AlAA 93-2948 Application of a Parabolized Navier-Stokes Technique for High-L/D, Hypersonic Vehicle Design

J.E. Daywitt Martin Marietta Astro Space King of Prussia, PA

B.A. Bhutta and C.H. Lewis VRA, Inc. Blacksburg, VA

AlAA 24th Fluid Dynamics Conference July 6-9, 1993 / Orlando, FL

For pemlsslon to copy or mpubllrh, contact the Amerbn Insthute of Aemnaullca and AstmnaUtlCS 370 L'Enfant Promenade, S.W., Washington, D.C. 20024

APPLICATION OF A PARABOLIZED NAVIERSTOKES TECHNIQUE FOR HIGH-L/D, HYPERSONIC VEHICLE DESIGN

lames E. Daywin' Martin Marietta Asuo Space?, King of Prussia, PA 19406

Bilal A. BhuttaS and Clark H. Lewis§ VRA, Inc., Blacksburg, VA 24063

Abstract

New wind-tunnel data, obtained on a wide variety of high lift-to-drag configurations, is used to systematically assess a novel pa rabo l i i NavierStokes (PNS) code as well as inviscid, boundary-layer, and approximate aerodynamic prediction techniques. The wind-tunnel data, spanning a range of supersonic Mach numbers, is used to support the design of a candidate flight configuration. Extensions of the PNS code are developed to enable design applications including predictions of yaw-stability an the pitch-and roll control characteristics of deflected elevons. Comparisons with the wind-tunnel data are. used to assess confidence in the PNS technique for the flight environment. The approximate techniques, calibrafed with the ground-test data, are also applied at design critical flight conditions. Applications of the PNS code for aerodynamic tailoring are used to assess its potential as a design tool.

Nomenclature

= Reference area, =planform area, ft? = Axial-force coefficient, = FdqmA= = Normal-force coefficient, = FN/q = Side-force coefficient, = F y / q m A m = Pitching-moment coefficient, =

= Axial Force, Ib. =Normal force, Ib. = Side force, Ib. = Vehicle length with pointed forecone, ft. = Reference length, = sharp-nose vehicle

= Freestream Mach number = Pitching moment about the moment

= Ressure, Ib.fft.2 = Freesueam dynamic pressure, Ib./ft? = Heat transfer rate, BTU/ft?-sec. = Nose radius, ft.

M-

length, ft.

reference center (= ,617 L m ) . ft.-lb.

* SupeMsmg Enginem, Arxolhcnnophysics. Sdor M a t e r AlAA. * Vice h s i h L senior ~ a b a m. ' Resident. Associate Fellow AlAA.

+ ~ o m e r ~ y GE AS^ Space. CopyightOl993 by Manin Marietta. Published by the Amencan I n s f l ~ t e of Aeronautics and ASUOMII~~CS, hc. with pimission.

VD = Freesueam velocity, ft./sec. XCP = Pitch center-of-pressure location.

measured from nosetip, ft. : x =Axial distance, ft.

y e = Yaw center-of-pressure location,

a = Angle of attack, deg. measured from nosetip, ft.

Introduction

The use of computational fluid dynamics (CFD) in the design of hypersonic lifting configurations, such as the National Aerospace Plane (NASP) and maneuvering reentry vehicles (MaRV's), is. by necessity, based on calibration of techniques at conditions that differ from flight. Incorporation of the relevant physics into the CFD formulation, and calibration with ground-test data, lends confidence in its ability to simulate the flight environment. In this paper a novel parabolized NavierStokes (PNS) technique. as well as other less- sophisticated methods, are assessed using new wind-tunnel data of various complex high-IiiVdrag &/D) -' reentry shapes over a range of angles of attack, Mach and Reynolds numbers. The wind-tunnel test data, described in Refs. 1 and 2, and the PNS technique. were used to develop aerodynamically tailored MaRV designs. The shapes that were tested and successfully analyzed with the novel PNS technique pose a challenge for conventional PNS methods. Application of the calibrated PNS code to flight conditions demonsuates the dangers of extrapolating wind-tunnel data to the flight environment and in relying solely on approximate aerodynamic techniques.

