[American Institute of Aeronautics and Astronautics 13th Computational Fluid Dynamics Conference -...

Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

A97-32509AIAA-97-2074

AN IMPROVED TREATMENT OF EXTERNAL BOUNDARY FORTHREE-DIMENSIONAL FLOW COMPUTATIONS*

Semyon V. Tsynkov* Veer N. Vatsa*

NASA Langley Research Center, Hampton, VA

Abstract

We present an innovative numerical approach forsetting highly accurate nonlocal boundary condi-tions at the external computational boundaries whencalculating three-dimensional compressible viscousflows over finite bodies. The approach is basedon application of the difference potentials methodby V. S. Ryaben'kii and extends our previous tech-nique developed for the two-dimensional case. Thenew boundary conditions methodology has been suc-cessfully combined with the NASA-developed codeTLHS3D and used for the analysis of wing-shaped con-figurations in subsonic and transonic flow regimes.As demonstrated by the computational experiments,the improved external boundary conditions allowone to greatly reduce the size of the computationaldomain while still maintaining high accuracy of thenumerical solution. Moreover, they may provide fora noticeable speedup of convergence of the multigriditerations.

1 Introduction

Preliminaries. External flows over finite bodiesor configurations of bodies represent a wide class ofimportant practical applications in fluid dynamics.To treat this type of problems numerically, one typ-ically truncates the original infinite domain. Theresulting truncated problem is obviously subdefiniteunless supplemented by the proper closing procedureat the external computational boundary. The lat-ter procedure is called the artificial boundary condi-tions (ABC's). In the ideal case, the ABC's wouldbe specified so that the solution on the truncateddomain coincides with the corresponding fragment

*This paper was prepared while the first author held aNational Research Council Resident Research Associateship.

'Correspondence author. Mail Stop 128, NASA LangleyResearch Center, Hampton, VA 23681-0001, U.S.A. Phone:(1-757)864-2150; Fax: (1-757)864-8816;E-mail: s. v. tsynkovSlarc. nasa. gov;URL: http: //f mad-WBH. larc. nasa. gov: 80/~t synkov/.

tMail Stop 128, NASA Langley Research Center, Hamp-ton, VA 23681-0001, U.S.A. Phone: (1-757)864-2236; Fax:(1-757)864-8816; E-mail: [email protected]. Mem-ber AIAA.

of the original infinite-domain solution. The issue ofABC's is significant in CFD and in other areas of sci-entific computing; theoretical estimates and compu-tational experiments by different authors show thatthe proper treatment of external boundaries has aprofound impact on the overall performance of nu-merical algorithms and interpretation of the results.

Different ABC's methodologies have been studiedextensively over the recent two decades. However,the construction of the ideal (i.e., exact) ABC's thatwould provide no error associated with the domaintruncation and at the same time be computationallyinexpensive, easy to implement, and geometricallyuniversal, still remains a fairly remote possibility.Among the variety of approaches proposed to dateonly a few can be regarded as the commonly usedtools in CFD. As a rule, these approaches are basedon the essential model simplifications (e.g., locallyone-dimensional treatment near the external bound-ary) and, therefore, often lack accuracy in compu-tations. This, in turn, necessitates choosing exces-sively large computational domains. On the otherhand, these simple methods usually provide for lo-cal ABC's, and, therefore, for cheap, geometricallyuniversal, and algorithmically simple numerical pro-cedures, which are attractive for practical use.

There are, of course, methods of another kind,which typically provide for highly accurate and ro-bust numerical algorithms. These methods, how-ever, are not used routinely because the correspond-ing ABC's in most cases appear global. As a conse-quence, these ABC's may be relatively cumbersomeand expensive; moreover, they can be derived easilyonly for the boundaries of regular shape.

The nonlocality is inherent for exact ABC's; onthe level of PDE's such boundary conditions typi-cally involve pseudodifferential operators. As con-cerns the approximate approaches, the basic trendis the following: higher accuracy for the boundaryprocedure requires more of the nonlocal nature ofthe ABC's to be somehow taken into account. Asurvey of methods for setting the ABC's in differentareas of scientific computing can be found in our re-

Copyright ©1997 by the American Institute of Aeronauticsand Astronautics, Inc. All rights reserved.

1139

American Institute of Aeronautics and Astronautics


cent work , as well as in the comprehensive reviewsby Givoli2 '3.

In this paper, we concentrate on constructing theABC's for the three-dimensional problems of compu-tational aerodynamics. We, however, mention thatthis area constitutes a fraction of the possible rangeof applications for the different ABC's techniques.Besides the hydro- and aerodynamic problems (ex-ternal flows, duct flows, boundary layers, free sur-faces, etc.), the entire range includes the flows inporous media, filtration, MHD flows, plasma (e.g.,solar wind), the problems of solid mechanics (in par-ticular, elasticity and aeroelasticity), and the prob-lems of wave propagation (electromagnetic, acoustic,seismic), just to name a few.Main Objective and Method of Approach.For external flow computations, our goal is to de-rive and implement the ABC's that would com-bine the advantages relevant to both global andlocal methodologies. Our approach to construct-ing the ABC's is based on usage of the finite-difference analogues to Calderon's generalized po-tentials and boundary projections (see the originalwork by Calderon and also work by Seeley ).Ryaben'kii (see Refs. 6, 7, 8) had modified the orig-inal Calderon's construction and proposed a numer-ical technique for the effective calculation of poten-tials and projections; this technique is known as thedifference potentials method (DPM).

