Journal of Computational Physics 412 (2020) 109442

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Journal of Computational Physics

www.elsevier.com/locate/jcp

An accurate and efficient computational method for

time-domain aeroacoustic scattering

Zhao-Huan Wang a,b,∗, Sjoerd W. Rienstra a, Chuan-Xing Bi b, Barry Koren a

a Department of Mathematics and Computer Science, Eindhoven University of Technology, Groene Loper 5, 5612 AE Eindhoven, the Netherlandsb Institute of Sound and Vibration Research, Hefei University of Technology, 193 Tunxi Road, Hefei 230009, People’s Republic of China

a r t i c l e i n f o a b s t r a c t

Article history:Received 19 March 2019Received in revised form 6 February 2020Accepted 27 March 2020Available online 1 April 2020

Keywords:Aeroacoustic scatteringConvective time-domain equivalent-source methodStability analysisMoving medium

This paper presents a convective time-domain equivalent-source method for determining the scattered acoustic pressure field in a uniform moving medium. The proposed method is based on the solution of the time-domain convective Ffowcs Williams–Hawkings (FW–H) equation while the strengths of equivalent sources are determined by the required pressure gradient boundary condition on the scattering surface. The scattered acoustic pressure can be calculated once the strengths of equivalent sources have been determined. The current paper adopts the recently published analytical time-domain formulation for the acoustic pressure gradient in a moving medium to evaluate the incident pressure gradient on the scattering surface. This makes the proposed method considerably more efficient and accurate than a direct method. The total acoustic pressure consists of the scattered and the incident components. The latter can be obtained by the time-domain acoustic pressure formulation of the convective FW–H equation. Causes of possible instability in the proposed method are analyzed and an effective stabilizing method is proposed. Three test cases are considered to demonstrate the validity of the proposed method: a point monopole source field scattered by: (1) a rigid sphere in a stationary medium, (2) an infinite flat plate in uniform flow parallel to its surface, and (3) a cylinder of infinite length in axial uniform flow. To demonstrate the usefulness of the proposed method in practical engineering applications, the scattering of a point monopole source field by a slender wing in uniform flow is considered.

© 2020 Elsevier Inc. All rights reserved.

1. Introduction

Aerodynamic noise sources from aerospace engineering applications usually exist in the neighborhood of scattering sur-faces. Hence the scattering effects should be considered for a complete aeroacoustic simulation. The scattered field may substantially modify the incident field both in magnitude and directivity [1]. For example, rotor noise scattered by a cen-terbody [2–4] or a fuselage [5–7], and propeller noise scattered by a wing [8–10] should all be taken into account for a comprehensive aeroacoustic noise simulation.

Hybrid computational aeroacoustics (CAA) methods based on acoustic pressure integral formulations of the Ffowcs Williams–Hawkings (FW–H) equation [11] have become the most common methods for aeroacoustic simulations in free

* Corresponding author at: Department of Mathematics and Computer Science, Eindhoven University of Technology, Groene Loper 5, 5612 AE Eindhoven, the Netherlands.

E-mail address: z.wang2@tue.nl (Z.-H. Wang).

https://doi.org/10.1016/j.jcp.2020.1094420021-9991/© 2020 Elsevier Inc. All rights reserved.

2 Z.-H. Wang et al. / Journal of Computational Physics 412 (2020) 109442

space. These acoustic-analogy-type methods use the flow solution in the near field obtained by computational fluid dy-namics (CFD) to compute the acoustic far field. This saves a lot of computational cost and time compared to direct noise computations [12]. The acoustic pressure integral formulations can be derived in time domain [13–15] as well as in fre-quency domain [16–18]. The time-domain formulations are theoretically equivalent to the frequency-domain formulations. However, as most of the CFD solvers are implemented in time domain, time-domain formulations are preferred in practi-cal applications. To take into account the effect of uniform flow on sound generation and propagation, the classic FW–H equation can be converted to the convective FW–H equation [19]. The corresponding acoustic pressure integral formulations have recently been derived in both time and frequency domains [20–22]. For nonuniform flow, the problem becomes much more complicated because there is no analytical Green’s function and because the possibly physically unstable solution will result in causality problems if the flow is assumed to be stable [23]. Therefore, the flow is assumed to be uniform. Some subtleties inside the boundary layer around the scatterer are ignored in this paper. In general, this is a good approximation for Re � 1.

Various numerical methods exist to determine the scattered acoustic pressure, for instance: direct CAA methods using finite differences or finite elements [24–26], boundary element methods (BEM) [27–29], and the equivalent-source method (ESM) [30–34]. Compared to the direct CAA methods, BEM and ESM have some advantages. Firstly, direct CAA methods re-quire the entire domain to be discretized, which is computationally expensive for the acoustic far field, while BEM and ESM only need discretization on the scattering surface. Secondly, the Sommerfeld radiation condition is automatically satisfied in BEM and ESM, while the direct CAA methods need extra treatments on the domain boundary [35,36]. Thirdly, the direct CAA methods suffer from numerical errors of dispersion and dissipation while there are no such problems in BEM and ESM, since the exact Green’s functions provide the direct relation between the acoustic pressure of a receiver and the strength of a source [28].

The ESM, also known as the wave superposition method, was first proposed by Koopmann et al. [37,38] in frequency domain and the time-domain ESM was first proposed by Kropp and Svensson [30] to solve acoustic radiation and scattering problems. The basic idea of this method is to locate a series of equivalent sources inside a body, approximating the radi-ating/scattered acoustic field outside the body surface, by imposing the required boundary condition on the body surface. Compared to BEM, ESM has some additional advantages when it is used for scattering problems. Firstly, as the equivalent sources are located off the actual body surface, singular integrals in BEM can be avoided while a unique solution can be obtained at the interior resonance frequencies. Secondly, the number of equivalent sources can be less than the number of surface collocation points, thus resulting in a smaller linear system to be solved using a least-squares algorithm. Thirdly, there is no need to perform surface integration for the ESM, which makes it easier to implement and also saves computing resources [39–42]. The applications of ESM in frequency domain and time domain for aeroacoustic scattering computa-tions have shown that ESM is computationally the most efficient of the boundary methods currently used in aeroacoustics [43–47]. For a comprehensive review on variants and applications of ESM, the reader is referred to the recent article by Lee [48]. The frequency-domain ESM is suitable for tonal noise signals while the time-domain ESM is more appropriate for broadband noise signals, such as the blade-vortex interaction noise, and moving sources with Doppler effects.

When it comes to an aeroacoustic scattering simulation in a moving medium, including aircraft engine duct scattering and airframe scattering in uniform flow, we can use the Lorentz transformation to transform the solution without flow to the corresponding solution with flow [32,49,50]. However, this transformation adds complexity to both the solution process and the process of imposing the boundary conditions. For example, each source field requires its own transformation. Moreover, the application of this transformation should be limited to low-speed flows as mentioned in [32]. An alternative method is to explicitly take into account the presence of the uniform flow, such that it will be easier to interpret and examine the convective effects in acoustic scattering [51–54]. Recently, Siozos-Rousoulis et al. [53] proposed a convective frequency-domain equivalent-source approach for aeroacoustic scattering simulation in a moving medium. In this approach, the uniform flow effects are considered directly when imposing the boundary conditions, as well as during scattered field propagation. It is clear that a method to solve aeroacoustic scattering problems completely in time domain is indispensable.

When solving acoustic scattering problems in a moving medium, the key point is imposing the boundary conditions. For hard-wall boundary conditions, zero acoustic normal velocity or zero acoustic pressure gradient normal to the scattering surface should be imposed [55]. Lee et al. [56] derived the analytical time-domain acoustic pressure gradient formulation in a stationary medium which is the first paper in terms of the vector form of the acoustic quantity with applications to acoustic scattering problems. Recently, Mao et al. [54] derived the semi-analytical acoustic velocity formulation in a moving medium, which still contains the observer time differentiation of the integrals. Fully analytical acoustic pressure gradient formulations were derived by Ghorbaniasl et al. [51] and Bi et al. [52] in frequency and time domains, respectively. The analytical time-domain acoustic pressure gradient formulation G1A-M [52] does not contain observer time derivatives of the integrals, which avoids complex and sensitive numerical differentiation algorithms. It is also an advantage that only time-dependent input data of the flow field, or (at most) numerical differentiation of these, are required.