Configurations

Candidate high-L/D designs were evolved in a two-stage wind-tunnel test program. The first test series] assessed 44 configurations at Mach 3 and 8 (Arnold Engineering Development Center (AEDC) Tunnels A and B, respectively) at 2.3 million Reynolds number/ft. The design of a unique modular model, shown in Fig. 1, readily accommodated parametric variations in wing span, dihedral angle, and Ieading-edge radius. A 6-degree / 4-degree expansion biconic forms the framework of this modular model. Effects of a vertical fin. winglets, uailing flaps, and yaw tabs were obtained using model attachments. In addition, a shim at the forebody/aftbody juncture canted the forebody upward Wegrees to permit testing of bent-axis

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configurations. The AEDC test matrix configurations are shown schematically in Fig. 2.

The second test series2 was performed at Mach 13 and 14.5 (Reynolds number/ft (in millions) of 0.5 and 3.8, respectively) in Tunnel 9 at the Naval Surface Warfare Center (NSWC). The baseline configuration, shown in Fig. 3, was developed using the AEDC test results, aerodynamic analyses of L D performance and stability, and packaging considerations. Fixed yaw tabs of 12, 14 and 16-degrees were tested. The effects of paired and differentially deflected elevons were measured over a range of -20 (upward) to 15-degrees.

u

A precise analytical representation of the wind-tunnel model geometry is required for application of CFD techniques. In addition to the body surface location (as a function of axial distance), accurate cross-section and axial slopes are needed. Analytical models were developed for three types of wind-tunnel configurations. Figure 4 shows the basic “bi-delta” geomeuy model. This model encompasses wing-diihedral and canted-forebody (Fig. 4b) options. The “flared-winglet” bi-delta analytical model is shown in Fig. 5. The NSWC tested “clipped bi-delta with yaw-stabilizer” geometry model is shown in Fig. 6. An extension of this model, shown in Fig. 7, provides deflected-elevon definition.

w Technique Description A modified version of the PNS scheme of Bhutta and

~ewis3 served as the primary aerodynamic analysis technique. Details on this method can be found in Refs. 3-1. The technique, referred to here as the GE-VRA/PNS code, is an iterative approach that is unique in not requiring a sublayer approximation. For this application to hypersonic vehicle design, the method was extended to ueat yaw angle of attack. This feature is critical because yaw stability is a prime concem for high-L/D vehicles.

The G E W N S code employs a general curvilinear coordinate system with an efficient elliptic grid-generation scheme. The elliptic grid approach is based on the techniques of Sorenson and Steger? Chaussee and Steger? and Kaul and Chaussee.’* Efficiency is achieved by not driving the grid panial-differential equations to complete convergence (since exact orthogonality is not needed) and by using a coarse body-t-hock grid. The final fine grid is obtained by interpolation. This grid-generation approach produces excellent grids on complex shapes and requires only 10-15% more computational time than the simplest cylindrical-grid scheme.

The GE-VRAPNS code contains perfect-gas and equilibrium-air thermochemisuy models. The two-layer, Cebeci-Smith,’l algebraic eddy-viscosity model is used for

i/

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turbulent flow wirh either instantaneous transition or transition development following the Dhawan-Narasimha’* model. The StioKg cross flows that arise on high-LD shapes at large angles of attack are resolved using a predictor-corrector~cheme~in thecross-flowdirectionanda cross-flow coupled, implicit bow-shock fitting scheme.

Minimal damping is needed for solution stability and the damping that is employed does not require user adjustment. A small amount of 4th-order smoothing is added in the cross-flow and axis-normal directions. The axis-normal smoothing is limited to the pressure field and turned off in the vicinity of embedded shocks. This approach has been dem~nstrated~to accurately capture embedded shocks. In the streamwise dimtion, a 2nd-order damping term, inversely propomonal to the step size, is added to the first-order, backward-difference approximation for the streamwise convective-flux derivatives.

