Domain decomposition technique for aeroacoustic simulations

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Applied Numerical Mathematics 49 (2004) 263–275www.elsevier.com/locate/apnum

Domain decomposition technique for aeroacoustic simulatio

Renzo Arina∗, Marco Falossi

DIASP – Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Abstract

A computational tool for the simulation of time harmonic sound propagation in open domains is propoconsists of a two step technique. In the first step the viscous flow field is calculated, solving the incompNavier–Stokes equations. In the second step the acoustic field is obtained solving the Lighthill’s wave equatwo calculations are performed on two different grids,both formed by non-conformingsubdomains. The Lighthill’swave equation is solved in the frequency domain, with a high order accurate compact finite difference scheme. Ttest the acoustic solver, the phenomenon of wave diffraction past an obstacle is calculated. The results cothe interface transmission conditions as well the non-reflecting boundary conditions do not affect the accuracthe scheme. The complete aeroacoustic technique is applied to the simulation of the noise radiation fromwith a grazing subsonic laminar boundary layer. The flow field presents a periodic vortex shedding past ththe vortical flow being a noise source. The numerical results show the existence of the directivity character of theacoustic radiation. 2004 IMACS. Published by Elsevier B.V. All rights reserved.

Keywords: Computational aeroacoustics; Wave equation; Domain decomposition

1. Introduction

The numerical simulation of aeroacoustic phenomena may be accomplished solving the compNavier–Stokes equations, which describe both sound generation and propagation. An accuratetation must preserve the wave form with minimum dispersion and dissipation. Moreover, the nelength scales that characterize the vortical flow are extremely small for moderate-to-high Reynoldbers. The far field has length scales associated with the acoustic waves, which are several omagnitude larger than those characterizing the vortical flow. The large variations in length scalescuracy requirements and the need for long time solutions strongly affect the cost of simulations,the direct computation of aeroacoustic fields in geometries of practical interest prohibitive.

* Corresponding author.E-mail addresses: [email protected] (R. Arina), [email protected] (M. Falossi).

0168-9274/$30.00 2004 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.apnum.2003.12.006

264 R. Arina, M. Falossi / Applied Numerical Mathematics 49 (2004) 263–275

In the context of low-speed flows, several attempts at overcoming these difficulties have been recentlyproposed, such that aeroacoustic numerical simulations become a viable tool. Amongst them, the most

d affectcoustiction andanalogy

well as

uations

tem ofepartures

ing thenvectionion is

orpresentdiation

ool forperforms flowe sound

rittennon-

n spacenction

promising approach is based on the assumption that the compressibility effects are small anonly the acoustical propagation. Consequently, splitting the flow field into incompressible and aparts, the computation is subdivided into a two steps procedure which is suitable for both generapropagation [4,7,10,12]. The advantage of the splitting approach, compared with the acoustictheories, is that the source strength is obtained directly and that it accounts for sound radiation asscattering.

In the first step the viscous flow field is calculated using the incompressible Navier–Stokes eqfor a constant property fluid,

∇ · U = 0, (1)∂U∂t

+ (U · ∇)U + 1

ρ0∇p0 = ν�U, (2)

whereU is the velocity field,ν the kinematic viscosity,p0 the mean pressure andρ0 the density of thefluid.

In the second step the acoustic radiation is obtained from the numerical solution of a sysperturbed compressible equations. The acoustic signals are assumed to be small enough, all dfrom a state in which the fluid has uniform densityρ0 and local pressurep0, to justify linearization,and neglecting viscous effects, the linearized Euler equations are obtained. In addition, assumsmall acoustic quantities propagate at hundreds of meters per second, compared with which coby relatively small flow velocities appears negligible, applying the Lighthill analogy, the propagatdescribed by the acoustic inhomogeneous wave equation

1

c20

∂2p

∂t2− �p = ∂2Tij

∂xi∂xj

, (3)

c0 being the ambient speed of sound,p the acoustic pressure fluctuation andTij the Lighthill tensor

Tij = ρ0uiuj + (p0 − c2

0ρ0)δij .

