JOURNAL OF AIRCRAFTVol. 29, No. 3, May-June 1992

Transonic Navier-Stokes Computations for an Oscillating WingUsing Zonal Grids

Neal M. Chaderjian* and Guru P. GuruswamytNASA Ames Research Center, Moffett Field, California 94035

Modern jet transports and maneuvering tactical fighters operating in the transonic regime often give rise totime-dependent fluid physics that interact with flexible structural components, e.g., vortical flow, shocks, andseparation. Efficient computational fluid dynamic methods are required to study such computationally intensiveproblems. A numerical method is presented to address this problem. Time-dependent, compressible, Navier-Stokes equations are used to simulate unsteady transonic flow about a three-dimensional rigid wing undergoinga forced periodic motion in angle of attack. An efficient, implicit, diagonal algorithm is utilized because of itslow operation count per time step compared to other methods that solve systems of block matrix equations. Theformal time accuracy is theoretically addressed and numerically demonstrated by comparison of computationalresults with experimental data. A zonal grid approach, capable of treating complex geometries, is presentedand its time accuracy is demonstrated by comparing two- and three-zone computations with a single gridcomputation and experimental data.

Introduction

J ET transports and maneuvering tactical fighters tend tooperate in the transonic regime where the flowfield is

primarily subsonic with regions of embedded supersonic flow.This often gives rise to complex fluid physics such as steadyand time-dependent vortical flow, shocks, and separation. Ithas been observed that these phenomena interact with theflexible components of an aircraft and can lead to sustainedaeroelastic oscillations and divergence.1"3 The ability to pre-dict accurately the time-dependent flowfield and aeroelasticresponse of an aircraft is essential to insure the integrity andsafety of the vehicle and can help extend the maneuveringenvelop to modern tactical fighters.

Computational fluid dynamics (CFD) can play an importantrole in the design process by providing detailed flowfield andstructural displacement information. This will help reduce thedesign cycle time and provide information that is comple-mentary to wind-tunnel and flight-test data. At present, themost advanced CFD codes for predicting the aeroelastic re-sponse of wing/fuselage configurations are based on potentialflow theory.4 Even though these equations have a limitedrange of applicability, they have been used because the morecomplete Navier-Stokes equations require significantly morecomputer memory and computer time. Navier-Stokes equa-tions, when coupled with the governing aeroelastic equations,can require two or more orders of magnitude of computertime. However, recent advances in numerical algorithms andcomputer technology make it possible to consider using theseequations in the research environment and eventually in thedesign environment.5-6 The literature confirms this trend, asseveral time-dependent, three-dimensional, Navier-Stokessolutions for wing geometries have begun to emerge, e.g.,Refs. 7 and 8, as well as aeroelastic computations, e.g., Refs.9 and 10. It is our eventual goal to simulate the aeroelastic

Presented as Paper 90-0313 at the AIAA 28th Aerospace SciencesMeeting, Reno, NV, Jan. 8-11, 1990; received March 26, 1990; re-vision received April 9,1991; accepted for publication April 17,1991.Copyright © 1991 by the American Institute of Aeronautics andAstronautics, Inc. No copyright is asserted in the United States underTitle 17, U.S. Code. The U.S. Government has a royalty-free licenseto exercise all rights under the copyright claimed herein for Govern-mental purposes. All other rights are reserved by the copyright owner.

* Research Scientist, Applied Computational Fluids Branch. Mem-ber AIAA.

tResearch Scientist, Applied Computational Fluids Branch. As-sociate Fellow AIAA.

response of a complete aircraft with the Navier-Stokes equa-tions. However, as a first step toward this goal, we will addresssome accuracy and efficiency issues regarding the use of anEuler/Navier-Stokes code for predicting the time-dependent,three-dimensional, flowfield about a rigid wing undergoing aforced periodic motion in angle of attack.

In order to simulate time-dependent viscous flow about arealistic aircraft geometry, it is important for a Navier-Stokescode to be able to treat complex geometries, have an efficientand robust numerical algorithm, and be time accurate. TheTransonic Navier-Stokes (TNS) code has been chosen as abase code because of its success in predicting the steady tran-sonic viscous flow about the complete F-16A fighter aircraft.11

This code uses a zonal grid approach for treating the geometriccomplexity of the aircraft. The flowfield is subdivided into anensemble of simply shaped grids. Each of these grids can berefined by adding clustered grid points as appropriate to ad-equately capture viscous effects near all body surfaces andwake regions. Flowfield coupling between zones is accom-plished by overlapping adjacent zonal grids. Moreover, thiscode uses an efficient and robust implicit approximate-fac-torization algorithm. This algorithm, due to Pulliam andChaussee,12 is a diagonal version of the Beam-Warmingalgorithm13 and uses central differencing. The operation countper time step is therefore low compared to most upwind schemesand methods requiring the solution of a system of block matrixequations. However, the TNS code has only been used forsteady flow computations.

