Vorticity Equation Solutions for Slender Wings at High Incidence

7/29/2019 Vorticity Equation Solutions for Slender Wings at High Incidence

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VOL. 29, NO. 4 AIAA JO U RN A L APRIL 1991

Vorticity Equation Solutions for SlenderWings at High IncidenceA. Dagan*RAFAEL, Haifa, Israeland

D. Almosnino1RAFAEL and Technion I s rael Institute of Technology, Haifa, IsraelA new approximate model for the viscous flow around slender wing or body shapes at high angles of attackis presented. The present theory assumes that the flow over a slender shape can be divided into an inner, viscouspart that is described by the vorticity equation and a matching outer potential part. A t present, incompressibleflow conditions are assumed. The parabolization of the problem that results from these assumptions allows oneto obtain a fast plane-marching solution scheme. The analysis shows that the Sychev geometric similarityparameter applies to the present viscous flow model and not only to the inviscid models. The present theory

supports the implusive start flow analogy. In the particular case of slender delta wings, the present theory yieldsexp ressions sim ilar to those of the Polham us suction analogy for the lift coefficient. The feasibility of the presentmethod is tested on a number of delta wings with flat elliptical cross sections, using several Sychev similarityparameters at Reynolds numbers of O (10 s) based on the wing root chord and the freestream velocity. Resultsinclude aerodynamic coefficients, surface pressure distributions, and velocity field as well as vorticity contoursin crossflow planes. Very good agreement is obtained between the calculated aerodynamic coefficients andexperimental data.

IntroductionH IGH angle of attack, subsonic flow calculations of slen-der wings and bodies that include separated vortex flowregions are one of the main challenges of present computa-tional fluid dynamics (CFD) methods. Computational meth-ods based on potential flow assumptions have been used ex-tensively since the early days of modern computers toapproximate separated vortex flows. These methods yield ac-ceptable engineering results as long as the separation is welldefined and not sensitive to viscous effects, as in the case of asharp leading edge of a slender delta wing. With the growingcapacity and performance of comp uters, inviscid solutions forthe Euler equations have become available. Euler solvers arealso limited by the inviscid flow assumption. As a result, theprocess of vorticity generation and diffusion from a smoothsurface is not well defined and the solution in such a case isshown to be controlled by the artificial viscosity introducedinto the numerical scheme.1'2 The inherent limitations of theinviscid flow models and the further increase in computerperformance encouraged the development of numerical podesthat solve viscous flow models. Within this framework, muchprogress has been achieved lately in solving fully developedvortex flows over slender wings, with the introduction ofsolvers to the full Navier-Stokes equations3 and also to theirthin-layer approximation.3"4 In Refs. 5 and 6, the incompress-ible flowfield over a delta and double delta wing is solved byadding an artificial time derivative of the pressure to thecontinuity equation. Consequently, the derived set of equa-tions is integrated like a conventional, parabolic time depen-dent set. A typical computation of a round-edge delta wing isreported to take about 37 ^s per grid point per iteration on a

Received May 17, 1989; revision received Jan. 19, 1990. Copyright 1990 by the American Institute of Aeronautics and Astronautics,

