AIAA 90-0313 Unsteady Transonic Navier-Stokes Computations for an Oscillating Wing Using Single and Multiple Zones

Neal M. Chaderjian and Guru P. Guruswamy NASA Ames Research Center Moffett Field, California

28th Aerospace Sciences Meeting January 8-1 1, 1990/Reno, Nevada

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 370 L'Enfant Promenade, S.W., Washington, D.C. 20024

UNSTEADY TRANSONIC NAVIER-STOKES COMPUTATIONS FOR A N

OSCILLATING WING USING SINGLE A N D MULTIPLE ZONES

Abstract

Modern jet transports and maneuvering tactical fight- ers operating in the transonic regime often give rise to time-dependent fluid physics that intrract wi th flexible structnral components, e . g . , vortical flow, shocks, and sep- aration. Eficicnt computational fluid dynamic (CFD) ine.thods are required to study such computationally in- tensive prohlcms. In this work a numerical mrthod is prcscntcd to address this problem. Time-depmdent, com- pressi lk , Navier-Stokes equations are used to simulate un- steady transonic flow vhout a threc-dimensional rigid wing ui~lergoing a forced periodic motion in angle of attack. An efficient, implicit, diagonal algorithm is utilized because of its low operation count per time step compared to other mrthods that solsc systrms of hlock matrix equations. T h e formal t imr accuracy is addressed theoretically and demon- strated numcrically b y comparison of computational re- sults with experimental data . A zonal grid approach, ca- pable of treating complex geometries, is presented and its time accuracy is dcmunst rated by comparing a zonal corn- putation with a single grid cnniputation and rxprriment~al dat.a.

J r t transports and maneuvering tactical figlitrrs tend to operate in the transonic regime where the flow field is primarily subsonic with regions of embedded supersonic flow. This often gives rise to complex fluid physics soch as steady and tiine dependent vortical flow, shocks, and sepa- ratiori. It has been observed that these plrenomma interact w i t h tlrc flcxiblc components of an airrraft and can lead to sustained arrorlastic oscillations and d i ~ c r g e n c e . ' - ~ The ability to accuratcly predict the time-dependent f low field and aeroclastic rrsporisr of an aircraft is essential to insure the intrgrity and safety of the vehicle and can help extend tiir maneuvering envelop of modern tactical fighters.

Computational fluid dynanlics (CFU) can play an i m portant role i n the design process by providing r lctai lcd flow field and structural displacement information. This will Iielp reduce thc design cycle time and provide infor- ~nat , ion that is complinvntary to wind t ~ i n m l and flight

~~ ~. ~~ ~~~~~ .,,. ~~~~ ~ ~~~

Research Scientist, Applied Conrpotntionnl Flnids Ihanrh, Mombcr A I A A .

tRcsearclt Scientist, Applied Comgutntionnl Fluids Branch, Asrnciiite Fellow A I A A .

Copyright @ 1000 by the American Institute of Aeronau- tics and Astrunnetirs, Inc. No copyright is asserted in tlir U n i t e d Sta ter nnder Title IT, U.S. Code. Thr U.S . Govern-

the copyright rlaiined herein for Governmental purposer. All other rights HIC rfsrrved by tho copyright owner.

- l l l C l l t has n royalty-frcr license t o oxerclsc all rights under

test data . At present, the most advanced CFD codes for predicting the aeroelastic response of wing/fuselagr config- urations are based on potential flow t h e ~ r y . ~ Even though these equations have a limited range of applicability, they havc hcen uscd because the mow complete Navier-Stokes equations require significantly more computer memory and computer time. Navier-Stokes equations, when coupled with the governing aeroelastic equations, can require mor? than one or two orders of computer time. However, re- cent advances in numerical algorithms and computer tech- nology make it possible to c.onsider using these equations in the research environment and eventually in the design e n ~ i r o n m e n t . ~ , ~ The literature c.onfirms this trend as 5 e v ~ cral time-dependcnt, three-dimensional, Navier-Stokes so- lutions for wing geometries have begun to emerge, e.g., References 7-8, as well as aeroelastic computations, e.g., References 9-10, It is our eventual goal to simulate the aeroelastic response of a complete aircraft with the Navier- Stokes equations. However, as a first step towards this goal we will address some accuracy and efficiency issues regard- ing the usc of an Enler/Navier-Stokes code for predicting the timedependent, three-dimensional, flow field about a rigid wing undergoing a forced periodic motion in angle of attack.

