲⫽a_n F_nⁱⁿ共r兲

F_nⁱⁿ共L_i兲, 共17兲 where F_nⁱⁿ(r) is that solution of Eq.共16兲which is bounded at the origin r⫽0, and

␳n

inter共r兲⫽b_n F_n^inter共r兲

F_n^inter共L_m兲⫹c_n G_n^inter共r兲

G_n^inter共L_m兲. 共18兲

The explicit form of the three functions F_nⁱⁿ, F_n^inter, and G_n^inter obviously depends on the details of the velocity pro- file.

3. Matching of the three solutions

To achieve the determination of the sound wave, we have to compute the five integration constants a_n, b_n, c_n, d_n, and e_n. They are fixed by requiring the continuity of both the density variation and the fluid velocity for the sound wave at the matching locations r⫽L_i and r⫽L_m. This im- plies the continuity of␳n for each PW, and from Eq.共4兲the velocity continuity leads to that of␳n

⬘

The fifth equation comes from a topological condition on the sound field at infinity: as pointed out before, far from the mean flow, kr→⬁, the sound wave should be the sum关39兴 of a scattered wave, decreasing like 1/冑kr, and a dislocated wave of constant amplitude 关28,34兴. This condition reads 关34兴

d_n⫽共⫺i兲^mJ_m共␤兲 共19兲 and the continuity conditions lead to the nonhomogeneous system

冢

^⬘

^⬘

^⬘

^⬘

^⬘

^⬘

冣 ^冉

^冊

^冉

^⬘

^冊

Then, the complete solution␳s is

␳s共r,␪^,t兲

␳s0 ⫽Re冋^{n⫽⫺⬁兺⬁ anFFninnin共共Lri兲兲ei(n␪⫺2␲␯t)}册^{, r⭐Li,}

共21兲

␳s共r,␪^,t兲

␳s0 ⫽Re冋^{n⫽⫺⬁兺⬁} 冉^{bnFFninterninter共L共rm兲兲}

⫹c_n G_n^inter共r兲

G_n^inter共L_m兲冊^{ei(n␪⫺2␲␯t)}册^{, Li⭐r⭐Lm,}

共22兲

␳s共r,␪^,t兲

␳s0 ⫽Re冋^{n⫽⫺⬁兺⬁} 冉^{共⫺i兲mJm共kr兲}

⫹e_nH_m¹共kr兲

H_m¹共␤兲冊^{ei(n␪⫺2␲␯t)}册^{, r⭓Lm, 共23兲}

where the constants a_n, b_n, c_n, and e_n are given by the solution of Eq. 共20兲. All our PW calculations are done using

MATHEMATICA. The symbolic computation capabilities of this software are useful to calculate the constants.

In the limit␣→0, m⫽兩n兩 and the outside sound wave is the exact sum of the incident plane wave and a wave scat- tered by a cylindrical distribution 关40兴. It should be pointed

out that we take here into account the inner structure of this distribution关see Eq. 共20兲兴.

4. Scattering amplitude

Assuming a linear incident plane wave of wave number k, wavelength ␭⫽2␲/k, celerity c, frequency ␯⫽␭/c, and density amplitude␳0i,

␳i⫽␳0iRe关e^{i(kជ•rជ⫺2␲␯t)}兴, 共24兲 two quantities of major importance for sound scattering stud- ies can be defined 关40兴: the scattered sound wave␳scat and the scattered amplitude f (␪). The former is deduced from␳s

and␳i:

␳s⬅␳i⫹␳scat 共25兲 and the latter, only defined for rⰇ2␲^L2/␭ 共far-field ap- proximation兲, is deduced from␳scat and␳i by

␳scat⫽Re冋^{␳0if共␪兲e⫺i(2␲␯t)e冑ikrr}册 ^共26兲

in two dimensions. The 1/冑r factor in Eq. 共26兲comes from the energy conservation in a two-dimensional problem. f (␪⁾ characterizes the angular structure of the sound wave result- ing from the scattering process. It is directly related to the scattering cross section:

␴scat⫽冕0 2␲

兩f共␪兲兩²d␪^, 共27兲

an important parameter to analyze the scattering efficiency.