The current paper builds on [52] and aims to develop a time-domain aeroacoustic scattering simulation method in a moving medium by coupling the formulation G1A-M to the convective time-domain ESM developed in the current paper. The geometry, flow field, and distributed physical source of the scattering model are depicted in Fig. 1.

The paper is organized as follows. The convective FW-H equation and its time-domain acoustic pressure formulation as well as the pressure gradient formulation G1A-M are first reviewed in section 2.1. Next, the convective time-domain ESM and its application to aeroacoustic scattering calculation in a moving medium are presented in section 2.2. In section 3,

Z.-H. Wang et al. / Journal of Computational Physics 412 (2020) 109442 3

Fig. 1. Sketch of the geometry and flow field of the scattering model.

three test cases are numerically calculated to validate the proposed method through comparisons with analytical solutions. Further, a practical aeroacoustic scattering case is considered. Additionally, numerical stability and how to achieve it are elaborated in the first test case. Finally, conclusions are drawn in section 4.

2. Theory

2.1. The convective FW–H equation in time domain

Starting from the convective FW–H equation for a uniform mean flow U ∞ and omitting the nonlinear quadrupole type source term, we obtain the following equation, with displacement and loading sources for a permeable surface,[

1

c20

D2

Dt2− ∇2

][p′ (x, t) H ( f )

]= D

Dt[Q δ ( f )] − ∂

∂xi[Liδ ( f )] , (1a)

with

D

Dt= ∂

∂t+ U∞i

∂

∂xi, (1b)

Q = ρ0 (vn − U∞n) + ρ [un − (vn − U∞n)] , (1c)

Li = [(p − p0) δi j − σi j

]n j + ρui [un − (vn − U∞n)] , (1d)

where δ ( f ) is the Dirac delta function and H ( f ) the Heaviside function of the surface function f (x, t), which is defined by f (x, t) > 0 in the medium and f (x, t) < 0 inside the surface, with |∇ f | = 1. Here, U∞n = U∞ini , with the local outer unit normal ni = ∂ f /∂xi in the direction xi (i = 1,2,3) and U∞i the velocity component of the uniform flow in the same direction.1 δi j is the Kronecker delta. ρ0, p0, and c0 are the undisturbed, uniformly constant density, pressure, and sound speed for the uniform flow, respectively. p and ρ are the local fluid pressure and density, respectively. p′ (x, t) is the acoustic pressure at the observer position x, at time t . The local fluid velocity component is denoted by ui . The local normal components of the fluid and acoustic source velocities are defined as un = uini and vn = vini , respectively. σi j is the viscous stress tensor.

Employing equation (1b) and a modification of the source term suggested by Ghorbaniasl et al. [51],

Fi = Li − Q U∞i, (2)

the acoustic pressure can be expressed in a more concise form as follows:

p′ (x, t, M∞) = p′α (x, t, M∞) + p′

β (x, t, M∞) , (3a)

with

1 Note that we use the Einstein summation convention.

4 Z.-H. Wang et al. / Journal of Computational Physics 412 (2020) 109442

4π p′α (x, t, M∞) =

∫S

[Q

R∗ (1 − MR)2

]e

dS

+∫S

[Q

R∗MR + c0(MR∗ − M2

)R∗2 (1 − MR)3

]e

dS

−∫S

[Q

c0MR∗(MR − γ 2MR∗

)+ c0γ2M2∞M

R∗2 (1 − MR)3

]e

dS, (3b)

and

4π p′β (x, t, M∞) = 1

c0

∫S

[F R

R∗ (1 − MR)2

]e

dS +∫S

[F R∗ − F M

R∗2 (1 − MR)2

]e

dS

+ 1

c0

∫S

[F R

R∗MR + c0(MR∗ − M2

)R∗2 (1 − MR)3

]e

dS

−∫S

[F R

MR∗ MR + γ 2(M2∞M − M2

R∗)

R∗2 (1 − MR)3

]e

dS

−∫S

[F R∗ MR + γ 2 (M∞M M∞F − F R∗ MR∗)

R∗2 (1 − MR)2

]e

dS. (3c)

The amplitude radius R∗ and phase radius R are defined as

R∗ = 1

γ

√r2 + γ 2 (M∞ · r)2, (4a)

R = γ 2 (R∗ − M∞ · r), (4b)

where

γ =√

1

1 − M2∞, (5a)

r = x − y. (5b)

The subscript e denotes that calculations inside the square brackets [·]e should be performed at the source emission time τ = t − R/c0. M∞ is the flow Mach number vector with components M∞i = U∞i/c0, and M is the magnitude of M , which is the acoustic source Mach number vector with Mi = vi/c0. Other variables are defined as follows: MR∗ = Mi R∗

i

with R∗i = ∂ R∗/∂xi , MR = Mi Ri with R i = ∂ R/∂xi , M∞M = M∞i Mi , MR = Mi R i , M∞F = M∞i F i , F M = Fi Mi , F R∗ = Fi R∗

i , F R = Fi R i , F R = F i R i , where the dummy indices follow the Einstein summation convention. A dot above each variable in this paper means the derivative to the source time τ , for example: F i = ∂ Fi/∂τ . A dot above the main variable does not mean the derivative of the associated subscript variable. For example, MR∗ denotes Mi R∗

i , which is different from the source time derivative of MR∗ .

In this paper, equations (3a)–(3c) are expressions for the free field acoustic pressure in a moving medium as the incident field. The scattered field will be calculated by the proposed convective time-domain ESM in the next subsection.

Recently, Bi et al. [52] gave an analytical time-domain formulation for the acoustic pressure gradient in a moving medium, which is conveniently split up into

∂ p′ (x, t, M∞)

∂xi= ∂ p′

α (x, t, M∞)

∂xi+ ∂ p′

β (x, t, M∞)

∂xi, (6a)

∂ p′α (x, t, M∞)

∂xi= I1 + I2 + I3 + I4 + I5, (6b)

∂ p′β (x, t, M∞)

∂xi= I6 + I7 + I8 + I9 + I10 + I11 + I12. (6c)

Equations (6a)–(6c), together with the definitions of I1 − I12, are referred to as formulation G1A-M, which is used to calculate the incident acoustic pressure gradient on a scattering surface. The detailed definitions of I1, I2, · · · , I12 and related variables can be found in [52].

Z.-H. Wang et al. / Journal of Computational Physics 412 (2020) 109442 5

2.2. The convective time-domain equivalent-source method

In this section, we present the convective version of the time-domain ESM to calculate the scattered component of the total acoustic field in a moving medium. We start with the mathematical derivation and the numerical implementation. The idea of the proposed method is that the scattered acoustic field can be expressed by the sum of the contributions from a series of convective time-domain equivalent sources. The equivalent sources are located outside the acoustic domain (inside the scattering surface for an exterior problem or outside for an interior problem) on a replica surface of the scattering surface. The strengths of the equivalent sources as a function of time are determined by fulfilling the hard-wall boundary condition at well-chosen collocation points on the scattering surface.

The approximated scattered acoustic field p′sc satisfies[

1

c20

D2

Dt2− ∇2

]p′

sc =NE∑

ne=1

qne (t) δ [x − xne (t)] , (7)

where ne is the index of the equivalent source, NE the total number of the equivalent sources, and xne (t) and qne (t) the location and strength, respectively, of the ne-th equivalent source at time t .

