[american institute of aeronautics and astronautics 16th aiaa applied aerodynamics conference -...

Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

A98-32410 AIAA-98-2412

STAGING OF A HYPERSONIC VEHICLE: NUMERICAL SIMULATIONS USING DYNAMIC

UNSTRUCTURED MESHES

Oktay Baysal* and Xiaobing Luo#

Aerospace Engineering DepartmentOld Dominion University

Norfolk, Virginia 23529-0247

e-mail: [email protected]

AbstractPrior research had shown that the dynamic unstructured meshtechnique (DUT) could simulate an unsteady flow around singleor multiple bodies, which are engaged in either a prescribed oran aerodynamically determined relative motion. In addition,some recent algorithmic improvements have largely alleviatedmost of the efficiency concerns that existed before. The presentpaper reports a demonstration of DUT by simulating theseparation of an air-breathing hypersonic research vehicle fromits booster vehicle. These very high-speed flows with strongshocks, however, required a modified flux-vector splittingmethod with a limiter to improve the numerical stability. Thesimulations were performed on a workstation with two-degrees-of-freedom dynamic motion on the longitudinal symmetryplane of the vehicle. The results demonstrated the dependenceof the unsteady forces and moments on the mutual interference.Further, they indicated that the impact of the interference was afunction of the flight Mach number. Finally, this proof-of-concept investigation could suggest that the numericalsimulations, if performed on the actual vehicle configurationwith the realistic conditions, could provide most of theinformation needed to reduce the matrix of the flight tests.

IntroductionA wide variety of engineering problems involves time-dependent flowfields. The forces and the momentsexerted on the surrounding boundaries by such aflowfield, therefore, are unsteady and they require oftenvery costly time-accurate computations. In the event of abody immersed in such a field, it is also desirable topredict accurately the trajectory of the resulting bodymotion. Following are some examples: rotor-statorinteractions, store separation sequences, escape-podejections, detachment of multistage rocket components,

rotorcraft dynamics, separation of booster tanks from thespace shuttle, actively controlled high-lift devices,moving interfaces in multiphase flow (Stefan problem),dynamic aeroelasticity problems, and heart-valve andblood flow interactions.

Considering the unsteady flowfield around amultibody configuration, with one or more of thecomponents engaged in a relative motion, there arevarious levels of simulations that can be made for theincident-flow and solid-surface interaction (Baysal, et al.,1996). Each moving component can be assigned its ownrigid-body motion, then either the motion is known so itcan be prescribed as input to the unsteady flowcomputations, or the trajectory is determined from theinstantaneous flowfield using the principles of rigid-bodydynamics, i.e. it is aerodynamically determined. Unlikethe dynamic domain decomposition methods (D3M), asreported by Baysal and Yen (1991), and Yen and Baysal(1994, 1995, 1997), the present method employs a singlemesh with deforming cells to adapt to the boundarymotion. Singh and Baysal (1995, 1996), and Singh, et al.(1995) previously reported this dynamic unstructuredtechnique (DUT). DUT has recently been significantlyimproved for computational efficiency (Baysal and Luo,1998). The present paper will report on theseimprovements and an application of DUT in simulatingthe separation of a research vehicle from its booster athypersonic speeds. The currently ongoing Hyper-Xprogram motivated this example, so some backgroundwill be presented next.

NASA, primarily to enhance hypersonic airbreathingtechnology for long term applications, has initiated the

Professor and Eminent Scholar. Associate Fellow, AIAA.Graduate Research Assistant.

Copyright © 1998 by the American Institute of Aeronautics and Astronautics, Inc. AH rights reserved.