In Ref. 9 we describe the foundations of theDPM-based approach to setting the ABC's for com-putation of two-dimensional external viscous flows(Navier-Stokes equations). In Ref. 10 we imple-ment this approach along with the multigrid Navier-Stokes algorithm by Swanson and Turkel n- 12' 13

and present some numerical results for subsonic andtransonic laminar flows over single-element airfoils.In Ref. 14 we show the results of subsequent numer-ical experiments and propose an approximate treat-ment of turbulence in the far field. Our work Ref. 15delineates the algorithm for solving one-dimensionalsystems of ordinary difference equations that arisewhen calculating difference potentials and the DPM-based ABC's. In Ref. 16 we extend the area of appli-cations for the DPM-based ABC's by analyzing two-dimensional flows that oscillate in time; we also pro-vide some solvability results for the linearized thin-layer equations used for constructing the ABC's. InRef. 17 we present a general survey of the DPM-based methodology as applied to solving externalproblems in CFD, including parallel implementationof the algorithm, combined implementation of non-local ABC's with multigrid, and entry-wise interpo-lation of the matrices of boundary operators with

respect to the Mach number and the angle of at-tack. Additionally, in Ref. 17 one can find some newtheoretical results on the computation of general-ized potentials, the construction of ABC's based onthe direct implementation of boundary projections(thin-layer equations), and some numerical resultsfor various airfoil flows: laminar and turbulent, tran-sonic and subsonic, including very low Mach num-bers (incompressible limit). Finally, in Refs. 18, 19we outline basic elements of the DPM-based ABC'sfor the case of three-dimensional steady-state exter-nal viscous flows; this case is undoubtedly the onemost demanded by the current computational prac-tice. Specifically, in Refs. 18, 19 we address theflows around wing-shaped configurations and showsome preliminary numerical results for the subsonicregime.

Below, we report on the subsequent develop-ment of the DPM-based approach for the three-dimensional external flows. We present the results ofcomputations for different flow regimes and comparethe performance of the DPM-based ABC's with theperformance of the standard boundary conditions.The assessment is conducted with respect to boththe quality of the resulting solution (accuracy as itdepends on the size of the computational domain)and the properties of the numerical procedure asinfluenced by the type of ABC's (efficiency, robust-ness, and convergence of the pseudo-time iterations).The results demonstrate the superiority of the DPM-based ABC's over the standard existing methods.

2 External Flow Problem

Formulation. We consider a steady-state flow ofthe viscous compressible perfect gas past a three-dimensional wing. The flow is uniform and sub-sonic at infinity; it is symmetric with respect to theCartesian plane z = 0, which, in particular, impliesthat the free-stream velocity vector is parallel to thisplane.

The flow equations are integrated on the grid gen-erated around the wing. This grid actually definesthe finite computational domain; the ABC's thatwould close the truncated problem should be set atthe external coordinate surface of the grid. Let usdesignate this surface F; for one-block curvilinearboundary-fitted grid around the ONERA M6 wingthe schematic geometric setup is shown in Figure 1.

The outermost coordinate surface of the grid isdesignated FI (see Figure 1); it represents the ghostnodes (or ghost cells for the finite-volume formula-tion). Clearly, when the stencil of the scheme usedinside the computational domain is applied to anynode from F, it generally requires some ghost cell

1140American Institute of Aeronautics and Astronautics


r,

Figure 1. Schematic geometric setup for the three-dimensional case. The wing on the left is enlarged.

data. Unless the required data are provided, thefinite-difference system solved inside the computa-tional domain appears subdefmite (i.e., it has lessequations than unknowns). From the viewpoint ofwhat the solution technique is, one can say thatwhen some iterative solver is employed to integratethe flow equations inside the computational domain,the values of the solution at the ghost cells shouldbe prescribed at each iteration in order to be able toadvance the next "time" step.

Therefore, in the practical framework the closureof the discretized truncated problem means the spec-ification of the solution values at the ghost cells.This will be done by means of the DPM-based ABC'sso that the boundary data used for the closure ad-mit an exterior complement that solves the problemoutside the computational domain (see below).

First, we assume that the flow perturbationsagainst the constant free-stream background aresmall in the far field and consider the accordinglysimplified problem outside the computational do-main (i.e., outside F). Clearly, for the small to mod-erate free-stream Mach numbers MO (purely sub-sonic flows) we can retain only the first-order termswith respect to perturbations in the governing equa-tions. Therefore, the aforementioned simplificationof the original problem will actually be the far-field linearization. However, in the transonic limit

the possibility of linearization in the far field re-quires, generally speaking, some additional analysis.For the model full-potential formulation, we presenthere a simple argument that the compressible three-dimensional far field remains linear for transonic flowregimes as well.

