Investigation of Nonreflective Boundary Conditions for Computational Aeroacoustics

Investigation of Nonreective Boundary Conditionsfor Computational Aeroacoustics

Mirela Caraeni

Fluent Inc., Lebanon, New Hampshire 03766

and

Laszlo Fuchs

Lund Institute of Technology, 221 00 Lund, Sweden

DOI: 10.2514/1.5807

Direct aeroacoustic computations require nonreective boundary conditions that allow disturbances to leave the

domain freely without anomalous reections. In the present paper we analyze a series of nonreective boundary

conditions already published in the literature and propose an improved outow nonreective boundary condition

with reection characteristic that is reduced greatly. For the solution of the linearized Euler equations, a sixth-order

compact nite-difference algorithm is used together with a sixth-order explicit digital lter that suppresses the high-

frequency spurious oscillations in the solution. A number of representative test cases are presented. The new outow

boundary condition is recommended for the simulation of sound produced by turbulence.

I. Introduction

S OLVING a problem formulated on an unbounded domainusually requires truncating the domain and solving the articialboundary conditions (ABCs) at the newly external boundary. ABCsappear in many areas of scientic computing such as acoustics,electrodynamics, solid mechanics, and uid dynamics (see reviewsby Givoli [1], Tsynkov [2], and Hagstrom [3]).The techniques used currently to set the ABCs can be classied

into two groups. The methods from the rst group, called globalABCs in Tsynkov [2], provide high accuracy and robustness of thenumerical procedure, but seem to be cumbersome and computationalexpensive. The methods from the second group, called local ABCs,are algorithmically simple, numerically cheap, and geometricallyuniversal, but usually lack accuracy.We review the ABCs for simulation of inow and outow prob-

lems with an emphasis on techniques suitable for compressible tur-bulent ows.Numerical simulations of sound propagation in the far eld are

based usually on the linearized Euler (LEE) or linearized NavierStokes (LNSE) equations together with appropriate boundaryconditions (BC) for inow and outow problems. These boundaryconditions should allow the ow disturbances, like the pressure(acoustic disturbances), vorticity, or entropy perturbations, etc., toleave the computational domain without signicantly affecting themand most importantly without reections. This is very important incomputational aeroacoustics (CAA), because the spurious (acoustic)waves generated by inadequate boundary conditions can mask thetrue sound eld, when the simulation is done far from the sources ofsound. For this type of simulation, state-of-the-art nonreectiveboundary conditions (NRBC) have been proposed by Giles [4],Colonius et al. [5], Tam [6], etc.Following the above classication, todays state-of-the-art

nonreective boundary conditions can be grouped as the following:1) Global ABCs, see, for instance, the boundary conditions

developed byThompson [7], Poinsot andLele [8], Colonius et al. [5],

Bogey and Bailly [9], etc. This group develops nonreectiveboundary conditions based on characteristics theory. The main ideais to cast the linearized Euler equations into a reference frame relativeto the boundary normal, in terms of the Riemann invariants. TheseRiemann invariants are the acoustic, entropy, and vortical invariants.The nonreective boundary condition will simply impose the value(zero) of the incoming invariants. Work by Watson and Zorumski[10] is extended to treat the one-dimensional time-periodic ductacoustic phenomena described by the linearized Euler equations inthe far eld (the one-dimensional methodology of Watson [10] doesadmit a multidimensional generalization, see Colonius [5]).2) Local ABCs, see boundary conditions suggested byBayliss and

Turkel [11], derive nonreective boundary conditions based on thefar-eld asymptotics for the solution of the two-dimensionallinearized Euler equations and construct the rst-order local ABCsfor the calculation of acoustic elds in uniformly moving media; seeTam and Webb [12]. Tams two-dimensional boundary conditionshave been extended for the case of nonuniform ow by Tam andDong [13].Colonius [14] discusses various models for an outow boundary

in the nonlinear case, such as absorbing layers and fringe methods.For an inow and outow boundary, Freund [15] provides a generalframework for addressing the generalized buffer zone technique. Forthe case of a turbulent jet where the articial boundary intersects thesource region, he suggests the use of an additional buffer layerscheme. Thus, an additional layer of several grid points is added tothe computational domain and the governing equations are modiedor amended in this buffer layer so as to absorb or dissipate the wavesand to prevent wave reection back to the solution domain. For thiskind of boundary treatment, we mention here the perfectly matchedlayer (PML) and the sponge-layer schemes.The PML scheme has been introduced byBerenger [16,17]. In this

scheme the governing equations are split according to the spatialderivatives. Furthermore, the dependent variables are also split intosubcomponents and an absorption coefcient is introduced into theseequations. The resulting PML equations are solved within the PMLdomain. The procedure originally developed by Berenger withapplications to electromagnetic waves has been extended tolinearized Euler equations by Hu [18], Tam [19] and Euler equationsby Hu [20].The sponge-layer scheme requires the introduction of an

additional layer where a source term is added to the governingequation with the objective to dissipate the wave within the spongelayer. The source term to be added to the right-hand side (RHS) of the

governing equation is x Q, where

Received 14 October 2003; revision received 24 April 2006; accepted forpublication 4 May 2006. Copyright 2006 by the American Institute ofAeronautics and Astronautics, Inc. All rights reserved. Copies of this papermay be made for personal or internal use, on condition that the copier pay the$10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 RosewoodDrive, Danvers, MA 01923; include the code $10.00 in correspondence withthe CCC.