The GE-31S/SCMI3 inviscid flow-field code (coupled to the GE-3DV14 boundary-layer technique) and the HABP” approximate aerodynamic code, were also assessed using the wind-mnnel data. The upwind, split-coefficient math differencing option in the GE-3IS/SCM code, described in detail in Ref. 13, was required because of the large cross-flow gradients around the wing leading edge. In the HABP code, the “delta-wing empirical” pressure option was found to provide the best match with data.

Code Calibration The primary objective of the wind-mnnel test programs

was to obtain an aerodynamic database on a broad range of high-L/D MaRV configurations. The data consists of force-and-moment measurements over a range of Mach and Reynolds numbers and angles of attack and yaw. In the NSWC tesfs, limited pressure and heat-transfer measurements were made in addition to the force-and-moment data. The heat-transfer gauges were placed only on the forebody, primarily to determine if the boundary-layer trip ring was effective. Pressure taps were added to the NSWC model for code calibration purposes. However, due to model constraints, the location of the Kulite pressure msducers was limited to the aftbody.

TheinitialGE-VRA/PNSpredictions weremadeforthe bi-delta wing configurations (straight and canted forebody) shown in Fig. 4a and 4b. The PNS code, with the clustered elliptic grid-generation scheme, proved capable of resolving the emergence of the wing on the bi-delta forebody. The forebody grid is shown in Fig. 8 and the corresponding flowfield for a high angleof-attack condition is shown in Fig. 9. A 50 (body+-hock) by 51 (circumferential) grid was used for all pitch-plane symmetric cases. The emerging wing generates an embedded shock that is further strengthened bycantingtheforebodyupward(Fig. 9b). Aft of

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the forebody/aftbody juncture the flow undergoes a snong expansion as showninFig. 10. The wingleading-edge radius is fixed, thus the wing-tip becomes “sharper” (rdrCone decreases) with increasing distance aft. The end-f-body isobarsandgridforthe bi-deltawingareshowninFig. 11. As shown in Fig. 1 Ib, the sharp leading edge is well resolved by the elliptic grid.

The newly developed yaw capability in the PNS code is demonstrated in Fig. 12. For yaw cases the circumferential grid was increased to 97 points. The skewness of the flowfield,due tothe2-degreeyawangle.isevidentinFig. 12.

As expected, the windward side pressures on the bi-delta wing dominate the vehicle forces and moments. As shown in Fig. 13,the windwardpressurelevelisremarkablyflatandfor this case ( IMegrees angle of attack) is an order of magnitude larger than the leeward pressure. Figure 13 also shows the strong expansion around the leading edge and subsequent recompression.

The axial heatdansfer distribution forthe bi-delta wing at IMegrees angle of attack is shown in Fig. 14. For this AEDC wind-tunnel case, the flow is expected to be turbulent. On theforebody the wing leading-edge(labeled”side”inFig. 14) heating exceeds the windward value.

Comparisons of preiest predictions with the measured bi-delta Configuration forces and moments are shown in Figs. 15-19. The axial force, Fig. 15. is well predicted by the PNS code. The addition of a simple viscous correction also brings the 3IS/SCM and HABP code predictions in-line with the data. All techniques compare favorably with the normal force,Fig. 16, butthe3IS/SCMcodestartstodeviatefromthe side-force data at high angles of attack (Fig. 17). The PNS code predictions providethebestmatchwiththecritical pitch, Fig. 18, and yaw, Fig. 19, center-of-pressuredata. Overall, the PNS code was found to always fall within the uncertainty of the force-and-moment measurements and, as expected, to be more accurate than the inviscid and approximate techniques.

The bi-delta configuration has excellent L/D capability. However, its yaw center-of-pressure (Fig. 19) is forward of the pitch center-of-pressure (Fig. 18) and therefore requires an excessively forward center-of-avity to maintain stability. The addition of a flared winglet on the aftbody, as shown in Fig. 5, greatly increases yaw stability but decreases L/D due to additional drag.