In order to simulate time-dependent viscous flow about a realistic aircraft geometry, it is important for r7

Navier-Stokes code to he able to treat romplrx geometries, have an efficient and robust numerical algorithm, and he time accurate. The Transonic Navier-Stokes (TXS) rude has been chosen as a base code because of its success in predicting the steady transonic viscous flow about the com- plete F-16A fighter aircraft." This code uses a zonal grid approach for treating the geometric complexity of the air- craft. The flow field is subdivided into an cnsenible of simply shaped grids. Each of these grids can he refined by adding clustered grid points as appropriate to adequately rapture viseous effccts near all body surfacrs and wake re- gions. Flow field c.oupling between zones is accomplished by overlapping adjacent zonal grids. Moreover, this code uses an efficient and robust implicit approximat,? factor- ization algorithm. This algorithm, due to Pulliain and Chaussee,12 is a diagonal version of th r Beam-Warming a l g o r i t h ~ n ' ~ and uses central differencing. The operation count per time step is therefore low compared to most upwind sc.hemes and methods requiring the solution of a system of hlock matrix equations. However, the TNS code has only been uscd for steady flow computations.

The Navicr-Stokes Simulation (NSS) code is a new and extended version of the TNS code for treating time- dependent flows. It allows for a moving zonal grid system with time-dependent coordinate transformation metrics. A Fourier-analysis routine has also heen included to facilitate comparison of computcd surface pressrire, force, and mo- ment coefficients with experimental data . Other features have also bfen included to enhance rode utility, e.g., air

1

improved restart rapaLility, additional input/output rou- tines, etc.

Tlrc Pulliarn-Chaussee diagonal algorithm is conser- vative in space but not conservative i n t i m e . This can affect the shock speed in a transonic computation. The form of this t~mporrrl causervation error will be prcsenttdin alatm scction, as well as ways to remove the error. The objective of this research is to verify the time-accuracy uf the NSS code algorithni and zonal boundary trcatment by rompar- ing computational results with exprrimental data for t ran- sonic now O V P ~ an oscillating tl,ree-dimcnsiond wing with moving shocks.

In the following sections a tlrroretical background is givrn describing the governing equations, numerical algo- r i thm, and formal time-accuracy. This is followed by a de- scription of the zonal grid approach, grid generation, and results. Finally, sonic ~onclc~ding icmarks are made.

T h e o r e t i c a l -- Ba&gro~!n_d_

G o v e r n i n g E q u a t i o n s ~- ...

The cornpressihle Reynolds-avcraged Navier-Stakes equations are used to numerically simulate time-dependent viscous flow about a wing with oscillating angle of a t - tack. For the high Rrynolds-numbrr flows considered in this paper, the thin-layer approximation is employed to- gether with body-fitted curvilinear coordinates. This im- proves the efliciency of the numerical simulation and sim- plifies the implementation of boundary conditions. As- suming a general coordinate transformation from physi- cal space (x,y,s) to computational space ( ( ,T , ( ) and that viscous terms arc only important in the (-coordinatc di- rection, thc governing equations can be rxpressed in the following strong conservation-law form:

where

Turbulence Model

Fluid turbulence is modeled by the BaldwimLornax algebraic eddy viscosity model." This isotropic model is used primarily because it is cornputationally efficient. AII viscous computations presented in this p p e r a5sumc fully turbulent flow at the leading edge of the wing, i c . , transi- tional effects are ignored. This approximation i s consistent with the high Reynolds-number assumption.

4

Numerical A l g o r i t h m -

An implicit approxjmate-factorization diagonal algo~ rithm is used in the NSS code to integrate Eq. (1). This al- gorithm is based on the Beam-Warming algorithmI3 given below.

( I + h6tA^- hD; l<)" ( l+ h6,& hDil,)" (6) ( r + f i ~ e -~ ~R~-'$(J-'MIJ -- = E",

where

in = -AI(6<j? t 6?$ t 46 - Re-'&S^-i Dc6)n, and

A@ = G n + l .. Q'.

In the above expression, the superscript n corresponds to the current time_ le+_tn a n d 2 + 1 to the new time l eve l I" + AI. Also, A , B , C, and M are the Jacobian matrices of E , F , G, and s^, respectively. The numerical metrics are evaluatcd so that uniform Row is an exact solution of the steady finite-difference equations, (see C h a d e r j i a r ~ ~ ~ ) The

^ ^ ^

2

Rram-Warming algorithm is second-order accurate in t ime when h = $ A t (trapezoidal nrle) and first-ordcr accurate in time when h = At (Euler implicit). The spatial oper-

in the implicit operator to be nominal for a variety of ap- plications with moving shocks. This will he demonstrated computationally in the "Results" section.

ators usc central differencing throughout, so fourth-order explicit (D,) and second-order implicit ( D , ) numerical dis- sipation ternis arc added to damp high irequency errors. O m drawback to this method, however, is the required so- lution of a block tridjagonnl system of equations which is computationally costly. For mow details of this algorithm