Ph.D.; [email protected], LTH; [email protected].

AIAA JOURNALVol. 44, No. 9, September 2006

1932

Dow

nloa

ded

by S

TAN

FORD

UN

IVER

SITY

on

Mar

ch 2

7, 2

013

| http:

//arc.

aiaa.o

rg | D

OI: 1

0.251

4/1.58

07

x ax xB=xB xEn; xB x xE0; otherwise

(1)

This expression is used for the sponge layer at the boundaryperpendicular to the x axis. A similar expression is written for theboundary perpendicular to the y axis. Terms xB and xE in relation (1)denote the x coordinates of the beginning and the end of the spongelayer, respectively. The constants a and n are specied to control theamplitude and distribution of the damping coefcient . Theimplementation of the sponge-layer concept is simple andstraightforward without any major modication of the governingequations.First, the numerical scheme used in this work is presented; more

details can be found in Caraeni [21]. Then, we investigate a series ofpublished nonreective boundary conditions and propose animproved outow boundary condition for computational aero-acoustics and the simulation of sound produced by turbulence. Thenewly proposed outow boundary condition reduces signicantlythe appearance of spurious pressure disturbances, abnormallyproduced in the outow by the exit of weak vortical structures (ofturbulence, etc.) or by oblique sound waves. Test cases are presentedto address the accuracy of the new boundary condition and also forthe other boundary conditions investigated.

II. Linearized Euler Equations

Nonreective boundary conditions are typically derived for linearhyperbolic systems. Let us consider the linearized Euler equationswritten in primitive variable formulation, in 2-D. Consider thefollowing decomposition of the primitive variable vector:

q q0 q (2)

where q, q0, and q are the instantaneous, mean (uniform), andperturbation values of the eld vector ;u; v;pT . Here,u, v are theinstantaneous velocities in the x and y directions, and p the(instantaneous) density and pressure, respectively. The linearizedEuler equations can be written in nonconservative form as

@q

@tA @q

@x B @q

@y S (3)

were q ; u; v; pT is the perturbed primitive variable vector,

A u0 0 0 00 u0 0

10

0 0 u0 00 p0 0 u0

2664

3775 (4)

B v0 0 00 v0 0 00 0 v0

10

0 0 p0 v0

2664

3775 (5)

are the convective-ux Jacobians, computed for q0 0; u0; v0; p0T . Note that the source term S can be a function ofspace and timeS Sx; y; t. Here, the termShas been introduced inthe linearized Euler equations for convenience. It is only used whenprescribing (space and time dependent) sources of perturbation forsound, entropy, etc., in the computational domain; see, for example,Sec. VII.C.Typically, the background conditions described by q0 represent

either uniform mean conditions or a solution of the steady linearizedEuler equations. For far-eld sound propagation we assume hereuniform mean conditions, for example, q0 const.

III. Spatial Discretization

Compact-nite differences are renowned for their high accuracyand computational efciency and have been traditionally employedin CAA. Here we use a compact nite-difference discretization

scheme on a uniform, structured Cartesian mesh. Let us considerEq. (3), written in semidiscrete form:

@q

@tAq0

@q

@x

num

Bq0@q

@y

num

Snum (6)

where the superscript num symbolizes the values of thecorresponding derivatives and the source term computed numeri-cally.In the present work we employ a sixth-order, ve-point stencil

compact nite-difference formula to compute these derivatives; seeHoffmann andChiang [22]. Thus, the formula for computing therstderivative in the x direction for the arbitrary eld , where is adiscrete eld in 2-D, for example, i;j, is given by an implicitrelation between the eld values and its (x) derivative, at someneighboring points in the stencil:

1

3

@

@xi1;j @@xi;j

13

@

@xi1;j

19

i2;j i2;j4dx

149

i1;j i1;j2dx

(7)

where i; j 1 Ni; 1 Nj. Thus, a tridiagonal linear systemhas to be solved for each j const line (j 1 Nj) of the mesh, tocompute @=@xi;j, where i 1 Ni. A similar procedure has beenapplied for computing the y derivatives, @=@yi;j. A directtridiagonal linear system solver has been used, as described inHoffmann and Chiang [22].The compact nite-difference formula presented above has been

applied for computing both x and y derivatives of the primitivevariable vector, q. Once the @q=@xnumi;j and @q=@ynumi;j terms havebeen computed, the RHS of Eq. (6) can be assembled:

RHS i;j Aq0@q

@x

num

i;j Bq0

@q

@y

num

i;j Snumi;j (8)

Note that we made the hypothesis that q0 is constant (uniformbackground ow conditions); the matrices Aq0 and Bq0 areconstant throughout the domain and have to be computed just once,leading to very efcient numerics for sound propagation when usingLEE.