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Hyper-X program. The goal is to demonstrate andvalidate the technology, experimental techniques andcomputational methods, and tools for design andperformance predictions of a hypersonic vehicle with anairframe integrated scramjet propulsion system (Rauschet al., 1997). Baysal et al. (1992) and Baysal andHoffman (1992) previously computed the mixing of thisscramjet nozzle jet with the hypersonic freestream. Fourresearch flights have been scheduled between 1999 and2002 to demonstrate, validate, and extend scramjettechnology readiness. The first flight of these 12-ft. long(the operational aircraft is conceptualized to be 200 ft.long) and 5-ft. wingspan research aircraft will be at Mach7, followed by a Mach 5 flight and two Mach 10 flights.Orbital Sciences Corporation is currently building aderivative of their Pegasus launch system as the Hyper-Xlaunch vehicle (HXLV), that will boost the Hyper-Xresearch vehicle to its test conditions. The desired testcondition for the test vehicle in free flight is a dynamicpressure of 1,000 lb/ft^. The flight trajectory starts witha B-52 aircraft carrying the HXLV-mounted researchvehicle to about 40,000 ft. (or less for lower flightspeeds). After separation from B-52, HXLV will ascendto 85,000 ft. for the Mach 5 flight, 100,000 ft. for theMach 7 flight and 110,000 ft. for the Mach 10 flight. Atthis point the research vehicle will separate from itsbooster and the flight test will resume.

The research vehicle will be mounted on an armattached to the nose of HXLV. The planned stageseparation will resume with the ignition of the explodablerivets fastening the vehicle to the arm. Then, the arm willswing down about the hinge connected to HXLV leavingthe research vehicle free and airborne at the desired flightspeed. However, whether this stage separation will endwith the Hyper-X vehicle having a favorable initialattitude for its flight is not certain at this time. Anaerodynamic interference is expected on the vehicle dueto the shocks generated by the arm. Therefore, it isdesirable to simulate this separation processcomputationally. An adequate prediction of this unsteadyhypersonic flowfield, until the aerodynamic interferenceis negligible, should allow a close scrutiny of theresulting loads on the vehicle.

MethodologyAn Eulerian approach to track the fluid-structure motiontreats the computational mesh as a fixed reference framethrough which the fluid moves. The coordinate system isstationary in the fixed reference frame or moves in aprescribed manner to account for the continual changingshape of the solution domain. Thus, the grid motion isindependent of the motion of the fluid.

In the present Eulerian method, the upwind-discretized form of the governing equations wereintegrated in time using an explicit four-stage Runge-

Kutta method. This method had a second-order temporalaccuracy, even in its earlier steady state and static version(Frink, 1992). To avoid grid-motion induced error, thegeometric conservation law had to be satisfiedconcurrently with the conservation equations (Singh et al.1995). For the adaptation of the mesh to the boundarymotion, the tension-springs analogy (Baysal, et al. 1992)was used, where each edge of a tetrahedron wasrepresented by a tension spring. To restrict the size ofthe adaptation region, a window was created around amoving body. The window points were allowed to beadapted, but the nodes exterior to the window werespatially fixed in time. Recently, several majorimprovements have been incorporated into the baselineOUT (Baysal and Luo, 1998). These will be brieflydiscussed next.

First, to enhance the quality of the conforming meshas a body moved, the mesh smoothing using the Jacobimethod followed mesh adaptation within a window.Then, the mesh was optimized for a target areadistribution (Pirzadeh, 1992). This has by and largealleviated otherwise conceivable mesh tangling problemsduring the relatively large increments of body motion permesh update. Hence, the motion could be covered withless number of mesh updates. Secondly, an $(2) -accurate implicit time integration, which employed dual-time steps (subiterations) for unsteady flows, wasextended for relative moving body problems.Venkatakrishnan and Mavriplis (1996) also reported thismethod. This allowed taking much larger time-accuratesteps than the explicit Runge-Kutta time integrationmethod of the baseline DUT. Denoting the vector ofconserved variables of fluid flow by Q and a finite-volume cell by V, the integral form of the Euler equationswere time-integrated using a three-point backward-difference formula (At denotes global, physical timestep):

L[(VQ)n+J] = (1)

where L and 5 denote an operator and a source,respectively:

(2)

(3)

The residual R was constructed as the sum of the fluxes Fthrough the cell surfaces, 5:

R, = Ij=kfi)

(4)

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The baseline DUT had employed either the van Leerflux-vector splitting or the Roe flux-difference splittingto obtain the upwind discretization, and no limiters wereused for the higher-order scheme (Singh et al., 1995).This method, however, experienced severe numericalinstabilities for hypersonic flows with strong shocks.Similar problems were reported by others, such as, Bibband Peraire (1997). Hence, the flux-vector splitting dueto Hanel (1987) with van Albada limiter was used toimprove the stability. This method splits a flux vector tofacilitate the upwind discretization as follows:

where

y

H

t±J±(un±c)2/4c ,if\un\<c" [ (MB±|«n |)/2 else '