- see Pulliam and Stegrr.'B

In order to reduce the overall execution t h e of the NSS code, a inore efficirnt diagonal form of thc Beam- Warming algorithm is used. This algorithm, due to Pul- l iam and C l i a o ~ s e e , ' ~ i s given by

( 7 ' ~ ) " ( 1 t . h6tAA i hD&)"

( f i ) " ( I -! h6,AB + hDiln)"

( p ) " ( I ~ i - h6(Ac ~! hDtlt)m(TG1)mAG" :: @, (7)

where A A is a diiigon;rl matrix consisting of the eigenvalues of A , and so o n Note that the viscous terms are treated explicitly. 'The right hand sidr of Eq. (7) is idrnt i ial to the right hand sidc of Eq. ( 6 ) . This algorithm uses both explicit and implirit fourth-order dissipation and only re- quirrs the solution of scalar pcntadiagonal equations. This results i n B Iowrr opeintion count per time step, and hence a rcducrd rxecution timc per soliltion. Further details of

tliis algorithm, inrhiding a description of the TA, N , and P matriccs, arc given in R e f e i e n ~ e s . ' ~ ~ "

A

- -

When a steady~stat r solution is desired, tlie rate of c o ~ ~ ~ c r g e n c e can he greatly improved by using the spatially varying time step drrrribed by Srinivasan et a1" and is given b y

v

where At, i s a user specified constant. l~loics , 'O using thr diagoiinl algorithm with variable timr step, reported ionvcrgr.iicc ratrs 40 t i m c s those ohtained using the Beam- Warming slgoritlim with constant time step. A constant time step is uscd for tiine-arcorate computations.

All of thc coniputations reported in t h i s paper use the diagonal algorithm. Duc to its diagonal forin, this algorithm is first-order accurate in time and trrnporally riorr~const.rvative. T h ~ s r two issues are addrcssed in the next srction and tire "llrsults" s r c t ion

T i m e A c c u r a c y

I , l l r r strong conservation-law form of the govcrning equations, I?qs. 1, whcn properly differenced, conserve nu- merical fluxes across flow discontinuities. Thcse so called "weak solntions", Le., piecewise continuous solutions, sat- isfy the proper integral relations irrespectivc of the grid sparing or time step. If a non~conservative form is used, the shock strcngth, position, and wave spred will he de- pcndcnt oil t he grid spacing and time step.

Although the diagonal algoritlim, Eq., (71, is conser- vative in space i t is not conservative in time. So tirne- accurate rompntations involving shocks can b r question- ablc. IIowevcr, s ince the explicit side is diffcrniccd ronser~ vativcly, one might expect the temporal error introduced

W

This section will address the form of the conservation error and the formal time accuracy of the algorithm. Two approaches for removing this error will also be presented in the event that temporal conservation F ~ - T O T S prove to be important for a sprcific application.

T h e temporal conservation error of the diagonal a l g o ~ rithm can be found by going back to its derivation from the Beam-Warming algoritbrn. Replacing tlie Jacobian matri- ces of Eq. (6) by

A := TAAaTi'

B = T B A B T ~ '

c = T ~ A ~ T ; ~ ,

and dropping the implicit viscous terms gives

(TATT' + h8cTAAaTi' + hTAn;ltT;')"

(TBTi l + ha,TBABTi' + hTBD;l,Ti')"

(TcT;' + hacTcAcTi-' + hTcD;l(Tg')"AG" = e. In the above expression, A A i s a diagonal matrix whose elements are the eigenvalues of A and TA is the matrix whose columns are the eigenvectors or A. Similar terms apply for the R and C matrices. Spatial finite-diffcrence operators have been replaced by differential operators to avoid unnecessary details that d o not affect thr analysis. Application of the chain rule results in

(9)

3

+ higher order terms

- O(AG")

So t h ~ Bmm-Warming algorithm can be expressed as the diagonal algorithm with an additional term. This extra term, e", coiresponds to the temporal conservation error of the diagonal algorithm. Now

so that

c " - O ( A t ) . (12)

So the diagonal algoritl~m is formally first-order accurate i n time, rven i f trapezoidal rule is used. This error can be important i f the shock strength and speed become too l u g e for agiven time s k p . However, this error can be cont,rolled by lowering the t i m c step for a given grid. This provides a inetlrod for rl ieckingif t h i s error is important for a spe- c.ific application. We also remark that the time step used in Navier-Stokes computations are generally small, due to stability considerations, and hence the temporal conserva- tion error should be small for a large class of applications.

There is another way of showing tha t the conserva- tion error vanishes with decreasing time step. A s the time step h .., 0, the dominant term in the implicit operator, (see Eq. 7) , is the identity matrix. So this method ap- proaches t h ~ E d e r explicit algorithm for very small t ime steps, which is fully conservative.