IV. Temporal Discretization

High order temporal discretization has to be used in conjunctionwith the high order spatial discretization, to reduce the number oftime steps, thus the computational effort for the simulation. A fourth-order explicit RungeKutta scheme has been used in the presentwork.Replacing (8) in the semidiscrete Eq. (6), one obtains

@q

@ti;jRHSi;j (9)

The fourth-order low-storage RungeKutta time advancementalgorithm employed here is

q0 qn (10)

qk qn k dt RHSqk1 k 1 4 (11)

qn1 q4 (12)

wherek are the fourth-order RungeKutta stage coefcients denedby

k 1

4 k 1 (13)

CARAENI AND FUCHS 1933

Dow

nloa

ded

by S

TAN

FORD

UN

IVER

SITY

on

Mar

ch 2

7, 2

013

| http:

//arc.

aiaa.o

rg | D

OI: 1

0.251

4/1.58

07

V. Filtering

Because the compact-nite difference scheme presented abovehas very small dissipation, efcient removal of spurious waves, thatis, waves which cannot be resolved on the computational grid, isnecessary. An option which is often used is the articial selectivedamping by Tam et al. [23]. This approach is quite expensive,because additional terms have to be calculated in the differentialequations. Amore convenientway is to use suitable digitallters; seeVasilyev et al. [24]. Because the numerical scheme has extremelysmall dissipation, to avoid unrestricted growth of high-frequency(short wavelengths smaller than the computational grid) spuriousoscillations to affect the solution, we apply a sixth-order explicitdigital lter, of the following form (in 1-D) (see Lummer et al. [25]):

i a0i X3&1

a&i& i&

2(14)

where the lter coefcients are a0 0:6875, a1 0:46875,a2 0:1875, a3 0:03125 and i is the ltered value of thequantityi. We apply the same lter in both i and j directions at thesame time, thus yielding the following formula in 2-D:

i;j a0i;j X3&1

a&4i&;j i&;j i;j& i;j& (15)

This digital lter has been applied in the entire computationalspace and preserves the sixth-order accuracy of the numericalscheme. To apply these lters at boundaries we used appropriateorder extrapolations.For a x const boundary, the x derivatives of the primitive

variable vector q have been computed using a high order one-sidednite difference, see Caraeni [21], whereas the y derivatives havebeen calculated using the above compact-nite differences.Note that throughout the paper, the vector q ; u; v; pT

denotes the perturbation vector, whereas q ;u; v;pT repre-sents the instantaneous eld vector; see relation (2).

VI. Boundary Conditions

Well-posed boundary conditions based on characteristic theoryhave been proposed in Caraeni [21]. In accord with this theory, thenumber of physical conditions must be equal to the number ofcharacteristics that enter the computational domain. Thus, atsupersonic inlets all characteristics enter the domain, therefore all(characteristic) variables have to be imposed; for LEE we need toimpose the values of q which is the perturbation of a primitivevariable vector. At supersonic outow boundaries all characteristicsexit the domain, hence the perturbations vector can be simplyextrapolated from inside the computational domain to the boundary.Thus, supersonic inow/outow boundary conditions pose no spe-cial problems in aeroacoustics. In the following we will concentrateonly on subsonic boundary conditions.Acoustic (sound) and aerodynamic (entropy, vorticity, total

enthalpy) disturbances, produced, for example, by turbulent ows,behave quite differently. Thus, in a subsonic regime the soundwavespropagate in all directions at a speed equal to the sum of the soundspeed and the local ow velocity, whereas entropic and vorticaldisturbances traveling with the ow speed, for example, are onlyconvected downstream by the ow.For brevity, let us consider here only the case of inow/outow

boundary conditions on a line at x const. Following Colonius [5],we can write the boundary conditions in terms of the characteristicvariable c, dened in this case as

c c1c2c3c4

2664

3775

p c200c0 u

p 0c0 up 0c0 u

2664

3775 (16)