The wave front dislocation is a nonperturbative effect due to the finite circulation of the vortical flow at infinity. Hence for a compact mean flow of typical size L 共no velocity cir- culation for r⭓L), there is no dislocated wave, and the in- cident plane wave is the exact solution of the scattering prob- lem far from the vortex. In the context of our calculations,

␣⫽0, m(n,␣⁾⫽兩n兩where n is an integer, and Eq.共19兲leads to the classical decomposition of plane waves in series of Bessel functions 共see Ref. 关41兴, formula 共8.511兲 and notice that this choice implies to locate the forward direction in ␪

⫽␲^):

␳i

␳0i

⫽Re关e^{i(kជ•rជ⫺2␲␯t)}兴

⫽Re冋^{e⫺i(2␲␯t)n⫽⫺兺⬁ ⬁共⫺i兲nei n␪Jn共kr兲}册^{. 共28兲}

Comparing this last expression with the expression of the outer solution共23兲, we conclude that the scattered wave␳scat

corresponds to the second term of the partial-wave solution:

␳scat共r,␪^,t兲

␳0i

⫽Re冋^{n⫽⫺⬁兺⬁ enHHm1m1共共kr␤兲兲ei(n␪⫺2␲␯t)}册^{. 共29兲}

Far from the vortex core, krⰇ1, we can expand in kr the Hankel functions H_m¹ :

H_m¹共kr兲⯝共⫺i兲^m冑_␲²kre^ikre^⫺i␲/4. 共30兲 We recover the e^ikr/冑r radial evolution, and can express the scattering amplitude f (␪^{) as}

f共␪兲⫽冑_␲²k_n⫽⫺⬁兺^{⬁ enei(n␪⫺␲}_H ^/4⫺m␲/2)

m

1共␤兲 . 共31兲

The scattering cross section can then be formally computed in terms of the partial wave coefficients e_n:

␴scat⫽4

k _n⫽⫺兺^⬁ _⬁冏^Hm^en

1共␤兲冏^{2. 共32兲}

5. Conservation of the acoustical energy

Another way to compute ␴scat is by using the two- dimensionnal optical theorem 关29,42兴

␴scat⫹␴abs⫽␴t⫽冑⁸k^␲ Im关f共␪⫽␲兲e^⫺i␲/4兴 共33兲 (␪⫽␲ locates the forward direction, see Eq.共28兲and Im(z) is the imaginary part of the complex number z).

It simply expresses the energy conservation during the scattering process. As we do not consider any source of ab- sorption,␴abs⫽0 and we expect

␴scat⫽␴t. 共34兲 From Eqs.共31兲and共32兲, it is a very hard task to demonstrate this equality because of the cumbersome expression of the coefficients e_n. Nevertheless, the quantum problem formally analogous to our acoustical problem is the interaction be- tween a particle and an axisymmetrical potential V(r): start- ing with the Schro¨dinger equation关30兴

⌬⌿⫹2m

ប² 关E⫺V共r兲兴⌿⫽0, 共35兲 one can introduce a partial-wave development of the wave function⌿,

⌿共rជ_{兲⫽n⫽⫺}兺^⫹⬁_⬁^{⌿l共r兲eil␪. 共36兲}

Then, each partial wave⌿l satisfies

⌿l

⬙

⬘

r ⫹冋^{k2⫺2m Vប2共r兲⫺rl22}册^{⌿l⫽0. 共37兲}

Equation共37兲is similar to Eq.共10兲. Moreover, if we con- sider a compact mean flow U(r) in acoustics, its quantum analog is an interaction potential V(r) which decays faster than 1/r 关43兴. In the large r region共corresponding to the far

field in acoustics and to the large distance scattering in quan- tum mechanics兲, both equations have the same asymptotic limits

X

⬙

␳n⬀冑_␲²krcos共kr⫺n␲^/2⫹␦n兲, 共39兲 where the phase shift ␦n depends on the velocity field U(r) 关for U⬅0, ␦n⬅0 and one gets the asymptotic expansion of the partial waves describing a monochromatic plane wave, Eq. 共42兲兴. Then the complete sound wave␳s reads

␳s共r,␪^,t兲

␳0i

⫽冑␲^{2kr Re}冋^{n⫽⫺⬁兺⬁ Bncos共kr⫺n␲/2}

⫹␦n兲e^{i(n␪⫺2␲␯t)}册^{, 共40兲}

where B_n are numerical constants. On the other hand, from Eqs. 共24兲–共26兲,␳s can be expressed as

␳s共r,␪^,t兲

␳0i ⫽Re冋^{ei(kជ•rជ⫺2␲␯t)⫹f冑共␪r兲ei(kr⫺2␲␯t)}册 ^共41兲

and the incident plane wave can be expanded into 关see Eq.