CDC Cyber 205 machine, with 572,000 total number of gridpoints and 400 global iterations needed to solve the case. Ref.1 solves the Navier-Stokes equations for a laminar compress-ible flow over round edge delta wings using the finite volumeapproach and a mesh of 411,000 points consuming 23 p ts pergrid point per time step on a similar machine (overall numberof time steps not reported). A different class of solutions isavailable via the thin-layer approximation of Navier-Stokesequations, w here the norm al derivative in the diffusion term isretained. Such a method is used in Refs. 4 and 7 to solvesimilar cases using a mesh of 854,000 grid points, w ith typi-cally 2000 global iterations that take 20 j u s per grid point periteration on a single processor Cray 2 machine or 8.6 / x s on anAm dahl 1200, putting it in the same class as the previou s twomethods.In view of these details, it seems that "the full Navier-Stokes solutions have not yet reached the stage at which therestrictions on computational speed and storage can be ig-nored."3 It turns out that, in order to bridge over these com-putational restrictions, a relatively fast and reliable approxi-mate viscous solution is needed, one that can be used as anengineering tool. The scope of such approximate solutions isto obtain relatively fast and satisfactory answers, usually sac-rificing some accuracy an d sometimes being limited in appli-cability to certain families of geometrical shapes (such asslender wings an d bodies). The main efforts should be devotedin this context to reduce the numb er of global iterations and tostore less data planes.The Parabolized Navier-Stokes (PNS) method, for exam-ple, is widely used for computing steady-state solutions be-cause by neglecting the derivatives of the streamwise diffusionterms, the reduced form resembles a set of parabolic equationsin space. This method is very useful w hen dealing w ith super-sonic flow, where the physical behavior of the equations be -comes parabo lic/hyperbolic. How ever the PNS method is al-most as expensive as the full Navier-Stokes code when solvinga subsonic flow regime. In that case, a global iteration method


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498 A. DAGAN AND D. ALMOSNINO AIAA JOURNAL

geometric shape.8 This limitation is removed in Ref. 9 byderiving a nonelliptic set of equations that can be treated as aninitial/boundary value problem, where only on e global itera-tion is needed to obtain the solution in the subsonic case. Thismethod yields satisfactory results for the inner flowfield in aduct and it encourages us to look for a similar approach tosolve certain types of external flow problems.Th e main motivation of the present study is to develop afast and reliable approximate method to solve the subsonicvortical flow around slender configurations that can be usedas an engineering tool. Th e present method is based on a newapproximate model for th e viscous flow around slender wingsor body shapes at high angles of attack. The main assumptionunderlying the present theory is that the flow over a slendershape can be divided into an inner, viscous part that is de-scribed by the vorticity equation and a matching outer, poten-tial part. Th e parabolization of the problem that results fromthese assumptions allows one to obtain a relatively fast plane-marching solution. One of the main features of the new modelthat emerges from the analysis is that the Sychev geometrysimilarity parameter10 used by Refs. 11-13 within an inviscidslender wing or body model is also valid for the new viscousslender wing (or body) theory. Moreover, it transpires that fo rthe first-order approximation, the inner, viscous flowfieldgenerated by a slender configuration is of nonelliptic nature.The ellipticity of the physical phenomena is weakly introducedby the outer potential part of the flow, through the thicknesseffect of the body. Th e nonelliptic equations derived in thepresent theory reduce the amount of computational work .They do not require an intensive global iteration process, andthe problem can be treated as an initial/boundary value prob-lem.At present the computations are for incompressible, steadyflow solutions. However, the present theory can be extendedto compressible as well as unsteady flows.Theory

The vorticity generated along the leading edge of a slenderwing is diffused an d advected by the flow, generating a well-developed vortical region above the wing w hose center may bedefined by r v = O(z tana)(Fig. 1). Slenderness of both wingand vortex (b ma x/L


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APRIL 1991 SOLUTIONS FOR SLENDER WINGS AT HIGH INCIDENCE 499

where A is the cross-sectional area and r = V*2 +y2. Thefunction g(z) is obtained by matching the inner and the outersolutions, resulting in 14

g(z) =X & --T 2 27T

i 27T (6)where a(z) = dzA. It is worth noting that no coupling existsbetween the free constant g(z) of < and the stream function of\ l/ for the first-order approximation.The asymptotic behavior of the pressure distribution in theinner region is found to be

fo r r > o o (7)This expression can be used as the far-field pressure boundarycondition.