where c20 p0=0 is the square speed of sound, for the background

oweld. The corresponding (1-D) characteristic entropy variable,c1 p c20, and vorticity variable, c2 0c0:u, are transporteddownstream by the ow. Thus, at an inow boundary these variableshave to be specied, whereas at an outow boundary these variablescan be extrapolated from inside the domain. The characteristicvariables c3 p 0c0 u, c4 p 0c0 u correspond toforward, respectively, backward propagating sound waves. Theirtreatment is essential for obtaining the nonreective characteristics atthe inow/outow boundaries.Two different inow boundary conditions have been tested:NRBC-I1, see Colonius [5]:

c1c2c3

24

35 0 (17)

c4 cint4 (18)NRBC-I2

c1r u; v

c3

24

35 0 (19)

c4 cint4 (20)where superscript int denotes a value that has been extrapolatedfrom inside the domain at the boundary. Note that the conditionc2 0 in NRBC-I1 has been replaced by r u; v 0 in NRBC-I2, which effectively enforces a (zero) vorticity eld at the inow. Ifthe incomingow is rotational, then the specic value of the vorticityhas to be imposed. This improvement produced a simple (non-PDE)inow BC with good nonreective characteristics, similar to theBCI2 condition by Colonius et al. [5].Outow boundary conditions with good nonreective properties

are more difcult to obtain. Four different outow boundaryconditions have been investigated here as follows:NRBC-O1, see Colonius [5]:

c4 0 (21)

c1c2c3

24

35 c1c2

c3

24

35

int

(22)

NRBC-O2, see Colonius [5]:

@c4@t

u0@c2@y

v0@c4@y

(23)

c1c2c3

24

35 c1c2

c3

24

35

int

(24)

NRBC-O3

c4 0 (25)

c1r u; v

c3

24

35 c1r u; v

c3

24

35

int

(26)

Note that, as compared to NRBC-O1, we propose to extrapolatethe vorticity from inside the domain to the boundary. With thisimprovement the outow BC will allow vortical structures to passthrough the outow boundary without perturbations. But, forNRBC-O3 boundary condition (and similar for NRBC-O1 andNRBC-O2), when a vortex reaches the exit boundary, a spuriouspressure disturbance is produced in outow and propagates inside

1934 CARAENI AND FUCHS

Dow

nloa

ded

by S

TAN

FORD

UN

IVER

SITY

on

Mar

ch 2

7, 2

013

| http:

//arc.

aiaa.o

rg | D

OI: 1

0.251

4/1.58

07

the domain, see the numerical tests below). To cure this problem wepropose a new PDE based boundary condition for outow:NRBC-O4

@p

@tu0 c0:signu0

@p

@x v0

@p

@y(27)

c1r u; v

c3

24

35 c1r u; v

c3

24

35

int

(28)

We propose a new Eq. (27) for the time evolution of the pressureperturbation at the boundary. This equation has been obtained bymodifying the equation for pressure valid inside the domain (29):

@p

@t u0

@p

@x v0

@p

@y p0

@u

@x p0

@v

@x 0 (29)

where the cross-derivative terms in pressure still remain, whereas thederivatives in the velocity perturbations have been dropped (thesewere the terms producing spurious pressure perturbations at theoutow boundary when a vortex was exiting the domain) and thenormal derivative in pressure has been modied so that the equationpreserves the correct speed of the outgoing pressure wave (e.g., ifu0 > 0, the outgoing pressure wave will have in outow a velocity ofu0 c0). The transport equation (27) is integrated in time with thesame fourth-order RungeKutta schemewe use for the interior of thedomain. Using Eq. (27) and extrapolating the vorticity r u; vfrom inside the domain (instead of extrapolating the c2 characteristicvariable), we obtained improved nonreective properties for ouroutow BC, as the numerical experiments presented here will show.In fact we found that while using NRBC-O4 there is no need toconsider an additional technique (such as absorbing layer, spongelayer, etc.) to additionally dump spurious pressure disturbancesproduced when vortical structures exit the domain.The extent to which the initial/boundary-value problem is well

posed is discussed in Caraeni [21].Note that all inow and outow boundary conditions investigated

here proved to have good nonreective properties for soundpropagation in an irrotational ow.

VII. Numerical Tests

To investigate the suitability of the above boundary conditions forcomputational aeroacoustics we propose four test cases: thepropagation of an initially Gaussian-shaped pressure pulse, theuniform transport of a vortex, the propagation of the sound producedby a rotating quadrupole, and sound generation by the Kirchhoffvortex. All tests are done considering a uniform, subsonicmeanow.

