Determination of Drag From Three-DimensionalViscous and Inviscid Flowfield Computations

David L. Hunt*Aircraft Research Association

Bedford, UK

Russell M. CummingsfCalifornia Polytechnic State University

San Luis Obispo, CA, USA

Michael B.Oxford University Computing Laboratory

Oxford, UK

Abstract

A momentum balance approach is used to extract the dragfrom flowfield computations for wings and wing/bodiesin subsonic/transonic flight. The drag is decomposed intovorticity, entropy, and enthalpy components which can berelated to the established engineering concepts of induceddrag, wave and profile drag, and engine power andefficiency. This decomposition of the drag is useful informulating techniques for accurately evaluating dragusing computational fluid dynamics calculations orexperimental data. A formulation for reducing the size ofthe region of the crossflow plane required for calculatingthe drag is developed using cut-off parameters forviscosity and entropy. This improves the accuracy of thecalculations and decreases the computation time requiredto obtain the drag results. The improved method isapplied to a variety of wings, including the M6, W4, andMl65 wings, Lockheed Wing A, a NACA 0016 wing, andan Elliptic wing. The accuracy of the resulting dragcalculations is related to various computational aspects,including grid type (structured or unstructured), griddensity, flow regime (subsonic or transonic), boundaryconditions, and the level of the governing equations(Euler or Navier-Stokes). The results show that dragprediction to within engineering accuracy is possibleusing computational fluid dynamics, and that numericaldrag optimization of complex aircraft configurations ispossible.

* Project Supervisor, CFD Research Group,email: dhunt@ara.co.uk. Member AIAA.

t Professor, Aeronautical Engineering Department,email: rcumming@calpoly.edu. Associate Fellow, AIAA.

$ Rolls-Royce Reader in CFD,email: giles@comlab.ox.ac.uk. Member AIAA.

Copyright © 1997 by D.L. Hunt, R.M. Cummings, andM.B. Giles. Published by the American Institute ofAeronautics and Astronautics, Inc. with permission.

Introduction

As computational fluid dynamics (CFD) has matured overrecent years, it has become a goal of CFD researchers tobe able to predict aerodynamic drag from numericalsimulations. Early attempts at accomplishing this wereusually met with frustration, as most approaches involvedintegrating the pressure and skin friction over the surfaceof the body in order to calculate forces (the computationalequivalent of force measurements in wind tunnels).Surface integration has met with difficulties, however,due to the need to approximate the curved surfaces of thebody with flat facets. Calculation of the pressure and skinfriction at the surface is also difficult at times. This hasled various researchers to look at the experimental wakeintegral methods of Betz [1], Maull and Bearman [13],Maskell [11], Wu et al [22], and Brune and Bogataj [2],and to attempt to apply them to CFD computations. Agood survey of drag computations methods was recentlyprepared by Takahashi [16] and may be useful to futureresearchers in understanding the uses and limitations ofthe experimental approaches. Computational methodsinvolving wake integration have been shown to bereasonably accurate at predicting profile and vortex drag,as shown by van Dam and Nikfetrat [20], Chatterjee andJanus [4], and Van Der Vooren and Sloof [21]. Anequivalent lifting-line approach by Mathias et al [12] hasalso been shown to be able to accurately compute induceddrag.

A reformulation of the momentum balance equations forlift and drag has shown that near-field calculations couldbe performed which were as accurate as the traditionalfar-field analysis methods [6]. These improved integralrelations need to be validated and applied to a variety ofcases, including both Euler and Navier-Stokescomputations, in order to determine the effect of thenumerical approach on the accuracy of the dragextraction. Various contributing factors, such as the typeof grid (structured or unstructured), the grid density in thewake region, the flow regime (subsonic or transonic),

boundary conditions, outer boundary distances from thebody, and the level of the governing equations (Euler orNavier-Stokes) need to be evaluated for their influence onthe accuracy of the drag estimation. Once methods aredeveloped which will make accurate drag calculationspossible, then these procedures can be applied to designoptimization algorithms which can be used to reduce thetotal drag of full configurations (including wings, bodies,and engine nacelles), including power effects. It will thenbe possible to optimally design an aircraft for overallreductions in total drag.

Review of Theory

By using a momentum balance approach, a system ofintegrals have been developed which reduce the task offorce computation to the integration of various flowparameters in a crossflow plane downstream of a body[6]. The lift of the body can be related to the streamwisevorticity in the far-field as

(1)

and the drag can be found to be made up of threecomponents related to the entropy (viscous and wavedrag), total enthalpy (engine thrust addition), andvorticity/stream function (induced drag)

detailed breakdown of the sources of drag in a CFDcalculation is required it is best to evaluate the threeintegrals in the near-field before numerical smoothing

D = £>! + D2 + D3

A =

D2 = -

(2)

(3)

(4)

(5)

In an experiment or a computation, each of the integralswill be a weak function of the streamwise position of thecrossflow plane on which they are evaluated. Whilemoving downstream D2 will approach a constant value of-E/U^, where E is the rate of energy addition in theengines. D3 will decay very slowly to zero as thestreamwise vorticity diffuses until the vorticity shed byone wing cancels the vorticity of the opposite sign shedby the other wing. In a CFD computation, because ofnumerical smoothing and coarse grids in the far-field, thiswill take place within the first 100 aircraft lengths; inreality it would take very much longer. As D3 decreasesthere is a corresponding increase in Dj since the totaldrag remains a constant. In fact, the sum of the threecomponents will be approximately constant well into thenear-field of the aircraft. This is fortunate because bothexperimental measurements and CFD estimations of dragwill usually have to be made in the near-field. Also, if a

causes a shift from £>3 to Dl.