Note that the chordwise free constant g(z) introduces theelliptic influence of the outer potential flow, including thepressure term. For the first-order approximation, the freeconstant g(z) is a function of the body thickness alone. Th epresence of the term w 6 breaks the similarity with respect tothe Sychev parameter because the pressure distribution isweakly dependent on 6. However, this dependency does notenter the expressions for the lift or pitching moment coeffi-cients (in a fashion similar to classical slender-body theory)since the pressure can be w ritten as p - p* Po(z)9 where onlyP o depends on g(z). In the present work, thickness effect isneglected, so g(z) is zero. In some sense, this constant isequivalent to the free constant obtained by Ref. 9, whenapplying the global mass conservation principle.The set of Eq. 1-5, together with the boundary conditions,completely determine the solution of a slender configurationat high angle of attack. The main advantage of this set ofequations is its nonelliptic form. In other words, only oneglobal iteration is required in order to obtain the solutionbecause the problem can be treated as an initial/boundaryvalue problem.9 Another point of interest can be found in thefact that both lift an d pitching momen t coefficients are func-tions of the Sychev parameter as well as the crossflow Rey-nolds number. This fact essentially extends the applicability ofSychev similarity also to a smooth surface separation, makingit subject for future studies.One further simplification is made at this stage to facilitatea fast feasibility study of the method. The axial velocity com-ponent w decreases to zero within the thin layer zone, which isof order O(Re ~1/2), due to the non-slip bounda ry condition onthe wing surface [see the axial momentum equations, Eqs.(1)]. Assuming that this happens only very close to the wingsurface, the axial velocity component can be approximated asw = 1 + O (62) throughout nearly all the field.11 Using thissimplification means neglecting the stretching mechanism ofthe vorticity described by the term Q jd x.w in Eqs. (2) and thesource term d zw in Eqs. (4) in the thin boundary-layer zone.These terms contribute, in general, to the vorticity intensity(stretching mechanism) and reduce its viscous core (the sourceterm). The last approxim ation is thus valid mainly for slenderwings or bodies having a leading edge with small radius ofcurvature that strongly defines the primary separation of theflow. (This is equivalent to the definition of a Kutta point.)This approximation is also valid for a slender configuration athigh angle of attack where X < 1. In such a case, it can beshown [Eqs. (2)] that the effect of the stretching as well as thesource term are small. It should be remembered that thepresent model of the flow still retains the diffusion terms in

ation on the surface and its normal diffusion along the surfaceare retained as well. As a result, it is expected that a satisfac-tory description of the primary vortex can be obtained, withsomewhat less accurate prediction of the secondary separationpoint.Using the last approximation and rewriting the vorticityequation in Eqs. (2), using dimensional notation for conve-nient physical interpretation, results in Eq. (8):v

with the prime denoting dimensional quantities. Presentingthe solution of Eq. (8) in the forml' z = Ql\ t,z' - ( t / o o C O S c x K */ y' (9)

reduces Eq. (8) to the classical two-dimensional parabolicequation:

d x r (10)Equation (10) describes the generation of the vorticity alongthe boundary of the wing (or body) due to the no-slip condi-tion. Th e generated vorticity is transported and diffuseddownstream at a velocity U ^cosa (see also Ref. 15). In otherwords, the solution of Eq. (10) may be obtained in a plane thatis advected downstream along the body axis. As a result, thesteady flow around slender wings or bodies at incidence can bedescribed by the unsteady, two-dimensional flow developmentin crossflow planes marching downstream along the body axis,as predicted by the impulsive start flow analogy.16'17 Equation(10) even extends the domain where the impulsive start flowanalogy is applicable by allowing for varying cross-sectionalshape while moving along the wing or body axis.

Solution ProcedureOverall Approach

The present solution adopts the ADI method coupled withthe vorticity stream function equation, as recommended inRef. 9. For the purpose, E q. (10) is rewritten nondimension-ally in cylindrical coordinates (r , 6) in the transform plane,using conformal mapping and stream function notation toobtain E q. (11) for slender delta wings in a nonconservativeform:+-r4/ r d O dr\dr

1 db 2 d \L 1 d d > dfi+- X - sin (26) + + - r L dr r V ' dr r 30J 30= V2Q n n^^(r) U1 J

Where 70 is the Jacpbian of the conformal mapping, b = b' /b m a x * 0 = Q /(T), ^ = \l//b(T), (f > = 0/Z?(r), and 5 - is defined as

The kinematic boundary conditions and the no-slip conditionon the wing (r = r0): .