Consider the Karman-Guderley equation (see 20)

(1)_

dx2 dy2 dz2 K

for the perturbation <f> the full potential <3> ofthe compressible gas flow around a thin three-dimensional wing. Here

(2)

y = z =

S is the wing thickness (6 — > +0 along with M0 — >I in the transonic limit), K is the parameter of tran-sonic similarity (the true linear theory correspondsto big values of K), u0 is the flow velocity at infin-ity, K is the ratio of specific heats, and the positivex direction coincides with the free stream.

The common practice 20 of developing the asymp-totic expansions for solutions of the equations that

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involve transonic nonlinearities consists of substi-tuting the leading linear term(s) into the nonlinearparts of the equation (right-hand side of (1)) and ob-taining the corresponding corrections by solving theresulting non-homogeneous problem. In the specificcase under study, the linear far-field expansion startswith the horseshoe vortex

y

Substituting the expression (3a) into the right-handside of (1) and solving the resulting Poisson equa-tion (this involves the Fourier transform in sphericalfunctions) one obtains the nonlinear correction

(3b)— 3

that decays at infinity two orders of magnitude fasterthan the source term (3a). Analogously, for the gen-eral doublet term <j>2 ~ f~2 the corresponding non-linear correction can be shown to be <f>2NL ~ f ~ 5 ,which is the three orders of magnitude difference.We therefore conclude that the transonic nonlin-ear corrections can be neglected when analyzing thecompressible far field in three space dimensions.

Of course, having established the fact of the far-field linearity, we cannot say in advance whether ornot the linearization outside F is possible for everyspecific configuration of the domains. Clearly, forthe very large computational domain one can lin-earize the flow outside F, and as we approach thewing (the source of perturbations), the validity oflinearization can be verified a posteriori (see, e.g.,our previous work ' ' ).

The linearized dimensionless thin-layer equationscan be written as

-TT- = 0

= 0

dp du dv dwdx dx dy dz

du dp I fd2u d2u\dx dx Re \dy2 dz2 J

dv , dp 1 (4d2v

dpdx

<9Vdz2 ,

dx

dwdx

IM2

) - -/ /

I r\ 7-) i f\ £\ Ody Re \6 ay*d2v I d2w \

' dz2 ' 3dydz)dp I (4d2w

' dz Re V3 dz2

d2w 1 d2v \1 dy2 SdydzJ

dp K \ f d 2 pdx RePr [\dy2

1 / Q2 a2 \ 11 / op op \^MQ \dy2 dz2 y j

= 0

= 0

(4a)

where p, u, v, w, andp are the perturbations of den-sity, Cartesian velocity components, and pressure,respectively, Re is the Reynolds number, and Pr isthe Prandtl number. System (4a) is supplementedby the homogeneous boundary condition at infinity:

u = (p,u,v,w,p) -

as (x2 + y2 + z2)

(0,0,0,0,0)

+ 00,

(4b)

which corresponds to the free stream limit of thesolution. Returning to the question of closing thediscretized truncated system, we clarify that theboundary data provided by the DPM-based ABC'swill admit an exterior complement that would solvethe discrete counterpart of (4a) and meet boundarycondition (4b) in a certain asymptotic sense.

DPM-based ABC's — Main Idea. We dis-cretize system (4a) on the auxiliary Cartesian gridwith the second order of accuracy; we use first-orderdifferences in x and second-order differences in yand z (see Refs. 16, 17 for the details in two di-mensions). The DPM will provide us with the com-plete boundary classification (in terms of the appro-priate traces) of all those and only those exteriorgrid vector-functions that solve the discrete coun-terpart of (4a) outside the computational domainand satisfy boundary condition (4b) in some approx-imate sense. The foregoing boundary classificationwill be obtained as an image of a special projectionoperator, which can be considered as a discrete ana-logue of Calderon's pseudodifferential boundary pro-jection ' . As we solve the Navier-Stokes equationsinside the computational domain iteratively, everytime we need to update the ghost cells we take acertain sufficient set of data from inside F (see be-low), project it onto the right manifold, i.e., onto thesubspace in the entire space of boundary data thatadmit the correct exterior complement, and obtainthe ghost cell values by calculating the trace of thiscomplement on FI.

DPM-based ABC's — Specific Implementa-tion. Let us split the nodes of the auxiliary Carte-sian grid into two distinct groups: those that areinside F and those that are outside. Applying thestencil of the scheme for (4a) to each node of bothgroups, we consider the intersection of the grid setsswept by the stencil. This intersection is called thegrid boundary 7; it is a multi-layered fringe of nodesof the auxiliary Cartesian grid located near F. Fig-ure 2 shows an example of the grid boundary 7 (sev-eral cross-sections in different directions).

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Figure 2. Grid sets for the three-dimensional case.