In the free field, the Green’s function, relating the source point y at time τ and the observer point x at time t in a moving medium, is given by [21,57]

G (x, t; y, τ ) =δ(τ − t + R

c0

)4π R∗ . (8)

For an observer located at xo, the scattered acoustic pressure at time t is thus obtained by using this Green’s function as

4π p′sc (xo, t) =

NE∑ne=1

t∫−∞

qne (τ )δ(τ − t + Ro-ne

c0

)R∗

o-nedτ . (9)

The subscript o-ne indicates that the corresponding variable is based on the distance between xo and yne , r = xo − yne . Using the identity of the generalized function [58]:

t∫−∞

h(τ ′) δ (τ ′ − t + Ro-ne/c0

)dτ ′ =

[h (τ )(

1 − MRo-ne

)]

e

, (10)

equation (9) can be simplified to

4π p′sc (xo, t) =

NE∑ne=1

[qne(

1 − MRo-ne

)R∗

o-ne

]e

. (11)

Using equation (11), we can calculate the scattered acoustic pressure at an observer position in the time domain as long as we know the strength of each equivalent source as a function of time. Note that equation (11) can also be seen as a generalization of the Liénard-Wiechert potential in a uniform flow [59].

Before determining the source strengths qne by imposing the hard-wall boundary condition at collocation points on the scattering surface, we need to express the scattered acoustic pressure gradient at an observer position with source strengths qne . Take the gradient operation on both sides of equation (9), and obtain

4π∂ p′

sc (xo, t)

∂xi= ∂

∂xi

NE∑ne=1

t∫−∞

qne (τ )δ(τ − t + Ro-ne

c0

)R∗

o-nedτ

=NE∑

ne=1

t∫−∞

qne (τ )∂

∂xi

⎡⎣δ

(τ − t + Ro-ne

c0

)R∗

o-ne

⎤⎦ dτ . (12)

The gradient operator has been moved inside the integral in equation (12), because the summation operation, the source strength function, and the integral variable τ are independent of the observer coordinates xi (i = 1, 2, 3). The spatial gradient in equation (12) can be replaced by the observer time derivative using the following equation [21,56]:

∂

∂xi

⎡⎣δ

(τ − t + Ro-ne

c0

)R∗

o-ne

⎤⎦= − 1

c0

∂

∂t

⎡⎣ R io-neδ

(τ − t + Ro-ne

c0

)R∗

o-ne

⎤⎦−

R∗io-ne

δ(τ − t + Ro-ne

c0

)R∗2

o-ne. (13)

6 Z.-H. Wang et al. / Journal of Computational Physics 412 (2020) 109442

By means of equations (13) and (10), equation (12) can be rewritten as

4π∂ p′

sc (xo, t)

∂xi= −

NE∑ne=1

{1

c0

∂

∂t

[qne Rio-ne

R∗o-ne

(1 − MRo-ne

)]

e

+[

qne R∗io-ne

R∗2o-ne

(1 − MRo-ne

)]

e

}. (14)

To eliminate the observer time differentiation ∂∂t in equation (14), the following identity can be used [21]

∂

∂t=[

1(1 − MRo-ne

) ∂

∂τ

]e

. (15)

Equation (14) then becomes

4π∂ p′

sc (xo, t)

∂xi= −

NE∑ne=1

{1

c0

[1(

1 − MRo-ne

)]

e

∂

∂τ

[qne Rio-ne

R∗o-ne

(1 − MRo-ne

)]

e

}

−NE∑

ne=1

{[qne R∗

io-ne

R∗2o-ne

(1 − MRo-ne

)]

e

}

= −NE∑

ne=1

{Eio-ne +

[qne R∗

io-ne

R∗2o-ne

(1 − MRo-ne

)]

e

}, (16a)

where

Eio-ne = 1

c0

[qne R io-ne

R∗o-ne(1 − MRo-ne)

2

]e

+⎡⎣qne

−Mine + γ 2(

MR∗o-ne

R∗io-ne

− M∞Mne M∞i

)R∗2

o-ne(1 − MRo-ne

)2

⎤⎦

e

+ 1

c0

[qne Rio-ne

R∗o-neMRo-ne + c0

(MR∗

o-ne− M2

ne

)R∗2

o-ne(1 − MRo-ne

)3

]e

− 1

c0

[qne Rio-ne

c0MR∗o-ne

(MRo-ne − γ 2MR∗

o-ne

)+ c0γ2M2∞Mne

R∗2o-ne

(1 − MRo-ne

)3

]e

. (16b)

The scattered acoustic pressure gradient at an observer position xo at time t can be expressed through equation (16a) by summing up all the contributions from NE equivalent sources.

Consider a stationary rigid scatterer in a moving medium, which means that all the equivalent sources are also stationary outside the acoustic domain on an equivalent-source surface. Then, with Mine = vine/c0 = 0, equations (11) and (16a) can be simplified to

4π p′sc(xo, t) =

NE∑ne=1

[qne

R∗o-ne

]e, (17a)

4π∂ p′

sc(xo, t)

∂xi= −

NE∑ne=1

[qne R io-ne

c0 R∗o-ne

+ qne R∗io-ne

R∗2o-ne

]e

. (17b)

The known incident acoustic pressure gradient ∇p′in calculated by equations (6a)–(6c) and the scattered acoustic pres-

sure gradient ∇p′sc expressed by the unknown equivalent-source strength qne in equation (17b) should fulfill the following

acoustically rigid surface boundary condition on the scattering surface:

∇p′sc · n = −∇p′

in · n, (18)

where n = (n1, n2, n3

)is the outward unit normal on the scattering surface. Assuming that there are NS collocation points

on the scattering surface, with the ns-th (ns = 1, 2 · · ·NS) collocation point located at xns (t), equation (18) can be written as

1

4π

NE∑[qne Rns-ne · nns

c0 R∗ns-ne

+ qne R∗ns-ne · nns

R∗2ns-ne

]= ∇p′

in,ns (t) · nns, (19)

ne=1 e

Z.-H. Wang et al. / Journal of Computational Physics 412 (2020) 109442 7

where subscript ns indicates that the corresponding variables are calculated at the ns-th collocation point xns . Subscript ns-ne of the acoustic radii and their spatial derivatives indicate that the corresponding distances are between xns and xne . ∇p′

in,ns (t) is the incident acoustic pressure gradient at position xns and observer time t , and the vector variables Rns-ne and

R∗ns-ne are defined as Rns-ne =

(R1,ns-ne, R2,ns-ne, R3,ns-ne

)and R

∗ns-ne =

(R∗

1,ns-ne, R∗2,ns-ne, R∗

3,ns-ne

), respectively.

For practical numerical computation, ∇p′in,ns (t) will be calculated at discrete observer times t j :

t j = t1 + ( j − 1)�t, j = 1,2, · · · , J , (20a)

where �t is the time step between adjacent discrete times of the observer time, and J the total number of time steps. Similarly, the discrete source times τ k are

τ k = τ 1 + (k − 1)�τ , k = 1,2, · · · , J , (20b)

where �τ is the time step of the source. We will calculate the strengths of equivalent sources at these source time steps. The relation between these two time coordinates is given by

τ 1 = t1 − min (Rns-ne)

c0, (21a)

�τ = �t, (21b)

where min (Rns-ne) is the minimum value of Rns-ne between all the collocation points and equivalent sources. In this pa-per, the time increment �τ for each time step is determined by ppp, the number of sampling points per period which corresponds to the highest frequency of interest. Additionally, the equivalent-source strength is assumed to be zero when τ � τ 0 = τ 1 − �τ , to satisfy the initial condition. Consider a collocation point xns at time t j , the corresponding source time τ

jns-ne for the ne-th equivalent source is determined by

τj

ns-ne = t j − Rns-ne

c0. (22)

In general, τ jns-ne will not be an integer multiple of the given time increment �τ , so the required source strength qne

(τ

jns-ne

)in equation (19) is expressed using the following time domain interpolation

qne

(τ

jns-ne

)=

j∑k=1

φk(τ

jns-ne

)qk

ne, (23)

where qkne is defined as the strength of the ne-th equivalent source at τ k . φk(τ ) is the linear interpolation function, given

by [31]

φk (τ ) =

⎧⎪⎨⎪⎩(τ − τ k−1

)/�τ, if τ k−1 � τ � τ k,(

τ k+1 − τ)/�τ, if τ k � τ � τ k+1,

0, otherwise,

(24)

while its source time derivative is

φk (τ ) =

⎧⎪⎨⎪⎩

1/�τ, if τ k−1 � τ � τ k,

−1/�τ, if τ k � τ � τ k+1,

0, otherwise.

