Wall boundary conditions for high-order finite-difference …acoustics.ae.illinois.edu/pdfs/tam-dong-1994.pdf · Theoret. Comput. Fluid Dynamics (1994) 6:303-322 Theoretical and Computational

Theoret. Comput. Fluid Dynamics (1994) 6:303-322 Theoretical and Computational Fluid Dynamics O Springer-Verlag 1994

Wall Boundary Conditions for High-Order Finite-Difference Schemes in Computational Aeroacoustics 1

Christopher K.W. Tam and Zhong Dong

Department of Mathematics, Florida State University, Tallahassee, FL 32306-3027, U.S.A.

Communicated by Jay C. Hardin and M.Y. Hussaini

Received 22 March 1993 and accepted 6 May 1993

Abstract. High-order finite-difference schemes are less dispersive and dissipative but, at the same time, more isotropic than low-order schemes. They are well suited for solving computational acoustics problems. High-order finite-difference equations, however, support extraneous wave solutions which bear no resemblance to the exact solution of the original partial differential equations. These extraneous wave solutions, which invariably degrade the quality of the numerical solutions, are usually generated when solid-wall boundary conditions are imposed. A set of numerical boundary conditions simulating the presence of a solid wall for high-order finite- difference schemes using a minimum number of ghost values is proposed. The effectiveness of the numerical boundary conditions in producing quality solutions is analyzed and demonstrated by comparing the results of direct numerical simulations and exact solutions.

1. Introduction

Recent studies in computational acoustics have shown that to obtain high-quality acoustic wave solutions by direct numerical simulation the use of high-order schemes is necessary. High-order finite-difference schemes, when properly formulated, are less dispersive and dissipative. Also the computed acoustic waves tend to be less anisotropic. These characteristics allow the computed wave solution to remain a good approximation of that of the original partial differential equations (Euler equations or Navier-Stokes equations) even for long-distance wave-propagation problems. Another important reason for using a high-order scheme is that the computed wave speed would generally be nearly equal to the true speed of sound whereas the wave speeds of low-order schemes usually are not.

It is well known that high-order finite-difference schemes support extraneous wave solutions (see Trefethen, 1982; Tam et al., 1993). These are solutions which bear no resemblance to the exact solutions of the original partial differential equations. In a sense they are contaminants or pollutants of the numerical scheme. Some of these spurious waves have very short wavelengths (grid-to-grid oscillations) and propagate with ultra fast speeds. They have been referred to as parasite waves. Other spurious waves are spatially damped. Their presence is, therefore, confined locally in space. To ensure

This work was supported by the NASA Lewis Research Center Grant NAG 3-1267 and in part by the NASA Langley Research Center Grant NAG 1-1479 and the Florida State University through time granted on its Cray-YMP Supercomputer.

303

304 C.K.W. Tam and Zhong Dong

a high-quality computational aeroacoustics solution it is important that these spurious waves are not excited in the computation.

For inviscid flows the well-known boundary condition at a solid wall is that the velocity component normal to the wall is zero. This condition is sufficient for the determination of a unique solution to the Euler equations. For a high-order finite-difference scheme the order of the difference equations is higher than that of the Euler equations. Thus the zero normal velocity boundary condition is insufficient to define a unique solution. Extraneous conditions must be imposed. Unfortunately these extraneous conditions would inevitably lead to the generation of spurious numerical waves as discussed above. The net result is the degradation of the quality of the numerical solution. The degradation may be divided into three types. First, if an acoustic wave is incident on a solid wall, perfect reflection (e.g., reflected-wave amplitude equal to incident-wave amplitude) may not be obtained. Second, parasite waves may be reflected off the solid surface. Third, a numerical boundary layer can form adjacent to the solid-wall surface due to the spatially damped spurious numerical waves.

The imposition of solid-wall boundary conditions, when a low-order finite-difference or finite- volume scheme is used for computation, can usually be carried out in a straightforward manner as in the works of Watson and Myers (1991) and Khan et al. (1987). For high-order schemes, the situation is entirely different and does not seem to have been carefully investigated. Treatment of this problem in computational fluid dynamics textbooks such as Anderson et al. (1984) or Oran and Boris (1987) is inadequate. On the other hand, there have been numerous studies on the related subject of non- reflecting boundary conditions, e.g., Bayliss and Turkel (1980, 1982), Engquist and Majda (1977, 1979), Jiang and Wong (1990), Thompson (1990), and Givoli (1991). It appears that the problem is over- looked because the advantages of using high-order finite-difference schemes to solve wave propagation problems have been recognized only recently.

The objective of this paper is to propose a way to specify solid-wall boundary conditions for high-order finite-difference schemes. The proposed boundary conditions have the minimum number of ghost values. The acoustic wave reflection characteristics of the proposed wall boundary conditions are analyzed in Section 3. It is shown that there is minimal reflection of parasite waves off the wall and that only a very insignificant numerical wall boundary layer would develop. A sequence of computational acoustics experiments has been carried out to test the effectiveness of the proposed solid-wall boundary conditions. Both inviscid and viscous fluids are considered. When compared with the exact analytical solutions, very satisfactory results are found. Both static cases and cases with a mean flow are included in the numerical experiments.

Results of these direct numerical simulations are reported in this paper, and additional results and analyses will be presented in a future paper.

2. Boundary Stencils and Ghost Values

In a recent paper Tam and Webb (1993) developed a high-order explicit-time-marching finite- difference scheme specifically designed for computational acoustics application. The finite-difference scheme has the unusual characteristic that for long waves it has practically the same dispersion relation as the original partial differential equation. Thus, except for the short-wave component, the computed waves are guaranteed to have the same propagation characteristics and speeds as those of the partial differential equations. For this reason the scheme has been referred to as the Dispersion- Relation-Preserving (DRP) scheme. In developing the finite-difference equations from the partial differential equations, the spatial derivatives are approximated by central differences with optimized coefficients. The size of the stencil is arbitrary, to be determined by the degree of spatial resolution required. In their numerical examples Tam and Webb, however, chose a seven-point stencil. Time is advanced by an explicit four-level scheme with optimized coefficients. Computed results in a number of test problems were found to agree well with the exact solutions. In this work the DRP scheme is used as a prototype of a high-order finite-difference scheme. The basic ideas and principles concerning solid-wall boundary conditions developed in this paper are not restricted to the DRP scheme alone. They are applicable to other computation schemes as well.