The Navier-Stokes Simulation (NSS) code is a new andextended version of the TNS code for treating time-dependentflows. It allows for a moving zonal grid system with time-dependent coordinate transformation metrics. A Fourier-analysis routine has also been included to facilitate compar-ison of computed surface pressure, force, and moment coef-ficients with experimental data.

The Pulliam-Chaussee diagonal algorithm is conservativein space but not conservative in time. This can affect the shockspeed in a transonic computation. The form of this temporalconservation error will be presented in a later section, as wellas ways to remove the error. The objective of this researchis to verify the time accuracy of the NSS code algorithm andzonal boundary treatment by comparing computational resultswith experimetnal data for transonic flow over an oscillatingthree-dimensional wing with moving shocks.

In the following sections, a theoretical background is givendescribing the governing equations, turbulence model, nu-

326

CHADERJIAN AND GURUSWAMY: OSCILLATING WING 327

merical algorithm, and formal time accuracy. This is followedby a description of the zonal grid approach, grid generation,boundary conditions, results, and concluding remarks.

Theoretical Background

Governing EquationsThe compressible Reynolds-averaged Navier-Stokes equa-

tions are used to numerically simulate time-dependent viscousflow about a wing with oscillating angle of attack. For thehigh Reynolds number flows considered in this paper, thethin-layer approximation is employed together with body-fitted curvilinear coordinates. This improves the efficiency ofthe numerical simulation and simplifies the implementationof boundary conditions. Assuming a general coordinate trans-formation from physical space (x,y,z) to computational space(f,77,f) and that viscous terms are only important in the (-coordinate direction, the governing equations can be ex-pressed in the following strong conservation-law form:

drQ (1)where

* = 7

ppupvpwe

PVPUV 4 T]zp

pvV 4 rjxppwV 4 T7zp

V(e 4 p) - 77,

pUpuU 4pv£/ 4pwU 4

tf(* + p)

G = T

pwpwiy 4 ^ppvW 4 £yppwW 4 ̂

W(* 4 p) - (,

Hwith

3 = 0.5(w2 4 v2 4

04 (M/3)m2£4 Gn/3)/n2&4 (M/3)m24M4" + ^v

4 Pr-\y -

In Eq. (1), the contravariant velocity components are definedby

= 4

W = fxu

(2a)

(2b)

(2c)

and the pressure is related to the other flow variables by theperfect gas law

p = (y - l)[e - \ p(u2 + v2 4- w2)]

The transformation metrics are defined as77, - J(z€yc -

(3)

(4a)

(4b)

(4c)

(4d)

(4e)

and the coordinate-tranformation Jacobian by1/7 - XffaZf - z^y^) + X^ZM - y€zc)

(4f)

(5)

Turbulence ModelFluid turbulence is modeled by the Baldwin-Lomax alge-

braic eddy viscosity model.14 This isotropic model is usedprimarily because it is computationally efficient. All viscouscomputations presented in this paper assume fully turbulentflow at the leading edge of the wing, i.e., transitional effectsare ignored. This approximation is consistent with the highReynolds number assumption.

Numerical AlgorithmAn implicit approximate-factorization diagonal algorithm

is used in the NSS code to integrate Eq. (1). This algorithmis based on the Beam-Warming algorithm13:

(/ 4 hSfA - hDt\€Y(I + hd^B - hD^Y

(6)where

Rn = - DeQ)"

and

- Q"In Eq. (6), the superscript n corresponds to the current timelevel t" and n 4 1 to the new time level r^ 4 Ar. Also, A, B,C, and M are the Jacobian matrices of E, F, G, and 5, re-spectively. The numerical metrics are evaluated so that uni-form flow is an exact solution of the steady finite differenceequation (see Ref. 15). The Beam- Warming algorithm is sec-ond-order accurate in time when h = 1/2 Ai (trapezoidal rule)and first-order accurate in time when h = At (Euler implicit).The spatial operators use central differencing throughout, sofourth-order explicit De and second-order implicit Di numer-ical dissipation terms are added to damp high frequency er-rors. One drawback to this method, however, is the requiredsolution of a block tridiagonal system of equations, which iscomputationally costly. For more details of this algorithm,see Pulliam and Steger.16

In order to reduce the overall execution time of the NSScode, a more efficient diagonal form of the Beam- Warmingalgorithm is used. This algorithm, due to Pulliam and Chaus-see,12 given is by

(PY(l 4 (7)

where A^ is a diagonal matrix consisting of the eigenvaluesof A, and so on. Note that the viscous terms are treatedexplicitly. The right side of Eq. (7) is identical to the rightside of Eq. (6). This algorithm uses both explicit and implicitfourth-order dissipation and only requires the solution of sca-lar pentadiagonal equations. This results in a lower operationcount per time step and, hence, a reduced execution time persolution. Further details of this algorithm, including a de-scription of the TA, N, and P matrices, are given in Refs. 12and 17.