(25)

Equation (19) can be rearranged with above discretization and interpolation processes as

1

4π

NE∑ne=1

j∑k=1

⎡⎣ φk

(τ

jns-ne

)Rns-ne · nns

c0 R∗ns-ne

+φk(τ

jns-ne

)R

∗ns-ne · nns

R∗2ns-ne

⎤⎦qk

ne = ∇p′in,ns

(t j)

· nns. (26)

By applying equation (26) to all NS collocation points at t j , the following matrix equation is obtained:

P j =j∑

k=1

Ψ jkϕk, (27)

where P j is a column vector composed of the incident acoustic pressure gradient normal to the scattering surface, at all NScollocation points at t j , i.e.,

8 Z.-H. Wang et al. / Journal of Computational Physics 412 (2020) 109442

P j =[∇p′

in,1

(t j)

· n1,∇p′in,2

(t j)

· n2, · · · ,∇p′in,ns

(t j)

· nns, · · · ,∇p′in,NS

(t j)

· nNS

]T. (28)

ϕk is a column vector composed of all NE equivalent-source strengths at τ k , i.e.,

ϕk =[

qk1,qk

2, · · · ,qkne, · · · ,qk

NE

]T, (29)

and Ψ jk is an NS × NE transfer matrix that can be expressed as

Ψjkns,ne = 1

4π

⎡⎣ φk

(τ

jns-ne

)Rns-ne · nns

c0 R∗ns-ne

+φk(τ

jns-ne

)R

∗ns-ne · nns

R∗2ns-ne

⎤⎦ . (30)

According to the property of the linear interpolation function given in equation (24), the non-zero elements of Ψ jk are determined by the following condition

τ k−1 � τj

ns-ne � τ k+1, (31)

where τ k−1 is the nearest source time smaller than τ jns-ne and τ k+1 is the nearest time larger than τ j

ns-ne . By using equations (21a) and (21b), equation (31) can be further rewritten as

t1 − min (Rns-ne)

c0+ (k − 2)�τ � t1 − Rns-ne

c0+ ( j − 1)�τ � t1 − min (Rns-ne)

c0+ k�τ. (32)

Equations (24) and (32) indicate that

φk(τ

jns-ne

)= φk−1

(τ

j−1ns-ne

)= φk−2

(τ

j−2ns-ne

)= · · · = φ1

(τ

j−k+1ns-ne

). (33)

Therefore,

Ψ jk = Ψ ( j−1)(k−1) = Ψ ( j−2)(k−2) = · · · = Ψ ( j−k+1)1. (34)

Accordingly, equation (27) can be rewritten as

P j =j∑

k=1

Ψ ( j−k+1)1ϕk. (35)

The non-zero elements of Ψ ( j−k+1)1 are found for

τ 0 � τj−k+1

ns-ne � τ 2. (36)

It can be derived from equation (36) that

1

c0�τ[Rns-ne − min (Rns-ne)] � j − k + 1 � 1

c0�τ[max (Rns-ne) − min (Rns-ne)] + 2, (37)

where max (Rns-ne) is the maximum value of Rns-ne between all the collocation points and equivalent sources. With the result of equation (37), we can further simplify equation (35) to

P j =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

j∑k=1

Ψ ( j−k+1)1ϕk, if j � Kt,

Kt∑λ=1

Ψ λ1ϕ j−λ+1, if j > Kt,

(38)

where Kt = ceil{

1c0�τ [max (Rns-ne) − min (Rns-ne)] + 2

}, with ceil (·) the round-up function.

In order to obtain the equivalent-source strengths at each discrete source time ϕk (k = 1, 2, · · · , J ), a time-marching algorithm is used to solve equation (38) at each observer time t j . For j = 1, with P 1 and Ψ 11 known, ϕ1 can be solved by

ϕ1 = (Ψ 11)+ P 1, (39a)

where the superscript + indicates the pseudo-inverse of a matrix. Next, for j = 2, ϕ2 can be solved as

ϕ2 = (Ψ 11)+ (P 2 − Ψ 21ϕ1

), (39b)

Z.-H. Wang et al. / Journal of Computational Physics 412 (2020) 109442 9

using the previously obtained vector ϕ1. The remaining ϕk (k = 3, 4, · · · , J ) are solved as

ϕk =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

(Ψ 11

)+(P k −

k−1∑λ=1

Ψ (k−λ+1)1ϕλ

), if k � Kt,

(Ψ 11

)+(P k −

Kt∑λ=2

Ψ λ1ϕk−λ+1

), if k > Kt .

(39c)

It can be seen from equations (38) and (39c) that for k > Kt , the number of terms for summation is constant (Kt and Kt −1, respectively), which is reasonable since the scattered acoustic pressure gradient at a collocation point at the current time is determined by equivalent-source strengths at a limited number of previous times, not all previous times. If the number of collocation points is larger than the number of equivalent sources, equation (39c) will be overdetermined. A least-squares solution may be determined then, yielding unique strengths of all the equivalent sources.

Once all the equivalent-source strengths qkne at each source time step (k = 1, 2, · · · , J ) are determined, the scattered

acoustic pressure at the observer at time t j can be calculated by

p′sc

(xo, t j

)= 1

4π

NE∑ne=1

j∑k=1

φk(τ

jo-ne

)R∗

o-neqk

ne. (40)

The total acoustic pressure of an observer located in a moving medium can be obtained by summing up the incident field from equations (3a)–(3c) and the scattered field from equation (40).

As is commonly the case in a time-marching method, the time-domain ESM may suffer from numerical instability. This means that when the matrix Ψ 11 in equation (39c) is ill-conditioned, the errors may accumulate and propagate in time. For the first test case in the following section, eigenanalysis is used to investigate the numerical instability in the time-domain ESM for aeroacoustic scattering. The effects of numerical parameters and the mechanism of using the truncated singular value decomposition (TSVD) regularization to obtain a stable solution are also explained by means of the eigenanalysis. Additionally, the convergence of the method is demonstrated.

3. Numerical tests

In this section, numerical simulation results for three test cases and one practical case are presented to validate the proposed acoustic scattering simulation method and show the usefulness of the proposed method in practical engineering applications, respectively. The numerical solutions are compared with corresponding analytical solutions, if available. Firstly, as a canonical test case, scattering of sound from a monopole source by a rigid sphere is considered. Subsequently, two test cases of acoustic scattering in uniform flow with the same monopole sources but alternative scatterers are considered, to show the validity of the proposed convective time-domain ESM. Lastly, the scattering by a finite body is considered, a slender parabolic wing in uniform flow, to show the engineering capabilities of the proposed method. Numerical stability and how to obtain it are investigated in the first test case. Common parameters used for all four cases are: the undisturbed medium density ρ0, which is assumed to be 1.213 kg/m3; the ambient speed of sound c0, which is chosen as 340 m/s; and ppp = 10. The radius of the spherical data surface S used for calculations of the incident acoustic pressure and pressure gradient is 0.1 m, with 15292 triangular elements uniformly distributed on S for a fine enough spatial resolution (using the same mesh data as in [52], albeit scaled). All the required flow-field quantities on the data surface, such as the pressure, density, and velocity are calculated analytically in order to avoid any error related to flow-field simulation.