Consider two-dimensional acoustic dis turbances super imposed on a uniform mean flow of density Po, pressure Po, and velocity Uo in the x-direction. The linearized N a v i e r - S t o k e s equat ions m a y be writ ten in the form

~U ~E 0F OG c~H ~3~- + ffxx + (3y ~x + ~ - + Q ' (1)

Where

po u + p u o po v 0 0

U = E = F = , G = ~ H =

' . o r ' I p / P o l OOL = ], 0OLd, j, uoP + 7po u LTpov]

~, , = 2 / x ~ , z , , = ~,= = /1 ~ + ~?x,/' r , r = 2# ~yy.

The first two terms on the r ight-hand side of (I) are viscous stress terms. They are to be discarded for inviscid fluids. The last term Q represents possible acoustic sources. Also viscous dissipation terms are omit ted in the energy equation. They are un impor tan t in mos t acoustics problems.

Let the x - y plane be divided into a mesh of spacings Ax and Ay in the x- and y-directions, respectively. The discretized form of (1) according to the D R P scheme is

1 3 1 3

=' l ,m = a x j ' = - 3 " j t - - l + j , m - - ~ l + j , m , - - ~yy j '= - 3 t~jt--l,m+j -- " ' t , , . + i , + ".~Z,,., (2)

~. " _ 2#

Txxll, m',(n) _ A x j = - 3 ajul + i ' " (")

a, { L ,,tn) . 1 (,) ) ~ I i , , . - ~o~ , t , , . = tz ~ = - 3 ~ \ A y "z'"+~ + ~ x x ~ l + i . . , ] , (3)

¢, ~(,) _ 2 # ~ ..,(.) t " y y l I , r a m y ~ W U l ' r n + j ~ "

j = - 3

3 U(, +~) i r(.) b K (n -j) ~,m = ' ~ t , , , + A t ~,, j t,,. , (4)

j=o

where l, m are the indices of mesh points and superscript n is the t ime level. Numer ica l values of the coefficients aj and bj are given in the Appendix. If U = U~.itia~ at t = 0 is the initial condit ion for the N a v i e r - S t o k e s or Euler equations, the appropr ia te initial condit ions for the D R P scheme (2), (3), and (4) are

U(O) = Uinitial ' IT(.) = 0 for negative n. (5) l , m ~-J l , r a

Suppose the fluid is inviscid and a solid wall lies at y = 0 as shown in Figure 1. Fo r points lying three rows or more away from the wall, the computa t ion stencil lies entirely inside the physical domain . They are referred to as interior points. Fo r the first three rows of points adjacent to the wall,

Figure 1. Seven-point computation stencil for an interior point. Also shown are the boundary points adjacent to a wall at wo. - y = 0 and ghost points outside the physical wall.

] interior polnts

i. boundary points

1-± ,=o i i [ i f i i i i i i i i i

_ 2 _ . 2 _ _ 2 _ _ 2 _ _ 2 _ _ 2 _ _ 2 _ _ 2 _ _ 2 _ _ ~ _ _ J _ _ 2 _ _ 2 _ G h o s t p o i n t s


their seven-point stencil will extend outside the physical domain. They are referred to as boundary points. The points outside the computation domain have no physical meaning, they are called ghost points.

Although ghost points are fictitious points with no physical meaning, they are necessary for high-order finite-difference schemes. To see this, recall that the solution of the Euler or Navier-Stokes equations satisfies the partial differential equations at every interior or boundary point. In addition, at a point on the wall the solution also satisfies the appropriate boundary conditions. Now the discretized governing equations are no more than a set of algebraic equations. In the discretized system, each flow variable at either an interior or boundary point is governed by an algebraic equation (discretized form of the partial differential equations). The number of unknowns is exactly equal to the number of equations. Thus there will be too many equations and not enough unknowns if it is insisted that the boundary conditions at the wall are satisfied also. This is, perhaps, one of the major differences between partial differential equations and difference equations. However, the extra conditions imposed on the flow variables by the wall boundary conditions can be satisfied if ghost values are introduced (extra unknowns). The number of ghost values is arbitrary but the minimum number must be equal to the number of boundary conditions. For an inviscid flow, the condition of no flux through the wall requires a minimum of one ghost value per boundary point on the wall. For two-dimensional viscous flow, the no-slip boundary condition requires a minimum of two ghost values per boundary point on the wall. In principle, more than the minimum number of ghost values can be introduced. However, for each extra ghost value, a condition must be imposed so that it can be uniquely determined. Clearly, the quality of the solid-wall boundary conditions in terms of acoustic wave reflection, acoustic boundary-layer characteristics, etc., would be affected by these extra ghost values. Since there is no compelling advantage to introduce extra wall boundary conditions, in this paper, only the minimum number of ghost values is used.

To fix the idea, let us first consider an inviscid fluid adjacent to a solid wall at y = 0 (see Figure 1). In this case the wall boundary condition is v = 0 at y = 0 where (u, v) are the velocity components in the x- and y-directions, respectively. Since there is one boundary condition, one ghost value is needed for each boundary point at the wall. Physically, the wall exerts a pressure on the fluid with a magnitude just enough to make v = 0 at its surface. This suggests that a ghost value of p (pressure) may be used at the ghost point immediately below the wall to simulate the pressure of the wall. As only one ghost value is adopted, the difference stencil for the y-derivative in the boundary region needs to be modified. Here it is proposed that the y-derivatives are to be computed according to the backward difference stencils shown in Figure 2. According to this scheme, the quantities ~p/Cy, Cu/Oy, and ~v/Cy are all calculated using values of the variables lying inside the physical domain. For the

r=o (re=o)

(a)

(b)

(~=6)

wolJ (,,,=o) Figure 2. Seven-point stencils used to compute (a) the y- derivative of p, and (b) the y-derivatives of p, u, and v in the boundary region near a wall at y = 0.

Wall Boundary Conditions for High-Order Finite-Difference Schemes in Computat ional Aeroacoustics 307

y-derivative of p, the stencils extend to the ghost point below the wall. Now the ghost value of p at -~") is zero for all n. This can be accomplished the ghost point (l, - 1 ) or ,,t") is to be chosen so that ~t,o Fl, - 1

through the discretized form of the y-momentum equation (the third equation of (1) which is used to calculate the value of v at the next time level). Upon incorporating the backward difference stencils of

a wall point (l, 0) is

3

+At ~ b K ~"-j) j z ,o, (6) j=O

Figure 2, the y-momentum equation at

v(n + 1 ) ,,(n) 1,0 ~ ~l, 0

U0 3 1 5 K ( n - j ) ~ ., ,,(n-j) - - ttit~l+i,O 2 ~51 r~(n-J)

- - - ~'i ~ t , i , ( 7 ) t,0 A x i = - 3 poAy i = - 1

where a f l are the optimized coefficients of the backward difference stencil. Their numerical values are given in the Appendix. Equations (6) and (7) are to be used to find ,,~,+1) after all the physical L,/, 0 quantities except pln,_l, the ghost value, are found at the end of the nth time level. To enure the wall boundary condition v ~"+1) = 0 is satisfied, the ghost value el,-1 i,o ~,o "(") may be found by setting v ~"+1) = 0 in (6) and (7). On solving this equation for pl")_l, it is easy to find

1 s pC., _ Z 1,--1 51 " i lYl, i" (8)

o-1 i=0

Equation (8) is tantamount to setting the ghost value such that @/Oy = 0 at the wall. t3p/t?y = 0 at the wall is an implied result of the governing partial differential equations. It is worthwhile to point out that if no ghost value is introduced and the boundary condition vl,% is imposed, (6) or its equivalent will, in general, not be satisfied. This means that t3p/Oy of the solution so computed would not necessarily be equal to zero at the wall.