When a steady-state solution is desired, the rate of con-vergence can be greatly improved by using the spatially vary-ing time step described by Srinivasan et al.18 and is given by

I 4 (8)

328 CHADERJIAN AND GURUSWAMY: OSCILLATING WING

where Af0 is a user specified constant. Flores,19 using thediagonal algorithm with variable time step, reported conver-gence rates 40 times those obtained using the Beam- Warmingalgorithm with constant time step. A constant time step isused for time-accurate computations.

All of the computations reported here use the implicit di-agonal algorithm. In its present coded form (NSS code), theimplicit operator only requires 70% more computer time pertime step than the explicit operator. Yet the implicit schemehas been used with a maximum Courant number of about900. Therefore, this algorithm is far more efficient (overallcomputer time) than its explicit counterpart. However, dueto its diagonal form, this algorithm is first-order^ accurate intime and temporally nonconservative. These two issues areaddressed in the next section and in the Results section.

Time AccuracyThe strong conservation-law form of the governing equa-

tions, Eq. 1, when properly differenced, conserves numericalfluxes across flow discontinuities. These so called weak so-lutions, i.e., piece wise continuous solutions, satisfy the properintegral relations irrespective of the grid spacing or time step.If a nonconservative form is used, the shock strength, posi-tion, and wave speed will be dependent on the grid spacingand time step.

Although the diagonal algorithm, Eq. (7), is conservativein space, it is not conservative in time. Therefore, time-accurate computations involving shocks can be questionable.However, since the explicit side is differenced conservatively,one might expect the temporal error introduced in the implicitoperator to be nominal for a variety of applications with mov-ing shocks. This will be demonstrated computationally in theResults section.

This section will address the form of the conservation errorand the formal time accuracy of the algorithm. Two ap-proaches for removing this error will also be presented in theevent that temporal conservation errors prove to be importantfor a specific application.

The temporal conservation error of the diagonal algorithmcan be found by going back to its derivation from the Beam-Warming algorithm. Replacing the Jacobian matrices of Eq.(6) with

B =

C =and dropping the implicit viscous terms gives

x (TcTcl + = R"(9)

In Eq. (9), A^ is a diagonal matrix whose elements are theeigenvalues of A and TA is the matrix whose columns are theeigenvectors of A . Similar terms apply for the B and C mat-rices. Spatial finite difference operators have been replacedby differential operators to avoid unnecessary details that donot affect the analysis. Application of the chain rule resultsin

and collecting terms give the following form for the Beam-Warming algorithm:

hDt

+ hD^Y x (/>)"(/ + H

where

" + higher order terms

~ O(A<2")

The Beam-Warming algorithm can therefore be expressed asthe diagonal algorithm with an additional term. This extraterm e" corresponds to the temporal conservation error of thediagonal algorithm. Now,

dOnA(> = Qn + l - Q» = -Z- fa .

so that

e" ~ 0(M) (12)

The diagonal algorithm is formally first-order accurate in time,even if trapezoidal rule is used. (This algorithm is also limitedto first-order time accuracy due to the explicit treatment ofthe viscous terms.) This error e" can be important if the shockstrength and speed become too large for a given time step.However, this error can be controlled by lowering the timestep for a given grid. This provides a method for checking ifthis error is important for a specific application. We alsoremark that the time step used in Navier-Stokes computationsare generally small because of stability considerations, hencethe temporal conservation error should be small for a largeclass of applications.

There is another way of showing that the conservation errorvanishes with decreasing time step. As the time step h — > 0,the dominant term in the implicit operator (see Eq. 7) is theidentity matrix. This method approaches the Euler explicitalgorithm for very small time steps, which is fully conserva-tive.

Another method for removing the temporal conservationerror is to use a Newton iteration method described by Chak-ravarthy.20 The diagonal algorithm then has the form

(TAy(i(Py(i= -(& - Qn) (13)

= R" (10)

In this expression, AQ1' -> 0 and Q -> Qn + l as / becomeslarge. This usually requires only a few subiterations to con-verge to the new time level. This approach not only removesthe conservation error, but also removes the approximate-factorization error and makes the scheme fully implicit. Thisapproach, however, can significantly increase the computa-tional time of the code.

Zonal ApproachGenerating a single grid about a complex geometric shape,

e.g., a fighter aircraft, with appropriate viscous clustering nearall body surfaces and wakes can be a formidable (if not im-possible) task. This difficulty can be overcome by adopting azonal grid approach. The NSS code uses a zonal grid proce-dure that subdivides the flowfield about a complex geometry

CHADERJIAN AND GURUSWAMY: OSCILLATING WING 329

Fig. 1 Perspective view of a C-H-mesh grid for a NACA 64A010rectangular wing.

Fig. 2 Symmetry plane view of a two-zone C-H-mesh grid for a NACA64A010 rectangular wing. One cell overlap between zones is shaded.