The integral relationships which were developedpreviously [6] used a Trefftz-plane concept and thefollowing assumptions: 1) the crossflow plane must bealigned perpendicular to the freestream velocity vector, 2)the plane must include both wing-tips unless appropriatemethods are used to compute the stream function/vorticityrelationships in a half plane, and 3) the crossflow planemust be downstream of the aircraft. The third assumptioncreates practical limitations to the ability of the method toaccurately predict the drag of a complex configuration(wing/fuselage/engine), in that the downstream crossflowplane may be far enough away from the lifting surfaces tohave allowed a significant transfer of energy from £>3 toD,. It may be useful, therefore, to place the crossflowplane relatively close to the trailing edge of the wing,which may cause the crossflow plane to include portionsof the fuselage. This portion of the fuselage will affectthe position and strength of the vorticity in the crossflowplane and lead to overpredicted values of lift, since lift isthe integral of the product of vorticity and lateral positionof the vorticity.

A possible method for correcting for the presence of thefuselage in a near-field crossflow plane is to replace thefuselage with a region of air with zero crossflow velocity.This would essentially create vorticity along the surfaceof the body, as in a wing vortex sheet. This vortex sheetcould be used to correct the lift calculations for thepresence of the body so that accurate predictions could bemade even in the vicinity immediately behind the wingtrailing edge. A correction of this sort will be shown insome of the following comparisons which include awing/body configuration with a simulated sting extendingdownstream from the base of the fuselage. In these casesthe crossflow plane has been placed behind the fuselage,but in the region which includes the sting. Thesecomputations show that it is possible to accurately correctfor the presence of a body or sting when extracting liftfrom a crossflow plane.

Basic ConceptsImplementation

For most CFD computations, especially for those usingunstructured grids, there is no crossflow plane in thecomputational grid, and so the most natural approach forthe evaluation of the crossflow drag integrals is to adopttechniques from flow visualization. A crossflow 'cuttingplane' can be defined orthogonal to the freestream flowand at a fixed distance downstream of the aircraft. For anunstructured grid the grid nodes on the cutting plane aredefined by the intersection of the plane and the edges of

the three dimensional grid, and all flow variables can bedefined at the new grid nodes by linear interpolationalong the cut edges. The nodes of the cutting planes canbe connected into triangles, based on the relationship ofthe cutting plane to the original cells. The full details forunstructured grids composed of tetrahedra, prisms,pyramids, and hexahedra are given in a paper by Gilesand Haimes [8]. For calculations on single-block andmulti-block structured grids, it is unlikely that there existsa suitable grid coordinate plane which is at a uniformstreamwise distance downstream of the aircraft. Oneoption is to use the same 'cutting plane' approach as wasused for the unstructured grids, creating an unstructuredtriangular grid on the crossflow plane, with flow variabledata interpolated along the cut edges of the structuredgrid.

The next issue is the interpretation of the values obtainedfrom the drag integrals. Using CFD methods, it ispossible to directly evaluate the aerodynamic forces onthe aircraft using a numerical approximation of thesurface integral. Almost all CFD methods areconservative, so if the surface force integration isperformed in a manner consistent with the CFDdiscretisation of the cells with surface faces, then it ispossible to sum over a very large number ofcomputational cells surrounding the aircraft and deducethat the numerical surface integral is exactly equal to thatwhich would be obtained by a numerical momentumintegral applied on the enclosing control surface. In thefar-field, numerical smoothing effects, like the realviscous effects, are very small. Therefore, the far-fieldasymptotic analysis remains valid, showing that thenumerical force integral on the aircraft surface can beequated to the drag integrals on the crossflow plane.

Need For Cutoffs

A variety of practical issues arise when the momentumintegrals for computing lift and drag are applied to eitheran experimental or computational set of data. Both sets ofusers want to reduce the size of the crossflow plane inorder to reduce the time it takes to extract the forces: thewind tunnel experimenter wants to decrease the size ofthe crossflow survey to reduce tunnel occupancy time,and the CFD user wants to be able to compute forces in asclose to real time as possible, especially when using thecomputations in conjunction with a numericaloptimization scheme. In the wind tunnel, the wakesurveys are restricted to a crossflow plane the size of thetest section, but further reduction in probe survey sizescan be obtained by knowing that the wake region,including the trailing vortices from the wing-tips, lies in afairly small region immediately behind the aircraft. Itwould be useful to be able to mimic this type of survey inthe evaluation of drag using CFD results. The problemlies in determining the correct parameters to be used todetermine when the crossflow plane is "large enough" to

give accurate results.

Initial applications of the crossflow plane integration toCFD results showed that if the outer boundaries were tooclose to the body, a large contamination to the integrationcould take place. Crossflow plane cells at the outerboundary are usually quite large, and since the integralsessentially sum the difference between local flowvariables and freestream values over a finite area, anysmall deviation in the flow variables from freestreamconditions near the outer boundary can lead to large errorsin the integration. Integrations which were performed in[6] showed that forces such as lift could be in error by asmuch as 50% if the outer boundary were included. Thislarge error is due to a magnification of small errors by themultiplication of the vorticity by the lateral distance (seeEqn. 1). A simple reduction in the size of the crossflowplane showed that accurate calculations of the lift anddrag could be obtained if the crossflow plane was as smallas a one chord radius around the wing surface. Furtherreductions in the size of the crossflow plane showed arapid decrease in the accuracy as regions of high vorticityand entropy changes would then be neglected in theintegration. However, the precise size and shape of therequired crossflow plane should be determined by theaircraft geometry and the resulting flowfield, whichmakes using a simple radius cutoff value for reducing thesize of the crossflow plane problematic. There would beno a priori way to determine the right size of thecrossflow plane for various geometries, i.e., a high aspectratio wing at high angle of attack would require a largerradius crossflow plane than a small aspect ratio wing atlow angles of attack. For this reason, a more suitable wayto reduce the size of the crossflow plane would be todelete various cells from the integration based on theamount of vorticity and/or entropy in the cell, rather thanon the distance of the cell from the wing.