Now let us turn our attention to a viscous compressible fluid. In this case the viscous stress terms of (1) and (2) are to be retained. The stresses are physical quantities and can be computed from the velocity field by the same seven-point difference stencil at an interior point or by the backward difference stencils of Figure 2 at a boundary point. For example, at an interior point (l, m) the shear stress zxy at the nth time level is given by

I~- ~(n) ,,(n) ~,~xyIl,m ~ /2 j j=-3 ~yy ~,m+j + Axx ~l+j, mJ, (9)

where # is shear viscosity coefficient. (Note: The stress terms are calculated and stored at each time level as a part of the solution. The derivatives of the stresses are computed by the first derivative stencil applied directly on the stresses. No second derivatives need be computed directly.) At the wall, the no-slip boundary condition u - - v - - 0 must be enfored. Two ghost values are, therefore, needed per each boundary point at the wall. On following the inviscid case, the ghost value pl,"kl will again be used to ensure that ~,o ''~"} = 0 for all n. Physically, in addition to applying a pressure on the fluid normal to the wall, the wall also exerts a shear stress %y on the fluid to reduce the velocity component u to zero at its surface. This suggests that a ghost value (%y)I"~_1 be included in the computation. To ensure that '-~,o"(") -- 0, it is noted that u is determined by advancing the x-momentum equation in time. By following the treatment of v -- 0, it is recommended that the y-derivative of Zxy in the x-momentum equation be approximated by the same backward finite-difference stencils as those for p (see Figure 2). It is easy to see that the ghost value (z~r)l"~_l can be found (in exactly the same way by which the ghost vaue pl"~l is determined) by imposing the no-slip boundary condition ut,+l) __ 0 on the x-momentum equation at the boundary point (l, 0) at the wall. In this way a unique 1,0 solution satisfying both the discretized form of the Navier-Stokes equations and the no-slip boundary conditions can be calculated.

3. Ref lect ion o f Acoust ic W a v e s f r o m a Solid Wal l

In this section an analysis of the reflection of acoustic waves from a wall using the solid-wall boundary conditions proposed in Section 2 is performed. Such an analysis provides a basic under-


standing of the difference between the reflection of acoustic waves as calculated by a high-order finite-difference scheme and by the original partial differential equatons. The analysis clearly shows that, in addition to the reflected acoustic wave, spurious waves are also reflected off the wall. Furthermore, a numerical boundary layer formed by spatially damped numerical waves of the computation scheme would be created at the wall surface. The results of the analysis offer a way to evaluate quantitatively the quality of the proposed numerical solid-wall boundary conditions.

3.1. Plane Waves on the Grid

Consider the propagation of inviscid plane acoustic waves in a static environment. In this case the density variable is uncoupled to the velocity and pressure fluctuations so that the first element of the vector finite-difference equations (2) and (4) may be ignored. In the absence of mean flow (Uo = 0) and acoustic source terms, these equations admit plane wave solutions of the form

~ r(,) e i(~tA~ + ~mAr - o~.~t), (1 O) ,~Fl, m ~

where (~, 0, ~0) are constants. Substituting (10) into (2) and (4), the following algebraic relations are readily found:

~ ^ c~p

•a = - - , Po

-~ tip (11) COY ~--- - - ,

Po

~p = (81 + ~)~'Po-

In (11) 8, fl, and ~ are the wave numbers and angular frequency of the finite-difference scheme (see Tam and Webb, 1993). They are related to the actual wave numbers (a, fl) and angular frequency ~ of the plane waves by

- i ~. .. = -~x aje" 'A~' (12)

j = - - 3

- i ~ a f i j~Ay ' (13)

_ i(e - u ° a t - 1) (14) f . 0 - - 3

At E bJ eij~At j = O

Equation (11) is a system of homogeneous algebraic equations. For nontrivial solutions, the determi- nant of the coefficient matrix must be equal to zero. This gives the dispersion relation

~2(co) = [82(~) + ~2(fl)]ao2" (15)

Dispersion relation (15) governs the propagation characteristics of all the plane waves supported by the numerical scheme, i.e., finite-difference equations (2) and (4). The corresponding eigenvector is

In (16) A is the amplitude of the plane waves. It is assumed that the incident acoustic wave frequency is low enough so that there is a spatial

resolution in the computed solution of not less than six mesh points per wave length. For the DRP scheme, these are long waves for which the stencil coefficients as. and bj have been chosen so that

"-~ ~, fl ~ fl, and N - co. Thus, for the incident acoustic wave, the dispersion relation (15) of the finite-difference equations is nearly the same as the exact acoustic wave dispersion relation o)2=


X Figure 3. Plane acoustic waves incident on a wall making an angle of incidence 0 with the normal

(~2 + f12)a2" This implies that the plane acoustic wave solutions of finite-difference equations (2) and (4) are for all intents and purposes identical to the exact solutions of (1).

Consider plane acoustic waves of angular frequency co impinging on a wall making an angle of incidence 0 with respect to the normal (the y-axis) as shown in Figure 3. The wave numbers are given by

= k sin 0, (17)

# = - k cos 0,

where k, the total wave number, is obtained from the dispersion relation (15), i.e.,

~z(co) = [~2(k sin 0) + f lZ(-k cos O)]a~. (18)

The wavelength 2 is related to the total wave number k by

2re 2 - k" (19)

By substitution of (17) into (16) and (10), the incident plane acoustic wave solution on the grid may be written as

"-q,ml Ttn)= __ (~2/a2 _ ~2)1/2/(p0 ~) ei(k sin OlA~-k cos 0,,ar-o.ao (20)

1

Here ~ = ~(k sin 0) and ~ = ~(co). For convenience, the pressure amplitude of the incident wave has been taken to be 1 as no confusion should arise.