Fig. 3 Symmetry plane view of a three-zone C-H-mesh grid for aNACA 64A010 rectangular wing. One cell overlap between zones isshaded.

into an ensemble of simple geometric shapes. Adjacent zonesoverlap with one another, typically by one or two grid cells,to couple the flowfields between zones. These zones are con-structed from a coarse global grid, and so they have a commoninterface surface. Each zonal grid is then clustered and refinedwith additional grid points as appropriate to accurately cap-ture the pertinent flow physics. This procedure ensures thatat an interface surface, the grid points of the coarser grid arecoincident with grid points of the finer grid. Flow variableinformation is transferred from the finer grid to the coarsergrid by direct injection. This step is fully conservative. Flowvariable information is transferred from the coarser grid tothe finer grid by interpolation along grid lines. This step isnot conservative. When conservation between zones is im-portant, e.g., when a shock passes through a zonal boundary,both zones are required to have identical grid points at theinterface. This will insure complete flow conservation acrossthe zonal boundary.

The flowfield is advanced to a new time level one zone ata time. The most recent data available at a zonal interfaceare used. Therefore, some zonal boundaries are updated ex-plicitly, whereas others are updated implicitly. Any effectsrelated to the explicit treatment of some zonal boundarieswill vanish as the time step is decreased. A comparison be-tween single, two-, and three-zone flow computations is givenin the Results section. For more details of the zonal gridprocedure and interface boundary treatment see Flores et al.21

Grid GenerationAll computations presented in this paper use a C-H grid

topology. A perspective view of a medium density grid isshown in Fig. 1. In particular, the surface geometry and partof the symmetry plane grid are highlighted. This grid consistsof 151 x 39 x 39 grid points (about 230,000). An additionalplane has been added inboard of the symmetry plane to fa-cilitate the symmetry plane boundary conditions.

The grid shown in Fig. 1 was constructed in the followingmanner. First, a surface grid for the wing was generated withthe S3D code.22 Additional grid clustering was provided nearthe wing nose and shocks. Then, a three-dimensional grid wascreated with 3DGRAPE,23 an elliptic grid generation pro-gram, with normal spacing to the wing surface appropriatefor inviscid computations. Finally, viscous clustering was ob-tained by redistributing grid points along coordinate lines nor-mal to the body with a clustering function described by Vi-nokur.24 This one-dimensional stretching function provides adistribution of points that minimize the truncation error of afinite difference code.

A two-zone grid was also constructed by splitting the single-zone grid, see Fig. 1, in the £ direction (body normal). Asymmetry plane view is shown in Fig. 2 with the shaded regionindicating the one cell overlap between zones. Zone 1 is ad-jacent to the wing surface and zone 2 extends from zone 1 tothe far field, about six chord lengths away. The grid pointsat the zone 1 and zone 2 interfaces match up exactly. Thiswas done in order to study the effect of the zonal boundarytreatment on time accuracy without the additional compli-cation of flow variable interpolation at zonal interfaces.

A three-zone grid was constructed from the two-zone gridby splitting zone 2 in the £ direction, see Fig. 3. The thirdzone is the outermost zone connecting zone 2 to the far field.Notice that every other grid point in the f direction (stream-wise) has been removed. This was done in order to demon-strate the capability of the NSS code to test grid interfacesthat do not have identical grid points.

Boundary ConditionsThe NSS code solves the Euler or Navier-Stokes equations

depending on the specific application. For inviscid flow, flowtangency is imposed on the wing surface while density andpressure are found by extrapolation. The total energy per unitvolume is then computed from the perfect gas law. The same

330 CHADERJIAN AND GURUSWAMY: OSCILLATING WING

boundary conditions apply for viscous flow except the no-slipcondition replaces flow tangency on the wing surface. Uni-form flow is imposed at the far field, and zero gradient con-ditions are used at the outflow boundaries. Boundary valuesalong the C cut (downwind of the wing trailing edge andoutboard of the wingtip) are obtained by averaging flow var-iables across the cut. Flow symmetry is obtained by reflectingthe flow variables about the symmetry plane in an explicitmanner.

The zonal interface boundary conditions have already beendescribed in the Zonal Approach section.

ResultsWind-Tunnel Model

All flow computations presented in this section correspondto a rectangular half-wing used in an AGARD wind-tunnelexperiment described by Mabey et al.25 The wing model wasmounted on a fuselage-like body to displace it slightly fromthe wind-tunnel wall and its boundary layer. No experimentaldata was taken at the wind-tunnel wall (wing symmetry plane).The wing's cross section consists of a NACA 64A010 airfoilwhose thickness/chord ratio has been modified to 10.6%. Thewing has a complete aspect ratio = 4.0. The experiment wasconducted in the Royal Aeronautical Establishment 8- x 8-ft tunnel with closed walls. (The model was originally designedfor a smaller tunnel that was not available when the experi-ment was conducted.) Various C-H grids were generated forthis wing, see for example, Fig. 1. No attempt was made tomodel the wind-tunnel test section.