Cutoff Formulation (Elliptic Wing Case)

In order to show that the lift and drag of an aircraft couldbe well predicted using reduced crossflow plane sizes, asimple test case was run. An elliptic planform wing withaspect ratio of seven M^ = 0.55 and a = 4° was chosenfor this study. Euler computations were performed usingthe SAUNA CFD system [16,17]. This wing has beencomputationally studied by van Dam and Nikfetrat [20],and computations were performed to match those cases.Bounding boxes were used to decrease the size of thecrossflow plane; a bounding box of N chords includeseverything within a box outlined by -N < Y < +N and-N < Z < +N. Figs. 1 and 2 show the effect of the sizeof the bounding box on the computation of lift and drag.

Figure 1 shows that the bounding box can be as small asthree chords without effecting the prediction of lift, butthat a bounding box which includes the outer boundary(fourteen chords) does not predict the lift accurately (as

was explained in the previous section and reported in [6]).In addition, the lift is consistently predicted withcrossflow planes which are as much as seven chordsbehind the wing (discrepancies in the results at crossflowplane conditions aft of seven chords are due to being inthe proximity of the outflow boundary). The vortex drag(Fig. 2) is not affected by the size of the bounding box,but the values do decease as the crossflow plane is movedfurther aft of the wing. This is due to numericaldissipation converting crossflow kinetic energy intoentropy (D3 to Dl).

These results show that the size of the crossflow plane canbe reduced significantly and still result in consistentpredictions of lift and drag. However, if the goal is tominimize the size of the crossflow plane in order toreduce computation time, then this method may not be thebest choice since the size of the bounding box will not beconsistent from configuration to configuration.

The inability of the simple radial cutoff to accuratelypredict the shape of the wake points to the need for someother type of cutoff parameter to determine which cellscontain significant levels of vorticity and entropy. Sincethe goal is for the integration to include cells whichcontain the wake, the first level of cutoff was needed toeliminate cells which had less than a certain proportion offreestream reference levels of entropy and vorticity, givenby

(6)

(7)

where Mx is the freestream Mach number, y is the ratioof specific heats, R is the universal gas constant, SK{ isthe wing reference area, and AR is the wing aspect ratio.The first entropy and vorticity cutoff parameters (Level1), Cs and C, , are defined as threshold levelsproportional to the freestream values (Eqns. 6 and 7). Theentropy and vorticity within a cell are included in theforce integration if the cell can be shown to be directlyconnected to the wake and have entropy or vorticity levelssuch that

(8)

cell (9)

Since these cutoff levels are relative to the freestreamvalues, they will be constant for all crossflow planes inthe wake.

A second set of cutoff parameters (Level 2) for the

entropy and vorticity, Cs and C* , were defined relative

to the maximum values in each crossflow plane, 5max and£max • These parameters were found to be necessary sincea global cutoff might not include smaller levels of entropyand vorticity in crossflow planes far downstream of thewing. The entropy and vorticity in the cells wereincluded in the integration if

cell

'

(10)

(11)

Values of the various cutoff parameters were determinedfrom computations using the elliptic wing.

Figures 3 and 4 show the variation of lift and vortex dragcoefficients for various values of C* . This figure showsthat the lift integration can be consistently made withcutoff values between 0.001 and 0.1, but that cutoffvalues either above or below these levels seriouslydegrade the accuracy of the prediction. The vortex drag isconsistently predicted for cutoff values between 0 and 0.1,with poor predictions occurring for cutoff values above0.1.

This points to the need for a combination of two levels ofcutoff values for vorticity: one relative to the freestream(Level 1) and one relative to local conditions in thecrossflow plane (Level 2). Figures 5 and 6 show theresults with two vorticity cutoff values, a Level 1 valuewhich varies as shown, and a constant Level 2 value ofC, =0.1. These results show that the lift and vortex

?2drag are consistently predicted for all combinations ofcutoff values and at crossflow plane positions within sixchords of the wing. This leads to choosing default valuesfor the two vorticity cutoff parameters of C, = 0.01 and

C, = 0. 1 . A similar determination of the entropy cutoffvalues (Fig. 7) shows that the entropy drag is consistentlypredicted for cutoff values of Cs = 0.003 and

The final model for deciding which cells to use in theintegration is to determine whether or not a cell has alevel of entropy or vorticity above the threshold levels inthe crossflow plane (Level 2). If the cell meets thiscriterion, and can be shown to be directly connected toother cells which form the wake, then it is used in theintegration. Level 1 thresholds are then checked to see ifother cells exist which have levels of entropy or vorticitywhich may be relatively large enough to be part of thewake, but the cell is only included if it also connects toother wake cells. In general, Level 1 thresholds are

lower than Level 2 thresholds, and provide a means forfine tuning the size and shape of the wake.