For any function u on the Cartesian grid we de-fine its trace Tr7u on 7 as merely a contraction.For any grid function u7 specified on 7 we introducethe generalized potential Pu7 with the density u7;the generalized potential is defined on the auxiliaryCartesian grid on 7 and outside it. The generalizedpotential is obtained as a solution of the special aux-iliary problem (AP); solution of the AP replaces andextends the operation of convolution with the funda-mental solution in the classical potential theory. TheAP is driven by the right-hand side that depends onu7, the formal construction of this right-hand sideis the same in two- and three-dimensional cases andwe refer the reader to our previous work ' ' fordetails. Boundary conditions for the AP should ap-proximate boundary condition (4b) and at the sametime ensure the finite-domain formulation for theAP. Therefore, we formulate the AP on a sufficientlylarge parallelepiped aligned with the Cartesian di-rections; the parallelepiped fully contains IV Wespecify the periodicity boundary conditions in the yand z directions and also assume symmetry at z — 0.After the Fourier transform with respect to y and zthe discrete counterpart to (4a) can be written as afamily of one-dimensional difference equations:

m,k + Bkum_!k = fm_1/2 ,k,

l,...,M, k=(ky,kz),0, . . . , Jy, Kz = 0, . . . , Jz,

(5)

where Ak and Bk are the 5 x 5 matrices and M + 1,1Jy + 1, and Jz + 1 are the numbers of grid nodesin the x, y, and z directions, respectively (symme-try is taken into account, as well as the fact that uand f are real-valued). Boundary conditions in thex (streamwise) direction are specified separately foreach pair of wavenumbers k:

(6a)

(6b)

(7a)

(7b)

Qk = Ak Bk, and p«(k) are the eigenvalues ofQk. The semi-analytic boundary conditions (6a)

S (k)u0ik = 0,

S+(k)uM,k = 0,

where

s-(k)= niS+(k)=

1143American Institute of Aeronautics and Astronautics


and (6b) (the eigenvalues for (7) are calculated nu-merically) explicitly prohibit growing modes of thesolution in the left and right directions, respectively.The periods in y and z should be chosen sufficientlylarge to ensure that the periodic solution considerednear F and FI is sufficiently close to the theoreticalnon-periodic solution; the latter can be thought ofas a limit when the periods approach infinity. Theapproximation of a non-periodic solution by the pe-riodic one on a finite fixed interval as the period(s)increase(s) is discussed in our work ' ' . InRef. 17, we also discuss the possibility to replace theFourier transforms by the non-unitary transforms.The latter appear when the grid in y or z is stretched(which provides for a drastic cost reduction) andthe corresponding eigenfunctions consequently forma skew basis.

The foregoing AP allows us to calculate the gener-alized difference potential P u7 for any grid densityu7 specified on 7. The composition of the operatorsTr7 and P, P7 = TrTP, is a projection, P7 = P7,and it is a discrete counterpart of Calderon's bound-ary projection for system (4a). The image of thisprojection, ImP7, contains all those and only thoseu7 's that are the traces of some exterior differencesolution to (4a) that satisfies the boundary condi-tions of the AP (periodicity in y and z and boundaryconditions (6) in x). The latter boundary conditions,in turn, approximate (4b).

Having constructed the procedure for calculatingthe potentials and projections for the discrete ver-sion of (4a), we can now close the system inside thecomputational domain, i.e., obtain the ABC's. First,we take u and du/dn on F, n is the normal, (thesedata are available from inside the computational do-main) and, using interpolation Rr along F and thefirst two terms of the Taylor expansion (denoted 7T7),obtain u-,:

u- = 7T7Rr (ll,du\dn) (8)

Then, we need to calculate the potential P v7 for thedensity v7 = P7u7 and interpolate it to the nodesFi:

u (9)

Finally, the ABC's are obtained in the operator form

u u, -T-an (10)

where T is composed of the operations (8) and (9).Boundary condition (10) is applied every time we

need to update the ghost cell values in the course ofthe iteration process. The implementation of ABC's(10) can either be direct or involve preliminary cal-culation of the matrix T. In the latter case, the run-time implementation of the ABC's (10) is reducedto a matrix-vector multiplication.

3 Numerical ExperimentsTwo-Dimensional Case. For the reason of com-pleteness, we first briefly comment here on the two-dimensional results from our previous work (seeRefs. 10, 14, 17). In that work, we calculated sub-sonic and transonic viscous flows past single-elementairfoils (NACA0012 and RAE2822).

The computational domain is formed by the C-type curvilinear grid generated around the airfoil.On this grid, the Navier-Stokes equations are in-tegrated using the code FLOMG by Swanson andTurkel n> 12' *3. The standard treatment of the ex-ternal boundary in the code FLOMG is based on the lo-cally one-dimensional characteristics analysis, whichmay or may not be supplemented by the point-vortex (p.-v.) correction .