A. Sound Propagating in Uniform Flow

First we study the sound propagation in a uniform ow. The soundwave is produced by a Gaussian-shaped pressure pulse introduced att 0 at the center of the 2-D computational domainD fx; yg f0 10g f0 10g. The uniform ow conditionscorrespond to a low free-stream Mach number,M1 0:04; 0:04.The initial pressure perturbation is given by the relation:

p p expx x0

2 y y02r20

(30)

where p is the pressure perturbation amplitude (here we takep 105p0), r0 is the characteristic dimension of the pulse, r0 1,and the point of coordinates fx0; y0g is the centroid of the 2-D domainD. There is no perturbation in any of the other eld variables, forexample, , u, v. The resulting propagating sound wave wassimulated using the numerical scheme presented above, until thewave has left the domain. We found that all boundary conditionspresented here performedwell in this test case. For brevitywe presenthere only the results obtained using our new boundary conditions:NRBC-I2 andNRBC-O4. Figures 15 present the soundwave atve

X

Y

0 5 100

1

2

3

4

5

6

7

8

9

10

Fig. 1 Pressure perturbation at t 0.

X

Y

0 5 100

1

2

3

4

5

6

7

8

9

10

Fig. 2 Pressure perturbation at tt.

X

Y

0 5 100

1

2

3

4

5

6

7

8

9

10

Fig. 3 Pressure perturbation at t 2t.


Dow

nloa

ded

by S

TAN

FORD

UN

IVER

SITY

on

Mar

ch 2

7, 2

013

| http:

//arc.

aiaa.o

rg | D

OI: 1

0.251

4/1.58

07

successive moments in time, up to the point when the wave totallyexited the domain. Notice that there are no spurious oscillationsproduced while the acoustic waves travel through boundaries.The analytical solution for this initial value problem, as given in

Bogey and Bailly [9]:

px; t p

Z 10

2 exp

2

4

costj0d (31)

where p is the pressure perturbation amplitude (see above), ln 2, 2 p ,

x Mxc0t2 y Myc0t21=2 (32)

and the spherical Bessel function of the rst kind and order zero j0 isgiven by j0z sinz=z.Figure 6 displays the comparison between the numerical results,

which have been obtained using the NRBC-I2 inow boundarycondition and the outowboundary conditionsNRBC-O1 toNRBC-O4, with the theoretical values for the pressure distribution on theoutow boundary at x 10 m, for a given instant in time(t 3:5t). Note that the results obtained with NRBC-O4 (solid

line) are practically indistinguishable from the theoretical solution(symbols), but we can see that all outow (and inow) boundaryconditions investigated here behave reasonablywell for this test case.

B. Vortex Propagating in Uniform Flow

Secondly, we investigate the transport of a vortex by uniform owand the suitability of the different nonreective boundary conditionsfor this kind of application. For brevity, we present here only theresults that have been obtained using the NRBC-I2 inow boundarycondition, with different outow boundary conditions. We considerthe same 2-D computational domain D fx; yg f0 10g

f0 10g and uniform ow conditions corresponding to a Machnumber of M1 0:5; 0:25. At time t 0, the oweld isperturbed such that a vortex is created, with a Gaussian-shapedvorticity distribution. The velocity perturbation u; v is given by

u Vvortex sin r exp r2=2 (33)

v Vvortex cos r exp r2=2 (34)where Vvortex M1c0=100, c0 is the sound speed, tan y y0=x x0, and r

x x02 y y02

p=r0.

The pressure and density perturbations have been determined byconsidering that the perturbation introduced has to preserve locallythe total enthalpy and entropy of the uniform ow. First, compute thelocal velocity, as a sum of mean-value plus perturbation:

u u0 u; v v0 v (35)

then, determine the local pressure and density perturbations using therelations:

T p0Rgas0

u20 v20 u2 v2

2cp(36)

p p0TRgas0p0

1 1

(37)

p p0RgasT

0 (38)

The solution is advanced in time until the vortex completely leavesthe domain, using the scheme described above. Figures 710 presentthe vorticity and pressure distributions at four successivemoments intime, when using the NRBC-O1 outow boundary condition.

X

Y

0 5 100

1

2

3

4

5

6

7

8

9

10


X

Y

0 5 100

1

2

3

4

5

6

7

8

9

10


Y

p

0 2 4 6 8 10-0.1

-0.075

-0.05

-0.025

0

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2 NRBC-O1NRBC-O2NRBC-O3NRBC-O4Theoretical

Fig. 6 Pressure perturbation on the outow boundary (x 10 m).Comparison between the numerical results and the theoretical values.


Dow

nloa

ded

by S

TAN

FORD

UN

IVER

SITY

on

Mar

ch 2

7, 2

013

| http:

//arc.