Results for the elliptic wing with the newly determinedcutoff values are presented in Fig. 8. These show thevorticity, entropy, and total drag coefficients as a functionof streamwise position downstream of the wing (in chordlengths). When the default levels of cutoff values areapplied to the total drag estimation various interesting, butnot unexpected results take place. First of all, the totaldrag does not vary a great deal at various positionsdownstream of the wing (total drag in the wake should beinvariant as was discussed previously in [6]). Thevorticity drag decreases continuously the fartherdownstream the crossflow plane is placed, and theentropy drag is continuously increasing. This is theinterchange of vorticity for entropy which is caused bynumerical dissipation as the vortex convects downstream.It is because of this transfer of drag that the near-field isthe best place for computation of drag from CFD results.This shows the value of the crossflow plane integralformulation, which was developed for use in the nearfield.

Applications

M6 Wing

The ONERA M6 Wing [15] was used as a basis fordetermining the effect of grid density and grid type ondrag prediction using the entropy and vorticity cutoffmethods. Both structured and unstructured grids wereused with the following grid densities: 1) structured gridwith 4,000 points (coarse grid), 27,000 points (mediumgrid), and 189,000 points (fine grid); 2) unstructured gridwith 33,898 nodes and 185,239 cells (coarse mesh),55,127 nodes and 314,518 cells (medium mesh), and with97,533 nodes and 576,234 cells (fine mesh). Note that thegrid densities between the two grid systems do not matchsince the computations were done with different Eulercodes: the unstructured grid used the Cindy code [5] andthe structured grid used the SAUNA code [16,17]. Bothsets of computations were performed for Mm = 0.84 anda = 6.06°. The entropy and vorticity cutoffs were set tothe default values which had been developed for theelliptic wing; Cf =0.01, C, = 0.1, Cs = 0.003, and

c " 2 = o . i .

Figures 9 and 10 show the results of the lift integration forthe two grid types (structured and unstructured,respectively) as a function of distance downstream fromthe wing leading edge (in wing root chords). The resultsshow that the lift is consistently predicted by thestructured medium and fine grids, but not for the coarsegrid. Likewise, the unstructured grids all show consistentresults for crossflow plane positions withn seven or eightchords of the wing. Figures 11 and 12 show similar

results for the entropy drag, and Figs. 13 and 14 show thevortex drag. These results show that the medium and finestructured and unstructured grids give consistent results,but that the structured and unstructured grids yielddifferent levels of entropy drag (the vortex drag resultsare consistent between the two grid types). This variationcould be due to different grid densities in the vicinity ofthe shock wave on the wing, which shows the need tocluster grids in this region in order to be able to accuratelypredict wave drag.

The total drag results for the structured and unstructuredfine grids are compared in Figs. 15 and 16, respectively.Each figure includes the entropy, vorticity, and total drag.Both codes yield total drag results of approximatelyCD = 0.06, with consistent values for the vortex drag.While the total drag seems fairly constant for both gridtypes, there is an obvious exchange of entropy drag forviscous drag as the crossflow plane is moved furtherdownstream. However, this trend stops after five chordlengths for the structured grid, while continuing for theunstructured grid.

W4 Wing-Bodv

The W4 configuration is a civil transport wing-bodywhich is used to show the ability of the method to handledrag estimation for a geometry which includes thepresence of a fuselage. This configuration has an aspectratio of eight and was experimentally tested atM^ = 0.78 and a = 1.523° by Fulker [7]; Eulercomputations were made to match these conditions usingthe SAUNA code [16,17].

If the computational geometry includes a fuselage, or asimulated sting extending from the base of the fuselage,problems can arise in the computation of drag. Thefuselage or sting pose a problem when applying thecrossflow plane integration concept since the originalintegral formulation assumed that the crossflow plane wasa Trefftz-plane. The sting creates a boundary surfacewithin the plane which acts to force the trailing vortexaway from the configuration centerline. This willadversely effect the prediction of lift, which is the integralof the product of the vorticity and the lateral location ofthe vorticity. When the vortex is forced further outboardthe lateral position is increased, which causes anoverprediction of lift. The corresponding calculation ofdrag components is not affected by the body since thedrag integrals are not a function of the position of thevorticity. As mentioned previously, this problem can berectified by assuming that the sting can be replaced by aregion of zero crossflow velocity fluid. This fluid gradientcreates the equivalent of a vortex sheet along the surfaceof the sting, the circulation of which can be determinedand included in the lift calculations. This should reducethe computed lift, effectively correcting the lift value forthe presence of the body in the crossflow plane. This

approach would also allow for placing the crossflow planeimmediately behind the trailing edge of the wing, so thatdetailed force information for the wing could be obtainedin spite of the presence of the body.

The W4 wing-body geometry was run without thefuselage correction in order to determine the feasibility ofpredicting drag within the region of the fuselage. Thegeometry was first run to determine the level of the C, i

waswhich would give consistent results; the value of C.S2set to the default value of 0.1, as was determined from theprevious test cases. Figure 17 shows that the entropy dragis consistent for C > 0.0001.

A study of the grid density in the wake of the wing wasalso conducted. Three grids were tested, each with adifferent number of axial planes aft of the wing: 13, 27,and 27 planes. The entropy drag and vortex drag resultsfor these three cases are presented in Figs. 18 and 19. Theentropy drag results are fairly scattered for the gridswhich contain 13 and 27 planes, with a smoother variationof entropy drag for the grid with 37 wake planes. Thevortex drag is well predicted by all three wake grid sizes,showing that the vortex drag (and therefore the lift)requires much lower grid density in the wake to giveaccurate and consistent results. The prediction of entropydrag in the wake, however, requires higher levels of griddensity.