Basic conclusions that could be drawn from ourtwo-dimensional numerical experience are the fol-lowing. The DPM-based ABC's are geometricallyuniversal, algorithmically simple and easy to imple-ment along with the existing solver. For the largecomputational domains (30-50 chords of the air-foil) , the performance of the standard methods andthe DPM-based ABC's is roughly the same. As,however, the artificial boundary approaches the air-foil the discrepancy between the corresponding so-lutions increases. The lift and drag coefficients ob-tained on the basis of boundary conditions (10) de-viate from their asymptotic (50 chords) values muchslighter (within fractions of one percent) than thecoefficients obtained using local ABC's do. In otherwords, the nonlocal DPM-based ABC's allow one touse much smaller computational domains (as smallas 2-3 chords) than the standard boundary condi-tions do and to still maintain high accuracy of com-putations. Moreover, if we compare three models:DPM-based, point-vortex, and standard local, thenit turns out that the DPM-based ABC's display thebest performance for small computational domains,the performance of the local characteristic boundaryconditions for the small domains is very poor, andthe point-vortex boundary conditions perform muchbetter for the lift than they do for the drag coef-ficient. This behavior seems reasonable since thepoint-vortex model is a lift-based treatment.

We also note that for the certain variants ofcomputation the DPM-based ABC's may noticeably

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Table 1. ONERA M6: M0 = 0.5; Re0 = 11.7 • 106; a = 3.06°.

Domain "radius"GridType of ABC'sFull lift, CLRelative errorFull drag, CD x 100Relative error

1.25 root chords 2 root chords 10 root chords197 x 49 x 33

standard0.22186.58%0.8173.8%

DPM0.20650.34%0.7910.38%

standard0.21855.0%0.7930.76%

DPM0.20650.34%0.7910.38%

standard0.2081

0%0.7870%

DPM0.2072

0%0.7880%

Table 2. ONERA M6: M0 = 0.84; Re0 = 11.7 • 106; a = 3.06

Domain "radius"GridType of ABC'sFull lift, CLRelative errorFull drag, CD x 10Relative error

3 root chords197 x 49 x 33

standard0.298±0.004

6.24%±1.43%0.168±0.008

8.95%±5.19%

DPM0.27980.43%0.15370.39%

10 root chords209 x 57 x 33

standard0.2805

0%0.1542

0%

DPM0.2786

0%0.1531

0%

speed up (by up to a factor of three) the conver-gence of the multigrid iterations, see Refs. 9, 10, 14.The discussion on combined implementation of theDPM-based ABC's with multigrid is contained inRef. 17.

Three-Dimensional Case. Here, we demon-strate some new results on computation of the ex-ternal flows around the ONERA M6 wing.

We use the code TLNS3D by Vatsa, et al. 22 to inte-grate the thin-layer equations on the curvilinear grid(see Figures 1 and 2) generated around the wing.This code is based on the central-difference finite-volume discretization in space with the first- andthird-order artificial dissipation. Pseudo-time iter-ations are used for obtaining the steady-state solu-tion; the integration in time is done by the five-stageRunge-Kutta algorithm (with the Courant num-ber calculated locally) supplemented by the resid-ual smoothing. For the purpose of accelerating theconvergence, the multigrid methodology is imple-mented; in our computations we used three subse-quent grid levels with V cycles; the full multigridmethodology (FMG) could be employed as well. Inaddition, we use the preconditioning technique ofRef. 23 to improve the convergence to steady state.We implement the DPM-based ABC's on the finestlevel of multigrid on the final FMG stage; the bound-ary data for coarser levels are provided by the coars-ening procedure. Moreover, even on the finest level

we implement the DPM-based ABC's only on thefirst and the last Runge-Kutta stages, which seemsto make very little difference compared to the im-plementation on all five stages; the boundary datafor the three intermediate stages are provided fromthe DPM-based ABC's on the first stage. Unlikethe two-dimensional case, the standard treatment ofthe external boundary in three dimensions is basedon merely the locally one-dimensional characteristicsanalysis and extrapolation (the point-vortex modelis not applicable).

In Table 1, we present the numerical results ob-tained with the two types of ABC's for the subcrit-ical flow around the ONERA M6 wing. As one canclearly see, the DPM-based ABC's in three space di-mensions outperform the standard approach as theydo in two dimensions: the new boundary conditionsare capable of producing accurate results on verysmall computational domains, whereas the perfor-mance of the standard technique deteriorates as thedomain shrinks. Analogously to the two-dimensionalcase, the three-dimensional DPM-based ABC's canbe calculated for any shape of T by means of thesame procedure (universality) and can be easily com-bined with the existing solver (TLNS3D). Let us ad-ditionally note that for this series of computations,the dimensions of all three grids are the same, gridsof smaller extent are obtained by scaling down theoriginal 10 chords grid. The concentration of nodesnear the wing that results from the down-scaling ob-

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viously contributes to the general improvement inthe accuracy of the solution.

We have also conducted some three-dimensionalcomputations for the standard transonic case forONERA M6 wing: M0 = 0.84; Re0 = 11.7 • 106;a = 3.06°. Here we could not bring the artificialboundary as close to the solid surface as in the fore-going subsonic case (see Table 1) because our far-field treatment is purely subsonic and the boundarycannot intersect the supercritical zone. We there-fore calculated the solution for two computationaldomains of the average radii of approximately 10and 3 root chords of the wing, respectively. Theresults summarized in Table 2 clearly demonstratethat for the small computational domains the DPM-based ABC's generate much more accurate solutionsthat the standard boundary conditions do. Note, inthis case the grids for the different domains havedifferent dimensions, and the smaller 197 x 49 x 33grid (3 chords) is now an exact subset of the big-ger 203 x 57 x 33 grid (10 chords). This is done inorder to eliminate any influence that the change ofthe grid in the near field could possibly exert on thesolution.