共28兲and Ref.关41兴, formula共8.451兲兴

␳i

␳0i

⫽Re关e^{i(kជ•rជ⫺2␲␯t)}兴

⫽冑␲^{2kr Re}冋^{e⫺i(2␲␯t)n⫽⫺兺⬁ ⬁共⫺i兲n}

⫻e^{i n␪}cos共kr⫺n␲^/2⫺␲^/4兲册^{. 共42兲}

The constants B_n are chosen in such a way that the scattered wave ␳s⫺␳i represents an outgoing wave, depending only on e^⫹ikr. Using Eqs. 共40兲–共42兲 and equating to 0 the coef- ficient of e^⫺ikr lead to

B_n⫽e^{i(␦n⫺n␲/2⫹␲/4)} 共43兲 and the scattering amplitude can be recast into

f共␪兲⫽冑2¹␲^k_n⫽⫺兺^⬁ _⬁^{ei n (␪⫺␲)e⫺i␲/4关ei(2␦n⫹␲/2)⫺1兴.}

共44兲 Similar expressions were derived for the scattering of sound by nonzero-circulation mean flows by Fetter in the large wavelength limit ␭ⰇL 关21兴, and by Reinschke 关22兴. How- ever, these authors did not discuss the validity of the optical

theorem because they considered problems in which the mean-flow velocity field decays too slowly.

From Eq. 共31兲, we get a complete analogy between our partial-wave method and the phase-shift method:

e_n⫽H_m¹共␤兲e^i(m␲/2)

2 e^{⫺i n␲}关e^{i(2␦n⫹␲/2)}⫺1兴. 共45兲 From Eq. 共44兲, we can compute the scattering cross section

␴scat,

␴scat⫽4

k_n⫽⫺兺^⬁ _⬁^{sin2共␦n⫹␲/4兲, 共46兲}

and the forward scattering amplitude

f共␪⫽␲兲⫽e^⫺i␲/4冑2␲^1k_n⫽⫺⬁兺^{⬁ 关ei(2␦n⫹␲/2)⫺1兴. 共47兲}

Thus, we prove the conservation of the acoustical energy 共33兲and find␴scat⫽␴t.

Therefore, the partial-wave analysis of Eq. 共6兲, valid at the first Mach order in the short wavelength, includes the conservation of the acoustical energy during the scattering process. If the interaction is analyzed in the first Born ap- proximation, it is well known that the optical theorem cannot be satisfied: f⬀M whereas ␴⬀M². The second Born ap- proximation 关30,44兴 or an asymptotical treatment 关45,46兴 must be considered to ensure energy conservation.

Energy conservation is satisfied here because we solve exactly Eq.共6兲, taking into account the full interaction phe- nomena, including both multiscattering events and geometri- cal coupling. The consequence is that our calculation fulfills the optical theorem for a compact mean flow 共quickly de- creasing at infinity兲, and that it describes properly the non- pertubative effect of wave front dislocation for a noncompact mean flow. This is clearly not the case in the first Born ap- proximation or the standard ray-tracing methods.

IV. APPLICATION TO AEROACOUSTIC PROBLEMS The PW method has been employed to solve the diffrac- tion of surface waves by a single vortex关37兴. In this context, the experimental vorticity distribution is very accurately de- scribed by a vortex core in solid rotation, and an outer part of the flow with a 1/r decreasing orthoradial velocity共see Ap- pendix A兲. A very good agreement was found between ex- perimental results and the PW method predictions 关37,47兴. Formally, the vorticity is a step function of the distance to the vortex center. In aeroacoustics, however, the solid rotation is a very poor approximation of the effective velocity field in the inner part of the vortex 关32兴. We thus show, in these examples, how to generalize the previous calculations关28兴to smooth 共polynomial兲vorticity distributions.

Another difficulty with the PW method is that the basic equation共6兲is only approximate, valid when␤Ⰷ1 where␤ is the product of the acoustic wave number by a characteris- tic length scale of the flow, typically the size of the vorticity distribution. But the analysis does not give any numerical

estimate of the minimum admissible value of ␤. In order to get more information, we study the acoustic scattering by a vorticity distribution of zero circulation outside a radius L_m. In this case, the scattered wave may be calculated in the Born approximation 共see Appendix B兲, which is valid if M␤Ⰶ1 where Mis the flow Mach number 关48,49兴. It is thus pos- sible to consider low Mach number flows for which the Born approximation is legitimate, and to compare the Born scat- tering amplitude with PW calculations. Then, we get numeri- cal estimates of the value of ␤ that gives good numerical agreement between the two approaches.