Steady Flow ComputationsSteady-state flow computations with different grid densities

are used to study their effect on the spatial accuracy of theNSS code. The three grids that are used are 1) a coarse grid(101 x 39 x 39), 2) a medium grid (151 x 39 x 39), and3) a fine grid (233 x 39 x 39). Each grid has the same gridpoint distribution in the spanwise and body-normal directions(77 and g directions). The streamwise distribution of points (£direction) downwind of the wing trailing edge are also iden-tical for all three grids (17 grid cells to the far field). However,each grid has 50% more points on the wing surface (in thestreamwise direction) than its previous coarser grid. Grid pointsare clustered near the wing leading edge and shock locationsto improve the flowfield resolution on each grid. The gridspacing in the viscous direction is Af = 5.0 x 10 ~6 chords.

Figure 4 shows a comparison between steady Navier-Stokespressure coefficients Cp computed on the three grids just de-scribed with experimental values at 17 = 50, 77, and 94%semispan locations. These computations correspond to afreestream Mach number Mx = 0.80, angle of attack a =0.0 deg, and a freestream Reynolds number based on thewing root chord Re = 2.4 x 106. At 17 = 50%, Fig. 4a, thecomputed pressure coefficients compare well with each otherand the experimental data, with the exception that the coarsegrid shock is slightly smeared. The suction peaks are alsoslightly underpredicted, presumedly due to wind-tunnel walleffects. All three grids have good Cp comparisons with eachother and experiment at 17 = 77%. At 17 = 94%, computa-tions with all three grids are in good agreement with eachother, but their suction peaks differ somewhat from the ex-perimental value. This is probably due to the use of a roundedwingtip in the computation vs a blunt (flat) tip in the wind-tunnel model. Overall, the medium and fine grids give thebest Cp comparisons with the experiment and do not differnoticeably from each other. Therefore, the medium grid den-sity (about 230,000 grid points) is used for the remainder ofthe viscous computations that are presented.

Unsteady Flow ComputationsTime-accurate Euler and Navier-Stokes computations for

a rigid wind with oscillating angle of attack are now presented.

A Exp (Mabey et al.)- Coarse grid (101 x 39 x 39)

- Medium grid (151 x 39 x 39)- Fine grid (233 x 39 x 39)

'A A A

r"

-.5(c) 94% Semi-span

.5X/C

1.0

Fig. 4 Comparison of steady Navier-Stokes pressure coefficients(computed with different grid sizes) and experiment—Mx = 0.80,a = 0.0 deg, Re = 2.4 x 106: a) 50% semispan; b) 77% semispan;c) 94% semispan.

Unless otherwise noted, the freestream Mach number is M^= 0.80 and the Reynolds number based on wing chord is Re= 2.4 x 106. The angle of attack varies periodically accordingto

where the amplitude is amax = 1.0 deg, the reduced frequencyis k = 0.27, and t is a nondimensional time based on wingchord and the freestream speed of sound. The reduced fre-quency is defined by k = a)C/Ux, where to is the circularfrequency (rad/s), c the wing chord length, and Ux the free-stream velocity. The computational grid undergoes a rigidbody rotation according to the angle-of-attack formula. Realand imaginary unsteady pressure coefficients are used to com-pare computational and experimental results. These in-phase

CHADERJIAN AND GURUSWAMY: OSCILLATING WING 331

10

oi

A Exp (Mabey et al.)— - - - Largest At -*• 360 steps/cycle— — Medium At -+- 720 steps/cycle——— Smallest At -̂ 1080 steps/cycle

otuQC

-5

(c) 94% Semi-span

10

.5x/c

1.0 .5X/C

1.0

Fig. 5 Comparison of Euler pressure coefficients with different computational time steps and experiment—Mx = 0.80, amax = 1.0 deg, k =0.27, a(t) = -amaxsin(Mxkt): a) 50% semispan; b) 17% semispan; c) 94% semis pan.

and out-of-phase pressures correspond to the time-dependentFourier coefficients normalized by amax (rad).

The time steps used in Navier-Stokes computations are usu-ally determined by stability considerations rather than timeaccuracy. Therefore, the Euler equations are used to deter-mine the time-step requirement for time accuracy with thefirst-order accurate diagonal algorithm. For these in viscidcomputations, the medium density viscous grid in Fig. 1 isunclustered by redistributing 21 grid points along coordinatelines normal to the wing surface. The inviscid and viscousgrids have identical spacing in the spanwise and streamwisedirections. Figure 5 compare computed real and imaginarypressures for three different time steps with experimental dataat three semispan locations. These time steps, beginning fromthe largest to the smallest, are 360, 720, and 1080 time steps/cycle, respectively. Pressure coefficients computed with themedium and smallest time steps are virtually indistinguishablefrom one another. However, significant differences between

the largest time step (360 steps/cycle) and the smaller timesteps can be noted, especially at 17 = 77 and 94%. Therefore,720 time steps/cycle is the largest usable time step for timeaccuracy.