Besides the improvement of accuracy, the appli-cation of the DPM-based ABC's to transonic flowcomputations on the small (3 chords) computa-tional domain yielded much higher convergence rateof the residual (continuity equation), as well asmuch faster convergence of other quantities, includ-ing those deemed as sensitive, e.g., the number of su-personic points in the domain. In Figures 3a and 3b,we show the convergence history for this supercrit-ical flow variant. One can see that the convergencefor the standard boundary conditions is poor; there-fore the corresponding force coefficients in Table 2are given with the error bands indicated.

For the 10 chords domain, the DPM-based ABC'salso provide for some convergence speedup, althoughthe difference between the two ABC's techniques isless dramatic here. This is reasonable because onecould generally expect that the bigger the computa-tional domain, the smaller is the influence that theexternal boundary conditions exert on the numeri-cal procedure. The convergence history for the 10chords computations is shown in Figures 4a and 4b.

Note, from Figure 3b one can conclude that on thesmall domain the two algorithms converge to quitedifferent solutions, whereas Figure 4b allows one toassume that on the big domain the final solutions areclose to one another. The data from Table 2 corrob-orate these conclusions. This behavior again fits intothe aforementioned concept that the impact of theABC's decreases as the domain size increases. In the

future, we plan on running some transonic compu-tations for the domain bigger than 10 root chords ofthe wing. This may provide for an experimental jus-tification of the statement that the solutions for thetwo different types of ABC's approach one anotheras the domain enlarges.

103

102

10'75

I 10°o>cc

10"'

10-*

10-*

- Standard ABC's• DPM-based ABC's

0.0 200.0 400.0Work

600.0 800.0

Figure 3a. ONERA M6: M0 = 0.84, Re0 = 11.7-106,a = 3.06°. Convergence history for the residual ofthe continuity equation. Average domain "radius"is 3 root chords of the wing; grid 197 x 49 x 33.

32000

28000

aI

24000

20000

16000

• Standard ABC'sDPM-based ABC's

800.0

Figure 3b. ONERA M6: M0 = 0.84, Re0 = 11.7 •106, a = 3.06°. Convergence history for the numberof supersonic nodes in the domain. Average domain"radius" is 3 root chords of the wing; grid 197 x 49 x33.

Miscellaneous Issues. Here, we will primarilydiscuss the computational cost of the DPM-basedABC's and possible ways for its reduction.

The cost of the DPM-based ABC's in two spacedimensions is modest; the ABC's may add about10% to the cost of the original integration proce-dure. In three space dimensions, the relative cost of

1146



10"

10=

10'

10°

,10

ID'2

10-*

10"4

Standard ABC'sDPM-based ABC's

0.0 200.0 400.0Work

600.0 800.0

Figure 4a. ONERAM6: M0 = 0.84, ReQ = 11.7-106,a = 3.06°. Convergence history for the residual ofthe continuity equation. Average domain "radius"is 10 root chords of the wing; grid 209 x 57 x 33.

32000

28000

24000

20000 -

16000 L

• Standard ABC's• DPM-based ABC's

0.0 200.0 400.0Work

800.0

Figure 4b. ONERA M6: M0 = 0.84, Re0 = 11.7 •106, a = 3.06°. Convergence history for the numberof supersonic nodes in the domain. Average domain"radius" is 10 root chords of the wing; grid 209 x57 x 33.

the DPM-based ABC's is so far higher; they typi-cally add from 25% to 30% of extra CPU time. Inmany cases, however, this increase can be compen-sated for by the convergence acceleration, see Fig-ures 3 and 4. Moreover, we expect that some changesto the ABC's algorithm can still be done so that thevectorization is exploited more effectively and there-fore the boundary operators are computed faster.Apart from the algorithm changes there are some"indirect" ways for reducing the computational costof the DPM-based ABC's.

Parallel implementation of the ABC's on themulti-processor machines is a relatively easy task

because the solution algorithm for the linear AP isreadily parallelizable. For two space dimensions, anup to five times speedup compared to the single-processor version has been achieved on an eight-processor CRAY Y-MP.

In three space dimensions, carrying out multiplecomputations on the same grid is very often the case(many different flow regimes in one geometric set-ting). In so doing, the external geometry that in-fluences the ABC's is fixed and the boundary con-ditions depend only on the aerodynamic parame-ters, e.g., Mach number MQ and the angle of at-tack a. Therefore, the entry-wise interpolation ofthe matrices of boundary operators T with respectto these parameters has a substantial promise fromthe standpoint of cost reduction. Indeed, after anoticeable startup expense for calculating the "ref-erence" T's for some selected values of the parame-ters, any other matrix needed for any specific regimewithin the prescribed range of the parameters canbe obtained for virtually no extra cost by means ofthe interpolation. This methodology has been testedfor the two-dimensional supercritical flow around theRAE2822 airfoil 17. The results seem very promis-ing: for the small computational domain the accu-racy provided by the operators T interpolated withrespect to MQ and a is only slightly worse than theaccuracy obtained using genuine DPM-based ABC's(i.e., the operator calculated for the given specificvalues of the parameters) and is still much betterthan the accuracy obtained on the basis of the point-vortex model.