3.2. Parasite Waves and the Numerical Boundary Layer

Upon impinging on the wall, the incident acoustic wave represented by (20) will be reflected. However, as will be shown, the reflected waves consist of not only acoustic waves but other intrinsic numerical waves of the finite-difference equations as well. To satisfy the wall boundary conditions, it is clear that the reflected waves must have the same x and t (or l and n) dependence as (20). That is, for the reflected waves,

= k sin 0, angular frequency = co,

so that ~ and ~ are the same as the incident wave. The wave numbers in the y-direction, fl, of these waves are to be determined by the dispersion relation (15) through the relationship fl(fl) of (13). Solving (15) for fl(fl), we find two families of roots,

[~2(co) ~2(k sin 0)1 u2

~(/b = - L ~ ~(k sin O)

and

(21)

(22)


~ . 0 -

0.5-

J I i 0.C -2.4 -1.6 -0.8

-0.5-

-1.0-

-1.5-

(~6)_2. 0 _

Im lax)

2 . 0 -

1 5 -

I I

0.8 1.6 ,D Re(15Ax)

Figure 4. Trajectories of the six roots of (21) in the complex fl-plane as o9 is pushed toward the real co-axis in the complex o>plane.

By substituting (13) into (21) and (22), it is easy to show that for a given co and 0 there are six independent roots for each family. Some of these roots represent waves propagating in the positive y-direction. The other roots represent waves propagating in the negative y-direction. To identify the direction of propagation of each root of the dispersion relation, the Briggs criterion may be used (see Briggs, 1964; Tam and Hu, 1989). According to Briggs, the angular frequency variable co in (21) and (22) is to be regarded as the Laplace transform variable of time t and fl as the Fourier transform variable of y. To begin the identification of the direction of propagation, the value of co in (21) and (22) is first regarded to have a large positive imaginary part (i.e., lying in the upper half co-plane) as is well known in the process of evaluating inverse Laplace transforms. Corresponding to this value of co, the roots of (21) and (22) are found in the complex fl-plane. The imaginary part of co is then slowly decreased until it is zero while real co is set equal to the angular frequency of the incident acoustic wave. As the imaginary part of co decreases, the roots of (21) and (22) move about in the fl-plane. Figure 4 shows the trajectories of the six roots of (21) in the complex fl-plane as co moves toward the real axis. For co real, the first and the fourth roots are real. All the other roots are complex. The first root (fl = fll - k cos 0) gives rise to the reflected acoustic wave. The fourth root f14, having a larger wave number, produces a parasite wave of very short wavelength. Roots 2, 3, 5, and 6 give spatially damped waves. It is important to point out that all except the first root are spurious solutions. They are numerical wave solutions of the finite-difference scheme but not the original partial differential equations. These numerical waves may be regarded as contaminants of the solution.

From the trajectory map of Figure 4, it is clear following Briggs' criterion that the three roots ill, f12, and f13 which originate from the upper half fl-plane give rise to waves propagating in the positive y-direction. The other three roots f14, fls, and f16 which originate from the lower half fl-plane give rise to waves propagating in the negative y-direction. Thus the reflected waves from (21) are

~l(po~) "~,,m'T("~ = (NZ/a~_~)l/Zl(poN)J(c~e',,,.a,+c,e'a2ma,+c3e'~3,.a,)e,kS'.o'ax-o,.ao, (23)

where Cl, c2, and Ca are unknown amplitudes to be determined by the wall boundary conditions. The central difference coefficients aj of the DRP scheme have the property a_~ = - aj (j = 0, 1, 2, 3).

It follows from (13) that f l ( - f l ) = -fl(fl). (24)

By means of this relation, it is easy to show that the roots of (22) are - f l j (j = 1 to 6), the negative of those of (21). The three waves propagating in the positive y-direction are, therefore, given by the roots -f14, -f15, and -f16. On combining these solutions with those of (23), the entire reflected wave


system is

U(") = f ( c l l,m

F ~/(po~) 1

d- (c4e -i#4may -F c s e -i#smAy + c6 e-i#6mAy) -(~2/a~ -- ~2)1/2/ (p0~ ) e i(ksinOIAx-c°nnt) (25)

1

Equation (25) is valid for all mesh points in the interior region with m > 3. The total solution is given by the direct sum of the incident wave, (20), and the reflected waves, (25).

Now, finite-difference equations (2) and (4) are not used for points in the wall boundary region. For points with m = 0, 1, 2, the y-momentum and the energy equations are discretized by the backward difference stencils proposed in Section 2. Clearly, then, the general reflected wave solution of (25) is not applicable here. The discretized x-momentum equation, however, remains unchanged so that the relationship between u and p on the grid found earlier (first equation of (11)) holds throughout the entire computation domain, that is,

1.1(71) m,,,"(") = p~eZ, m" (26)

The l and n dependence of the unknown variables in the boundary region must be the same as those of the incident and reflected waves. These variables may be written out as

/.(.,Fv(") 9 ' ' " / = [~"~ei(ksinO'Ax-°'"AO' (27)

LYl, mJ LP~J

where m = 0, 1, 2 and (t~,,,/~m) are six as yet unknown constants. In order to be able to satisfy the difference equations, the ghost value P[")-I must also have the same dependence on 1 and n, i.e.,

pl,) l = p_l e,k si. OtAx-o,,A,). (28)

Including the ghost value P-l, there are altogether 13 unknowns. Six are associated with the amplitudes of the reflected waves; cj (j = 1, 2 , . . . , 6) of (25). The other six are the pressure and velocity amplitudes,/~m and 13 m (m = 0, l, 2), in the boundary region. To find these 13 unknowns, 13 equations are required. Six of these equations are obtained from the diScretized y-momentum equation at m = 0, 1, . . . , 5. Another six equations are obtained from the discretized energy equation at m = 0, 1, 2 . . . . ,5. The 13th equation is given by the boundary condition z3 o = 0. To save journal space, these equations are not written out explicitly here. By means of these equations, it can be shown that the unknowns form a system of linear algebraic equations. These equations can easily be solved numerically. The numerical results provide a basis for evaluating the quality of the proposed solid-wall boundary conditions.