A. Range of validity of the partial-wave method To analyze the range of validity of our method, we want to make comparisons with usual calculations in the first Born approximation. To this end, we use two zero-circulation flows, because Born approximation breaks down in the case of flows with circulation 关19,50,51兴. Setting b_i the vortex size and a_i⫽x_ib_i(x_i⬍1), the first one is

U₁共r兲⫽␻1

2 r, 0⭐r⭐a₁,

U₁共r兲⫽ ␻1x₁²

2共x₁²⫺1兲冉^r⫺br12冊^{, a1⭐r⭐b1, 共48兲}

U₁共r兲⫽0, b₁⬍r.

⍀1共r兲⫽␻1, 0⭐r⭐a₁,

⍀1共r兲⫽ ␻1x₁²

x₁²⫺1, a₁⭐r⭐b₁, 共49兲

⍀1共r兲⫽0, b₁⬍r.

The velocity field is continuous at r⫽a₁, but the vorticity is not. This is corrected in the second case,

U₂共r兲⫽␻2

2 r冉^{1⫺x22b22共r22⫺x22兲}冊^{, 0⭐r⭐a2,}

U₂共r兲⫽ ␻2x₂²

2共2⫺x₂²兲r冉^br222⫺1冊^{, a2⭐r⭐b2, 共50兲}

U₂共r兲⫽0, b₂⬍r.

⍀2共r兲⫽␻2冉^{1⫺x22b222r共22⫺x22兲}冊^{, 0⭐r⭐a2,}

⍀2共r兲⫽⫺␻2

x₂²

2⫺x₂², a₂⭐r⭐b₂, 共51兲

⍀2共r兲⫽0, b₂⬍r.

This second velocity profile is thus smooth, which is in prin- ciple a more favorable situation since the velocity gradients are lower.

The analytical expressions of the velocity fields are cho- sen to make the PW calculations easier. In order to facilitate the comparison, we impose the same flow size b₁⫽L_m⫽b₂ and we determine the value of the parameter ␻2 (␻1 fixes the velocity scale兲by the following physical requirement: we require for the two flows the same maximal velocity, which means the same Mach number, and that this maximum ve- locity occurs at the same radius r⫽L*.

Straightforward algebra leads to

a₁⫽L*, 共52兲

a₂⫽b₂冉^{1⫺冑1⫺3Lb*222}冊^{1/2 共53兲}

␻2⫽3␻1

2 . 共54兲

In the following, we compare the scattering amplitudes f de- fined by Eq. 共26兲, computed using either the first Born ap- proximation applied directly on Eq. 共6兲or the PW solution, Eq. 共31兲.

1. Partial-wave computations

With the first vortical flow U₁, Eqs. 共15兲 and 共16兲 are Bessel equations, with solutions

F_nⁱⁿ共r兲⫽J_兩n兩共冑^kkn

0r兲, F_n^inter共r兲⫽J_p共冑^kkn 1r兲, G_n^inter共r兲⫽N_p共冑^kkn

1r兲, 共55兲

with

k_n⁰⫽k⫺n␻1/c₀, k_n¹⫽k⫺ n c₀

␻1x₁² x₁²⫺1,

p²⫽n²⫺nk c₀

␻1x₁²L_m²

x₁²⫺1 . 共56兲 In the region 0⭐r⬍a₁, we have kept the Bessel function of positive index only, since it is the only solution of Bessel equation which is finite for r⫽0.

If c₀k/␻1 is an integer n₀, k_n⁰vanishes for n⫽n₀. In this special case, the inner solution can be found by assuming F_n

0

in(r)⬀r^p. This leads to p⫽⫾n₀ and we only keep the solution p⫽n₀ to ensure a bounded solution at the origin:

F_n

0

in共r兲⫽rⁿ⁰. 共57兲

If k_n¹vanishes for some n⫽n₁, the intermediate solution can also be found ⬀r^p. This leads to p⫽⫾n₁ and the interme- diate solution becomes, for this particular value of n,

F_n

1

inter共r兲⫽rⁿ¹, G_n

1

inter共r兲⫽r^⫺n1. 共58兲