Dr' cty= _i^* (13)


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and < /> is given ex plicitly as- db (14)

In this particular case, both d r \l / a nd d e < j > at r = r0 are identi-cally zero. Fa r from the wing, the velocity disturbances mustvanish. In general cases, the inner solution for large r ismatched to the outer solution in the vicinity of r > 0 byapplying the asymptotic matching technique to obtain the freeconstant g(z) described in Eqs. (5) and (6). In the presentstudy, a delta wing with very small thickness is computed sothat g(z) is taken to be zero (similar to the slender-wingassumption).Expressions for Aerodynam ic Coefficients

Some further development of the equations leads to anexpression for the pressure on the wing surface in the presentcase:(15)Reb(r)

where c$ denotes a pressure coefficient defined by a referencepressure on the surface and the crossflow dynamic pressure.The expression for the normal force coefficient in this caseis then obtained as

Re1 P27T dtoor (16)

where r0 is the local radius in the transformed plane thatrepresents the wing surface. It should be noted that Eq. (16),obtained for incompressible f low, is identical in its form to theequivalent one presented in Ref. 13, for compressible flow:(17)

As pointed out in Ref. 13, it is interesting to compare thePolhamus suction analogy18 with the results of the presenttheory. According to the suction analogy, the lift coefficient isgiven byCL =K p sina?cos2a: +Kv sin2acosa (18)

For slender delta wings, the theoretical values K p = irAR/2 = 47rXtancx and K v = T T may be assumed, and so the normalforce coefficient takes the formC^ = 7r[4X + I]sin2a:

Equation 19 is similar in its form to Eq. 16.(19)

Numerical Computation SchemeTh e computations are carried out using second-order accu-racy in space and first-order accuracy in time. Consideringthat \l /k+ 1 = \l / k + ih, an d Q*+! = 0* + Q I, where the subscript1 denotes the correction at time level k + 1, then the vorticityequation and the stream function equation can be w ritten in anonconservative form in the transformed plane19 as

where [/ + AsA] [I + AsC] X =/

A =

(20)

(21)

c =

X =

(22)

(23)here, / is the finite difference approximation of Eq. (11) attime level k, whereas v* and v* are defined as

(24a)ve=- V-sin (20)r \ dr r (24b)

A small stream wise derivative (/3Asds\l/) is added to the streamfunction equation in order to give it a parabolic type nature,which is necessary for the ADI formulation. The parameter /3is chosen arbitrarily. All the partial derivatives have beenreplaced with their central difference operator, where 6 / and S /represent central difference with respect to the r coordinateand the 6 direction.The time step is found to be sensitive to the smallest radiusof curvature and also the Reynolds number.In order to prevent numerical dispersion waves that aretypical to central differencing, a first-order upwind differencescheme of the advection can be used; however, this techniquetends to smear the results since it introduces an artificialdiffusion. On the other hand, a fourth-order artificial viscos-ity that has been added explicitly has severely reduced therequired time step.A stretched coordinate is used in the radial direction asfollows

r, = r0exp (% - /o)4 1

(rmax - r0)3 J (25)where r / is evenly spaced in the transform plane and r is the


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APRIL 1991 SOLUTIONS FOR S LE NDE R WINGS AT HIGH INCIDENCE 501

LEGENDE XP . RR-.25 (Ref. 20)CflLC. RR-.25_________E XP . FIR-.35 (Ref. 21)RLC . RR-.35________E XP . RR-.5+ CRLC. flR

10.0 20.0a ( D EGREE )Fig. 3 Lift coefficient of slender delta w ings.