The GE-VRA/PNS code was applied to the bi-delta with flared-winglet configuration and calibrated with AEDC data. Thebehavior of the elliptical gid-generation technique foraflared-wingletcrosssectionis showninFigs. 20and21. On the basis of grid-sensitivity studies, the 97 radial-grid lines for the yaw case, shown in Fig. 20, were more than

~

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adequate to resolve the complex flared-winglet geomeuy. Details of the grid shucture near the winglet leading edge are shown in Fig. 21. The end-of-body pressure field for a small angle-of-attack and yaw case (corresponding to the grid shown in Figs. 20 and 21) is shown in Fig. 22. Figure 23 w shows the dramatic effects of high angle of attack (15-degrees) on the structure of the pressure field. Angle-of-attack effects on the aft-station surface pressure and heating are shown in Figs. 24 and 25, respectively. The peak heating shown in Fig. 25 occurs on the flared-winglet leading edge at low angle of attack (highest on the side facing &e yawed wind vector). Force-and-moment comparisons with the AEDC data are shown in Figs. 26-30. The PNS predictions for all cases are in excellent agreement with the measurements. The gain in yaw stability due to the flared-winglet is demons!rated by comparing Fig. 30 with Fig. 19.

The high Mach number NSWC tests focused on the clipped bi-deltaconfiguration (Figs. 3 and6). Thismodel has the same forebody as the basic bi-delta and flared-winglet bi-deltaconfigurations tested at AEDC. The aftbody wing is sliced (removing the rounded leading edge) and a yaw tab has been added. In addition, elevons, shown in Figs. 3 and 7, provide pitch-ankoll control.

The end-of-bcdy radial grid lines for this configuration are shown in Fig. 3 1. The grid is slightly clustered in the yaw plane to capture flow gradients near the slice and on the y a w u tab. PNS pre-test predictions for a high angle-of-attack case with 12-degrees yaw-tab deflection are shown in Figs. 32-34. The axial surface-pressure distribution, Fig. 32, clearly shows the shock formed by the wing emerging from the forebody and the shock/boundary-layer interaction effects on the yaw tab. The PNS computations assumed that the trip ring on the forecone was effective. However, Fig. 33 indicates that for high angle of attack the flow on the windward side of the forebody (the aftbody juncture is at XRn= 144) is transitional. Comparisons of aftbody surface-pressure predictions with NSWC data are shown in Fig. 34. Theagreementisexcellent. me two predicted spikes in the aft-station pressure distribution correspond to grid points on the underside of the yaw tab which is expected to be ahigh-pressureregion. Unfortunately, pressuretaps werenot placed at this location.

Force-andqoment comparisons with the PNS predictions are shown in Figs. 35-39. In general, the “ agreement is quite good. The largest deviation from the data occurs in the side-force prediction (although the prediction is within the data uncertainty). Additional computations, with post-test hindsight, could be made with varior Uansition-front locations to determine if that would yield a closer match with the side-force data.

Application to Flight Design

Comparisons with the ground-test measurements, and previous experience with the GE-VRAPNS code, provides confidence in the application of the technique for high-LD shapes at flight conditions. Candidate flight designs, to be assessed with this PNS code, were derived using the wind-tunnel data and analyses that encompassed other design disciplines.

A proposed flight configuration is compared with the NSWC wind-tunnel model in Fig. 40.The most noticeable difference is in the forebody. For the flight design the cone-forebodylwing juncture has been eliminated by extending the wing to the spherical nosetip. The all-wing forebody design removes the embedded shock that occurs on the wind-tunnel model (see Fig. 32). A matrix of cases, spanning design critical trajectory conditions, was performed with the PNS and approximate HABP codes. The flight cases employed the PNS code’s equilibrium-air capability (via an efficient table-interpolation approach). A typical high-Mach number, high-altitude solution is shown in Figs. 41-43. For this case, an embedded shock is evident in the pressure field, Fig.41,duetothecombinedeffectsofyaw-tabdeflectionand sideslip. The corresponding surface pressures, around two-forebcdy and twc-aftbody stations, are shown in Fig. 42. The heating in this laminar flow environment is shown in Fig. 43. As expected, the forebody leading edge is subject to the highest heating.