aiaa.o

rg | D

OI: 1

0.251

4/1.58

07

Figures 1113 and Figs. 1416 display the pressure and vorticitydistributions for the case when using NRBC-O2 and NRBC-O3,respectively.Note for NRBC-O2 that the use of a PDE for the fourth charac-

teristic variable c4 did improve the behavior of the scheme, by reduc-ing the pressure reections at the outow boundary. At the same timethe use of vorticity extrapolation instead of c2 extrapolation, whenusing NRBC-O3, reduced signicantly the schemes effect on thevorticity distribution at boundary. The new outow conditionNRBC-O4 combines the two ingredients presented above to producean improved scheme: uses a PDE for the pressure perturbation at the

boundary and extrapolates the vorticity from inside the domain to theboundary. Figures 1719 present the vorticity and pressuredistribution when using the new outow condition, NRBC-O4.Please note that when using NRBC-O4 the spurious reection of

pressure in outow, due to the vortex exiting the domain, hasdisappeared and the vortex also leaves the domain freely withoutbeing perturbed.To quantify the numerical error introduced by the use of the four

different boundary conditions investigated here, the same owsituation is simulated on a much larger domain, D fx; yg f45 55g f45 55g and the pressure, velocity (vorticity)distribution on the boundary of our original (much smaller) domain ismonitored. Note that the same grid resolution and Courant numberhave been used. We use the solution on the large domain as thereference solution, to assess the quality of our results obtained whenusing the four different nonreective boundary conditions. Pressuredata have been collected on the outow boundary of the smalldomain at x 10. Note that the size of the large domain is sufcientso that there are no spurious reections from its boundaries during theactual simulation. This setup allows the errors from the boundaryconditions to be isolated from the other discretization errors and to bequantied.The numerical error is dened here as the normalized difference

between the pressure perturbation computed on the large domain onthe outow boundary of the small domain at x 10 and the pressureperturbation on the boundary of the small domain. The normalization

X

Y

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

X

Y

0 5 100

1

2

3

4

5

6

7

8

9

10

11

Fig. 7 Initial perturbed eld. Vorticity and pressure eld.

X

Y

0 5 100

1

2

3

4

5

6

7

8

9

10

11

X

Y

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

Fig. 8 Perturbed eld. Vorticity and pressure eld at tt. UsingNRBC-O1 boundary condition in outow.

X

Y

0 5 100

1

2

3

4

5

6

7

8

9

10

11

X

Y

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

Fig. 9 Perturbed eld. Vorticity and pressure eld at t 2t. UsingNRBC-O1 boundary condition in outow.

X

Y

0 5 100

1

2

3

4

5

6

7

8

9

10

11

X

Y

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10


X

Y

0 5 100

1

2

3

4

5

6

7

8

9

10

11

X

Y

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10


XY

0 5 100

1

2

3

4

5

6

7

8

9

10

11

X

Y

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10


X

Y

0 5 100

1

2

3

4

5

6

7

8

9

10

11

X

Y

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10



Dow

nloa

ded

by S

TAN

FORD

UN

IVER

SITY

on

Mar

ch 2

7, 2

013

| http:

//arc.

aiaa.o

rg | D

OI: 1

0.251

4/1.58

07

factor used here is Vvortex2=2. Figure 20 shows the normalizederror versus normalized time t ta=L, where a is the speed ofsound, and L is half the size of the small domain. Our numericalresults clearly show that NRBC-O4 gives the smallest errorcompared with the two boundary conditions by Colonius and Lele,that is, NRBC-O1 and NRBC-O2 or the modied boundarycondition, NRBC-O3.

C. Propagation of Sound Produced by a Quadrupole

We study the propagation of the sound produced by a rotatingquadrupole in uniform ow. The computational domain isD fx; yg f0 10g f0 10g. The uniform ow conditionscorrespond to a lowMach number,M1 0:01; 0:01. The pressureperturbations are introduced in the form of a source term for thepressure equation, given by the relation:

s p p r exp r

2

2

sin2 !Rt cos!pt (39)

Here p is the pressure source amplitude (here we take p 103p0),r

x x02 y y02

p=r0 and r0 is the characteristic

dimension of the pulse, taken r0 1, the point of coordinatesfx0; y0g is the centroid of the 2-D domainD, tan y y0=x x0,!R is the angular velocity of the rotating source, and!p is the angularvelocity of the pulsation of the source, computed as a function of T.

T jDj=c0 is a characteristic time, jDj is the norm [1] of D, andc20 p0=0 is the sound speed. Thus, one can compute !R T and!p 8!R. There is no perturbation source introduced in any of theother eld variables, for example, , u, v. The soundwaves producedwere simulated using the above presented algorithm. The resultspresented herewere obtained using the new inow conditionNRBC-I2 and the new outow condition NRBC-O4.Figures 2125 display the sound waves at ve successive

moments in time. Notice that there are no spurious oscillations, as theacoustic waves travel through boundaries without being perturbed.

D. Sound Propagation by the Kirchhoff Vortex

Finally, we study the propagation of sound generated by aKirchhoff vortex. The Kirchhoff vortex is a patch of constantvorticity ! inside an ellipse:

x2

a2 y

2

b2 1 (40)

and zero vorticity outside, where a and b denote the semimajorand semiminor axes, respectively, of the ellipse. The Kirchhoffvortex rotates without change of shape with constant angularfrequency around the origin. The computational domain is D fx; yg f0 10g f0 10g.