For isentropic flows, for whichp0 = c20ρ0, we haveTij = ρ0uiuj . This analogy does not account f

acoustic feedback onto the underlying flow field. Therefore, phenomena like mean flow effectsin sheared duct flows, for instance, are not correctly represented; however, it is well suited for raproblems.

The main objective of the present work is to develop a fast and accurate computational tsimulating time harmonic sound propagation in open domains. The split technique enables us tothe viscous and the acoustic computations on two different grids, one optimized for the viscouphenomena and the other suitable for representing the acoustic signals propagating outside thgeneration region.

The flow field is firstly determined solving the incompressible Navier–Stokes equations (1), (2) win the stream function-vorticity formulation. A domain decomposition method is employed, withconforming Cartesian subdomains, in conjunction with second-order accurate discretizations iand fourth-order accurate time discretization [1]. In this way, it is possible to evaluate the stream fuapplying a cyclic-reduction fast direct solver for the resulting Helmholtz problem.

R. Arina, M. Falossi / Applied Numerical Mathematics 49 (2004) 263–275 265

As a second step, the acoustic source terms related to the flow field are evaluated. Finally, assumingharmonic dependence of the solution of Lighthill’s wave equation (3), the time dependent problem is

volvingnumberfourth-matrix

niqueons, theed andnditionsmissiontizationnditions

tions 3e and the

,

and the

of waveltzis

ithhemefor any

reduced to a steady state problem for each frequency (the complex Helmholtz equation), incomplex quantities. Because of the wave propagation accuracy requirement demands that theof points per wave length increases as the frequency increases, it is necessary to adopt aorder accurate scheme in order to contain the necessary number of grid points. The resultingto be inverted is non-symmetric, indefinite and ill-conditioned; a preconditioned GMRES techhas been proved to be efficient for the present case. To avoid very large systems of equaticomputational domain is divided in subdomains, and the boundary-value problem is formulatsolved independently for each subdomain. On the artificial subdomain interfaces, transmission coensuring the continuity of the solution are imposed. An iterative solution scheme updates the transconditions after each iteration. The technique has been developed in [2], where different discreschemes, the effectiveness of the linear system solver and the accuracy of the transmission cohave been investigated, for one-dimensional and two-dimensional cases.

In Section 2, we present the numerical method for the solution of the wave equation. In Secand 4 we apply the present technique to the analysis of the wave propagation around an obstaclsound generated by the flow past a two-dimensional cavity.

2. Numerical solution of the wave equation

Assuming harmonic time dependence of the formeiωt , wherei = √−1, andω the angular frequencyEq. (3) is reduced to the complex Helmholtz equation

−�p̂k − k2p̂k =(

∂̂2Tij

∂xi∂xj

)k

, (4)

with the wave numberk = ω/c0. We will refer to Eq. (4) as thereduced wave equation.The acoustic pressure can be computed after the flow field calculations are completed,

inhomogeneous term of the wave equation (3)

f (xi, t) = ∂2Tij

∂xi∂xj

= ∂2

∂xi∂xj

(ρ0uiuj ),

has been evaluated, and a discrete Fourier transform has been performed. In this way, the rangenumberk ∈ [0, kmax] is illustrated, and for each value ofk we must solve the associated Helmhoproblem with the corresponding inhomogeneous forcing termf̂k(x). The final acoustic pressure fieldrecovered performing the inverse discrete Fourier transform

p(xi, t) =kmax∑k=0

p̂k(xi)eikc0t .