The resulting values of entropy, vortex, and total drag areshown in Fig. 20. These results use the grid with 37planes in the wake and the default entropy and vorticitycutoff values of C, =0.01, C, =0.1, C, =0.003,

M ^2 *1and C =0.1. The results show similar trends to the

S2previous cases: an interchange of vortex and entropy drag,with a resulting total drag which is nearly constantthroughout the wake. The corresponding lift for this casewas predicted to be CL=0.75, and is constantthroughout the wake.

M165 Canard-Wing

The Ml65 configuration is a simple canard-wing whichwill show the effect of computing the forces on multiplelifting surfaces which are at different longitudinalpositions in the configuration. The configuration has anaspect ratio of 2.1 and was wind-tunnel tested at^=0.9 and a = 6° by Stanniland [17]; Eulercomputations from the SAUNA code [16,17] were madeto match the experimental case. The default values forthe cutoff parameters were used for this case.

The crossflow plane is positioned at various positionsbehind the trailing edge of the wing. Since the wing is

behind the canard, the transfer of vorticity drag to entropydrag for the canard has already begun to take place at theposition of the wing. In spite of this, the results in Fig. 21show that the overall drag of the configuration isconsistently predicted. As was seen in previous cases, thetotal drag is constant at CD = 0.033, with the majority ofthe total coming from vortex drag. The interchangebetween vortex drag and entropy drag is also apparent forthis case.

NAG A 0016

This is a simple rectangular wing model using anuntwisted NACA 0016 airfoil section which wasexperimentally tested by Brune and Bogataj [2]. The testmodel had an aspect ratio of six with rounded wing tipfairings, and was tested at Mm =0.18, a = 8.22°, andRe = 1.26 x 106. An incompressible Navier-Stokes CFDprediction of the flowfield around this wing wasconducted by Mathias et al [12], including lift and dragestimates using both surface integration and an equivalentlifting line model (results are presented in Table 1). Thecomputations utilized a single zone structured C-O gridwith three grid densities: coarse (61 x 61 x 51), medium(61x61x81), and fine (81x81x81). Results for allthree grid densities were previously presented [6], but thevorticity/entropy cutoff methods have now been appliedto the fine grid.

Figure 22 shows the drag results for the wing as afunction of distance downstream from the wing leadingedge. Since this is the first viscous case where the cutoffparameters were applied it was nbecessary to see if thevorticity/entropy cutoffs used for the inviscid cases stillyielded reasonable results. This wing does not haveshock waves, but does have viscous effects included, sothe general trends are somewhat different; in general, itwill not be possible to determine the difference betweenwave drag and viscous drag within the entropy integral.

The total drag is overpredicted in the region immediatelybehind the wing, and continues to decrease at variouspositions downstream of the wing. It is not possible todetermine if a general exchange of entropy drag forvortex drag takes place as the crossflow plane is movedfurther downstream, since the entropy drag already existsat fairly high levels due to the shed boundary layer behindthe wing's trailing edge. The general trends hold fartherthan one chordlength downstream of the trailing edge, butthe region close to the trailing edge exhibits a rise invorticity drag where the boundary layer is shed into thewake. The computed lift-to-drag ratio at threechordlengths behind the wing is L/Z)=14.9, whichunderpredicts the experimental values of L/D-17.6from a wake survey and LID = 17.9 from a balancemeasurement. The total drag is not constant as a functionof distance from the crossflow plane, which may point toproblems with artifical entropy which is created at the

surface of the wing has been convected into the wake,which makes it very difficult to determine the actual drag.This type of problem will need to be investigated further.

Lockheed Wing A

The Lockheed Wing A has a planform which isrepresentative of an advanced commercial transport wingand was extensively wind-tunnel tested by Hinson andBurdges [10]. The wing has an aspect ratio of eight, ataper ratio of 0.4, a leading edge sweep of 27 degrees, anduses a 12% thick airfoil which is continuously twistedfrom the root to the tip. The flowfield about the wing wascomputed by Greenman et al [9] using the OVERFLOWcompressible Navier-Stokes solver [3] and a single blockstructured grid. The computations were made atA^^O.82, a = 1.5°, and Re = 6.0xl06, which isrepresentative of a cruise flight condition for acommercial transport.

Figure 23 shows the entropy, vorticity, and total drag forWing A. These results show trends which are similar toprevious results for the various inviscid cases, namely thatthe total drag is constant for as much as 30 chordsdownstream of the wing. The effects of the outflowboundary then begin to contaminate the crossflow planedata and cause inaccuracies in the prediction. Theentropy drag is seen to increase throughout much of thewake, with a corresponding decrease in vortex drag. Thetotal lift-to-drag ratio for the crossflow plane integrationis L /D = 14.5, which compares well with theexperimental value of L / D = 13.4. The lack ofdifficulties with the entropy drag for this case may pointto different types of artificial viscosity having effects onthe quality of the drag prediction.

Conclusions

A previously developed wake integral formualation forthe prediction of lift and drag (both entropy drag andvortex drag) has been used on a variety of test cases.Methods for reducing the computational time required forintegrating the wake crossflow planes have beendeveloped so that the process can be used in conjunctionwith numerical optimization algorithms. Entropy andvorticity cutoff parameters have been developed relativeto freestream reference levels and maximum levels in thecrossflow plane. Cells in the crossflow plane whichcontain small proportions of these two levels of entropyand vorticity are not included in the integration. Themethods have been applied to a variety of inviscid andviscous cases in order to determine the levels of the cutoffparameters for consistent prediction of lift and drag.Further applications of these concepts may lead to theaccurate and efficient extraction of forces from CFDcomputations.