EXP. REFS. 20-24SUCTION ANALOGYPRESENT CALC

0:0 0.4 0.8 1.2 1.6 2.0 2.4AR

Fig. 4 Lift and pitching m om ent coefficient vs aspect ratio at a = 20deg.Results

A erodynam ic CoefficientsTh e feasibility and accuracy of the present method aretested by computing a series of slender delta wings with aspectratio of 0.25, 0.35, 0.50, 0.70, and 1.0 in steady incompress-ible flow, at angles of attack up to 30 deg. For an accuratecomparison with experimental data and because the presentnumerical code has to be modified to include a sharp leadingedge, the overall lift and pitching moment coefficients (com-puted for cross sections having a maxim um ellipticity ratio of

10) are first-order extrapolated to an infinite ellipticity ratio(zero leading-edge radius). The extrapolated coefficients arethen compared with experimental data for various delta wings

LEGENDa PRESENT CALC. EXP. (REF. 25)

Fig. 5deg). Surface pressure distribution fo r A = 0.33 (AR = 0.7 at a = 15

The calculated results of the present method show an excellentagreement with experimental values for the first four wings,up to an aspect ratio of 0.7, but tend to be higher by about1 0 7 o for the AR = 1 wing. Th e difference between the calcu-lated and experimental values in the last case is mainly due tothe fact that trailing-edge effect is not included in the presentmethod. As a result, the typical decrease in wing load is notcaptured there, causing an over-prediction of the lift an dpitching moment coefficients. Th e overprediction is expectedto be minor for very slender wings, an d becomes more pro-nounced as the slenderness ratio of the wing decreases (see alsoRef. 13). A similar behavior can be seen in Fig. 4, where thevariation of lift an d pitching mom ent coefficients with aspectratio for various delta wings at a. = 20 deg is show n. Thepresent results for low aspect-ratio wings show a good agree-ment with experimental data obtained from Refs. 20-24, beingalso very close to the results computed using the Polhamussuction analogy shown in the same figure. The present resultsshow a deviation from the experimental data as the aspectratio increases above 0.7.

Pressure D istributionTh e first example of the spanwise pressure distribution,

computed at z/L = 0.40 of a delta wing is presented in Fig. 5,using X = 0.33 at Re 103 (corresponding to a streamwiseReynolds number of the order of 105). Note that the pressurecoefficient C p in Fig. 5 is defined by Eq. (15), using thecrossflow dynamic pressure and a reference pressure that ishalf of the pressure difference between lower and upper wingsurface points at its root. A sharp spike of pressure is observedat the leading-edge tip. This spike is typical of slightly roundedleading edges,22 due to the strong acceleration of the flowaround the slightly blunted edge, before its detachment. Thispressure spike does not exist in the case of a sharp leadingedge. Excluding the edge effect, the main pressure peak in Fig.5 is due to the effect of the primary vortex. A small secondarypeak due to the effect of the secondary vortex is also apparent.The present results underpredict the primary pressure peakwhen compared with experimental results of Ref. 25, for thesame X, testing a wing of AR =0.7 at a = 15 deg and lowspeed of 80 ft/s. This is possibly due to differences in the exact


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502 A. DAGAN AND D. A L M O S N I N O AIAA J O U R N A L

LEGEND 0 PRESENT CALC. EXP. (REF. 26)

FULL N.S. SOLUTION 1ST ORDE R (REF. 5) FULL N.S. SOLUTION 2ST ORDER (REF. 5)

Fig. 6 Surface pressure distribution of X = 0.33 (AR = 1.0 at a = 20.5deg).