Considering the rectangular domain[0,L] × [0,H ], on which an uniform rectangular mesh wN × M grid points is defined, Eq. (4) is discretized on the uniform grid. Any discretization scbased on three nodes along each coordinate direction, leads to the following discrete relationinterior node(j, l), with j = 1, . . . ,N − 1 andl = 1, . . . ,M − 1,


Ax

[(p̂k

)j−1,l

+ (p̂k

)j+1,l

] + Ay

[(p̂k

)j,l−1 + (

p̂k

)j,l+1

] + Bxy

(p̂k

)j,l[( ) ( ) ( ) ( ) ]

s have

ectingalong

tion to

r

ns. Iton is

+ Dxy p̂k j−1,l−1 + p̂k j+1,l−1 + p̂k j−1,l+1 + p̂k j+1,l+1 = Fj,l . (5)

Applying the fourth-order compact scheme proposed by Harari and Turkel [6,2], the coefficientthe following expressions:

Ax = �y2 − 1

6

(�x2 + �y2) + 1

12(k�x�y)2,

Ay = �x2 − 1

6

(�x2 + �y2

) + 1

12(k�x�y)2,

Bxy = −5

3

(�x2 + �y2) + (k�x�y)2, Dxy = 1

12

(�x2 + �y2).

The forcing term is discretized as follows:

Fj,l = �x2�y2

{1

12

[(f̂k

)j+1,l

+ (f̂k

)j−1,l

+ (f̂k

)j,l+1 + (

f̂k

)j,l−1

] − 1

3

(f̂k

)j,l

}.

Along the far field boundaries, to avoid incoming spurious reflections, appropriate non-reflboundary conditions must be specified. In order to formulate non-reflecting boundary conditionsthe boundaries, it is useful to write the one-dimensional analog of Eq. (4) along the normal directhe boundary. Considering the boundaryx = L, beingx the normal coordinate, we have

−d2p̂k

dx2− k2p̂k = f̂k.

Following [2], it is possible to factorize the left side operator as follows

− d2

dx2− k2 = −

(d

dx+ ik

)(d

dx− ik

)= −(

D+D−),

and the solution of the homogeneous equation may be expressed in the form

p̂k(x) = p̂ +k + p̂ −

k .

It is possible to remark that the solution is the sum ofp̂ +k = ake

−ikx , solution of the complex first-ordereduced wave equation

D+p̂k =(

d

dx+ ik

)p̂k,

and ofp̂ −k = bke

ikx , solution of

D−p̂k =(

d

dx− ik

)p̂k.

Consequently, the solution̂pk(x) is the sum of waves travelling from left to right (p̂ +k ) and from right

to left (p̂ −k ). This remark is useful for formulating appropriate non-reflecting boundary conditio

follows that atx = L, to avoid incoming spurious reflections, the appropriate boundary conditiD+p̂k = 0. In our case we have to enforce the relation(

∂

∂x+ iξ

)p̂k =

(∂

∂x+ ik

√1−

(µ

k

)2 )p̂k = 0, with k2 = ξ2 + µ2. (6)


ginary

n, andh ILUT[9]. Inalso for

d solvinge

rmare

is well

issionertical

ringissionpriately

Fig. 1. Solution of Eq. (4); instantaneous pressure along the horizontal (y = 0, –�–) and the diagonal (–�–) cuts.

Similar conditions apply along the other external boundaries.The compact discretization of the above conditions leads to the coupling of the real and ima

parts of the solution̂pk [2].For the Helmholtz problem it is not possible to apply fast direct methods, such as cyclic reductio

therefore it is necessary to resort to an iterative method. We have applied the GMRES method witpreconditioning, consisting of an ILU decomposition with threshold and diagonal compensationRef. [2] it has been shown that the performances of this GMRES linear solver remain the samevery high wave numbersk, on coarse and fine grids.

To assess the effectiveness of the discretization scheme, a numerical test has been performeEq. (4), on the domain(x, y) ∈ [−4,4] × [−4,4], with 81 × 81 uniform grid and 64 modes. Thoscillations are induced by a source placed atx = y = 0, represented by the periodic forcing tef (xi, t) = 10sin(4t) for xi = 0, andf (xi, t) = 0, elsewhere. Non-reflecting boundary conditionsimposed on the boundaries. In Fig. 1 the instantaneous pressure along the horizontal liney = 0 is shown,as well as the solution along the diagonal cut, showing that the circular symmetry of the solutionpreserved on the Cartesian geometry.