Acknowledgments

The authors wish to thank Karlin Roth, RoxanaGreenman, and David Baker of NASA Ames ResearchCenter, Donovan Mathias of Stanford University, and G.Shrinivas of the Oxford University Computing Laboratoryfor making available crossflow plane information for theLockheed Wing A, NACA 0016, and unstructured M6wing solutions, respectively. This work was partiallyfunded by Rolls Royce pic and DRL.

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[14] S.E. Rogers, "Numerical solution of theincompressible Navier-Stokes equations," NASA TM102199, Nov. 1990.

[15] V. Schmitt and F. Charpin, "Pressure distribution onthe ONERA-M6-Wing at transonic Mach numbers,"AGARDAR-138, 1979.

[16] J.A. Shaw, A.J. Peace, J.M. Georgala, P.N. Childs,"Validation and evaluation of the advanced aeronauticalCFD system SAUNA—a method developer's view,"European Forum on Recent Development and Applicationin Aeronautical CFD, Bristol, U.K., 1993.

[17] J.A. Shaw, A.J. Peace, N.E. May, M.K. Pocock,"Verification of the CFD simulation system SAUNA forcomplex aircraft configurations," AIAA Paper 94-0393,Jan. 1994.

[18] D.R. Stanniland, "A selection of experimental testcases for the validation of CFD codes—Case D5",AGARD AR 303, 1994.

[19] T.T. Takahashi, "On the decomposition of drag fromwake survey measurements," AIAA Paper 97-0717, Jan.1997.

[20] C.P. van Dam and K. Nikfetrat, "Accurate predictionof drag using Euler methods," /. Aircraft, Vol 29, No. 3,1992, pp. 516-519.

[21] J. Van Der Vooren and J.W. Sloof, "CFD-based dragprediction: state-of-the-art, theory, prospects," NLRTechnical Report TP 90247, 1990; also "Lecture notes forthe AIAA professional studies series course on dragprediction and measurement," 1990.

[22] J.C. Wu, J.E. Hackett, and D.E. Lilley, "Ageneralized wake-integral approach for dragdetermination in three-dimensional flows," AIAA Paper79-0279, Jan. 1979.

U.D

0.5

O"1 0.4 \"c.

!§ n T i(D

8§ 0.2

0.1

n

i i i j i i i

isjga fK Tf a

i t

o 14 chordsn 11 chordso 8 chordsx 5 chords

I * 3 chords

i : :

| I -_3 X /*! X T&" X •

i i D

; j

^ ° 6 i o -i ' <?• i; i; :j ; :

i i i i . , i i , , i "

2 4 6 8Crossflow Plane Position (chords)

Fig. 1 Elliptic Wing—Effect of Bounding BoxSize on Lift Coefficient.

10

OOc.2

8oS?0

u.uuo

0.007 1

0.006

0.005

0.004

u.uuo

U.UU&

0.001

n

| , , , I , , , | , , , I , . , .

? § ! 8 i 8 £ a ;i ! i \ '-• • ; : -i! i ! ! J

I i : i 1

^ O 14 chords i i :

: o 8 chords j ! :: x 5 chords i ; :r ••" -t- 3 chords ....................... .|..........................j...........~..-....._

i i _ L I i _i __ _i __ i i i i i —— 1 —— t —— I —— ; —— i —— I

2 4 6 8Crossflow Plane Position (chords)

Fig. 2 Elliptic Wing—Effect of Bounding BoxSize on Vortex Drag Coefficient.

10

8

o'c<o'ocoo

U.D

0.5

r0.4 1

.0 ^

0.2

0.1

n

8 10.001.000

i • ; 0 0.100: i : X 0-030; • ; - 0.010

_.......................i__........._...........;...........................l............. £ 0.003: : : • 0.001j I ! • 0.000

•»-. * : * f • . *

L. ! D 4 ! i .................? • • i a i :

o i " • i g a B: o \ \ \ \

o \ i i ;'-_ f I i i

:

-a

O

1

1

Q

0.008

0.007

0.006

0.005

0.004

0.003

0.002

0.001

n

0 2 4 6 8Crossflow Plane Position (chords)

Fig. 3 Elliptic Wing—Effect of Level 1 CutoffParameter on Lift Coefficient.

10

0.6

0.5 —

0.4

.2iE 0.3

H

...J

O0.2

0.1 -

L.....

--r i>

on0X~

, i

2

0.1000.0300.0100.0030.001

j j ! 4

I ! ! 1i i i ~\

\ i i 1I i i . I . . i ! . . , 1

4 6 8 10Crossflow Plane Position (chords)

Fig. 5 Elliptic Wing—Effect of Level 1 CutoffParameter on Lift Coefficient; C, =0.1.

U.U14

0.012

| 0.008

!2 S

Q 0.004 '

0.002 ,

n I

. ' ' '

r— ••••----—

I

r-&§-•• . 1

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ft .........

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• j

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i . . I i . .

O 0.10000n 0.010000 0.00100X 0.000304- 0.00010A 0.00001• 0.00000

4-

§ i §_l —— i —— I —— \ —— I —— : —— L

--:

"

.