Finally, we mention that the nonlocality of theDPM-based ABC's is expressed by the fact that thematrix T (see (10)) is dense. However, this ma-trix is typically structured. From the standpointof physics, the structure of T reflects the simpleconsideration that each specific node influences itsclose neighbors stronger than it influences the re-mote points. For system (4a), the operator T from(10) is composed of 5 x 5 blocks as shown in Fig-ure 5 (provided that the vectors of variables alongthe boundaries F and FI are arranged properly: firstp's for all the nodes, then u's, etc., and, finally, p's).The qualitative meaning of each block in Figure 5 ishow one variable of the corresponding pair influencesanother one along the entire boundary. Stronger in-terdependence between closer nodes results in thefact that the near-diagonal entries for each blockare considerably larger than the off-diagonal ones (if,additionally, the order of nodes for F and FI is thesame). The latter observation has, in fact, been cor-roborated computationally (for the two-dimensionalcase).

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p ~ p

p~u

p~v

p~w

p ~ p

u~ p

u ~u

U~ V

u ~ w

u ~ p

v~ p

v~ u

V ~ V

V ~ W

v ~ p

w~ p

w ~u

W ~ V

w ~ w

w ~ p

p~ p

p~u

p ~ v

w~p

p ~ p

Figure 5. Qualitative block structure of the bound-ary operator T for three-dimensional flow computa-tions.

To effectively calculate matrix-vector products forthe structured matrices of the foregoing type, onecan use multiresolution-based data compression al-gorithms, e.g., the one proposed by Harten andYad-Shalom . In Ref. 19, we applied this algo-rithm to the model example of the matrix T fortwo-dimensional scalar Yukawa equation. As couldbe seen from the nested multiresolution representa-tion 24 for the model operator T (see Ref. 19), allthe nonzero entries are concentrated near the maindiagonal of each nested block; as we recede the diag-onal the corresponding values rapidly fall below thedouble-precision machine accuracy (10~13) thresh-old. Therefore , this nested structure can be usedfor the efficient calculation of matrix-vector productswith consecutively improving resolution. The con-centration of nonzero entries near the main diagonalalso implies effective localization of the boundaryconditions. This feature may have certain promisefor the future use in the view of the multi-block grids.

The next major challenge from the standpointof treating numerically the infinite-domain formula-tions is presented by the time-dependent problems.As mentioned in the beginning, the issue of ABC'sfor the time-dependent problems is significant notonly in CFD but also in other areas of scientific com-puting, e.g., those originating from acoustics andelectrodynamics.

The primary difficulty that distinguishes the time-dependent problems from the steady-state ones isthat here the exact ABC's are nonlocal not only inspace but also in time (except in the simplest one-dimensional problems). As concerns the approxi-mate approaches, the same basic trend that is rele-vant to the steady-state problems also holds for the

time-dependent ones: more accurate boundary pro-cedure requires more of the nonlocal nature of theABC's to be taken into account.

Of course, the nonlocality of the ABC's in timepresents a very serious computational obstacle, be-cause as the solution evolves such boundary condi-tions would become more and more expensive fromthe standpoints of both memory and computer timerequirements. Therefore, the most crucial numeri-cal issue in constructing the time-dependent ABC'sis how to effectively restrict the nonlocality of theboundary conditions in time. This is an interestingsubject for the future research.

4 Conclusions.

The new global ABC's for calculating steady-stateexternal viscous flows in three space dimensions havebeen constructed on the basis of the difference poten-tials method. The approach generalizes and extendsour previous two-dimensional results.

The new ABC's are capable of greatly reducingthe size of the computational domain (compared tothe standard methods) while still maintaining highaccuracy of the numerical solution. This size re-duction amounts to either the possibility of refiningthe grid in the near field, which potentially leadsto the improvement in accuracy or usage of thesmaller-dimension grids without compromising theaccuracy. Moreover, the DPM-based ABC's may no-ticeably improve the convergence rate of the multi-grid iteration procedure. Finally, the new boundaryconditions appear geometrically universal and easyto incorporate in the structure of the existing flowsolvers. The properties of the new ABC's have beencorroborated experimentally by computing the sub-sonic and transonic flows past the ONERA M6 wingusing the NASA-developed code TLNS3D.

A more detailed description of the theoreticalfoundations of the DPM-based algorithm, as well asa wider selection of the computational results, willbe presented in the forthcoming paper Ref. 25.

Acknowledgments

We are most grateful to Dr. E. B. Parlette of Vi-gyan, Inc., for his valuable help in grid generation.We are also very thankful do Dr. R. C. Swanson andDr. J. L. Thomas of NASA Langley for the usefuldiscussions.