It is to be pointed out that c x of (25) is the amplitude of the reflected acoustic wave. c4 is the amplitude of the reflected parasite wave. c 2, c 3, c5, and c 6 are the amplitudes of the spatially damped numerical waves. These waves decay away from the wall and thus effectively form a spurious numerical boundary layer adjacent to the wall surface. Also Po is the pressure amplitude at the wall. To estimate the thickness of the numerical boundary layer, we start with the spatial distribution of the amplitude of the j th spatially damped wave:

Ipjl = Icjl e -lm(~J)y, (29)

where Im(flj) is the imaginary part of fir" The amplitude of the reflected acoustic wave is nearly equal to unity (the same as the amplitude of the incident wave which has been set equal to 1). Suppose the numerical boundary layer thickness, 6z, is defined to be the distance between the wall and where the spurious numerical wave solution drops to z times the magnitude of the reflected acoustic wave amplitude, e.g., 60.005 denotes the boundary-layer thickness based on 0.5~ of the reflected wave amplitude. From (29) we find

Z = ICj[ e -Irn(#-/)oz. (30)

z = 0 . 1 ~

. . . . = . 5 ~ *

to ~5

C.K.W. Tam and Zhong Dong

i r i I i I , l I ~ I ~ I t

0.0 10.0 2 0 . 0 5 0 . 0 4 0 . 0 5 0 . 0 6 0 . 0 7 0 . 0 8 0 . 0

e ( d e g )

(a)

~wJ

,5

_ _ z = 0 . 1 ~ .......... z = 0 . 5 7 ,

(b)

i I i I i I i

9 0 . 0 0 . 0 I 0 . 0 2 0 . 0 JO.O 4 0 . 0 5 0 . 0 6 0 . 0 7 0 . 0 8 0 . 0 9 0 . 0

Figure 5. Thickness of the numerical boundary layer as a function of angle of incidence 0. (a) 2/Ax = 6, (b) 2/Ax = 10.

Therefore,

5= = max - - - I n . (31) j Im(/ j)

Figure 5(a) shows the calculated numerical boundary-layer thickness as a function of the angle of incidence if a spatial resolution of Z/Ax = 6 (Ay = Ax) is used for computing the acoustic waves. 2 is the wavelength. Two thicknesses are provided, one corresponds to z = 0.005 (0.5%), the other to z = 0.001 (0.1%). This figure indicates that the numerica boundary layer is thickest for normal incidence. In this case 6o.ool is almost equal to 2{Ax. Figure 5(b) shows the corresponding numerical boundary-layer thickness if a spatial resolution of 2/Ax = 10 is used in the computation. It is clear that with better spatial resolution, the numerical boundary-layer thickness decreases.

312

N

d

X

_ _ X / & x = 6 . . . . . X,/k~x = 8

.......... ~,/&x = 10 c~

- ~ . c 5

0 .0 10.0 2 0 . 0 3 0 . 0 40 .0 5 0 . 0 00 .0 70.0 8 0 . 0 9 0 . 0

Figure & M a g n i t u d e of the reflected parasite wave as a function of the angle of incidence.

7 "E

x / t , x = 6 . . . . . X / A x = 8

......... ;~/AX = 10

........................ 0 .0 10.0 2 0 . 0 3 0 . 0 4 0 . 0 5 0 . 0 6 0 . 0 zo.o 8 0 . 0 9 0 . 0

8 ( d e g )

Figure 7. Magnitude of the reflected acoustic wave as a function of the angle of incidence.


Figure 8. Wall pressure as a function of the angle of incidence.

o ~5

c ~

d "G

d

d

. . . . . X / 6 x = 8

" • " I i I ~ I i I i I M I i

o.o 1o.o 2o.o 30.0 40.0 50.0 60.0 70.0 eo.o 90.0

e (d~g)

Figure 6 shows the dependence of the magnitude of the reflected parasite wave (grid-to-grid oscillations), IPincident/Pp . . . . itel or Ic4i, on the angle of incidence for different spatial resolution. At 2/Ax = 6, the reflected parasite wave can be as large as over 1% of the incident wave amplitude at normal incidence. At glazing incidence, i.e., beyond 0 = 50 °, there is generaly very little parasite wave reflected off the wall. The magnitude of the reflected parasite wave would be greatly reduced if the spatial resolution in the computation was increased. By using 10 or more mesh points per acoustic wave length in the computation, the reflected parasite wave is so weak that it may be ignored entirely.

Figure 7 is a plot of the deviation from perfect acoustic reflection. It is a measure of the quality of the numerical solid-wall boundary condition proposed in Section 2. For perfect reflection, the ratio of the reflected wave amplitude to that of the incident wave, IPreflected/Pineid~,tl, is unity. This is given by Ic~l of (25). Figure 7 shows that by using the proposed numerical solid-wall boundary condition the deviation from perfect reflection is small. It is of the order of 1% even when only six mesh points per acoustic wavelength are used in the computation. When acoustic waves impinge on a wall, the pressure at the wall is twice the incident sound pressure. This phenomenon is usually referred to as pressure doubling. Figure 8 shows that the computed wall pressure, I~ol, would, indeed, be close to twice the incident sound pressure level for 2lAx = 6. Again when a better spatial resolution is used in the computation, e.g., 2lAx = 8, the computed results would reproduce the pressure doubling phenomenon better.

4. Numerical Simulations

To assess the effectiveness of the solid-wall boundary conditions proposed in Section 2, a number of direct numerical simulations have been carried out. Comparisons between numerical and analytical solutions are made. This permits an evaluation of the fidelity of both the proposed inviscid as well as viscous boundary conditions. In all the examples discussed below, dimensionless variables are used. The characteristic scales are:

length scale = Ax = Ay (mesh size),

velocity scale --- ao (speed of sound),

Ax time scale = - - ,

ao

density scale = Po (ambient density),

pressure scale = poa~.

314 C.K.W. T a m and Z h o n g D o n g

4.1. Reflection of a Transient Acoustic Pulse by a Wall

Consider the reflection of a two-dimensional acoustic pulse by a plane wall as shown in Figure 9. The fluid is inviscid and is at rest at time t = 0. An acoustic pulse is generated by an initial pressure disturbance with a Gaussian spatial distribution centered at (0, 20). The wall is located at y = 0. The initial conditions are:

In 2[x 2 + ( y - 20)21t p -- 0.01 exp 9 ' ( 3 2 )

p = p, u = v = 0. (33)

In the numerical simulation the time step At is set equal to 0.07677. This value of At not only satisfies the numerical stability requirement but also ensures that the amount of numerical damping is insignificant.

Figure 9 shows the calculated pressure contour patterns associated with the acoustic pulse at 100, 300, and 500 time steps. The corresponding contours of the exact solution are also plotted in this figure. To the accuracy given by the thickness of the contour lines, the two sets of contours are almost indistinguishable. At 100 time steps, the pulse has not reached the wall so the pressure contours are circular. At 300 time steps, the front part of the pulse reaches the wall. It is immediately reflected back. At 500 time steps, the entire pulse has effectively been reflected off the wall creating a double pulse pattern; one from the original source and the other from the image source below the wall.