-"

:

) 2 4 6 8 10

Crossflow Plane Position (chords)

Fig. 7 Elliptic Wing—Effect of Level 1 CutoffParameter on Entropy Drag Coefficient; C^ =0.1.

u.uua ,

0.007

0.006a

O ,H 0.005

1 0.004 ,3g> 0.003Q

0.002

0.001

00

]]

0.008

0.007 i

0.006a

O•S 0.005

I 0.004

o> 0.003Q

0.002

0.001

0

0.01

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0°| 0.006

1CT 0.0042Q

0.002

0

; ' ' ' ' ' ' ' ! ' ' ' ! ' ' ' :

*8S ? « » i 1 i » . jis '" r ----i------ r---«-— • .: 1 i o 10 :........................r.........n.........,..........................r............. g ^ .i a. i ? o? :i '• : A .003 -i i a i : r :I i f a •? a :

0 i i i I :

^ i I i ]-2 4 6 8 10

Crossflow Plane Position (chords)

Fig. 4 Elliptic Wing— Effect of Level 1 CutoffParameter on Vortex Drag Coefficient.

i i i i ! 1- ^ i s S a Q a d ) J

: ! i l l :; | | ! j

: i i l :: o 0.100 i i i •

a 0.030 ......I........................;. 1: 0 0.010 i I i -: X 0.003 ! I I :

D 2 4 6 8 1CCrossflow Plane Position (chords)

Fig. 6 Elliptic Wing— Effect of Level 1 CutoffParameter on Vortex Drag Coefficient; C, =0.1.

1 ' i . . I , , ,i O Entropy Dragi D Vortex Drag_.......................|.......................... ...................... o Total Qrag

i B Q Q f t o o o o o O / N -- a D D D a S d, 2 -

i l l !

i l l -

<«,,oo . Q . o . c ^ . o c b o o o ;0 2 4 6 8

Crossflow Plane Position (chords)

Fig. 8 Elliptic Wing—Total Drag Coefficient;Cfi =0.01, Cf =0.1, C =0.003, and

10

o"§_o

oO

1

0.8

0.6

0.4

0.2

n

: I i , , , , i | , , , , i

_.............s.H ...6....r........0..........r.......0..........0....... ...... .

0° i ! i i

tSfeicra Q on >4 EI Q Q .. . 0.............. .........••••- ••••^g*SCny6'"v"'iCJ""G]"" "' •••sir— •- ••(#•— — ..^.~— —

i I • I i

O Coarse i iD Medium i ~" '" ;O Fine i i

" , . , ! , , , , , , I , , , i , , , "

0 2 4 6 8 10

Crossflow Plane Position (chords)

Fig. 9 M6 Wing—Effect of Grid Density on LiftCoefficient; Structured Grid.

0.8 —-

~ 0.6.2.1CD

(3 0.4

0.2

OQ... ; : ; o... ..... ....Q g i Q ; 8 i 0

10 2 4 6 8 10

Crossflow Plane Position (chords)

Fig. 10 M6 Wing—Effect of Grid Density on LiftCoefficient; Unstructured Grid.

U.I

0.08

0°2 0.06CD

" 0.04Ol

I0.02

0(

0.05

0.04

O1 0.03

i<DO

0) O-02toQ

0.01

n

— i —— i —— i —— • — r — r- i — i i i j < i i i : i i

: : : _.; i ! O Coarsei i i n Mediumi ; ; o Fine

i i i i -

o Q?00 o 6 o 0 o QO i i i i

Q n i I ... ;..... ......... .......—O ' '

) 2 4 6 8 1Crossflow Plane Position (chords)

Fig. 11 M6 Wing — Effect of Grid Density onintropy Drag Coefficient; Structured Grid.

- , . , | , , . , . . ' ' ' -

O i 1 ; i

i i i :

I I o o § o g o 9

: ! I ° I i! o Coarse i iQ Medium .4.........-——— f. ......... — ......j.......................

O Fine i I i

' i i i i — i — i — i — — i — i — i — i — i — i — i — — i — i — i —

0.1

0.08

0°

I 0.06

i8 004

Q

0.02

00 C

0.05

0.04

0°

» 0.022

0.01

o

i i i | . , . j , , , j , i i | i , i

O Coarse i i i

O Fine i i i -

1 ! 1 I| .,, , ., ;, ,. , .. | ,., ... . , , , |

i : : i O dI 1 Q ! g ! ^ 6

. BtfJ .

, , i , , , i , , , i , , , i , : ,

) 2 4 6 8 1CCrossflow Plane Position (chords)

Fig. 12 M6 Wing — Effect of Grid Density onEntropy Drag Coefficient; Unstructured Grid.

. i , | , . , j , . , j i , , j , i i

O Coarse i i ;

O Fine : i i .

i 1 i i

i 1 i i -

i i i o a <i i i O E

i l l

~ , , , i , , , i , , , i , , , i , , , ~0 2 4 6 8

Crossflow Plane Position (chords)

Fig. 13 M6 Wing—Effect of Grid Density onVortex Drag Coefficient; Structured Grid.

10 0 2 4 6 8Crossflow Plane Position (chords)

Fig. 14 M6 Wing—Effect of Grid Density onVortex Drag Coefficient; Unstructured Grid.

10

10

aO

Coef

ficie

r

aa

U.I

0.08

0.06

0.04

0.02

n

— i —— i —— i —— | —— i — , —— i —— [- ' -"T- '<•—[• i i i | i ' < J

O Entropy Drag I iD Vortex Drag i ;O Total Drag ..............................................................