(5)

(6)

(7)

±\i((un/c)±l)2(2±(un/c)) ,if\un\<c" [ p(un±\un\)l2un ,else

For the higher-order upwind scheme, the flux values at acell interface were obtained using the following nonlinearlimiter. Considering an interface between two triangularcells, with the left cell center i and the right cell centerj+7, and the left vertex./ and the right vertex j+1, Q~and Q+ components at this interface, respectively, are:

(9)

fi+=fi i+/-f[(/ + w)4_+y-w)4 t]/+7 , (10)where

2A_A.+e« ̂ ~____

A2_+A2++eand

A-M = QM ~ Qi A+M = QJ+I ~ QM

Considering U as an approximation to Q, anunsteady residual was defined from eq. (1) as,

/? (V

(11)

(12)

(13)

The resulting nonlinear system could be first augmentedby a pseudo-time term,

d(VU) n+l

dt*-+R*(Un+') = (H)

then solved by a low-storage, t?(2) -accurate, m-stageimplicit Runge-Kutta scheme (At denotes local,pseudo-time step)

0 =Vn+lQM _a>Af

rn, Un+I

where, Q(0)

(15)

(16)

To accelerate the convergence of the subiterations ineq. (15), an overset-type and full-approximation-scheme(FAS) multigrid method was developed. Bonhaus (1993)and Ollivier-Gooch (1995) previously reported similarapproaches. The purpose of any multigrid scheme(Mavriplis, 1995, Baysal et al., 1991) is to transform theslow decaying, low frequency error component to ahigher frequency error by reconstructing the residual on acoarser mesh, then using this correction in the fine-meshsolution. By referring to any pair of sequentiallycoarsened meshes as fine and coarse meshes anddenoting them by the subscript/and c, respectively, themethod may be summarized as follows. When eq. (15)was solved on a coarser mesh, the source term wasmodified to include the residual of the fine meshsolution:

**f(Uf) , (17)

where //_*; is a fine-to-coarse restriction operator.Upon obtaining the coarse mesh solution, the fine-meshsolution was then corrected as,

(18)

where 7C_^ is a coarse-to-fine prolongation operator.After locating a coarse mesh node n in a fine-mesh cell(or vice versa) with nodes numbered as 7,2,3, finding thevalue of a transferred quantity q at the coarse mesh noderequired solving the equation of the plane at that node.This plane was mathematically defined as,

Ax+By + C = (19)

and solved for coefficients A, B, C. Then, the value ofthe transferred quantity at the coarse mesh node was,

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(20)

The geometric weights Wj, W2, W3 were found in termsof the coordinates of the coarse mesh node and thevertices of the surrounding fine mesh cell. To interpolatethe residual from the fine mesh to a coarse meshconservatively, the weights used for the distribution werethe same weights used in the linear interpolation from thecoarse mesh to the fine mesh. This ensured that all fine-mesh residuals contributed to the coarse mesh and thatthe total residual was conserved, since the sum of theweights multiplying the residual at any fine mesh nodewas unity.

ResultsPrior to embarking on the staging problem, themethodology was first tested through a dynamic flowcase: an NACA-0012 airfoil was immersed in aM^ = 0.6 flow and sinusoidally pitched about its three-quarters chord with a mean incidence of o ,̂ = 4.86-deg.The amplitude and the reduced frequency were Oo =2.44-deg and k =0.081, respectively (fig. 1). Both theexperimental data by Landon (1982) and independentcomputational predictions by Singh et al. (1995), andYen and Baysal (1994) were available for this benchmarkcase. The flowfield was first computed using thebaseline DUT, which was an explicit scheme on a singlemesh (case 1), then with the present implicit (case 2).Finally, they were repeated using the multigridacceleration on one fine mesh with one coarse mesh (case3), then on one fine mesh with two coarser level meshes(case 4). As shown in fig. 1, the present methodpredicted the instantaneous pressure distributionssuccessfully. More importantly, significantcomputational timesavings (an order of magnitude) wererealized over the baseline DUT.