The velocity potential for a monopole point source with uniform flow of an arbitrary direction is given by

ϕ(x, t) = A

4π R∗ exp

[iω0

(t − R

c0

)], (41)

where A is the velocity potential amplitude and ω0 the source pulsation angular frequency. The acoustic particle velocity can be obtained from the gradient of the velocity potential:

u (x, t) = ∇ϕ (x, t) . (42)

The corresponding acoustic pressure and disturbed density are obtained by the unsteady linearized Bernoulli equation:

p′ (x, t) = −ρ0

[∂ϕ (x, t)

∂t+ U∞i

∂ϕ (x, t)

∂xi

](43a)

and

ρ ′ (x, t) = p′ (x, t)

c2. (43b)

0

10 Z.-H. Wang et al. / Journal of Computational Physics 412 (2020) 109442

3.1. Test case 1: scattering of a monopole source by a sphere in fluid at rest

The first test is to consider the benchmark test case of acoustic scattering by a rigid sphere in a quiescent medium, which means M∞ = 0. The analytical solution of the incident and scattered acoustic pressure for this case is given by Crighton et al. [60]. In this test case, the velocity potential amplitude A = −1/ (iω0ρ0). The sound from a pulsating monopole source is scattered by a sphere of radius a = 1 m, with its center placed at the origin of a Cartesian coordinate system. The source is located at x = (2, 0, 0) m, and has wave number κ = ω0/c0 = 5 m−1. The equivalent-source surface is an 80% replica of the spherical scattering surface. Both the scattering surface and the equivalent-source surface are discretized by 12 × 12uniformly spaced points in the azimuthal and the polar directions, respectively, yielding NS = NE = N = 144 points on each of these two surfaces. We first employ this simple scattering model to investigate numerical stability by means of an eigenanalysis and then present some numerical results to prove the validity of the proposed method for aeroacoustic scattering simulation.

For the purpose of analyzing numerical stability, equation (39c) is written as an iterative scheme (assume k > Kt ) [61,62]:

Y k = H Y k−1 + Xk, (44a)

where

Y k =[ϕk,ϕk−1, · · · ,ϕk−Kt+3,ϕk−Kt+2

]T, (44b)

Y k−1 =[ϕk−1,ϕk−2, · · · ,ϕk−Kt+2,ϕk−Kt+1

]T, (44c)

Xk =[(

Ψ 11)+ P k,0, · · · ,0,0]T

, (44d)

H =

⎡⎢⎢⎢⎢⎢⎣

− (Ψ 11)+

Ψ 21 − (Ψ 11)+

Ψ 31 · · · − (Ψ 11)+

Ψ (Kt−1)1 − (Ψ 11)+

Ψ Kt 1

I 0 · · · 0 00 I · · · 0 0...

.... . .

......

0 0 · · · I 0

⎤⎥⎥⎥⎥⎥⎦ . (44e)

Since the stability of a system is determined by its homogeneous equation [62,63], the stability analysis is based on

Y k = H Y k−1. (45)

To formulate the eigenequation, the equivalent-source strengths vector ϕk is assumed to be a multiplication of the expo-nential time function ηk and an arbitrary spatial vector ζ [61], as follows:

ϕk = ηkζ 1, (46a)

η = μeiξ . (46b)

Here, η is the change rate per single time step, with decay rate μ and phase change ξ . By substituting equation (46a) into equation (45), a polynomial eigenvalue problem can be formed as[

ηk + (Ψ 11)+ Ψ 21ηk−1 + (

Ψ 11)+ Ψ 31ηk−2 + · · · + (Ψ 11)+ Ψ Kt 1ηk−Kt+1

]ζ 1 = 0. (47)

Making use of the transformations

ζ λ = ηλ−1ζ 1, λ = 1,2, · · · ,k, (48)

equation (47) can be converted into an equivalent linear eigenvalue problem as

H Z = ηZ , (49a)

where

Z = [ζ k, ζ k−1, · · · , ζ k−Kt+3, ζ k−Kt+2

]T, (49b)

and η and Z are an eigenvalue and eigenvector of the iterative matrix H . The eigensolutions of equation (49a) contain de-tailed information about the modal characteristics of this scattering model. To make the time-marching scheme in equation (39c) stable, it is required that |η| � 1 for all eigenvalues in equation (49a) [61]. It should be noted that this condition is a necessary but not a sufficient condition for stability because the iterative matrix H is not a normal matrix, which means that the matrix H does not have a complete eigenspace composed of orthogonal eigenvectors [64]. Fig. 2(a) shows the dis-tribution of eigenvalues in equation (49a) by decomposing the iterative matrix H . It should be noted that here the matrix

Z.-H. Wang et al. / Journal of Computational Physics 412 (2020) 109442 11

Fig. 2. (a) Eigenvalues of the iterative matrix H in the complex plane; (b) Comparison between the analytical scattered acoustic pressure and the numerical result calculated by the proposed method at fixed observer position xo = (−1.2, 0, 0) m: analytical ( ) and numerical ( ).

(Ψ 11

)+is inverted directly. A unit circle (red dash-dotted line) is used to identify those eigenvalues with magnitudes larger

than 1. According to the discussion above, if there are eigenvalues located outside the unit circle, the calculated numerical result will become unstable. It can be seen that there are many eigenvalues located outside the unit circle in Fig. 2(a), especially in the left half plane, yielding the unstable result shown in Fig. 2(b).

As shown in Fig. 2(a), the complex plane can be divided into three parts [65,66]: the physics area, the Shannon-Nyquist numerical area and the area in between the former two. The eigenvalues in the physics area represent the characteristic modes of the discretized model. Among them, the eigenvalues of which the magnitudes are larger than one correspond to internal resonances that lead to physical instability. The eigenvalues in the Shannon-Nyquist numerical area are associated to histories with oscillations approximately at the Shannon-Nyquist frequency, which is the highest possible frequency that equals 1/(2�t). Among these, the eigenvalues of which the magnitudes are larger than one correspond to the Shannon-Nyquist modes that lead to the numerical instability.

It is known that some numerical parameters affect the stability of the system, so special attention must be paid to the choice of the numerical parameters, including the number of collocation points and equivalent sources, the position of the equivalent-source surface, and the time step size. By means of eigenanalysis, the influences of these numerical parameters on the numerical stability are analyzed here. Only the value of one numerical parameter is changed for each analysis case while all other parameters are kept the same as in the above simulation model. Given that the spectral radius indicates the stability of the system, the relation between the spectral radius and each of aforementioned numerical parameters is studied. Fig. 3(a) shows the spectral radius for different numbers of collocation points and equivalent sources. It can be found that the spectral radius increases with the number of collocation points and equivalent sources, which indicates that an increasing number of collocation points and equivalent sources may lead to a deteriorating stability. On the other hand, the procedure of discretizing the scattering surface to generate the collocation points should ensure that the mesh spacing is small enough to capture the incident pressure gradient fluctuations. Fig. 3(b) shows the spectral radius for different positions of the equivalent-source surface. The position of the equivalent-source surface is determined by the value of a ratio ae/a with fixed a, where ae is the radius of the equivalent-source surface. It can be seen that there is no monotonic relationship between the spectral radius and the position of the equivalent-source surface. If the equivalent-source surface is positioned closer to the scattering surface (for cases with ae/a > 0.5 in this simulation), the number of equivalent sources contributing to the collocation point at the current time will increase, making the solution more stable (the decrease of the spectral radius for larger value of ae/a). However, this may cause a numerical singularity when the distance between the scattering surface and the equivalent-source points tends to zero. For general cases, the position of the equivalent sources is recommended to be at an 80% to 90% replica of the scattering surface. Fig. 3(c) shows the spectral radius for different time sampling frequencies. The spectral radius decreases when the time sampling frequency increases, which indicates that a smaller time step will lead to a more stable result. From Fig. 3, it can be seen that proper values of numerical parameters may give numerical stability. It can be seen, for example, that for small values of N or big values of ppp, stability can be achieved, but the computational accuracy or efficiency will be reduced then. Therefore, it is not ideal to achieve numerical

12 Z.-H. Wang et al. / Journal of Computational Physics 412 (2020) 109442

Fig. 3. Spectral radii for different values of parameters. (a) different numbers of equivalent sources and collocation points; (b) different positions of the equivalent-source surface; (c) different time sampling frequencies.

stability by only optimizing the numerical parameters. Ill-conditioning of the transfer matrix Ψ 11 in equation (39c) is the main cause of instability. This ill-conditioning is caused by the inherent physical characteristics of the time-domain equivalent-source model, including the number and position of the equivalent sources, the number and position of the collocation points, and the time step size. To quantitatively estimate the ill-conditioning, the condition number of this matrix can be calculated by

cond(Ψ 11) = wmax

wmin, (50)

where wmax and wmin are the maximum and minimum singular values of matrix Ψ 11, respectively. In this test case, cond(Ψ 11) = 28.94. As the condition number is much larger than 1, a severe amplification of numerical errors may arise in ϕk in equation (39c).