Allother method for removing the temporal conser- vation ~ r r o r is to use a Newton iteration niethod desrribed by Chakravartliy.zo The diagonal algorithm then Iizs the form

( T a ) ' ( l + h6cAa + hDilt)'

(.$)'(It h S l i A ~ t hD,l,)'

( F ) ' ( I + h6(Ac + hD&)'(T;')'Atj'

~ ~~~(4, .~. @) + 3, (13)

~n this euprrssion, A @ -+ o and Q - @ + I as i be- comes large. This usually requires only a few sub-iterations to converge to the new time level. This approach not only SFIIIUYCS thr conservation error, but also removes the ap l ,~ox i ina t e~f~r tu r i aa t ion error and makes the scheme fully iniplicit. This approach, howevcr, can significantly increase the computational time of the code.

Zonal Ajg.r&r&

Generating a single grid about a complex geonietric shape, e.g., a fighter aircraft, with appropriate viscous clus- tering near all body surfacer and wakes can be a formidable (if not impossible) task. This difficulty can be overcome

All computations presented in this paper use a C-I1 grid topology. A perspective view of a medium dcnsity grid is shown in Fig. 1. In particular, the surface geometry and part of the symmetry plane grid are highlighted. This grid consists of 151 x 39 x 39 grid points (about 230,000). A n additional plane has been added inboard of the sym metry plane to facilitate the symmetry plane boundary conditions.

by adopting a zonal grid approach. The NSS code uses a zonal grid procedure which subdivides tlre flow field about a complex geometry into an ensemble of simple geometric shapes. Adjacent zones overlap with one another by one or two grid cells to couple the flow fields betwern zones. These zmes are constructed from a coarse global grid, so they have a common interface surface. Each zonal grid is then clustered and refined with additional grid points as appropriate to accurately capture the pertinent flow physics. This procedure ensures that at an interface sur- face, the grid points of the coarser grid are coincident with grid points of the finer grid. Flow variable information is transferred from the finer grid to the coarser grid by direct injection. This step is fully conservative. Flow variable i w formation is transferred from the coarser grid to the finer grid by interpolation d o n g grid lines. This-step is not c o w servative. When conservation between zones is important. e.g., when a shock passes though a zonal boundary, both zones are required to have identical grid points at the iii- terface. This will insure complete flow conservation across the zonal boundary.

The flow field is advanced to a new time level one zone at a time. T h e most recent da ta available at a zonal interface is used. So some zonal boundaries are updated explicitly while others are updated implicitly. A n y cffects related to the explicit treatment of some zonal bound;rrirs will vanish as the t ime step is decreased. A comparison between single and two zone flow simulations is given in the ''Results" section. For more details of the zonal grid procedure and interface boundary treatment see Flores et a l . 2 '

- G r i d G e n e r a t i o n

d

The grid shown in Fig. 1 was constructed in the fol- lowing manner. First, a surface grid for the wing was gen- crated with the S3D code.'* Additional grid clustering was provided near t he wing nose and shocks. Then a tlrree- dimensional grid was created with 3DGRAPE,23 an elliptic grid generation program, with normal spacing to the wing surface appropriate for inviscid computations. Finally, vis- cous clustering was obtained by redistributing grid points along coordinate lines normal to the body with a cluster- ing function described by Vinokur.2' This one dimensional stretching function provides a distribution of points that minimizes tlie truncation error of a finite-difference code.

A two zone grid was also constructed by splitting the single zone grid in Fig. 1 in the viscous direction A symmetry plane view is shown in Fig. 2 with the shaded region indicating the one cell overlap between zones. Zone one is adjacent to the wing surface and zone two extends from zone one to the far field, about six chord lengths away. This was done in order to study thc effect of the zonal boundary treatment on time-accuracy.

B o u ~ n d ~ a q C o n d i t i o n s

The VSS code solves the Euler or Navier-Stokes equa- tions depending on thr specific application. For inviscid flow, flow tangency is iniposed on the wing surface while density and pressure are found h y extrapolation. Thc total energy per u n i t volume i s then computed from the perfect gas l a w . The same boundary conditions apply for viscous flow except the no-slip condition replaces flow tangency 011 the wing surfare. Uniform flow is imposcd at the far field and aero gradient conditions are used a t the outflow boundaries. Boundary values along the C-cut (downwind of the win(: trailing edge and outboard of the wing tip) are obtained by averaging flow variables across the cut. Flow symmetry is obtained by reflecting the flow variables about the synimctry plane i n an rxplicit manner.

'The $mal intrrface boundary conditions have already heen descrihed in the "Zonal Approach" section.