The multicomponent configuration used in thepresent demonstrations (fig. 2) was inspired by theHyper-X research vehicle, which was discussed in theIntroduction section. For simplicity, only the two-dimensional longitudinal symmetry plane wasconsidered. Hence, although the spanwise effects wouldin reality be rather significant, they were not included inthe present simulations. Also, the engine cowl wasassumed as blocked, that is, no through flow. Thestaging was simulated as a two-degrees-of-freedommotion. The booster and the arm were assigned atranslational motion and superimposed was the arm'srotation:

AX=Vxt e=cot (21)

where the values for translational and the rotationalvelocities were arbitrarily assigned as Vx =9.8 ft I s and(O =7.1 rod I s , respectively. The results presented arefor t= 12.2ms .

Presented in fig. 2 are the close up view of thedynamic unstructured mesh and the instantaneous Machnumber contours for the M^=5 flight at the instantwhen the distance was AX. = 0.12 ft. and 6 =5-deg. Thesame instant is depicted via its pressure contours in fig. 3.The mesh consisted of 69,936 cells, of which only 11,356were inside the dynamic window. First the steady-statesolution was obtained for the static carriage position onan R 10000 workstation. Then, the time-accuratecalculations were started for the dynamic separation withunsteady flowfield, and theywere obtained by using aboutten hours of CPU time for each case.

As observed in these figures, a very complexnetwork of strong shocks, expansions, and theirinteractions dominated the predicted flowfield. Asubsonic, high pressure region was formed at the inboardbetween the vehicle and the arm, resulting in two jets,one bleeding into the base region of the vehicle and theother to the lower surface of the booster nose. The sameinstant but for the Mach 10 flight was also captured viaits Mach contours and pressure contours, as presented infigs. 4 and 5, respectively. As expected, the shocks werestronger and blown closer to the surfaces by the fasterfreestream.

To demonstrate the interference effects on thevehicle for the Mach 5 and Mach 10 flights, the pressurecoefficient distributions on the vehicle's surface atvarious instants were plotted in figs. 6 and 7,respectively. The time variation was confined to only 5%length on the upper surface, but it stretched to 20% onthe lower surface. For the Mach 5 case, as the armrotated, the pressure rise on the upper surface was largerthan that on the lower surface. This resulted in adecreasing normal force, decreasing pitching moment(about its area centroid), but the thrust (negative axialforce) kept increasing (fig. 8). This trend was not,however, quite similar for the Mach 10 case. Thepressure rise was slightly greater on the lower surfaceduring the separation as compared to the upper surface.Hence, the normal force increased, and with increasingthrust contribution, axial force decreased (fig. 9). Theincrease in the pitching moment was attributed to thelonger torque arm on the upper surface to the point wherethe resultant pressure was effective.

ConclusionsA dynamic unstructured mesh technique (DUT) was usedto simulate an unsteady flow around a multiplecomponent configuration, which was engaged in a

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prescribed relative motion. The recent algorithmicimprovements have largely alleviated most of theefficiency concerns that had existed before. Asdemonstrated via a benchmark case, the presentimprovements decreased the computer time needed for acycle of the motion by about an order of magnitude. Thiswas realized by the multigrid accelerated, implicit dual-time stepping scheme on the dynamic unstructuredmeshes.

The present proof-of-concept demonstration wasinspired by the separation of an air-breathing hypersonicresearch vehicle from its booster vehicle. These veryhigh-speed flows with strong shocks, however, required amodified flux-vector splitting method with a limiter toimprove the numerical stability. The simulations wereperformed on a workstation with two-degrees-of-freedomdynamic motion on the longitudinal symmetry plane ofthe vehicle. The results demonstrated the dependence ofthe unsteady forces and moments on the mutualinterference. Moreover, they indicated that the impact ofthe interference was a function of the flight Machnumber.

Further research, however was deemed neecessaryfor a number of outstanding issues. Among these wereadaptive remeshing scheme (e.g. h-refinement) for large-amplitude and low-frequency motion problems andparallel algorithms for faster processing. Finally, thisproof-of-concept investigation suggested that thenumerical simulations, if performed on the actual vehicleconfiguration with the realistic conditions, could providemost of the information needed to reduce the matrix ofthe planned flight tests.

ReferencesBaysal, O., Fouladi, K., and Lessard, V.R., 1991,

"Multigrid and Upwind Viscous Flow Solver on 3DOverlapped and Embedded Grids," AIAA Journal,Vol. 29, No. 6, pp. 903-910.