L E G E N DD PRESENT CALC. EXP. (REF. 28)

Fig. 7deg). Surface pressure distribution for X = 0.50 (AR = 0.7 at a = 10

delta wing having X = 0.33 at z/L = 0.3, computed also at achord-based Reynolds number of the order of 105. The resultsare compared with the experimental data of Ref. 26 , obtainedfor a delta wing of AR = 1, a. = 20.5 deg, at a low speed anda chord-based Reynolds number of 9 x 105. Figure 6 alsoincludes the calculated results of the full Navier-Stokesmethod of Ref. 5 for the same case. Th e second-order, fine-

a) Present calculation

1.0

0.0-1.0

b ) Experiment290.0 1.0

Fig. 8 Crossflow velocity field for AR = 1 delta wing, z/L =0.9 ata = 20 deg.

Th e difference between the present results and the other dataclose to the leading edge in the windward side is caused by thedifference in the cross-section edge shape between the cases.Reference 27 shows that the actual elliptical influence of thetrailing edge for this wing becomes significant already at z/L = 0.5, and indeed the calculated results of the presentmethod overpredict the experimental data beyond this station,as expected (not shown). A good comparison turns out for thethird case, shown in Fig. 7. Th e calculation is carried out usingX = 0.5 at z/ L =0.4 and at a streamwise Reynolds number of105. Good agreement exists between the computed results andexperimental data of Ref. 28 , obtained for a delta wing ofAR =0.7 at a= 10 deg and a low speed of 80 ft/s. The out-board location of the secondary suction peak is typical to thespanwise pressure distribution of delta wings at low Reynoldsnumber.27

Crossflow Velocity FieldA typical velocity vector field in the crossflow plane isshown in Figs. 8 for an AR = 1delta wing at a = 20 deg and anaxial station of z/ L =0.9. Th e calculation is carried out atstreamwise Reynolds number of 105. The experimental mea-surements of Ref. 29 for the same wing (at streamwise Rey-nolds number of 9x 105) are added to the same figure fo rqualitative comparison (note the different cross-section

shape). Th e position of the primary vortex core seems to besimilar in both cases, but the secondary vortex is more devel-oped in the computed results. The centerline saddle point is


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APRIL 1991 SOLUTIONS FOR SLENDER WINGS AT HIGH INCIDENCE 503

Vorticity ContoursA sequence of crossflow planes that shows the developmentof the vorticity contours along the wing appears in Figs. 9a-cat axial stations z/L = 0.2, 0.53, 0.83. Thecomputed vorticitycontours are obtained for A = 0.5, at a crossflow Reynoldsnumber Re = l t f > using a time step of minC/'VlOO and amesh of 60 grid points in the radial direction and 108 points inthe circumferential direction (half plane).In Fig. 9a (z/L 0.2), the primary vortex is not yet fullydeveloped and it lies close to the wing surface, with signs of asecondary vortex staring to emerge from the surface. Follow-ing the evolution of the flow along the wing axis, Fig. 9b(z/L = 0.53) shows that the primary vortex at this station ismuch more developed, moving up and occupying a larger partof the wing span. The secondary vortex at this station isstronger and larger in size, clearly affecting the separatedshear layer that feeds the primary vortex. The interactionbetween the secondary and the primary vortices is caused byviscous effects (shear and entrainment of the fluid particles).As a result, the primary vortex feeding sheet gets thicker,losing some of its momentum, and possible becoming lessstable. Some fluid entrainment also occurs between the al-ready rolled-up primary vortex sheet and the secondary vor-tex. Traces of a possible weak tertiary vortex appear in Fig!9b and 9c, just beneath the secondary vortex. Figure 9c furtheidescribes the development of the vortex flow in the crossflowplane, at z/ L =0.83, with essentially the same features thatappear in Fig. 9b. The wavy disturbance in the primary vortexouter contour line in Figs 9c is probably the result of insuffi-

cient grid resolution in this region of the flow. It should benoted that the description of the flow phenomena appears tobe more sensitive and informative using vorticity contoursinstead of classical stream function contours. Figures 10 showthe experimentally measured vorticity contours obtained byRef. 29 for a 7 5 deg delta wing at a. = 20.5 deg and a chordwiseReynolds num ber of 106, at z/L = 0.7. The present calculatedresults shown in the same figure capture the flow details,including secondary separation and its influence on the pri-mary shear layer.