References

1. Tsynkov, S. V., "Artificial BoundaryConditions Based on the Difference Poten-tials Method," NASA Technical Memorandum

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No. 110265, Langley Research Center, July1996.

2. Givoli, D., "Non-reflecting Boundary Con-ditions," Journal of Computational Physics,Vol. 94, 1991, pp. 1-29.

3. Givoli, D., Numerical Methods for Problems inInfinite Domains, Elsevier, Amsterdam, 1992.

4. Calderon, A. P., "Boundary-Value Problemsfor Elliptic Equations," in Proceedings of theSoviet-American Conference on Partial Dif-ferential Equations at Novosibirsk, Fizmatgiz,Moscow, 1963, pp. 303-304.

5. Seeley, R. T., "Singular Integrals and BoundaryValue Problems," American Journal of Mathe-matics, Vol. 88, 1966, pp. 781-809.

6. Ryaben'kii, V. S., "Boundary Equations withProjections," Russian Mathematical Surveys,Vol. 40, 1985, pp. 147-183.

7. Ryaben'kii, V. S., Difference Potentials Methodfor Some Problems of Continuous Media Me-chanics, Nauka, Moscow, 1987 (in Russian).

8. Ryaben'kii, V. S., "Difference PotentialsMethod and its Applications," Math. Nachr.,Vol. 177, 1996, pp. 251-264.

9. Ryaben'kii, V. S. and Tsynkov, S. V., "Arti-ficial Boundary Conditions for the NumericalSolution of External Viscous Flow Problems,"SIAM Journal on Numerical Analysis, Vol. 32,1995, pp.1355-1389.

10. Tsynkov, S. V., "An Application of Nonlo-cal External Conditions to Viscous Flow Com-putations," Journal of Computational Physics,Vol. 116, 1995, pp. 212-225.

11. Swanson, R. C. and Turkel, E., "A Multi-stage Time-Stepping Scheme for the Navier-Stokes Equations," AIAA Paper 85-0035, Jan-uary 1985.

12. Swanson, R. C. and Turkel, E., "Artificial Dis-sipation and Central Difference Schemes for theEuler and Navier-Stokes Equations," AIAA pa-per 87-1107, June 1987.

13. Swanson, R. C. and Turkel, E. "MultistageSchemes with Multigrid for the Euler andNavier-Stokes Equations. Volume I: Compo-nents and Analysis," NASA Technical Paper3631, Langley Research Center, 1996.

14. Tsynkov, S. V., Turkel, E., and Abarbanel,S., "External Flow Computations Using GlobalBoundary Conditions," AIAA Journal, Vol. 34,1996, pp. 700-706; also AIAA Paper 95-0564,January 1995.

15. Ryaben'kii, V. S. and Tsynkov, S. V., "An Ef-fective Numerical Technique for Solving a Spe-cial Class of Ordinary Difference Equations,"Applied Numerical Mathematics, Vol. 18, 1995,pp. 489-501.

16. Tsynkov, S. V., "Artificial Boundary Condi-tions for Computation of Oscillating ExternalFlows," to appear in SIAM Journal on Scien-tific Computing; also NASA Technical Memo-randum 4714, Langley Research Center, August1996.

17. Ryaben'kii, V. S. and Tsynkov, S. V., "An Ap-plication of the Difference Potentials Method toSolving External Problems in CFD," to appearin M. Hafez and K. Oshima (eds.), CFD Review1996.

18. Tsynkov, S. V., "Nonlocal Artificial Bound-ary Conditions for Computation of ExternalViscous Flows," to appear in P. Le Tallecand J. Periaux (eds.), Proceedings of ECCO-MAS'96. Pans, September 9-13, 1996. JohnWiley & Sons, 1996.

19. Tsynkov, S. V., "Artificial Boundary Condi-tions for Infinite-Domain Problems," to ap-pear in M. D. Salas et al. (eds.) Proceedings ofthe ICASE/LaRC Workshop on Barriers andChallenges in Computational Fluid Dynamics.Hampton, August 5-7, 1996. Kluwer AcademicPublishers, 1996.

20. Cole, J. D. and Cook, L. P., Transonic Aerody-namics, Elsevier, Amsterdam, 1986.

21. Thomas, J. L. and Salas, M. D.: "Far-FieldBoundary Conditions for Transonic Lifting So-lutions to the Euler Equations," AIAA Paper85-0020, 1985.

22. Vatsa, V. N., Sanetrik, M. D., and Parlette,E. B., "Development of a Flexible and Effi-cient Multigrid-Based Multiblock Flow Solver,"AIAA Paper 93-0677, January 1993.

23. Turkel, E., Vatsa, V. N., and Radespiel,R., "Preconditioning Methods for Low-SpeedFlows," AIAA Paper 96-2460-CP, June 1996.

24. Harten, A. and Yad-Shalom, I., "Fast Multires-olution Algorithms for Matrix-Vector Multipli-cation," SIAM Journal on Numerical Analysis,Vol. 31, 1994, pp. 1191-1218.

25. Tsynkov, S. V., "External Boundary Condi-tions for Three-Dimensional Problems of Com-putational Aerodynamics," in preparation.

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