In the case of multidomain partitioning, along the interior boundaries we have to apply transmconditions. For the sake of simplicity, but without loss of generality, we treat the case of the vinterface, placed atx = const . We have to impose the conditions, along an interface at constantx = xint,

p̂ Lk = p̂ R

k anddp̂ L

k

dx= dp̂ R

k

dx, for x = xint,

wherep̂ Lk andp̂ R

k are the restrictions of̂pk on the left and right sub-domains respectively. Considethe structure of the solution, composed by opposite traveling waves, it follows that the transmconditions must also take into account the signal traveling across the interface. Therefore, approcombining the above interface constrains, we have [2], forx = xint,

dp̂ 1k

dx+ ikp̂ 1

k = dp̂ 2k

dx+ ikp̂ 2

k ,dp̂ 2

k

dx− ikp̂ 2

k = dp̂ 1k

dx− ikp̂ 1

k .


ontal cut

is

ns.

t solutiondof the

an edge,nted byximity

mains.e solid

Fig. 2. Solution of Eq. (4); instantaneous pressure contour levels (left), and comparison of the solution along the horiz(y = 0), on the single domain (–�–) and on the multidomain (–�–) (right).

The above conditions must be imposed iteratively as follows, withξ = k√

1− (µ/k)2,[(∂

∂x+ iξ

)p̂ L

k

]n+1

= θ

[(∂

∂x+ iξ

)p̂ R

k

]n

+ (1− θ)

[(∂

∂x+ iξ

)p̂ L

k

]n

,

[(∂

∂x− iξ

)p̂ R

k

]n+1

= θ

[(∂

∂x− iξ

)p̂ L

k

]n

+ (1− θ)

[(∂

∂x− iξ

)p̂ R

k

]n

.

The updating of the wave signal is relaxed imposing 0� θ � 1. In Ref. [2] it has been shown that thiterative approach is optimal in the sense that it converges exactly in a finite number of iterations.

The interface conditions are discretized in a way similar to the non-reflecting boundary conditioThe previous test case has been solved on the same domain with a 3× 3 subdomain partitioning, with

interior interfaces atx = ±2 andy = ±2, and the uniform grid arrangement of globally 81×81 points. InFig. 2, we report the instantaneous isopressure contours (left), and the comparison of the presenwith the solution on the single domain (right) along the horizontal cut aty = 0, The interfaces are placeat x = ±2. It is possible to remark that the transmission conditions do no affect the accuracyscheme.

3. Wave diffraction past an obstacle

Wave phenomena, such as the diffraction due to the interaction between incident waves andare represented by solutions of the wave equation (3). A prototype of this phenomenon is represethe interaction of a wave signal, generated by a linear source, with a reflecting solid wall in the proof the ground (Fig. 3).

The reduced wave equation (4) is solved in the domain shown in Fig. 3, divided into five subdoAt the interfaces (dotted lines) transmission boundary conditions are enforced, while along thwalls and the ground (thick solid lines) the reflecting boundary condition∂p/∂n = 0 is applied. Along


6), areform

rom

the solidportantbut onlyular, weproper

displayse frontbetween

s beenalnt of thedo not

Fig. 3. Domain decomposition of the wave diffraction past an obstacle problem.

the external boundaries (thin solid lines) non-reflecting boundary conditions, given by relation (specified. The linear source (S1 in Fig. 3) is placed at the point of coordinates (0.5, 0.5) and has the

f (t) = A

32∑n=1

sin(2πnf t),

with A = 5000 [Pa/m2], andf = 100 [Hz], corresponding to a superposition of waves spanning f100 to 3200 [Hz].