X

Y

0 5 100

1

2

3

4

5

6

7

8

9

10

11

X

Y

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10


X

Y

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

X

Y

0 5 100

1

2

3

4

5

6

7

8

9

10

11


X

Y

0 5 100

1

2

3

4

5

6

7

8

9

10

11

X

Y

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10


X

Y

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

X

Y

0 5 100

1

2

3

4

5

6

7

8

9

10

11


XY

0 5 100

1

2

3

4

5

6

7

8

9

10

11

X

Y

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10


X

Y

0 5 100

1

2

3

4

5

6

7

8

9

10

11

X

Y

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10



Dow

nloa

ded

by S

TAN

FORD

UN

IVER

SITY

on

Mar

ch 2

7, 2

013

| http:

//arc.

aiaa.o

rg | D

OI: 1

0.251

4/1.58

07

The perturbations are introduced in the form of source terms.Using the lowMach number approximation, the perturbations in thenear eld are given by the relations by Muller [26]:

s pr; ; t 1

2"juellipsej2

a

r

2

cos2 t (41)

s ur; ; t 2 a"sin2 t cos (42)

s vr; ; t 2 a"sin2 t sin (43)

s r; ; t spc20

(44)

where juellipsej 2 a is the modulus of the velocity of the Kirchhoffvortex on the ellipse, r

x x02 y y02

p=r0 and r0 is the

characteristic dimension of the pulse, taken r0 1, the point ofcoordinates fx0; y0g is the centroid of the 2-D domain D, andtan y y0=x x0. !R is the angular velocity computed asfunction of T.

-5.00E-02

0.00E+00

5.00E-02

1.00E-01

1.50E-01

2.00E-01

2.50E-01

3.00E-01

3.50E-01

4.00E-01

0.00 2.00 4.00 6.00 8.00 10.00

time*a/L

Erro

r

NRBC-O1

NRBC-O2

NRBC-O3

NRBC-O4

Fig. 20 Vortex propagating in uniform ow. Normalized error on

outow boundary as a function of time.

X

Y

0 5 100

1

2

3

4

5

6

7

8

9

10

Fig. 21 Propagation of sound produced by a rotating quadrupole

source. t 0.

X

Y

0 5 100

2

4

6

8

10


source. tt.

X

Y

0 5 100

2

4

6

8

10

Fig. 23 Propagation of sound produced by a rotating quadrupolesource. t 2t.

X

Y

0 5 100

2

4

6

8

10


source. t 3t.


Dow

nloa

ded

by S

TAN

FORD

UN

IVER

SITY

on

Mar

ch 2

7, 2

013

| http:

//arc.

aiaa.o

rg | D

OI: 1

0.251

4/1.58

07

T jDjc0is a characteristic time, and jDj is the norm ofD. One can

compute!R 2T , and then compute 5!R. The perturbations areintroduced inside a circular region with a < r < 5 a, where a 0:1and " 0:01. a a1 ", and b a1 ".The results presented here were obtained using the new inow

condition NRBC-I2 and the new outow condition NRBC-O4. Thesound waves spiral outwards from the rotating source and exit thedomain smoothly. Figure 26 displays the soundwaves at an instant intime.

VIII. Conclusions

A series of nonreective boundary conditions have beeninvestigated and a new outownonreective boundary condition hasbeen proposed with certain improved characteristics as comparedwith the other nonreective boundary conditions investigated here.This nonreective boundary condition is recommended for the directcomputation of sound produced by turbulence.

References

[1] Givoli, D., Nonreecting Boundary Conditions, Journal ofComputational Physics, Vol. 94, No. 1, 1991, pp. 129.

[2] Tsynkov, S. V., Numerical Solution of Problems on Unbounded

Domains: AReview,Applied Numerical Mathematics, Vol. 27, No. 4,1998, pp. 465532.

[3] Hagstrom, T., Radiation Boundary Conditions for the NumericalSimulation of Waves, Acta Numerica, Vol. 8, Cambridge Univ. Press,Cambridge, MA, 1999, pp. 47106.

[4] Giles, M. B., Non-Reecting Boundary Conditions for EulerEquations Calculations, AIAA Journal, Vol. 28, No. 12, 1990,pp. 20502058.

[5] Colonius, T., Lele, S. K., and Moin, P., Boundary Conditions forDirect Computation of Aerodynamic Sound Generation, AIAAJournal, Vol. 31, No. 9, 1993, pp. 15741582.

[6] Tam, C. K. W., Advances in Numerical Boundary Conditions forComputational Aeroacoustics, Journal of Computational Acoustics,Vol. 6, No. 6, 1998, pp. 377402.