The real and imaginary pressure coefficients for the Eulerequations, Navier-Stokes equations, and experimental dataare compared in Fig. 6. These pressures correspond to thesame freestream conditions and reduced frequency in Fig. 5.At 50% semispan location, the real Cp shock position andstrength computed by the Navier Stokes equations are in goodagreement with the experiment. Reducing the time step inhalf or using a two-step Newton subiteration has no perceiv-able effect on any of the pressure coefficients. This indicatesthat the temporal conservation error is negligible for this tran-sonic-flow example. It is evident from this figure that theshock moves about 25% of a chord length during a completecycle. This provides a good test for the time accuracy of thediagonal algorithm. The Euler equations, on the other hand,

332 CHADERJIAN AND GURUSWAMY: OSCILLATING WING

10

OuiDC

(a) 50% Semi-span

10

5 -

A Exp (Mabey et al.)- - - - Euler, At -*• 720 steps/cycle——— Navier Stokes, At -+• 1440

steps/cycle

5 -

5 -

.5X/C

1.0

Fig. 6 Comparison of Euler pressure coefficients with Navier Stokes and experiment—Mx = 0.80, Re = 2.4 x 106, amax = 1.0 deg, k = 0.27,a(0 = - crmaxsin(M xkt): a) 50% semis pan; b) 77% semispan; c) 94% semispan.

predict a much stronger shock with a slightly downwind po-sition than the experiment and the Navier-Stokes equations.Even for this relatively easy case that does not have anysignificant separation, the Euler equations would predict thewrong wave drag. The drop in pressure before the shock isnot predicted very well by either the Euler or Navier-Stokesequations. This is attributed to inadequate grid resolution.Although the clustered grid gave good shock resolution forthe steady-state case, there is inadequate grid resolution inthe shock excursion region. With hindsight, it is obvious thattime-dependent flows involving moving shocks require greatergrid resolution than steady-state flows. This can be accom-plished by adding more grid points in the shock excursionregion or using a solution adaptive technique. The latter wouldbe more efficient.

The real and imaginary pressure coefficients computed bythe Euler and Navier-Stokes equations compare well witheach other and the experimental data for 17 = 77 and 94%semispan locations. The time step used in the Navier-Stokes

computations at Af = 0.02, or 1400 steps/cycle. Although thistime step was chosen to maintain code stability, it is only onehalf of the time step required for time accuracy in the Eulercomputations. This is very encouraging, especially since theviscous grid spacing normal to the wing was two orders ofmagnitude smaller than the Euler grid spacing. This indicatesthe robustness of the diagonal algorithm.

In order to verify the time accuracy of the zonal boundaryconditions, the single grid, shown in Fig. 1, is split into twozones, shown in Fig. 2. The first zone is close to the wingsurface, whereas the second zone extends from the first zoneto the far field. The Navier-Stokes equations are solved inthe first zone and the Euler equations are solved in the secondzone. The zones overlap by one grid cell as indicated by theshaded region in Fig. 2. The interface surfaces between thezones have identical grid points. The zonal overlap positionwas determined as the closest inviscid position, i.e., wherethe turbulent eddy viscosity is zero. The close proximity ofthe zonal interface to the wing surface and leading edge pro-

CHADERJIAN AND GURUSWAMY: OSCILLATING WING 333

10

(a) 50% Semi-span

10A Exp (Mabey et al.)

——— One zone- - - - Two zones......... Three zones

oiuoc

(c) 94% Semi-span

.5X/C

1.0 .5X/C

1.0

Fig. 7 Comparison of pressure coefficients for one-, two-, and three-zone Navier-Stokes computations with experiment—Mx = 0.80, Re = 2.4x 106, amax = 1.0 deg, k = 0.27, a(0 = -amaxsin(AOO: a) 50% semispan; b) 77% semispan; c) 94% semispan.

vides an extreme test for time accuracy of the zonal boundarycondition treatment. A three-zone grid, shown in Fig. 3, isalso used to demonstrate the capability of the NSS code totreat zonal interfaces in which all grid points do not coincide.Zone 3, the outer grid, has every other grid point removedin the streamwise direction. The Navier-Stokes equations areapplied in zone 1, near the body, and the Euler equations inthe other two zones, away from the body.