Adequate yaw stability is difficult to achieve while maintaining high-L/D performance. Comparison of the HABP yaw center-of-pressure predictions with wind-tunnel data (see Fig. 19) shows that for ground-test conditions HABP underprdcts yaw stabfity. However, application of the HABP and PNS codes in the flight environment indicates that HABP overpredicts yaw stability, as shown in Fig. 44. The PNS code thus provides a valuable check on the approximate technique and averts potential design problems.

In addition to the analyses of candidate flight designs, the PNS code can be used directly to aerodynamically tailor design features. For example, as shown in Fig. 45, the PNS code was used in the restart mode to rapidly determine the effects of changing an aftbcdy slice angle. The predictions show thatoverarangeofMachnumbers,altitudes,andangles of attack an inboard cant provides the largest yaw stability margin.

Deflection of the aft-rnounted elevons provides pitch (paired deflections) and roll (differential deflections) control. The ability of the PNS code to address the more complex geometry and flowfield suucture due to elevon control surface deflection is demonstrated in Fig. 46. Additional analyses are needed, including comparisons with the NSWC

wind-tunnel data, to calibrate PNS hinge-moment and control-effectiveness predictions.

Computational Requirements

The applicability of a 0 technique in the design process depends on its accuracy and its computational demands. Wind-tunnel data provides a measure of accuracy. The accuracy of the PNS code is achieved at some expense relative to the HABP code. The 97-circumferential by .5O-body-teshock grid used for yaw cases required about 1 to 1.5 hours on a Cray 2 (and only half that time on a Cray XMP). This grid requires about 20megabytes local memory. However, the PNS time/memory requirements are far less demanding than conventional Navier-Szokes solvers.

Conclusions

An extensive series of wind-tunnel tests were conducted to obtain parametric tradeoff data for high-L/D vehicle design. Emphasis was placed on measuring six-component forces and moments. This data, together with limited pressure and heat transfer measurements, were used to assess prediction techniques that are needed to determine aerodynamic characteristics in the flight environment. Inviscid and boundary-layer techniques, approximate methods, and a novel parabolued Navier-Stokes (PNS) code were examined. Analytical geometry models were developed for a representative selection of the numerous configurations that were tested. Overall, the PNS results were superior to the other techniques and in good agreement with the ground-test data. In order to address these complex shapes, and obtain vital design information, the PNS code was extended to include yaw anglwf-anack and differentially deflected elevon capability. Details of computed PNS flowfields were examined. The code was shown to resolve suong expansions and embedded shocks characteristic of high-L/D maneuvering vehicles. The PNS surface pressure distribution was shown to be in good agreement with the limited test data. The extended PNS code was applied to predict vehicle performance at design-xitical trajectory points and surface pressures (loads) and heating rates were generated to support vehicle design aadeoffs. It was found that application of the approximate rapid-design codes in the flight environment can yield misleading results. Use of the PNS code to aerodynamically tailor a candidate flight vehicle was demonstrated. The efficiency of the PNS code, together with confidence gained through detailed comparisons to ground-test data, establishes the code as a viable design tool.

Acknowledgments

This work was primarily sponsored by GE Re+ntry Systems Independent Research and Development Program.

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Technical guidance and direction was provided by Darius Brant. Dave Szostowski, Yick Chan, Benny Prats, and Dan Szostowski made significant contributions in developing geometry models and applying the PNS code. Jerry Berman directed the wind-tunnel activities and vehicle aerothermodynamic design efforts. We gratefully acknowledge the support of Lt. Mark Sardelli in obtaining computer time at the Air Force Super Computer Center at Kirtland AFB.