[7] Thompson, K. W., Time Dependent Boundary Conditions forHyperbolic Systems, I, Journal of Computational Physics, Vol. 68,Jan. 1987, pp. 124.

[8] Poinsot, T. J., and Lele, S. K., Boundary Conditions for DirectSimulations of Compressible Viscous Reacting Flows, Journal ofComputational Physics, Vol. 101, No. 1, 1992, pp. 104129.

[9] Bogey, C., and Bailly, C., Three-Dimensional Non-ReectiveBoundary Conditions for Acoustic Simulations: Far-Field Formula-tions and Validation Test Cases, Acta Acustica United with Acustica,Vol. 88, No. 4, 2002, pp. 463471.

[10] Watson, W. R., and Zorumski, W. E., Periodic Time Domain Non-Local Non-Reecting Boundary Conditions for Duct Acoustics,NASA TM-110230, Langley Research Center, March 1996.

[11] Bayliss, A., and Turkel, E., Far-Field Boundary Conditions forCompressible Flows, Journal of Computational Physics, Vol. 48,Nov. 1982, pp. 182199.

[12] Tam, C. K. W., and Webb, J. C., Dispersion-Relation-PreservingFinite-Difference Schemes for Computational Acoustics, Journal ofComputational Physics, Vol. 107, No. 2, 1993, pp. 262281.

[13] Tam, C. K. W., and Dong, Z., Radiation and Outow BoundaryConditions for Direct Computation of Acoustic and Flow Disturbancesin a Nonuniform Mean Flow, Journal of Computational Acoustics,Vol. 4, No. 2, 1996, pp. 175201.

[14] Colonius, T., Modeling Articial Boundary Conditions forCompressible Flow, Annual Review of Fluid Mechanics, Vol. 36,2004, pp. 315345.

[15] Freund, J. B., Proposed Inow/Outow Boundary Condition forDirect Computation of Aerodynamic Sound, AIAA Journal, Vol. 35,No. 4, April 1997, pp. 740742.

[16] Berenger, J. P., A Perfectly Matched Layer for the Absorption ofElectromagneticWaves, Journal of Computational Physics, Vol. 114,No. 2, 1994, pp. 185200.

[17] Berenger, J. P., Perfectly Matched Layer for the FDTD Solution ofWave-Structure Interaction Problems, IEEE Transactions onAntennas and Propagation, Vol. 44, No. 1, Jan. 1996, pp. 110117.

[18] Hu, F. Q., On Absorbing Boundary Conditions for Linearized EulerEquations by a Perfectly Matched Layer, Journal of ComputationalPhysics, Vol. 129, No. 1, 1996, pp. 201219.

[19] Tam, C. K.W., Auriault, L., and Canbuli, F., PerfectlyMatched Layerfor Linearized Euler Equations in Open and Ducted Domain, AIAAPaper 98-0183, Jan. 1998.

[20] Hu, F. Q., On Perfectly Matched Layers as an Absorbing BoundaryConditions, AIAA Paper 96-1664, May 1996.

[21] Caraeni, M.-L., Developing Computational Tools for ModernAeroacoustic Research, Ph.D. Thesis, Lund Institute of Technology,ISBN 91-628-5869-6, Sweden, 2003.

[22] Hoffmann, K. A., and Chiang, S. T.,Computational Fluid Dynamics,3rd ed., ISBN 0-9623731-2-5, Vol. 3, Engineering Education System,Wichita, KS, 2000, Chap. 22.

[23] Tam, C. K.W., Webb, J. C., and Dong, Z., A Study of the Short WaveComponents in Computational Acoustics. Journal of ComputationalAcoustics, Vol. 1, No. 1, 1993, pp. 130.

[24] Vasilyev,O., Lund, T., andMoin, P., AGeneral Class of CommutativeFilters for LES on Complex Geometries, Journal of ComputationalPhysics, Vol. 146, No. 1, 1998, pp. 82104.

[25] Lummer, M., Delfs, J., and Lauke, T., Simulation of the Inuence ofTrailing Edge Shape on Airfoil Sound Generation,AIAA Paper 2003-3109, 2003.

[26] Muller, B., OnSoundGeneration by theKirchhoff Vortex,Rept. 209/1998, Uppsala University, 1998.

K. GhiaAssociate Editor

X

Y

0 5 100

2

4

6

8

10

Fig. 25 Propagation of sound produced by a rotating quadrupolesource. t 4t.

X

Y

0 5 100

1

2

3

4

5

6

7

8

9

10

Fig. 26 Propagation of sound produced by Kirchoff vortex. tt.


Dow

nloa

ded

by S

TAN

FORD

UN

IVER

SITY

on

Mar

ch 2

7, 2

013

| http:

//arc.

aiaa.o

rg | D

OI: 1

0.251

4/1.58

07

Documents

Investigation of Nonreflective Boundary Conditions for Computational Aeroacoustics