Figure 7 compares the real and imaginary pressure coeffi-cients for the single-, two-, and three-zone Navier-Stokescomputations with the experimental data. The single-zonecomputation applied the Navier-Stokes equations and Eulerequations in the same region as the zonal cases. Once again,the time step used for these viscous computations was Af =0.02. Overall, the comparison of the single- and two-zonecomputations are very close. Slight differences can be notedin the imaginary pressure coefficients. Reducing the time stepin half does not alter the results. This indicates that theseslight differences are not due to the partially explicit zonal

boundary conditions. These differences are more likely dueto differences in numerical dissipation. Although the diagonalalgorithm uses fourth-order dissipation, it must drop to secondorder near grid boundaries. This means that the single-zonecomputation uses fourth-order dissipation in the zonal overlapregion, whereas the two-zone computation uses second-orderdissipation there. The close proximity of the zonal boundaryto the wing surface and leading edge can easily account forthese differences. Note that both solutions compare well withthe experimental data. The real and imaginary Cp for thethree-zone computation differs only slightly from the two-zone computation at the 50% semispan location. They arevirtually identical at the other two semispan locations. Theseslight differences are most likely due to the different trun-cation errors of the zonal grid systems. Recall that zone 3, ofthe three-zone system, is twice as coarse as zone 2, of thetwo-zone system (see Figs. 2 and 3). Overall, the zonal gridapproach is capable of simulating time-dependent flow witha moving grid system.

334 CHADERJIAN AND GURUSWAMY: OSCILLATING WING

10-6

10~7

10"8

Steady Unsteady1st cycle

Unsteady2nd cycle

Unsteady3rd cycle

ZonelZone 2

1200 2640Time step

4080 5520

Fig. 8 Convergence history of steady and unsteady three-zone Navier-Stokes computation—Mx = 0.80, Re = 2.4 x IO6, amax = 2.0 deg,k = 0.27, a(t) = -a

Fig. 9 Instantaneous pressure coefficient contours at the symmetryplane near amax. Two-zone Navier-Stokes computations with Mx =0.80, Re = 2.4 x 106, omax = 2.0 deg, k = 0.27, a(t) =

Figure 8 shows the convergence history of the three-zonecomputation. A steady solution was first obtained for themean angle of attack (a = 0.0 deg). A reduction of the L2norm of the residual by four orders of magnitude was obtainedin 1200 time steps. A variable time step was used, accordingto Eq. (8), to accelerate the rate of convergence. The steadysolution was then used as a starting point for the time-accurateoscillating wing case. The flow was completely periodic in thesecond cycle. This was verified by computing a third cycleand comparing unsteady pressure coefficients. When oper-ating in steady-state mode, the NSS code requires about 14/i-s/grid point/time step on a Cray Y-MP supercomputer. Whenoperating in a time-accurate mode, with a forced periodic gridmotion, it requires about 17 /x-s/grid point/time step. Overall,a complete time-accurate computation for the present ex-ample requires about 4 h of Cray Y-MP time. This includesthe steady solution and two cycles of forced periodic motion.

A final example of a time-accurate zonal computation isshown in Fig. 9. The same freestream conditions and grids ofthe previous two-zone example are used, except now amax =2.0 deg. Although experimental data is not available for thiscase, the stronger shock provides a qualitative check of thezonal boundary treatment. Figure 9 indicates instantaneouspressure coefficient contours when the upper surface shockis its strongest. The zonal overlap region is also indicated inthe figure. Notice that the Cp contours are extremely smoothacross the zonal boundaries, even near the shock and the wingleading edge. This shows that there is no significant flowdisturbances introduced by the zonal boundary conditions.

ConclusionsThe zonal, steady-state, Transonic Navier-Stokes code has

been extended to treat unsteady flows. This new Navier-StokesSimulation code has the capability of treating a moving zonalgrid system. The Pulliam-Chaussee diagonal algorithm hasbeen shown to be an efficient and robust time-dependentalgorithm. The NSS code requires approximately 17 ^t-s/gridpoint/time step on a Cray Y-MP supercomputer for a time-dependent flow with a forced periodic grid motion. A typicaltime-accurate result for 230,000 grid points requires about 4h of computer time. Although this algorithm is conservativein space, it has a temporal conservation error 0(Af). SinceNavier-Stokes algorithms require small time steps for nu-merical stability, this error should be small for a large classof transonic applications.

The NSS code has been used to simulate unsteady transonicviscous flow about a three-dimensional rectangular wing withoscillating angle of attack. Computed real and imaginary pres-sure coefficients compared well with the experimental valuesand demonstrates the time accuracy of the diagonal algorithm.The temporal conservation error was shown to have no no-ticeable effect on the speed, strength, and position of theshock wave (for the present computations) through reductionof the time step and application of a Newton-iteration pro-cedure. Two- and three-zone transonic computations for theoscillating wing compared well with the single-grid compu-tation and the experimental data. This demonstrates the time-accurate capability of the zonal interface procedure. The zonalgrid approach provides a way of simulating time-dependentflow about a complete aircraft with the Navier-Stokes equa-tions.

References'Guruswamy, G. P., Goorjian, P. M., Ide, H., and Miller, G. D.,

"Transonic Aeroelastic Analysis of the B-l Wing," Journal of Air-craft, Vol. 23, No. 7, 1986, pp. 547-533.