To stabilize the calculation of the scattered acoustic pressure in the time-domain ESM, the TSVD is adopted to regularize the ill-conditioned matrix Ψ 11, by setting very small singular values below the cut-off singular value to zero [67,68]. A parameter σp = wc/wmax is introduced to determine the cut-off singular value wc . By using TSVD, the size of the inverse matrix is also reduced, which is a by-product of this choice. In the following, eigenanalysis is applied to demonstrate the effectiveness of this stabilizing approach.

Fig. 4(a) shows the distribution of eigenvalues in equation (49a) by decomposing the iterative matrix H . It should be noted that here the TSVD is incorporated in inverting matrix

(Ψ 11

)+. Based on previous experience, σp is determined as

0.2. After the TSVD regularization, the condition number of Ψ 11 is 4.89. Compared with the distribution of eigenvalues shown in Fig. 2(a), nearly all the eigenvalues located outside the unit circle have moved into the unit circle. Therefore, the calculated scattered acoustic pressure shown in Fig. 4(b) and the corresponding analytical solution are of the same order of magnitude. However, instability still occurs after 60 time steps, because there are still some eigenvalues in the Shannon-Nyquist numerical area located outside the unit circle. Usually the value of the cut-off parameter σp is specified before simulation, based on previous experience [31,39,44]. As small singular values lead to numerical instability, increase of the value of σp will yield a stable solution. On the other hand, discarding many singular values will decrease the accuracy of the solution. It is a challenge of the time-domain ESM to properly set this parameter for different problems. Recently, Zhang et al. [61] used the golden section method to tackle the difficulty of choosing the appropriate value of the cut-off parameter σp . However, in some cases, it is numerically difficult to use the golden section method when the iterative matrix H is too large to be decomposed and we determine the value of σp based on previous experience. Fig. 5(a) shows the distribution of eigenvalues in equation (49a) by decomposing the iterative matrix H . Here, the TSVD is used to regularize the transfer matrix Ψ 11 and the cut-off parameter σp is determined by the golden section method as 0.517. After the TSVD regularization, the condition number of Ψ 11 is 1.91. It can be seen in Fig. 5(a) that all the eigenvalues are located inside the unit circle to meet the stability criterion with the given set of numerical parameters. Therefore, the calculated scattered acoustic pressure shown in Fig. 5(b) is stable. For such a configuration of parameters (NS = NE = N = 144), the maximum and minimum singular values are 16.3756 and 0.5659, respectively, resulting in a severely ill-conditioned transfer matrix Ψ 11. The value of σp is determined by the golden section method to be a relatively large value 0.517, retaining only 23 out of the 144 singular values of the transfer matrix Ψ 11. Although a large part of the singular values are discarded, the main information of the transfer matrix Ψ 11 is still retained. The rather large difference between the numerical solution and the analytical solution is due to the fact that such a small number of collocation points and equivalent sources (three points per wavelength) can not very accurately resolve the scattering sound field. In the following, we will demonstrate the convergence of the time-domain ESM. Fig. 6 shows the root-mean-square (RMS) error in one source period between the calculated scattered acoustic pressure and the corresponding analytical solution. The cut-off parameter σp for each pair of NS and NE is determined by the golden section method. The number of equivalent sources for each black point is chosen

Z.-H. Wang et al. / Journal of Computational Physics 412 (2020) 109442 13

Fig. 4. (a) Eigenvalues of the iterative matrix H in the complex plane with the TSVD; (b) Comparison between the analytical scattered acoustic pressure and the numerical result calculated by the proposed method with the TSVD at fixed observer xo = (−1.2, 0, 0) m: analytical ( ) and numerical ( ).

Fig. 5. (a) Eigenvalues of the iterative matrix H in the complex plane with the TSVD; (b) Comparison between the analytical scattered acoustic pressure and the numerical result calculated by the proposed method with the TSVD at fixed observer xo = (−1.2, 0, 0) m: analytical ( ) and numerical ( ).

such that it is the closest square number to NS/2, in order to avoid a nonuniform distribution error of equivalent sources. The convergence behaviors of both NE = NS and NE = NS/2 are shown in solid lines by using the least-squares fitting approach. In each case, two orders of convergence can be extracted, a lower order one and an asymptotic higher order one, corresponding to the start-up part and the asymptotic part of the result, respectively. The first section of the solid line (the start-up part) in each case is relatively flat and not important for extracting the convergence rate. The asymptotic convergence rates are extracted from the second sections of the solid lines, as 2.35 and 2.48, for NE = NS and NE = NS/2, respectively. Although the error is not decreasing monotonically in Fig. 6, the slopes shown in solid lines indicate that for both NE = NS and NE = NS/2, the proposed method is about second-order convergent in NS. It is worth mentioning that the linear order interpolation function in equation (24) is used for the time-domain discretization, so it has no bearing on the convergence rate in space domain. The cause of the nonmonotonicity is probably that when the number of collocation points and equivalent sources increases, the structural symmetry also changes, which has an effect on the convergence behavior. Note also that the two black solid line sections do not indicate less good accuracy for cases with NE < NS, on

14 Z.-H. Wang et al. / Journal of Computational Physics 412 (2020) 109442

Fig. 6. RMS error ε as a function of the number of collocation points NS and the number of equivalent sources NE at fixed observer position xo =(−1.2, 0, 0) m: NE = NS (•: numerical result, : least-squares fitting) and NE = NS/2 (•: numerical result, : least-squares fitting). (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

the contrary, accuracy is even better, at least for NE = NS/2. Moreover, the calculation time reduces dramatically when the number of equivalent sources is half of the number of collocation points, especially for large values of NS (almost one fourth of the calculation time for cases NE = NS). Without any matrix regularization, the condition number of the matrix Ψ 11 is large, which may result in unstable aeroacoustic scattering computations, as some eigenvalues of the iterative matrix Hare located outside the unit circle, violating the stability criterion. When the number of equivalent sources and collocation points is increased, as shown in Fig. 3(a), the spectral radii of the iterative matrix H will also increase, leading to a further deteriorated solution. The process of obtaining a stable and accurate solution may be summarized as follows. By using the TSVD regularization and by selecting the appropriate value of the cut-off parameter σp , the condition number of the transfer matrix Ψ 11 is decreased drastically and its ill-conditioning problem is fixed; all the eigenvalues of the iterative matrix H are located inside the unit circle to meet the stability criterion. Once the solution method is stable, any prescribed accuracy of the numerical solution can be achieved by increasing the number of equivalent sources and collocation points, as demonstrated in the convergence test. The general strategy is that we first ensure the solution method to be stable, after which the prescribed accuracy of the numerical solution can be achieved. Note that in this process the value of the cut-off parameter σp varies for different configurations of parameters. As long as the main information of the transfer matrix Ψ 11

is retained, the accuracy of the numerical solution can be assured.In the following of this test case, we choose the parameters of the lowest point in Fig. 6 with NE = NS = 32 × 32 = 1024.