Figure 10 shows the computed pressure wave forms along the line x = y. The distance measured along this line from the origin is denoted by s. The computed wave forms at 400, 700, and 1000 time steps are shown together with the exact solution. As can be seen, there is excellent agreement between the exact and computed results. At 400 time steps, the pulse has just been reflected off the wall. At 700

75.

50,

25.

0 .0 75.

50.

25.

0.0 75.

50 .

25.

0.0

- 7 5 . 0

f / / ~ l = I 0 0 . 0 0

@ t i i t

t / / & t = 3 0 D . O 0

t / A t = 5 0 0 . 0 0

, /

i i i i [ , i . . . .

- 5 0 . 0 - 2 5 . 0 0.0 25 .0 5 0 . 0 75.0

X

Figure 9. Pressure contour patterns associated with the reflec-

tion of an acoustic pulse by a solid wall at y = 0. - - , p = 5 x 10-'~; - - - , p = 10-'~; . . . . . . , exact solution.

12.

5 .0

P.o.o

- 6 . 0

- 12. 12.

6 .0

P.o.o ,10 q

- & O

12. 12,

t /&t = 4 0 0 . 0 0

t/tat = 700.00

6 0

,~4 0.0

-&O

t / ta t= 7 0 0 0 . 0

- 7 2 . i i i , ] . . . . i . . . . I . . . . J . . . . I . . . . I , I J I

o o ,oo 2 0 0 Joo ,~oo 5 0 0 ~o.o 700 s/~/2Ax

Figure 10. Pressure waveforms of an acoustic pulse reflected off a solid wall a long the line x = y. , numerical solution;

. . . . . . , exact solution.


Figure 11. Pressure contour patterns associated with the reflection of an acoustic pulse by a solid wall in the presence of a Mach 0.5 uniform mean flow. - - , p = 5 x 10-4; - - - , p = 10-4; ...... , exact solution.

75.

50.

25.

0 .0 75.

50.

25.

0 .0 75.

50.

25.

0 .0

- 4 0 . 0 - 2 0 . 0

At = 200.00 ]

j L i

t t

I t / A t = 500.00 ]

i t i i

t c t

t t

' " " " . . . . . . . . . . . ~ ' ; ' = ~oo.oo I

, - - . . . . I , , , , , , , I . . . . . h h , ,

0.0 20 .0 40 .0 60 .0 8 0 . 0

X

time steps, the double pulse characteristic wave form is fully formed. Both pulses propagate away from the wall with essentially the same wave form. The amplitude, however, decreases at a rate inversely proportional to the square root of the distance.

4.2. Effect of Mean Flow

To test the efficacy of the inviscid solid-wall boundary condition in the presence of a mean flow, we consider the above acoustic pulse problem again but in a uniform stream of Mach number 0.5 flowing parallel to the wall. In this case the reflection process is modified by the effect of mean flow convection. Figure 11 shows the computed pressure contour patterns of the acoustic pulse at 200, 500, and 800 time steps. Again, the computed pressure contours are indistinguishable from those of the exact solution. Geometrically, the pressure contour patterns are identical to those without mean flow (see Figure 9). The uniform mean flow translates the entire pulse downstream (M = 0.5). The good agreement between the computed and the exact solutions suggests that the proposed numerical boundary conditions of Section 2 can be relied upon to simulate correctly a solid boundary even in the presence of a flow.

4.3. Reflection of a Time Periodic Acoustic Wave Train by a Solid Wall

We now consider the reflection of a periodic acoustic wave train by a solid wall in the absence of a mean flow. Here the acoustic wave train is assumed to be generated by a time-periodic source in the energy equation. Physically, the source represents a heat source or sink. U p o n writing out in full, the energy equation including the source term is

~Pt Ou ¢9v { l n 2 [ x 2 + ( y - - 2 0 ) 2 ] } + fix + ~yy = 0.01 exp 9 cos ~t. (34)

75.

Y 50.

25.

0.0 10C

75.~

25.

316

10~

C.K.W. Tam and Zhong Dong

Y 50.

0.0

-75.0 -50.0 -25.0 0.0 25.0 50.0 75.0 -75.0 -50.0 -25.0 0.0 25.0

X X

(a) (b)

Figure 12. Pressure c o n t o u r pa t te rns ad jacent to a solid wall genera ted by a t ime-per iodic acoust ic source. - -

- - - , p = - 1 0 - 3 ; . . . . . . , exact solution.*'

50.0 75.0

, P = 10-3;

The center of the source is located at (0, 20) and the angular frequency is co. The simulation is carried out in time with zero initial condition. After the transient solution has propagated out of the computational domain, the pressure fluctuation is time periodic with angular frequency co. The spatial pressure distribution has the form of an interference pattern created by the reflected wave train and the wave train generated directly by the source.

Figure 12 shows the computed pressure contour patterns adjacent to the solid wall during one cycle of oscillation. The period of oscillation is equal to 120 time steps. The corresponding acoustic wavelength is approximately equal to nine mesh spacings. Each of the four figures gives the pressure pattern at each quarter-period interval. The pressure contour patterns of the exact solution are also plotted in these figures. Again they are indistinguishable from the computed solution. Figure 13 gives the corresponding pressure waveforms along the line x = 0. The exact waveforms are shown in dotted lines. Clearly, there is excellent agreement between the computed and the exact solution providing further support to the reliability of the proposed numerical solid-wall boundary condition.

4.4. Oscillating Viscous Boundary Layer Adjacent to a Solid Wall

To demonstrate the effectiveness of the proposed numerical viscous boundary conditions for a solid wall, the case of an oscillating viscous boundary layer is simulated. The oscillating boundary layer is generated by the same time-periodic acoustic source described above except at a much lower frequency. All the viscous stress terms of (2) are included in the computation. To ensure that there are at least seven to eight mesh points in the boundary layer (so that it can be resolved computationally), the mesh Reynolds number R = poaoAx/# is taken to be 5.0 and the oscillation period set equal to 1250 time steps (At = 0.07677). An exact solution of this problem is not available. However, far from the source, the boundary layer resembles the well-known Stokes oscillatory boundary layer. In Stokes' boundary layer the oscillatory velocity component u is given by the following formula:

u---e cos(cot + 6 ) - e x p l - / z - - / y / c o s / / ~ - J y - c o t - 6 . (35)


2#.

12.

. ~ j 0 .0

-12,

-24, 24.

12.

P,O.O °1o °

-12.

-24. 24.

f2.

P 0.0 ~fO J

- - 12.

--24. 24.

12.

P.O.O , io "~

-12.

-24.