The diffraction depends upon several length scales, such as the source position with respect towalls and the ground, the wall height and its thickness. This last dimension being the most imone. In the present case we are not interest in a detailed parametric study of the phenomenon,to assess the ability of the proposed approach to correctly represent its main features. In particare interested in verifying the behaviour of the non-reflecting boundary conditions, enabling asimulation even with a relatively narrow computational domain.

Fig. 4 displays three snapshots at three different instants of the time cycle. The upper snapshotthe reflection from the wall and the ground, the middle one shows the diffraction of the upper wavdue to the wall edge, and the last one exhibits the complex wave structure, due to the interactionthe two reflected fronts.

To verify the effectiveness of the non-reflecting boundary conditions, further computation haperformed for the same case, but with anenlarged computational domain (Fig. 5). Two additionsubdomains have been added at both sides, and the solution, corresponding to the same instalast snapshot of Fig. 4, is shown in Fig. 6. It is possible to remark that the boundary conditionsaffect the far field wave propagation.


ver anwalls ofarmonic

onment.ty with

ar layerorticesic waves

Fig. 4. Wave diffraction past an obstacle: snapshots at three different times.

4. Cavity noise

In this section we study the noise radiated by the flow past a two-dimensional cavity. The flow oopen cavity is usually unsteady, and large fluctuations may occur in the pressure acting on thethe cavity and on the surrounding surfaces. These intense, periodic, but not necessarily simple hpressure fluctuations induce a noise radiation and acoustic oscillations in the surrounding envirDespite the large amount of studies on cavity flows, few deal with the radiated noise from a cavia subsonic grazing flow [8,11,3,5].

The cavity resonance is thought to arise from a feedback loop involving several steps: the sheinstability and the growth of vortical structures inside the shear layer, the impingement of the vat the downstream edge and the scattering of acoustic waves, and the transmission of acoustupstream and their conversion to vortical perturbations at the leading edge.


d two

in thediationcousticain in

Fig. 5.Enlarged computational domain decomposition of the wave diffraction past an obstacle problem.

Fig. 6. Wave diffraction past an obstacle withenlarged computational domain.

Recent numerical experiments [3], with laminar upstream boundary layers, have individuatefundamentally different modes of cavity oscillations termedshear layer mode and wake mode. Thecontrol parameters seem to be the ratioL/θ andReθ (L being the cavity length andθ the momentumthickness of the boundary layer at the cavity leading edge).

In the present analysis, we study a shallow cavity withL/D = 4 (D is the cavity depth). Theincoming flow is a laminar boundary layer withL/θ ≈ 200 andReθ ≈ 30, corresponding to awakemode configuration. The numerical procedure is based on the two step calculation describedprevious sections, firstly the incompressible viscous flow calculation and then the acoustic ranumerical simulation. Fig. 7 shows a close-up of the two grids used in the viscous (top) and a(bottom) calculations, where it is possible to note the decomposition of the computational domnon-conforming subdomains.


ations,fourthheddingdicallyontoursiently

tainedmples,ition, assent thensitivitytaneousctivityr rangehe

Fig. 7. Close-up of the grids used in viscous (top) and acoustic (bottom) calculations.

The incompressible viscous flow calculations are obtained solving the Navier–Stokes equwritten in stream function-vorticity form, with second order accurate spatial discretization andorder accurate time integration [1]. The wake mode is characterized by a large scale vortex sfrom the cavity leading edge. The shed vortex, with dimension of nearly the cavity size, is perioejected from the cavity abruptly. Fig. 8 shows the instantaneous streamlines (up) and vorticity c(down) at an instant of the oscillation cycle. The viscous flow solution was first carried out to a sufficlong non-dimensional time to assure the cycle oscillations were well established.