Resnlts

W i n d - T " n n e _ l M o d d

A l l flow computations presented in this section c o r w spond to s rectangular Iralf-wing used in an AGARD wind- tunnel experiment describrd by Mabey et al." The wing model was inountcd on a fuselage-like body to displace it slightly from the wintl-tunnel ivall and its boundary layer. No expcrimental data was taken ut the wind-tunnel wall (wing symmetry plana). The wing's cross-section consists o f a NACA 64.4010 airfoil whose thirknessfchord ratio has heen modified to 10.6%. The wing has a complete aspect ratio = 4.0. 'Yhp experiment was conducted in the RAE 8 x 8 foot tnnnel with closed walls. (The modd was origi- nally dcsigncd for a s m a l l ~ r tunnel which was not available when thc: cxperimnit mas conducted.) Vanons c-11 grids WCTP gcneraird for t h i s wing, see for exampk Fig. 1. No attempt w a s madc to inotlrl t he wind-tunnel test section.

'v

Steady Flow C o m p u t a t i o n s

Steady-state flow computations with different grid densitirs are used to s t u d y their effect on the spatial ac- curacy of the KSS code. Three grids are used: 1) a coarse grid (101 x 39 x 39); 2) a medium grid (151 x 39 x 39); 3) and a finc grid (233 x 311 x 39). Each grid has t h e same 81id point distribution in the spanwise and body-nonnal dirrctions (q- and C-diwctions). The streamwise distrihu- tion of points ((-direction) downwind of the wing trailing edge are also idcritical for all threc grids (17 grid cells to t,hr far field). However, each grid has 50% niore points on t h e wing surface (in thr streamwise dircction) than its yre- &us C O ~ T S P T grid. Grid points are clustered near the wing leading edge and shock locations to improve the flow-field resolution on each grid. T h e grid spacing in t h e viscous direction i s A{ = 5.0 x

Figure 3 shows a comparison between steady Navier- Stokcs pressure coefficients ( C p ) computed on the three grids described above with experimental values at 7 = 5096, 7 -= 77%, and 7 = 94% semiLspan locutions. These computations corrcspond to a freestream Mach numbrr M , = 0.80, angle of attack a :: O . O " , and a frerstream Reynolds number based on the wing root chord Re =

chords.

W

2.4 x IO6. At 7 = SO%, Fig. 3a, the computed pressure coefficients compare well with each other and the experi- mental data , with the exception that the coarse grid shock is slightly smeared. The suction peaks are also slightly u n ~ derpredicted, presonicdly due to wind-tunnel nzall effects. All. three grids have good C, comparisons with each other and experiment at 7 = 77%. At q = 94%, computations with all three grids are in good agreement with each other, but their suction peaks differ slightly from thr experimen- tal value. This is probably due to the use of a rounded wing tip in the computation VEISUS a blunt (flat) tip in the wind-tunnel model. Overall, the rncdium and fine grids give the best C , comparisons with the experiment and do not differ noticeably from each other. So the medium grid density (about 230,000 grid points) is used for the rcmain~ der of the viscous computations presented hclow,

U n s t e a d y Flow c o m p u t a t i o n s -

Time-acciirate Fuler and Navier-Stokes computa- tions for a rigid wing with oscillating angle of attack are ow presented. Unless otherwise noted, the freestream Mach number i s M , = 0.80, and the Reynolds number based on wing chord is Re = 2.4 x 10'. T h e angle of a t - tack vsries periodically according to

a( t ) = -a,,,sin(M,lct)

where the amplitude is amor = 1.0 degrees, the reduced frequency is k = 0.27, and t is a. nondimensional time based on wing chord and the freestream speed of sound. The reduced frequency is defined by k = wc/U, , where w is the circular frequency (radians/sec), c the wing chord length, and U, the frieestream velocity. The computa- tional grid undergoes a rigid body rotation according to the above angle-of-attack formula. Real and iinaginary unsteady pressure coefficients are used to comparc com- putational and experimental rrsi*lts. These in-phase and out-of-phase pressurcs corrzspond to the time dependent Fourier cocficients normaliard by arnar (in radians).

The time steps used in Navier-Stokes computations are iisually dcternlined by stahility considerations raiher than time accuracy. So the Euler equations arc used to drternline the time-step requirement for time accuracy with the first-order accurate diagonal algorithm. For these inviscid computations, the mrdium density viscous grid i n Fig. 1 is uncliistered by rcdistribihng 21 grid points along coordinate lines normal to the wing surface. The inviscid and Y ~ S C O U S grids have identical spacing in thv spanwise and streamwise di~cct ions. Figure 4 compares coniputed real and imaginary pressures for three different time steps with rvperimental da ta at three semi-span loca- tions. Thcse time steps, beginning from the largest to the smnllest, are 3F0, 720, and 1080 t i m e steps/cycle, r e s p r c ~ tively. Pressure coefficients computed with the medium and smallest time steps arc virtually indist.inguisliable from one another. However, significant differences between the largest time step (360 steps/cycle) and the smaller time steps can be noted, especially at q = 77% and 7 = 94%. Therefore 720 time steps/cycle is the largest usable time step for time-accuracy.