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0

y / A y

6.0

3.0

,70 3 0 . 0

- 3 . 0

- 6 . 0 6.0

5 .0

u 0.0 *fO 3

- ,3 .0

- 6 . 0 6.0

3.0

,1050.0 - 5 . 0

- 6 . 0 6.0

,3.0

, i O 3 0 " 0

- ,3 .0

- 6 . 0

0 .0 1.0 I I I

2.0 3.0 I [ [ I I I

4 0 5.0 6 0 7 0 6 o 9 0 l o o

y/ay

Figure 13. Pressure waveforms generated by a t ime-periodic I Figure 14. Velocity profiles of an osci l latory viscous bound- acoust ic source adjacent to a solid wall. , numerical a r y layer at each quarter cycle. , numerical solution; solution; . . . . . . , exact solution. • . . . . . , Stokes' solution.

In (35) the amplitude and phase factors e and 6 are constants of the Stokes solution. For the purpose of comparison with the numerical solution, these two constants are determined by fitting solution (35) to the numerical solution at the point of maximum velocity fluctuation at two instants of time.

Figure 14 shows the computed velocity profile of the oscillating boundary layer along the line x = 60. In this figure, T is the period of oscillation. Also shown are the velocity profiles of (35) at every quarter period. The agreement between the results of the numerical simulation and the approximate analytical solution appears to be good. On considering that only seven to eight mesh points are used to resolve the entire viscous boundary layer, the performance of the numerical viscous boundary conditions and the DRP scheme must be regarded as remarkable.

4.5. Diffraction of Acoustic Waves Around a Thin Flat Plate

When a train of acoustic waves impinges on a thin flat plate of finite length as shown in Figure 15, the waves are scattered and diffracted. In the shadow region behind the plate, acoustic waves radiated directly from the source are blocked. The measured sound pressure, therefore, consists only of the contributions from waves diffracted by the two sharp ends of the plate. Except in the direction of symmetry, the ray paths or distances to the two ends of the plate are usually different. This results in a phase difference between the diffracted waves from the two ends. When the phase difference is zero, the two waves reinforce each other. When the phases differ by 180 ° , the waves tend to cancel each other. Thus a diffraction pattern of maxima and minima is formed. The complexity of this problem offers an excellent opportunity to test the accuracy of the proposed solid-wall boundary condition.

Let L be the width of the plate. To keep the computation boundary far from the plate, a computational domain of 8L by 8L is used in the numerical simulation. Acoustic waves are generated by a simple harmonic heat source located at a distance of L from the midpoint of the plate. The time-periodic heat source is incorporated as a nonhomogeneous term of the energy equation. The


4 L

A

iC

0 V S o u r c e

\ O u t p u t

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . " ~

18L

t Figure 15. Schematic diagram of the scattering and diffraction of acoustic waves from a localized source,by a thin fiat plate. Also shown is the computation domain.

source term has the form

~p ~u c~v { 2)[x2 + (y + L)2]}coscgt" ~-7 + ~xx + Yyy = 0.01 exp - ( In (L/8)2 (36)

Here, the dimensionless angular frequency ~o is related to the acoustic wavelength, 2, by co = 2rc/2. To minimize possible reflection of outgoing acoustic waves at the boundary of the computaton domain back into the computational domain itself, radiation boundary conditions given in Tam and Webb (1993) are employed. Numerical experiments indicate that low-level spurious waves (grid-to-grid oscillations) could be generated at the sharp edges of the plate. To avoid contaminating the computed solution, they are eliminated by adding a small amount of artificial selective damping in the discretized equations following the work of Tam et al. (1993). These artificial damping terms are so constructed that only the short waves are damped. The acoustic waves having longer wavelengths are essentially unaffected.

The plate is assumed to be infinitesimally thin lying on the grid line along the x-axis from x = - L / 2 to x = L/2. The pressure, density, and velocity of the gas on the two sides of the plate are different. Since a seven-point stencil is used to calculate the derivatives, it follows that an approximation is needed to compute the x-derivatives at the three grid points on the x-axis closest to the ends of the plate. For simplicity, the average values of the physical quantities on the two sides of the last three grid points on the plate closest to the ends are used to calculate these derivatives. In the physical problem the solution satisfies both the linearized Euler equations and the wall boundary condition right at the endpoint of the plate. This is, however, not possible for the discretized equations. In this work the tip of the plate is, therefore, considered to be midway between the last point where the solid-wall boundary condition is imposed and the next grid point where the linearized Euler equations are satisfied. In other words, the width of the plate for the exact solution is taken to be L + Ax.

Numerical simulations of the above-described acoustic-wave diffraction problem have been carried out for the case where the plate width is equal to four times the acoustic wavelength, i.e., L = 42. Computed results of two simulations, one with spatial resolution 2 = 6Ax (Ay = Ax) and the other with ;t = 10Ax are reported here. Figures 16(a)-(d) show the pressure distribution along the upper left half boundary of the computation domain behind the source. Figure 16(a) is at the beginning of a cycle of oscillation. Figures 16(b)-(d) are at a successive quarter-period time interval later. A resolution of six mesh points per wavelength is used in the computation. Also plotted on these figures is the exact solution. The exact solution is constructed by the method of separation of variables in an elliptic coordinate system centered at the plate. As can be seen, there is reasonably good agreement

Wall Boundary Condit ions for High-Order Finite-Difference Schemes in Computat iona l Aeroacoust ics 319

' c 3 c ~ ~ c b ~

c ~

. . . . i . . . . . . . . I . . . . I . . . . I , , , r I , ¸ ¸ ¸ ' .

0.0 0 .5 1.0 1.5 2 .0 2.5 3 .0 ,5.5

(at

? 4~0

l

. . . . I I . . . . I I . ~ i i i i I i ' •

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

(b)

4.0

¥,.,

c~

c3

c3 Q - d

c~

7

<b

. . . . i . . . . i . . . . i . . . . i . . . . ~ . . . . , . . . . i , ,

0.0 0 ,5 1.0 1.5 2 ,0 2,5 3 .0 3 .5

{c)

• .: ili t

. . . . . . i . . . . i . . . . q . . . . i . . . . i . . . . r . . . . , . . . .

4 .0 0.0 0 ,5 1.0 1.5 2 .0 2 .5 3 .0 3.5 4 .0

x/L (d)

Figure 16. Pressure distribution along the upper right half boundary of the computat ion domain. - - , numerical solution

using 2/Ax = 6; . . . . . . , exact solution. (a) Beginning of a cycle of oscillation: t - to = 0, (b) t - to = 7"/4, (c) t - to = T/2, (d) t - t o = 37"/4.

between the numerical and the exact results. Obviously at a coarse spatial resolution of 2/Ax = 6 the difference equations are unable to simulate accurately the wave diffraction phenomenon at the two sharp ends of the plate. Figures 17(a)-(d) show the results of an identical calculation with a spatial resolution of 2/Ax = 10. It is easy to see that the agreement between the numerical and exact solution is greatly improved. This is true both in terms of the phase and amplitude of the diffraction pattern. It is believed that the remaining discrepancies are due primarily to the approximation used in calculat- ing the x-derivatives at the three grid points closest to the sharp edge s of the plate and that the numerical solid-wall boundary condition of Section 2 has been effective in simulating the exact wall boundary condition.