References

Berman, R. J., “GE IR&D T-32 High Performance MaRV Wmd-Tunnel Tests,” General Elechic Re-entry Systems Department, GE-90SDS2023, Philadelphia, PA, Jan. 1990.

Berman, R. J. and h t s , B. D.. “GE IR&D T-32 High Performance MaRV Wmd-TunnelTests - NSWC,” General Electric R-ntry Systems Department, GE-90SDS2141, Philadelphia, PA, Oct. 1990.

3 Bhutta, B. A. and Lewis, C. H., “Aerothermodynamic Performance of 3-D and Bent-Nose RVs under Hypersonic Conditions,” AIAA Paper 90-3068, Aug. 1990.

Bhutta, B. A. and Lewis, C. H., ‘*Large Angle-of-Attack Viscous Hypersonic Flows over Complex Lifting Configurations,”-, Vol. 27, March - April 1990, pp. 194 - 204.

Bhutta, B. A. and Lewis, C. H., “PNS Predictions of Three-Dimensional Hypersonic Flows with Strong Crossflow Effects,” AIAA Paper 88-2696, June 1988.

Bhutta, B. A. and Lewis, C. H., “Prediction of Three-Dimensional Hypersonic Flows Using a Parabolized NavierStokes Scheme,” p, Vol. 26, Jan. - Feb. 1989, pp. 4 - 13.

Daywin J. E. and Bhutta, B. A,, “A Robust Parabolized NavierStokes Technique for Aerothermodynamic Analyses.” ASE 91 Conference Proceedings, Computational Mechanics Publications, Aug. 1991.

Sorenson, R. L. and Steger, J. L., “Simplified Clustering of Nonorthogonal Grids Generated by Elliptic Partial Differential Equations,” NASA TM-73252,1978.

Chaussee, D. S. and Steger, J. L.. “Three Dimensional Viscous Flowfield Program; Part 2; A Curvilinear Grid and Body Generation Program for Generalized Configurations (Interim Report), ” Flow Simulations, Inc., Sunnyvale, CA, March 1981. IW

lo Kaul, U. K. and Chaussee, D. S.,“A Comparative Study of the Parabolized Navier-Stokes (PNS) Code Using Various Grid Generation Techniques,” AIAA Paper 84-0459, Jan. 1984.

l1 Cebeci, T., Smith, A. M. 0.. and Mosinskis, G., “Calculation of Compressible Adiabatic Turbulent Boundary-Layers,” Vol. 8, Nov. 1970, pp.

l2 Dhawan, S. and Narasimha, R. “Some Properties of Boundary Layer Flow During the Transition from Laminar to Turbulent Motion,”- ’ , VOl. 3, Pt. 4. Jan. 1958, pp. 418 -436.

l3 Daywin J. E., Szostowski, D. J., and Anderson, D. A,. “A Split CoefticientlLocally Monotonic Scheme for Multishocked Supersonic Flow,” Vol. 21, June 1983, pp. 871 - 880.

l4 Hecht, A. M. and Nestler, D. E., “A Three-Dimensional Boundary Layer Computer Program for Sphere-Cone Type Re-entry Vehicles, Vol. I, Engineering Analysis and Code Description,” Air Force Flight Dynamics Laboratory, AFFDL TR-78-67, Wright-Patterson AFB, OH, June 1978.

Is Gentry, A. E., Smyth, D. N., and Oliver, W. R.. “The Mark IV Supersonic-Hypersonic Arbitrary Body Program: Vol2 - Program Formulation, Technical Report,” Air Force Flight Dynamics Laboratory, AFFDL TR-73-159, Wright- Patterson AFB, OH, Nov. 1973.

1974- 1982.

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Fig. 4 Analytical model of bi-delta configuration. I .-

Fig. 1 Modular model components.

.Ioo.LIcrmlDu " mwoIpY-

Fig. 6 Analytical modcl of clipped bi-delta.

Fig. 3 Configuration for NSWC tests.

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Fig. 7 Analytical model of clipped bi-delta with differentially deflected elevons.