2Farmer, M. G., and Hanson, P. W. "Comparison of Supercriticaland Conventional Wing Flutter Characteristics," NASA TM X-72837,May 1976.

3Ashley, H., "Role of Shocks in the Sub-Transonic Flutter Phe-nomenon," Journal of Aircraft, Vol. 17, No. 3, 1980, pp. 187-197.

4Guruswamy, G. P., and Goorjian, P. M., "Unsteady TransonicFlow Simulation on a Full-Span Wing-Body Configuration," AIAAPaper 87-1240, Jun. 1987.

5Kutler, P. and Gross, A. R., "CFD in Design: A GovernmentPerspective," AIAA Paper 89-0094, Jan. 1989.

6Goldhammer, M. I., and Rubbert, P. E., "CFD in Design: AnAirframe Perspective," AIAA Paper 89-0092, Jan. 1989.

7Nakamichi, J., "Calculations of Unsteady Navier-Stokes Equa-tions Around an Oscillating 3-Wing Using Moving Grid System,"AIAA Paper 87-1158, Jun. 1987.

8Srinivasan, G. R., and McCroskey, W. J., "Navier-Stokes Cal-culations of Hovering Rotor Flow Fields," Journal of Aircraft, Vol.25, No. 10, 1988, pp. 865-874.

9Guruswamy, G., "Numerical Simulation of Vortical Flows onFlexible Wings," AIAA Paper 89-0537, Jan. 1989.

10Guruswamy, G., "Vortical Flow Computations on Swept FlexibleWings Using Navier-Stokes Equations," AIAA Paper 89-1183, Apr.1989.

HFlores, J., and Chaderjian, N. M., "A Zonal Navier-Stokes Meth-odology for Flow Simulation About a Complete Aircraft," Journalof Aircraft, Vol. 27, No. 7, 1990, pp.583-590.

12Pulliam, T. H., and Chaussee, D. S., "A Diagonal Form of AnImplicit Approximate-Factorization Algorithm," Journal of Com-putational Physics, Vol. 39, No. 2, 1981, pp. 347-363.

13Beam, R. M., and Warming, R. F., "An Implicit Finite-Difference Algorithm for Hyperbolic Systems in Conservation LawForm," Journal of Computational Physics, Vol. 22, No. 1, 1976, pp.87-110.

14Baldwin, B. S., and Lomax, H., "Thin Layer Approximation andAlgebraic Model for Separated Turbulent Flow," AIAA Paper 78-257, Jan. 1978.

15Chaderjian, N. M., "Transonic Navier-Stokes Wing SolutionsUsing a Zonal Grid Approach: Part 2. High Angle-of-Attack Sim-ulation," AGARD CP 412, Applications of Computational Fluid Dy-

CHADERJIAN AND GURUSWAMY: OSCILLATING WING 335

namics in Aeronautics, Paper 30B, Nov. 1986; also NASA TM 88248,April 1986.

16Pulliam, T. H., and Steger, J. L., "Implicit Finite DifferenceSimulations of Three-Dimensional Compressible Flow," AIAA Jour-nal, Vol. 18, No. 2, 1980, p. 159.

17Pulliam, T. H., "Artifical Dissipation Models for the Euler Equa-tions," AIAA Paper 85-0438, Jan. 1985.

18Srinivasan, G. R., Chyu, W. J., and Steger, J. L., "Computationof Simple Three-Dimensional Wing-Vortex Interaction in TransonicFlow," AIAA Paper 81-1206, Jun. 1981.

19Flores, J., "Convergence Acceleration for a Three-DimensionalEuler/Navier-Stokes Zonal Approach," AIAA Journal, Vol. 24, No.9, 1986, pp. 1441-1442.

20Chakravarthy, S. R., "Relaxation Methods for Unfactored Im-

plicit Upwind Schemes," AIAA Paper 84-0165, Jan. 1984.21Flores, J., Chaderjian, N. M., and Sorenson, R. L., "Simulation

of Transonic Viscous Flow Over Fighter-Like Configuration Includ-ing Inlet," AIAA Paper 87-1199, June 1987.

22Luh, R. C. C., "Surface Grid Generation for Complex Three-Dimensional Geometries," NASA TM 101046, Oct. 1988.

23Sorenson, R. L., "The 3DGRAPE Book: Theory, Users' Man-ual, Examples," NASA TM 102224, July 1989.

24Vinokur, M., "On One-Dimensional Stretching Functions forFinite Difference Calculations," NASA CR 3313, Oct. 1980.

25Mabey, D. G., Welsh, B. L., and Pyne, C. R., "A Summary ofMeasurements of Steady and Oscillatory Pressures on a RectangularWing," Aeronautical Journal of the Royal Aeronautical Society, Vol.92, No. 911, Jan. 1988, pp. 10-28.

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