The real and imaginary pressure coeficients for the Euler equations, Navier-Stokes equation&, and exp?rimen- tal data are compared in Fig. 5 . These pressures corre- spond to the same freestream conditions and reduced fre- quency in Fig. 4. At 50% semi-span location, the real

5

C, shock position and strength computed by the Navier- Stokrs equations are in good agreement with the experi- ment. Reducing the time step in half or using a two-step Newton subiteration has no perrrivable effect on any of tbe pressure corfficients. This indicates that the temporal conservation error is negligible for this transonic-flow ex- ample. It is evident from this figure that the shock moves ahout 25% of a chord length during a complete cycle. So this provides a good test for tlie time accuracy of the diag- onal algorithm. The Eder equations, on the other hand, predict a much stronger shock with a slightly downwind position than the experiment and the Navier-Stokes equa- tions. Eveii for this relatively easy case which does not have any sigiiifirant separation, the Eulcr equations would predict tlre wrong wave drag. T h e drop in pressure before the shock is w t pndicted very well by either tlie Euler or Navier~Stokes equations. This is attributed to inade- quate grid resolution. i\lthough the clustered grid gave good shock resolution for the steady-state case, there is inndequate grid resolution i n the shock excursion region. With hind sieht, it is obvious that time-dependent flows in- volving nioving shocks require greater grid resolution than steady-state flows. This can he ac.romplished by adding more grid points in the shock excursion region or using a solution adaptivr kchnique. The latter would be more efficirnt.

The real and imaginary pressure coefficients c.om- prated by the Euler and h’avier-Stokes equations com- pare well with each otlier and tlie experimental data for 7 = 77% and 7 = 94% seiN-span locations. The time step used in the Navier-Stokes computations was At = 0.02, or 1440 steps/cycle. Although this time step was chosen to maintain code stability, it is only one half of the time step required for time-accuracy i n the Euler computations. This is “cry mcouraging, especially since the viscous grid sparing normal to the wing was two orders of magnitude smaller that the Euler grid spacing. This indicatrs the robustness of tlre diagonal algorithm.

In order to verify t h r t ime~arcuracy of the zonal boundary conditions, the single grid shown in Fig. 1 is split into twc zones shown in Fig. 2. The first zone is close to tlre wing surface while tlie second zone extends from the first zone to thc far field. T h e Navier-Stokes equa- tions are solved in tlie first zone and the Euler equations are solved in the sec.ond zone. The zones overlap by one grid cell as indicated by the shaded region in Fig. 2. The zonal overlap position was determined as the closest in- viscid position, i .e. , where the turbulent eddy viscosity is zero. The close proximity of the zonal interface to the wing surface and leading edge provides an extreme tcst for time accuracy of the zonal houndary condition treatment. Figure F compares thc real and imaginary pressure roeffi- rients for a one zone Navier-Stokes simulation, a t w o zone Navier-Stokes simulation, and experimental data. The sin- gle zone computation applied tlre Navier-Stokes equations and Euler equations i n the same regions as the zonal case. Once again the time step used for these viscous computa- tions was At = 0.02. Overall, the comparison of the single zone and t w o zone computations are very close. Slight differences can he noted i n the imaginary pressure coeffi- cients. Reducing tlie time step in half does not alter the results. This indicates these slight differences are not due to the partially explicit zonal boundary conditions. These diffrrencer arc more likely due to differences in numerical dissipation. Although the diagonal algorithm uses fourth- order dissipation, it must drop to second order near grid

boundaries. This means that the single zone computation uses fourth-order dissipation in the zonal overlap region whereas the two zone computation uses second-order dis- sipation there. T h e close proxiniity of the zonal honndary to tlie Wing surface and leading edge can easily account for these differences. Note that both solutions compare ‘LJ

equally well with the experimental data . So the zonal grid approach can he used for time-accurate computations.

A final example of a time-accurate zonal computa- tion is shown if Fig. 7. T h e same freestream conditions of the previous example are used except now arnni = 2.0 degrees. Altliough experimental da ta is not available for this case, the stronger shock provides a qualitative chmk of the zonal boundary treatment. Figure 7 indicates iri- stantaneous pressure coefficient contours when tlie upper surface shock is its strongest. The zonal overlap rcgion i s also indicated in the figure. Notice tha t the G, contours are extremely smooth across the zonal boundaries, even near the shock and the wing leading edge. This shows that there are no significant flow disturbances introduced by the zonal boundary conditions.