a) Present calculation

10.0 20.0 -160.0,-;:

-1.0b) Experiment29

0.0 1.0

Fig. 10 Crossflow iso-vorticity contours, 75 deg delta wing ato r = 20.5 deg, z/L = 0.7.

bH/= 0.533

c)*/0.866

Fig. 9 Chordwise developm ent of crossflow iso-vorticity contours,b)

Fig. 11 Crossflow iso-v orticity contours at two different crossflow


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Figures 11 show an example of the vorticity contours ob-tained at z/L=0.99, for X =0.25 and crossflow Reynoldsnumber of Re = 102 (Fig. lla) and 103 (Fig. lib). Both casesshow a well-developed primary vortex. As explained in theprevious case, the higher Reynolds number calculation suffersprobably from insufficient grid density in the primary vortexregion, as appears from Fig. lib. The Reynolds number seemsto affect the location and development of the secondary vor-tex. It is clear that the secondary vortex in Fig. lib, calculatedat Re =103, is much more developed than the one in Fig. lla,calculated at Re = 102. It also seems that the secondary vortexobtained at Re = 102 is located more inboard than the oneobtained at Re= 103.

ConclusionsThe new computational method can be used to calculate theflow about slender wing or body shapes. Accurate results areobtained as long as the basic slenderness assumption is met.The theory developed supports the impulsive start flow anal-ogy as a valid approximationfor slender body vortical flow,while extending the computational domain to more complexgeometries as well. The present theory arrives at an expressionfor the lift coefficient, which is very similar to the one derivedin the Polhamus suction analogy for delta wings. Th e parabo-Hzation of the problem in the inner, viscous flow domainmakes the new method very attractive to use for solving in-compressible or subsonic flow cases, due the fast plane-march-ing scheme it employs. The new method show s high sensitivityin capturing flow phenomena such as primary and secondaryvortex separation, roll-up, and mutual interaction. The vortic-ity equation formulation used in the present method offersgood insight into, an d convenient physical interpretation of,the flow.The present calculations prove the feasibility of the newmethod while applying some simplifying assumptions to theflow model. One of the assumptions, namely, neglecting the

axial velocity gradient within the thin boundary-layer zone onth e wing surface, could be removed to obtain a more accuratemodel. Such a model would describe the mechanism ofsmooth surface separation more exactly, thus improving th eprediction of secondary separation on delta wings and alsoallowing for wider domain of geometries to be calculated.Future studies should include an effort to calculate the ellipticinfluence of the trailing edge on the pressure field upstream.Further development should also include topics such as com-pressible flow (see Ref. 13), unsteady flow, and asymmetricflow cases.References

*Rizzi, A., and Muller, B. , "Large-Scale Viscous Simulation ofLaminar Vortex Flow over a Delta Wing," AIAA Journal, Vol.27,No. 7, 1989, pp. 833-840.2Kandil, O. A., and Chuang, A. H., "Influence of NumericalDissipation on Computational Euler Equations for Vortex-Domi-nated Flows," AIAA J ou rn a l , Vol. 25, No. 11, 1987,pp. 1426-1434.3Pulliam, T. H., "Euler and Thin Layer Navier-Stokes Codes:Arc2d, Arc3d" Note s for Comp u tat ion a l Fluid Dynamics, Universityof Tennessee Space Institute, Tullahoma, TN, M arch, 1984 (unpub-lished material).4Fujii, K., Gavali, S., and Hoist, L. T., "Evaluation of Navier-Stokes and Euler Solution for Leading-Edge Separation" I n terna -tional Journal for Numerical Me thod s in Fluids, Vol. 8, No. 10, 1988,pp. 1319-1329.5Hartwich, P.M., Hsu,C. H., and Liu, C. H., "VectorizableImplicit Algorithms for the Flux-Difference Split, Three-Dimensional

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