320 C.K.W. T a m and Z h o n g D o n g

. . %

. . . . i . . . . i . . . . i . . . . i . . . i . . . . i . . . . i .

0 . 0 0 . 5 1.0 1.5 2 .0 2 .5 3 .0 3 .5

(a)

i " ' 1 ' i i i i I i i i i r l l l l l l l l l l . . . . ' . r . . . . ~ . . . .

4 .0 0 .0 0 .5 1.0 1.5 2 .0 2 .5 3 .0 3 .5 4 .0

~/L (b)

• . . /

. . . . i • . I i i i I i i ¸ • I . . . . i . . . . i . . . . r . . . .

0 .0 0 .5 t.O 1.5 2 .0 2 ,5 J.O .~.5 4 .0

~/L (c)

I C 3 c ~

c )

c b

c ~ _ c 5

c ~

?

c b

o

0 .0 0 .5 1.0 1,5 2 .0 2 .5 3 .0 3 .5 4 .0

~/c (d)

Figure 17. Pressure dis tr ibut ion a l o n g the upper right ha l f b o u n d a r y of the c o m p u t a t i o n domain . - - , numer ica l so lut ion us ing 2 / A x = 10; . . . . . . , exact so lut ion. (a) Beg inn ing of a cycle of osci l lation: t - t o = 0, (b) t - to = T/4 , (c) t - t o = T /2 ,

(d) t -- to = 3 T / 4 .

5. Summary

High-order finite-difference schemes support extranous wave solutions which bear no relationship to the exact solution of the original partial differential equations. These extraneous wave solutions inevitably pollute the quality of the numerical solution. They are invariably generated when solid-wall boundary conditions are imposed on the numerical solution. This results in the radiation of parasite waves from the wall and the formation of a numerical boundary layer adjacent to its surface.

Wail Boundary Conditions for High-Order Finite-Difference Schemes in Computational Aeroacoustics 321

The solution of a system of partial differential equations satisfies the equations at every interior and boundary point. In addition, the solution also satisfies the no-slip or no flux (if flow is inviscid) boundary condit ions at a boundary point on the wall. This, however, is not possible for the corresponding discretized system of finite-difference equations. In order to satisfy both the discretized governing equations and boundary condit ion at a boundary point on the wall, ghost values at ghost points must be introduced. The min imum number of ghost values per boundary point on the wall is equal to the number of physical boundary conditions. A set of numerical wall boundary conditions using the min imum number of ghost values is proposed. The effectiveness of these numerical boundary condit ions in simulating the presence of a solid wall adjacent to an inviscid or viscous fluid is tested by compar ing the results of direct numerical simulations with exact solutions. Very favorable agreement is found in all the test cases.

Addit ional case studies are planned and these will be the subject of an upcoming paper.

Appendix. Optimized Coefficients of Central and Backward Difference Stencils for the DRP Scheme

The coefficients for the four-level t ime-marching schemes are:

b o = 2.3025580884, b I = - 2.4910075998, b2 = 1.5743409332,

The coefficients for the seven-point central difference stencil are:

a t -- 0.799266427 = - a - l , a2 = -0 .189413142 = - a _ 2 , a 3 = 0.026519952 = - a _ 3 ,

The coefficients of the various seven-point backward difference stencils are:

a~ 2 -- - -0 .026369431,

a 4 2 = 1.273274737,

42 _ 0.049041958, a_ 2

a~l = 0.048230454,

a251 = - 1.388928322,

a5_~1 -- - 0.209337622,

a66 ° = -0 .203876371,

a 6° = 4.461567104,

a 6° = - 2.192280339,

a~ 2 = 0.166138533,

a~ 2 = -0 .474760914,

a] 1 = -0 .28181465,

a~ 1 = 2.147776050,

a~ ° = 1.128328861,

a~ ° = -5 .108851915,

b a = -0.3858914222.

a~ 2 = -0.518484526,

aS] = --0.468840357,

a~ 1 = 0.768949766,

a~ I = -- 1.084875676,

a~ ° = -2 .833498741,

a~ ° = 4.748611401,

a o = 0.0.

References

Anderson, D.A., Tannehill, J.C., and Pletcher, R.H. (1984). Computational Fluid Mechanics and Heat Transfer. McGraw-Hill, Washington.

Bayliss, A., and Turkel, E. (1980). Radiation Boundary Conditions for Wave-Like Equations. Comm. Pure Appl. Math., 33, 707-725.

Bayliss, A., and Turkel, E. (1982). Far Field Boundary Conditions for Compressible Flows. J. Comput. Phys., 48, 182-199. Briggs, R.J. (1964). Electron-Stream Interaction with Plasmas. MIT Press, Cambridge, MA. Engquist, B., and Majda, A. (1977), Absorbing Boundary Conditions for the Numerical Simulation of Waves. Math. Comp., 31,

629-651. Engquist, B., and Majda, A. (1979). Radiation Boundary Conditions for Acoustic and Elastic Wave Calculations. Comm. Pure

Appl. Math., 32, 313-357. Givoli, D. (1991). Non-Reflecting Boundary Conditions. J. Comput. Phys., 94, 1-29. Jiang, H., and Wong, Y.S. (1990). Absorbing Boundary Conditions for Second-Order Hyperbolic Equations. J. Comput. Phys.,

88, 205-231. Khan, M.M.S., Brown, W.H., and Ahuja, K.K. (1987). Computational Aeroacoustics as Applied to the Diffraction of Sound by

Cylindrical Bodies. AIAA J., 25, 949-955.


Oran, E.S., and Boris, J.P. (1987). Numerical Simulation of Reactive Flow. Elsevier, New York. Tam, C.K.W., and Hu, F.Q. (1989), Three Families of Instability Waves of High Speed Jets. J. Fluid Mech., 201, 447-483. Tam, C.K.W., and Webb, J.C. (1993). Dispersion-Relation-Preserving Schemes for Computational Acoustics. J. Comput. Phys.,

107, 262-281. Tam, C.K.W., Webb, J.C., and Dong, Z. (1993). A Study of the Short Wave Components in Computational Acoustics.

J. Comput. Acoustics, 1, 1-30. Thompson, K.W. (1990). Time-Dependent Boundary Conditions for Hyperbolic Systems, II. J. Comput. Phys., 89, 439-461.

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