\ I O i I<SS<X>O^ O O ; <> ;I i 1 i o <f ;

i i o 6 i^fi 0 0 9 i o Q

• , , , i , , , i , , , i . , , i , , , "2 4 6 8

Crossflow Plane Position (chords)

Fig. 15 M6 Wing—Total Drag Coefficient;Structured Grid; C, = 0.01, C, =0.1,

'1 *2C =0.003, and C =0.1.

10

0.1

0.08 -

0.04 -

0.02 -

-

-

—— L-

, i | i i . | i .

O Entropy Drag 1D Vortex Drag 1O Total Drag I

«K> i <> O i O

]- n, Q i °....... ..,...e..........w.........................

^^ 0 n ^

L ——— 1 1 ———— 1 ———— 1 ———— 1 ———— 1 ———— 1 ———— \ ———

| 1 1 1 | I

:

i |

i O i O <

|o |°

\° \ a c1 1 1 1 . i 1 1 — 1 —

2 4 6 8

Crossflow Plane Position (chords)

Fig. 16 M6 Wing— Total Drag Coefficient;Unstructured Grid; C, = 0.01, C, =0.1,

'1 '2 •C, =0.003, and C =0.1.

• S

10

aO

<DOO01aQ

U.UO

0.025

0.02

0.015

0.01

0.005

n

. , | , , , | ,

'• ..

' • i

1 1 1 ,

onoX-t-A••

. ........ . . .. — ,

: ^iSBBBBBBB B Bj j l l

1.1.01.003.001.0003.0001.00001

'-:

1

a « : B |i \

L . . ! . . . I , . . I . . , , , ^2 4 6 8

Crossflow Plane Position (chords)10

Fig. 17 W4 Wing-Body—Effect of Level 1 CutoffParameter on Entropy Drag Coefficient; C, = 0.1.

3O)

0.03

0.025

0.02

0.015

0.01

0.005

n

—————— I ——————— I ——————— I —————— —————— t —————— I ————— "I ——————— p 1 1 1 1 • 1 1 1 :

a°faaBoai{OOD o i B | O ;

• : :

: : i :

- o 13 wake planes 1 ! ID 27 wake planes I.............;..........................;.......................O 37 wake planes 1 i

i i j _ __i — i — i — 1 — i — i — i — — i — i — i — i — i — i — i —

o>

0.025

0.02

0.015

0.01

n f\f\gf

n

• ' • | • ' • ! • • • i • • • i ' ' • j

D 27 wake planes ; iI o 37 wake planes i ! ;

j BaiaBaDa^ga 1 g o j ° :

: i ! ! 1 -

, , 1 , , . i , , , i , . i , , , -0 2 4 6 8

Crossflow Plane Position (chords)

Fig. 18 W4 Wing-Body—Effect of Wake GridDensity on Entropy Drag Coefficient; C, = 0.01,

C. =0.1, C =0.003, and C =0.1.?2 s\ "1

10

0.05

ent,

C D

o

oS

2o S

o 2

I

O EntropyD VortexO Total

oo<>oooooc| o o o <> o o i

nniinnnanqna n D a n i n

..o.o.d>2P.?.°..2..9...0....0.....2.....!L...£_....°.J.; O

0 2 4 6 8 10Crossflow Plane Position (chords)

Fig. 19 W4 Wing-Body—Effect of Wake GridDensity on Vortex Drag Coefficient; C, =0.01,

C, =0.1, C. =0.003, and Cs =0.1. 11^2 1 2

0 2 4 6 8 10Crossflow Plane Position (chords)

Fig. 20 W4 Wing-Body—Total Drag Coefficient;Cfj =0.01, C^ =0.1, CSi =0.003, and

c*2 = o.i.

0.05

0.04

0.03

o • 0.02

0.01

O EntropyD VortexO Total

o <>

a en a P a

,._o..

0 1 2 3 4

Crossflow Plane Position (chords)Fig. 21 Ml65 Canard-Wing—Total DragCoefficient; C, =0.01, Cf = 0.1, Cs =0.003,

and C_ = 0.1.

QO"c

§oza

0.06

0.05

0.04

0.03

0.02

0.01

n

— , — i — i — | — i — i — i — r1""1 • ' < ] i i ' ] • • ' i

i i o vortex 1 1_..... — .. — ...I..........................;. — ................ Q entropy |-----^i ! o total 1 1

_^__^_^__j_._j

: a "? a n i 1

i °°° ° ° o a io 0 ! o i

1 , , ' , , , , i , , , j , , a i , : r-, -

0 2 4 6 8 10Crossflow Plane Position (chords)

Fig. 22 NACA 0016 Wing—Total DragCoefficient; C, =0.01, C, = 0.1, C =0.003,

M ?2 S\

and C; = 0.1.

0.06

OUei

8O)SQ

0.05

0.04

0.03

0.02

0.01

n

O vortex I ; I : I

0 total I ! i I I

- ! i I \ ° I \ '-: < x > < 3 > o i < > ! < > ^ I n ! -

: m A a a n f °| ;• ...............i.................J...................L....._..........l.............._.L........Q....J...... ...... .. •- o o o o i o j o 6 ° ; i- , , , . i , , , , i , , , . i . , , , i , , , , i , , , , j ,o , , -

10 15 20 25 30 35 40 45Crossflow Plane Position (chords)

Fig. 23 Lockheed Wing A—Total DragCoefficient; C, =0.01, C, = 0.1, Cs =0.003,

M *2 iand Cs = 0.1.

12