The corresponding values of the cut-off parameter σp and the RMS error ε at fixed observer position xo = (−1.2, 0, 0) mare: 0.088 and 5.84 × 10−4 Pa (about 1% of the amplitude of the scattered pressure), respectively. With this small cut-off parameter, σp = 0.088, only 111 out of the 1024 singular values are retained. The maximum and minimum singular values are 184.6947 and 0.0024, respectively. The values of the discarded singular values are very small compared with the large singular values. Although about only 10% of the singular values are retained, the main information of the transfer matrix Ψ 11 is still preserved to assure the accuracy of the computational result. The scattering sound field is well resolved by such a large number of collocation points and equivalent sources (eight points per wavelength); the difference between the numerical solution and the analytical solution is rather small. Fig. 7 shows the incident, scattered, and total acoustic pressure time histories at two observer positions: xo1 = (−1.2, 0, 0) m and xo2 = (−5, 0, 0) m for Helmholtz number He = κa = 5. It can be seen that all the numerical acoustic pressure time histories are in excellent agreement with the analytical solutions. The RMS error of the total acoustic pressure at xo2 is 7.85 × 10−4 Pa (about 5% of the amplitude of the total pressure). As the first observer is located near the surface of the scattering sphere in the core shadow zone, the amplitude of the total pressure is less than that of the incident pressure. The second observer is no longer located in the strong shadow zone and away from the sphere, so the amplitude of the total pressure is larger than that of the incident pressure. It is clear that the diffraction effects modify the total acoustic pressure to a large extent, in both amplitude and phase.

Fig. 8 shows the normalized acoustic pressure magnitudes at 40 observer positions along the circle of radius 1 m around the origin, in the x1 − x2 plane. In other words, all the observers are located on the scattering surface. The numerical results for He = 1 and He = 5 are compared against the analytical solutions. The acoustic pressure is normalized such that the amplitude of the incident pressure is equal to 1 at xo = (1, 0, 0) m. Again it can be seen that the numerical results agree very well with the analytical solutions for different observer positions. The RMS errors of all the observers for He = 1and He = 5 are 0.0144 and 0.0190, respectively. The change of noise magnitude and directivity due to the presence of the scattering body is clear. It can also be observed that for larger Helmholtz numbers, the magnitudes of the scattered pressure become larger in both the shadow zone (around 180◦) and the area near the source (around 0◦ or 360◦), due to a stronger

Z.-H. Wang et al. / Journal of Computational Physics 412 (2020) 109442 15

Fig. 7. Time histories of acoustic pressure at two observer positions, in case of monopole source near sphere in fluid at rest: analytical ( : incident, : scattered, : total) and numerical (•: incident, •: scattered, •: total).

Fig. 8. Azimuthally angular distributions of acoustic pressure in case of monopole source near sphere in fluid at rest, for two Helmholtz numbers: analytical ( : incident, : scattered, : total) and numerical (•: incident, •: scattered, •: total).

scattering effect. The results shown in this test case prove the validity of the proposed method for prediction of incident and scattered acoustic pressure in a stationary medium.

3.2. Test case 2: scattering of a monopole source by an infinite flat plate in uniform flow

The second test case considers the acoustic scattering by an infinite flat plate in a uniform flow. The analytical solution for the incident acoustic pressure can be obtained by using equation (43a) while the analytical scattered acoustic pressure can be obtained by the image source method [69].

For implementation of the convective time-domain ESM, the infinite flat plate is represented by a finite plate with dimensions Lx = L y = L = 20 m, centered at the origin in the x1 − x2 plane. The monopole source is located above the plate at (0, 0, 1) m with A = −1/(iω0ρ0) and source frequency f s = 100 Hz (the corresponding Helmholtz number is He =2π Lx fs/c0 ≈ 37). If the finite plate is bent into the wall of a cylinder, the corresponding radius Rp can be calculated as Rp = L/(2π) = 3.18 m. As found in the first test case, the position of the equivalent sources is recommended to be at an 80% to 90% replica of the scattering surface. So the equivalent-source surface is chosen parallel to the scattering surface with its center at (0, 0, −0.51) m and with the same dimensions as the scattering surface. The scattering surface is uniformly discretized with NS = 26 × 26 collocation points while NE = 21 × 21 points are uniformly distributed on the equivalent-source surface. Currently, the length of the finite flat plate is set to be nearly 6 times the wavelength of the source, in

16 Z.-H. Wang et al. / Journal of Computational Physics 412 (2020) 109442

Fig. 9. Time histories of acoustic pressure at fixed observer position, in case of monopole source near infinite flat plate in uniform flow: analytical ( : incident, : scattered, : total) and numerical (•: incident, •: scattered, •: total).

order to reduce the diffraction effects from the edges of the plate. Based on previous experience, the cut-off parameter σp

is chosen as 0.36 in this test case, which looks relatively large compared with that of test case 1, however, it does not mean that a lot of singular values are discarded. Only 25 out of the 441 singular values are discarded and the main information of the transfer matrix Ψ 11 is still retained. As the maximum and minimum singular values are 0.8983 and 0.3161, respectively, the parameter σp should be relatively large.

Fig. 9 depicts the numerical incident, scattered, and total acoustic pressure time histories at the observer position xo =(0, 9, 0) m, in a uniform flow with Mach number M∞ = (0, −0.4, 0), together with the corresponding analytical solutions. Also here, the numerical solutions are in excellent agreement with the analytical solutions for all discrete times considered. The RMS error of the total acoustic pressure is 0.0064 Pa (about 4% of the amplitude of the total pressure). Fig. 10 shows the total acoustic pressure for different flow Mach numbers at the scattering plane with x3 = 0 m. In this test case, the flow Mach number varies from 0 to 0.4 in the negative x2-direction, in order to examine the convection effects on the acoustic field. The numerical fields and the analytical fields are in excellent agreement for different flow Mach numbers. The RMS errors for these three configurations are 0.0020 Pa, 0.0011 Pa, and 0.0027 Pa, respectively. Due to the Doppler effect caused by the moving medium, the pattern is no longer symmetric and shows amplification in the upstream direction.

3.3. Test case 3: scattering of a monopole source by a cylinder of infinite length in uniform flow

In this benchmark case, aeroacoustic scattering by an infinite cylinder with a uniform axial flow is considered, to investi-gate the capability of the time-domain convective ESM for acoustic scattering problems in a moving medium. This case is a simplified representation of the engine noise scattered by the fuselage of an aircraft. As the flow is assumed to be uniform and inviscid, the thickness of the boundary layer around the cylinder is zero. Subtleties such as acoustic refraction and scat-tering inside the boundary layer are ignored [70]. A harmonic monopole source, with unit strength and source frequency f s = 100 Hz, is placed at x = (0, 0, 2) m, in the vicinity of a rigid cylinder of infinite length and radius Rc = 1 m (Fig. 11). The cylinder axis is coincident with the x1-axis and the uniform axial flow is in the positive x1-direction with Mach number M∞ = 0.4. The analytical solution of this problem is given in [71]. For implementation of the convective time-domain ESM, the scattering cylinder surface is truncated at x1 = ±10 m and has 8040 uniformly distributed collocation points on it. These collocation points are uniformly placed on 201 equidistant rings along the truncated cylinder surface (with 40 collocation points per ring). The equivalent-source surface radius is 0.85 times the radius of the scattering cylinder surface, on which 2020 equivalent sources are uniformly placed on 101 equidistant rings along the equivalent-source surface (with 20 equiva-lent sources per ring). Based on previous experience, the cut-off parameter σp is chosen as 0.1. Fig. 12 depicts the numerical incident, scattered, and total acoustic pressure time histories at the observer position xo = (1, 3, 0) m and the corresponding analytical histories. The numerical results are in excellent agreement with the analytical solutions at all discrete times. The RMS error of the total acoustic pressure is 3.42 × 10−4 Pa (about 1% of the amplitude of the total pressure). Fig. 13 shows the real part of the total acoustic pressure field on the scattering surface of the cylinder. Also here, the numerical solutions agree very well with the analytical solutions. Insight to be obtained from Fig. 13 is that the amplification appearing in the upstream field further distorts the total field. The RMS error for all observers on the scattering surface is 0.0011 Pa.