Baysal, O., Yen, G-W., 1991, "Kinematic DomainDecomposition to Simulate Flows Past MovingObjects," Paper No. 91-0725, AIAA 29th AerospaceSciences Meeting, Reno, NV.

Baysal, O., Eleshaky, M. E., and Engelund, W. C., 1992,"Computations of Multispecies Mixing BetweenScramjet Nozzle-Afterbody Flows and HypersonicFreestream," Journal of Propulsion and Power, Vol.8, No. 2, pp. 500-506.

Baysal, O., and Hoffman, W. B., 1992, "Simulation of 3-D Shear Flows Around a Nozzle-Afterbody at HighSpeeds," Journal of Fluids Engineering, Vol. 114,No. 2, pp. 178-185.

Baysal, O., Singh, K.P., and Yen, G-W., 1996, "DynamicCFD Methods for Prescribed and Aerodynamically-Determined Relative-Moving Multibody Problems,"

Proceedings of First AFOSR Conference onDynamic Motion CFD. New Brunswick, NJ, pp. SI-44.

Baysal, O., and Luo, X-B., 1998, "ComputationalAeromechanics Method with Multigrid AcceleratedDual Time Stepping on Unstructured Meshes,"Paper No. FEDSM98-4939 CP, Proceedings ASMEFluids Engineering Division Summer Meeting.Washington, DC.

Bibb, K.L., and Peraire, J., 1997, "Hypersonic FlowComputations on Unstructured Meshes," AIAAPaper 97-0625, 35th Aerospace Sciences Conference,Reno, NV..

Bonhaus, D.L., 1993, An Upwind Multigrid Method forSolving Viscous Flows on Unstructured TriangularMeshes. Master's Thesis, George WashingtonUniversity, Hampton, VA.

Frink, N.T., 1992, "Upwind Scheme For Solving theEuler Equations on Unstructured TetrahedralMeshes," AIAA Journal, Vol. 30, No. 1, pp. 70-77.

Hanel, D., Schwane, R., and Seider, G., 1987, "On theAccuracy of Upwind Schemes for the Solution of theNavier-Stokes Equations," AIAA Paper 87-1005,25th Aerospace Sciences Conference, Reno, NV..

Landon, R., 1982, "NACA 0012 Oscillatory andTransient Pitching," Compendium of UnsteadyAerodynamic Measurements. AGARD Report No.702, pp. 3.3-3.25.

Mavriplis, D.J., 1995, "Multigrid Techniques forUnstructured Meshes," ICASE Report 95-27, NASACR 195070, Hampton, VA.

Ollivier-Gooch, C.F., 1995, "Multigrid Acceleration ofan Upwind Euler Solver on Unstructured Meshes,"AIAA Journal, Vol. 33, No. 10, pp. 1822-1827.

Pirzadeh, S., 1992, "Recent Progress in UnstructuredGrid Generation," Paper No. 92-0445, AIAA 30thAerospace Sciences Meeting, Reno, NV.

Rausch, V.L., McClinton, C.R., and Hicks, J.W., 1997,"Scramjets breathe new life into hypersonics,"Aerospace America, Vol. 35, No. 7, pp. 40-46.

Singh, K.P., and Baysal, O., 1995, "3D UnstructuredMethod for Flows past Bodies in 6-DOF RelativeMotion," Proceedings of 6th InternationalSymposium on CFD. Vol. 3. pp. 1161-1168. Also,to appear in AIAA Journal.

Singh, K.P., and Baysal, O., 1996, "Development of aDynamic Unstructured Euler Method for MovingRigid Bodies," Unsteady Flows 1996. (Eds. Wei,Baysal, Keith) FED Vol. 312, ASME, New York,NY, pp. 32-38.

Singh, K. P., Newman, J. C. Ill, and Baysal, O., 1995,"Dynamic Unstructured Method for Flows PastMultiple Objects in Relative Motion," AIAAJournal, Vol. 33, No. 4, pp. 641-659.

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Venkatakrishnan, V., and Mavriplis, D.J., 1996, "Implicit Dynamic Domain Decomposition Method,"Method for the Computation of Unsteady Flows on Unsteady Flows 1995. (Eds. Keith, Tsukamoto,Unstructured Grids," Journal of Computational Baysal, Wei) FED Vol. 216, ASME, New York, NY,Physics, Vol. 127, pp. 380-393997. pp. 21-28.