Conclus ions

The zonal, steady-state, Transonic Navier-Stokes (TNS) code has been extended to treat unsteady flows. This new Navier-Stokes Simulation (NSS) code has the capability of treating a moving zonal grid system. The Pulliam-Chaussee diagonal algorithm has been shown to be an efficient and robust time-dependent algorithm which re- quires approximately 17p-s/grid point/time step on a Cray Y-MP supercomputer. Although this algorithm is conser- vative in space it has a temporal conservation error O ( A t ) . Since Navier-Stokes algorithms require small time steps for numerical stability, this error should he small for a large

The NSS code has been used to simulate nnstrurly transonic viscous flow ahout a three-dimensional rectan- gular wing with oscillating angle of attack. Computed real and imaginary pressure coefficients compared well with the experimental values and demonstrates the time accuracy of the diagonal algorithm. The temporal conservation er- ror was shown to have no noticeable effect on the spred, strength, and position of the shock wave (for the present computations) through reduction of the time step and ap- plication of B Newton-iteration procedure. A two-zone transonic cornputation for the oscillating wing compared well with the single zone computation and the experimen- tal data. This demonstrates the time accurate capability of the zonal interface procedure. The zonal grid approach provides a way of simulating time-dependent flow about a complete aircraft with the Navier-Stokes equations. Future work will include a time-dependent computation ahout a wing/fuselage configuration.

class of transonic applications. d

R e f e r e n c e s

’Guruswamy, G.P.,Goorjian, P.M., Ide, H. and Miller, G.D., “Transonic Aeroelastic Analysis of the B-I Wing,” AIAA J o u r n a l of AircraR, vol. 23, 547-553, 1986.

‘Farmer, M.G. and Hanson, P.W., “Comparison of Supcrcritical and Conventional Wing Flutter Chararteris- tics,” NASA T M X-72837, 1976.

3Ashley, II., “Role of Shocks in the Sub-Transonic d’

6

Fluttcr I’hcnomenon,” A I A A J o u r n a l of Aircraf t , Vol. 17, 187-197, 1980.

‘Gurusnamy, G.P. and Goorjian, P.M., “Unsteady Transonic Flow Sinlidation on a Full-Span Wing-Rody Configuration,” AlAA .87-1240, 1987.

5 K u t l ~ r , Paul, Gross, Anthony R., “CFD in Design: A Government Pcrspcrtive,” A I A A Paper 89-0094, Jan. 1989.

EGoldhammrr, Mark I., Rubbert, Paul E., “CFD in 1)~sign: A n Airframe Perspective,” A l A A Paper 89-0092, Jan. 1989.

’Nakamichi, J. , “Calculations of Unsteady Navier- Stokes Equations Around an Oscillating 3-D Wing Using Movillg Grid SystPm,” AIAA-87-1158-CP, 1987.

8Srinivasan, G.R. and McCroskey, W.J., “Navier- Stokes Calculations of l lovrring Rotor Flow Fields,” A I A A Jonrnnl of Aircraf t , Vol. 25, 865-874, 1988.

Flows on Flexible Wings,” AIAA-89~0537, 1989.

’”Guruswarny, G. , “Vortical Flow Computations on Swrpt Flrxihle Wings \‘sing Navicr-Stokes Equations,” A I A A ~ 8 9 - 1 183, 1989.

"Flares, J . arid Chaderjian, N.M., “A Zonal N a v i e r ~ Stokes Methodology fQr Flow Simulation Ahout a Com- pletr Aircraft,” A I A A Papcr 88-2506, 1988.

‘ZPulliam, T . 11. and Chaussee, D. S., “A Diago- nal Form of An lrnplicit Approximate-Factorization Algo- rithm.” J o u r n a l of C o m p u t a t i o n a l Phys ics , Vol. 39,

-

gGuruswamy, G . , “Numerical Simulation of Vurt,iral

No. 2, 1981, pp. 347-363. - ‘3Ream, R. M. and Warming, R. F., “An Implicit Finite-Differcnce Algorithm for Hyperbolic Systems in Conscwation Law Form,” J o u r n a l of C o m p u t a t i o n a l Physics, Vol. 2 2 , 197G, pp. 87-110.

Baldwin, iJ.S., and Lomax, H., “Thin Layer A p ~ proximation and Algebraic Model for Separated Turhulent Flow,“ 1 1 4 A Paper 78.257, 1978.

Chaderjian, Neal M., “Transonic Navier-Stokes Wing Solutions Using a Zonal Grid Approach: Part2. High Angle-of-Attack Simulation,” AGARD Conf. Proc. No. 412, Appl ica t ions of C o m p u t a t i o n a l Fluid Dynam- ics in Aeronaut ics , Paper 300, No”. 1986; also N A S A TM 58248, April 1986.

’6Pullian~, T. H , and Steger, J . L., “Implicit Finite Difference Simulations of Three-Dimensional Compressible Flow,” A I A A J o u r n a l , Vol. 18, 1980, p. 159.