Then, the Lighthill’s wave equation is solved for an oscillation cycle, the fluctuating pressure obin the viscous calculation as the wall boundary condition. During an oscillation cycle, 128 saequally spaced in time, have been employed for describing the wall pressure boundary condwell as to evaluate the stress tensor. The present number of samples was found to fully repreflow behaviour within the cycle. No further tests have been made in order to assess the seof the aeroacoustic calculation upon the number of time samples. Fig. 9 presents the instanisopressure contours of the acoustic field at an instant of the oscillation cycle. In Fig. 10 the direpattern of the cavity noise radiation is shown. The directivity pattern is measured over the angula0deg< θ < 180deg, on the circular arc of rayr = 3D, centered at the cavity downstream edge. T


Fig. 8. Instantaneous streamlines (up) and vorticity contours (down) at an instant of the oscillation cycle (L/θ ≈ 200,Reθ ≈ 30).Close-up of the computational domain.

Fig. 9. Instantaneous isopressure contours of the acoustic field at an instant of the oscillation cycle.


e

s beensolvingsolving

rmed, withnterfacey of the

cavitye of the

nternal

he

..ymp.

(1995)

ut. Fluid

Fig. 10. Directivity pattern of the cavity noise radiation.

levels range from 95 to 105 dB with a peak radiation near the angleθ = 130deg, showing the existencof a directivity effect of the acoustic radiation [8,5].

5. Conclusions

A computational tool for the simulation of time harmonic sound propagation in open domains haproposed. It consists of a two step technique. In the first step the viscous flow field is calculated,the incompressible Navier–Stokes equations. In the second step the acoustic field is obtainedthe Lighthill’s wave equation. The two calculations are performed on two different grids, both foby non-conforming subdomains. The Lighthill’s wave equation is solved in the frequency domaina high order accurate compact finite difference scheme. Numerical tests have shown that the itransmission conditions as well the non-reflecting boundary conditions do not affect the accuracscheme. Finally, the technique has been applied to the simulation of the noise radiation from awith a grazing subsonic laminar boundary layer. The numerical results have shown the existencdirectivity character of the acoustic propagation, in accordance with previous numerical results.

References

[1] R. Arina, Multi-block method for the numerical solution of the incompressible Navier–Stokes equations, DIASP INote, Torino, 1999.

[2] R. Arina, E. Ribaldone, Aeroacoustic modeling of complex flow problems: I—Domain decomposition method for treduced wave equation, Comput. Visual. Sci. 4 (2002) 139–146.

[3] T. Colonius, A.J. Basu, C.W. Rowley, Numerical investigation of the flow past a cavity, AIAA Paper 99-1912, 1999[4] J.A. Ekaterinaris, New formulation of Hardin–Pope equations for aeroacoustics, AIAA J. 37 (9) (1999) 1033–1039[5] X. Gloerfelt, C. Bailly, D. Juve, Direct calculation of cavity noise and validation of acoustic analogies, in: RTO/AVT S

Development in Comput. Aero- and Hydro-Acoustics, Manchester, October 2001.[6] I. Harari, E. Turkel, Accurate finite difference methods for time-harmonic wave propagation, J. Comput. Phys. 119

252–270.[7] J.C. Hardin, D.S. Pope, An acoustic/viscous splitting technique for computational aeroacoustics, Theoret. Comp

Dynamics 6 (1994) 323–340.[8] R.J. Hardin, D.S. Pope, Sound generation by flow over two-dimensional cavity, AIAA J. 33 (3) (1995) 407–412.


[9] Y. Saad, Krylov subspace methods on supercomputers, SIAM J. Sci. Statist. Comput. 10 (1989) 1200–1232.[10] W.Z. Shen, J.N. Sorensen, Aeroacoustic modeling of low-speed flows, Theoret. Comput. Fluid Dynamics 13 (1999) 271–

s using

289.[11] C.M. Shieh, P.J. Morris, Parallel numerical simulation of subsonic cavity noise, AIAA Paper 99-1891, 1999.[12] S.A. Slimon, M.C. Soteriou, D.W. Davis, Development of computational aeroacoustic equations for subsonic flow

a Mach number expansion approach, J. Comput. Phys. 159 (2000) 377–406.

Domain decomposition technique for aeroacoustic simulations

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