., uDIN.LuDLnm 01 LBIDwoQaBImxm

Fig. 8 Clustered-elliptic grid resolves forebody leading edge.

Fig. 9 GE-VRAPNS code captures embedded wing-shock and expansion around leading edge.

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Fig. 10 Isobars on bi-delta wing aftbody reveal suong expansion emanating from forebody juncture.

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Fig. 11 Isobars at end of bi-delta. Clustered-elliptic grid resolves sharp leading edge.

Fig. 12 Yaw capability in GE-VRA/F"S code Isobars at end of bi-delta.

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tl

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Fig. 13 Surface pressure at end of bi-delta configuration. Suong leeside expansion and recompression.

M..I. c1.,o..T~.yT IURELROHT

SRUOMRRPBODl 80

0 0 0 '

* / ............... ................... .......

*n*LrpIANx Q.

Fig. 14 Axial heat-transfer distribution on bi-delta wing. High leading-edge heating on forebody.

Fig. 15 Axial-force comparison for bi-delta. 'U'

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Fig. 16 Normal-force comparison for bi-delta.

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Fig. 17 Side-force comparison for bi-delta.

a w n m s

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Fig. 18 Pitch center-of-pressure comparison for bi-delta.

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DATA lJ?&slTAmn

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I A W OF A n a . D (Oon

Fig. 19 Yaw center-of-pressure comparison for bi-delta.

Fig. 20 Elliptic grid-genmtion technique well-suited for flared winglet.

Fig. 21 Details of grid smctun over winglet.

Fig. 22 Isobars at end of flared-winglet at small angle-of-attack.

Elg. 23 Isobars at end of flared-winglet at high angle-of-attack.

Fig. 24 Angle-of-attack effects on pressure distribution at the end of the flared-winglet.

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Fig. 25 Angle-of-attack effects on heating at end of flared-winglet.

Fig. 26 Axial-force comparison for bi-delta, flared winglet.

0 I MTAUNCURTAIM*

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uyu 0s mu-x a aw Fig. 27 Normal-force comparison for bi-delta. flared-winglet.

DATA UNCUTAIMI I 4 ,I s3

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iw of mm. 0 mi Fig. 28 Side-force comparison for bi-delta, flared-winglet.

Fig. 29 Pitch center-of-pressure comparison for bidclta, flared-winglet.

DATA uIIyT*IHn I 10 I

om -* ANUXoIATTKX.EQ3

Fig. 30 Yaw center-of-pressure comparison for bi-delta, flared winglet.

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Fig. 31 End-of-body elliptic grid for clipped bi-delta.

'.. ............... ~' \ :E%% ....

u I.. I.. m * LUAL W A I I L F . WR,,

Fig. 32 Axial surface prcssun distribution for clippcd bi-delta. Embedded shocks at windcone juncture and yaw stabilizer.

Fig. 33 Comparison of axial surface heat-transfer distribution on the bi-delta forebody. Predictions indicate windward flow is aansitional.

Fig. 34 Comparison of clipped bi-delta circumferential pressure disuibution.

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Fig. 35 Axial-force comparison for clipped bi-delta.

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Fig. 40 Comparison of flight design with wind-tunnel model.

.6 w " I I I Fig. 41 Flight design end of body pressure field with I 0 s

N u 8 WrnAcK.3 mcq embedded shock due to yaw tab. High altitude conditions. Fig. 38 Pitch center-of-pressure comparison for

clipped bidelta.

Fig. 39 Yaw center-of-pressure comparison foi clipped bi-delta.

M, = 20. a = 7.O. p = ID. 2OOKV. LAMINAR

e - . x.,- + &,.I

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Fig. 42 Circumferential pressure distribution on flight design. Peak pressure on forebody leading edge.

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Fig. 43 Circumferential heating. Peak heating on forebody leading edge.

Fig. 44 HABP overpredicts yaw stability for flight conditions.

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Fig. 45 Prediction of aft-slice effects on stability.

Fig. 46 Isobars at end of clipped bi-delta with differentially deflected elevons.

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