17Pnlliam, T. H,, “Artificial Dissipation Models for the Euler Equations,” AIAA Paper 55.0438, Jan. 1985.

“ S h i v a s a n , G.R. , Chyu, W.J., Steger, J.L., “Com- putation of Simple Three-Dimensiond Wing-Vortex Inter- action in Transonic Flow,” AIAA-81-120G, 1981.

’’ Flores, J., “Convergence Acceleration for a Three- Dimensional Euler/Navier-Stokes Zonal Approach,” A I A A J o u r n a l , Vol. 24, No. 9, Sept. 1986, pp. 1441-1442.

z‘ Chakravarthy, S. R., “Relaxation Methods for Un- factored Implicit Upwind Schemes,’’ A l A A Paper 84-0165, Jan. 1984.

z1 Flores, Jolen, Chaderjian, N e d M., Sormson, Reese I,., “Siniulatiun Of Transonic Viscous Flow Over A Fighter-Like Configuration Including Inlet,” AIAA Paper 57-1199, June 1987.

22 Luh, R. C. C., “Surface Grid Generation for Com- plex Three-Dimensional Geometries,” N A S A TM 101046, Oct. 1988.

23 Sorenmn, Reese L., “The 3DGRAPE Book: The- ory, Users’ Manual, Examples,” NASA TM 102224, July 1989.

2 4 Vinokur, M., “On One-Dimcnsional Stretching Functions for Finite Differrncr Calculations,” N A S A C R 3313,1980.

25Mal,ey, D.G., Welsh, B.L., and Pyne, C.R., “A Summary of Measurements of Steady and Oscillatory Pres- s u w s on a Rectangular Wing,“ T h e Aeronau t i ca l Jour- nal of t h e R o y a l Acronau t i ca l Soc ie ty , Jan. 1988.

7

A EXP (MABEY ET AL)

- - COARSE GRID (101 x 39 x 39)

. ~ MEDIUM GRID (151 x 39 x 39) d

__ FINE GRID (233 x 39 x 39)

1.0

e .v'.: e.sHo .n

/No 0

.n

-.5

XIC

a) 50% Seni~Span.

Fig. (1) A perspective view of a C - I i - n r d ~ grid for a N A C A 64A010 rcctairgolar wing.

1.0

-.5

X/C

b) 77% Sern-Span.

1.0

Fig. (2) Symmetry plane view of a two zone C-H-mesh grid for a NACA 64A010 rectangular wing. -.5

x/C

c) 94% Srmi-Span.

Fig. (3) Comparison of steady Navier-Stokes pressure c o ~ efficients (computed with different grid sizcs) and experi- rnent. M , = 0.80, a = 0.0", Re = 2.4 x IO6.

~-'

8

A EXP (MABEY ET AI,)

- . - L . A R G E ~ T ~t - 360 STEPS/(:YCI,E ....- ~. -Mb;I)IUM Ai ~+ 720 S'I'EPS/CYCLE

-~ -SMALLEST Ai - + 1080 STEPSiCYCLE

a) 50% Semi-Span.

b) 77% Semi-Span

-5 WC

E ) 04% Semi-Span

fcrent computnt iwal t i m e steps and experiment. M, = 0.80, v

= 1.0", k 7- 0.27, n( t ) -: -e,nazsin(M,ki).

A EXP (MABEY E?' AL) - EIJLER, At ~ + 720 STEPS/CYC:l.l?

~ - - NAVIER STOKES, At -+ 1440 STEPS/CYCI,E

a) 50% Semi-Span

h ) 77% Semi-Span

3 W K \~+-

c ) 94% Serni-Span.

Fig. ( 5 ) Comparison of Eulcr pressure coefficients w i t h Navier Stokes a n d experiment. M , :- 0.80, Rc = 2 . 4 ~ IO6, emol = L O " , k = 0.27, a ( t ) = n,,,sin(M,kt).

9

A EXP (MABEY E T AL)

ONE ZONE

.... TWO ZONES

5 XIC

a) 50% SemiLSpan

WC

b) 77% Semi-Span.

'OT

c ) 94% SerniGSpan.

k

Fig. (6) Comparison of pressure coefficients for a one zone Navier-Stokes simulation, a two zone Navier-Stokes sim- ulation, and experiment. M , = 0.80, Re = 2.4 x lo6 , amar = 1.0", k = 0.27, n ( t ) = -am,,zsin(M~kt).

Fig. (7) Instantaneous pressure coefficient contours a t the symmetry plane near amo.. Two zone Navier-Stokes com- putations with M , == 0.80, Re = 2.4 x l o6 , .: 2.0", k = 0.27, a ( ( ) = -n, , , ,sin(M,kt).

10