3.4. Practical application case: scattering of a monopole source by a slender wing in uniform flow

In this case, aeroacoustic scattering by a wing at zero angle of attack with a uniform flow is considered, to show the practical engineering usefulness of the proposed method. The considered problem is depicted in Fig. 14. The upper and

Z.-H. Wang et al. / Journal of Computational Physics 412 (2020) 109442 17

Fig. 10. Total acoustic pressure distributions on the scattering plane, in case of monopole source near infinite flat plate, in uniform flow, for three different Mach numbers: analytical (left) and numerical (right).

Fig. 11. Cylinder in uniform axial flow with monopole source.

18 Z.-H. Wang et al. / Journal of Computational Physics 412 (2020) 109442

Fig. 12. Time histories of acoustic pressure at fixed observer position, in case of cylinder in uniform axial flow (M∞ = 0.4) with monopole source: analytical ( : incident, : scattered, : total) and numerical (•: incident, •: scattered, •: total).

Fig. 13. Total acoustic pressure distributions around the cylinder, in case of uniform axial flow (M∞ = 0.4) with monopole source.

Z.-H. Wang et al. / Journal of Computational Physics 412 (2020) 109442 19

Fig. 14. Slender wing in uniform flow at zero angle of attack with monopole source.

Fig. 15. Distribution of the collocation points and the equivalent sources for aeroacoustic scattering calculation: Collocation points (blue) and equivalent sources (red).

lower surfaces of the wing are defined as follows:

x3 = ±0.1L(

1 − x21/L2

), −L � x1 � L, −L � x2 � L, (51)

where L = 0.5 m. Hence, the wing has the same double-parabolic, symmetric airfoil over its entire span, and a square plan form, i.e., wing span and chord length are the same (wing aspect ratio = 1). Its 10% thickness makes the wing sufficiently slender to apply the uniform flow assumption in our method, just as in [32,49]. The test case is a simplified model of engine noise scattered by a wing or tail plane of an aircraft. The uniform flow is pointing in the positive x1-axis with Mach number M∞ = (0.4, 0, 0). The monopole source is located above the wing at (0, 0, 0.5) m with A = −1/(iω0ρ0) and source frequency f s = 500 Hz (the corresponding Helmholtz number is He = 2π L fs/c0 ≈ 4.6). The equivalent-source surface is constructed by shrinking the scattering surface and the corresponding upper and lower equivalent-source surfaces are defined as follows:

x3 = ±0.1L(

0.8 − x21/L2

), −Les � x1 � Les, −Les � x2 � Les, (52)

where Les = 1√5

≈ 0.4472 m. The 5730 collocation points and the 4588 equivalent sources are uniformly distributed over their surfaces, as shown in Fig. 15. The cut-off parameter σp is chosen to be 0.1.

Fig. 16 shows the total acoustic pressure on the scattering surface of the wing. Due to the flow, Doppler shifts in wave length in x1 direction occur, which can indeed be observed in Fig. 16. Fig. 17 shows the total acoustic pressure in the plane x2 = 0 (midspan). The white disk indicates the spherical data surface used for calculating the incident acoustic pressure at observer points, as well as the incident acoustic pressure gradient at collocation points. The black area indicates the wing section. Fig. 17 shows qualitatively proper shielding for and redirection of noise.

An aspect that has not been an issue for the previous smooth scattering surfaces, is the possible singularity at sharp edges, and the associated non-uniqueness of the solution. For this, we distinguish two types of problems. Firstly, there is the problem that a boundary condition that involves a normal derivative ∇p · n, is not valid at a sharp edge because the normal vector n is not defined there. As a result, a line source (of for instance delta-function type) may “hide” itself at the

20 Z.-H. Wang et al. / Journal of Computational Physics 412 (2020) 109442

Fig. 16. Total acoustic pressure on the scattering surface.

Fig. 17. Total acoustic pressure in the plane x2 = 0.

edge to create an additional field from an unspecified source of non-physical origin, making the mathematical solution non-unique. This problem exists both with and without flow, and can be solved by adding the so-called edge condition to our boundary conditions [72,73]. This edge condition says that no acoustic energy should radiate from any small neighborhoodof the edge. In numerical practice, it is common to check afterwards if the edge condition is satisfied. Most numerical methods have numerical diffusion, to an extent which is sufficient to satisfy the edge condition.

Secondly, there is the problem that with inviscid mean flow, unsteady vorticity may be shed from the trailing edge. The amount of vortex shedding is a priori undetermined in our linear, inviscid model, because it is based on a non-uniform limit: a vanishing amplitude versus a vanishing viscosity. The vanishing viscosity yields a thin (steady) boundary layer and wake as well as a thin (unsteady) Stokes layer, while the vanishing amplitude limits the size of the neighborhood of the sharp edge where a singularity may be felt. As a result, in one extreme, if the effect of viscosity is much smaller than the effect of amplitude, the influence of the singularity may reach through the boundary layer to the uniform mean flow such that viscosity enforces the shedding of vortices to reduce the singularity. In the other extreme, when amplitude is much smaller than viscosity, the singularity remains well inside the mean flow boundary layer and there is no need for the shedding of vorticity. Anything in-between is also possible. The condition where the amount of shed vorticity is just enough to remove the singularity completely is the Kutta-Joukowski (or just Kutta) condition [74].

In an inviscid model, the amount of vortex shedding should be a boundary condition of the problem. In numerical practice, if the contribution of the shed vorticies is assumed to be small and therefore no Kutta condition (or otherwise) is explicitly imposed, the solution method “chooses” itself the amount of vortex shedding and the level of singular behaviormay be checked afterwards.

In the above problem of scattering by the thin wing, we made a provisonal check by plotting the difference of the pressure between top and bottom side of the wing. We found a smooth behavior near the edge.

The contribution of the shed vortices to the diffracted sound field is of scientific and engineering interest [75]. In future, it might be investigated more extensively.

Z.-H. Wang et al. / Journal of Computational Physics 412 (2020) 109442 21

4. Conclusion

In this paper, a convective time-domain equivalent-source method has been proposed, to numerically simulate aeroa-coustic scattered pressure in a moving medium. The developed method is based on the time-domain convective Ffowcs Williams–Hawkings equation, to represent the scattered acoustic field by adding all the contributions from equivalent sources located outside the acoustic domain. For a hard-wall scatterer, the strengths of equivalent sources are solved, im-posing the rigid wall boundary condition. In this process, the analytical time-domain acoustic pressure gradient formulation G1A-M [52] is used for evaluation of the incident acoustic pressure gradient on the scattering surface, which makes this process easier to implement and computationally less intense. The total acoustic pressure field can be obtained by coupling the proposed method with a numerical solver for the acoustic pressure in a moving medium, to gain more insight into scattering problems in a moving medium. The mathematical derivation and the numerical algorithm have been described in detail. The cause and a fix for numerical instability have been analyzed for a simple sphere scattering case. Besides, the convergence of this method is demonstrated.

Compared with the existing numerical method for a stationary medium, the present method allows direct solution of acoustic scattering problems in a moving medium. It is easier now to examine the convection effect. Three numerical test cases have been considered to validate the proposed method for solving aeroacoustic scattering problems in a moving medium. For these test cases, the numerical solutions agree perfectly well with the analytical solutions, which proves that the proposed method provides an alternative way to solve scattering problems by explicitly taking into account the presence of the moving medium. In addition, aeroacoustic scattering by a slender wing in uniform flow yields qualitatively correct results, indicating that the method probably can be applied for engineering purposes. Singularity problems in sound diffraction at sharp edges are an interesting topic for future research. We found smooth pressure behavior near sharp edges.

Another future challenge is to extend the current method to solve acoustic scattering problems in a moving medium, with soft-wall boundary conditions in the time domain.

Declaration of competing interest

The authors declare that they have no conflict of interest.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (grant nos. 11674082 and 51875147). The authors gratefully acknowledge the financial support from the China Scholarship Council (grant no. 201706690024). The authors would additionally like to acknowledge dr. Xiao-Zheng Zhang and dr. Arris S. Tijsseling for their valuable and constructive advice over the period of this research. This work is part of an inter-university collaboration between Hefei University of Technology and Eindhoven University of Technology, for a joint PhD thesis research project.

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