Yen, G.W., and Baysai, O., 1994, "Computing Unsteady Yen, G.W., and Baysal, O., 1997, "Effects of EfficiencyHigh-Speed Flows Past An Oscillating Cylinder Techniques on Accuracy of Dynamic OverlappedNear a Vertical Wall," Journal of Spacecraft and Grids for Unsteady Flows," Journal of FluidsRockets, Vol. 31, No. 4, pp. 630-635. Engineering, Vol. 119, No. 3, pp. 577-583.

Yen, G.W., and Baysal, O., 1995, "Computing StoreSeparation and its 6-DOF Trajectory using 3-D

90

a = 4.28° T

to

a =6.87° T

a =5.41° i

a = 7.28° T

a =4.23° i

a =7.13° i

Fig. 1 Comparisons of pressure coefficient on airfoil with prescribed motion.motion equation : a = 4.86° + 2.44° sin(0.0486t). M_ = 0.6. data from Landon (1982).case 1 : explicit scheme , case 2 : implicit scheme (single mesh)case 3: implicit scheme (two level mesh) , case 4: implicit scheme (three level mesh).

a = 2.56° i

oo-a^ca'̂

©CDCD00

(D̂

o'£D

r-l-(DO— H

>(D330)^ov>0)3a.

0)

o'y>

o


Close up view of dynamic mesh Mach contours

Y/L

0.3

0.2

0.1

Y/L 0.0

-0.1

-0.2

-0.3

-0.4

Nose close up

-1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.2 -0.1

Fig. 2 Staging during Moo = 5.0 flight: instantaneous (0 = 5°) Mach contours when vehicle ( L=12 f t ;separates from booster at 85,000 ft altitude ( qM = 1000 Ibf /ft2 ).

Close up view of mesh Pressure contours

1.0 -0,8 -0.6 -O.4 -0.2 0.0 0.2

0.3

0.2

0.1

Y/L o.o

-0.1

-0.2

-0.3

Nose close up

-0.6 -0.4 -0.2 0.0

X/L

Base close up

•1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2

X/L

Fig. 3 Staging during M_ = 5.0 flight: instantaneous (9 = 5°) pressure contours when

vehicle (L = 12ft) separates from booster at 85,000ft altitude ( q^ = 1000 Ibf / ft2 )

92


Close up view of dynamic mesh Mach contours

Y/L

•1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2

X/L

-0.1

-0.2

-0.3

Nose close up Base close up

Y/L

-1.0 -0.8 -0.6 -0.4 -02X/L

Fig.4 Staging during M_=10. flight: instantaneous (6 = 5°) Mach contours when vehicle (L=12ft)separates from booster at 110,000 ft altitude (qM= lOOOIbf/ft2).

Close up view of dynamic mesh Pressure contours

Y/L

-0.6 -0.4 -0.2 0.0 0.2

X/L

Base close up

Fig. 5 Staging during M_=10. flight: instantaneous (0 = 5°) Pressure contours whenvehicle ( L = 12 ft) separates from booster at 110,000 ft altitude (q_ = 1000 Ibf / ft2).

93


Upper surface

-O.8 -0.6 -0.4 -0.2 O.O

X/L

Lower surface

0.30

0.25

0.2O

O" 0.15

0.10

0.05

O.i

Upper surf ace

O 1.1

-0.100

X/L

Lower surfaceCtoa* up view of tail

'-1.0 -0.8 -0.6 -0.4 -0.2 0.0X/L

-O.10 -O.OSX/L

Fig. 6 Staging during M_ = 5 flight. Pressure coefficient along vehicle's surfaceat various instants . L = 12 ft. q_ = 1000 Ibf / ft2 , PMAX = 1828 Ibf / ft2 , Pmin = 63 Ibf / ft2 .

Upper surfaceUpper surface

-1.0 -0.9 -O.8 -0.7 -O.6 -0.5 -0.4 -O.3 -0.2 -0.1 0.0X/L

Lower surface

' -O.20 -0.18 -O.16 -0.14 -0.12 -O.10 -O.08 -O.06 -O.04 -0.02 O.OOX/L

Lower surfaceClos* up of view of tail

0.00

Fig. 7 Staging during M_= 10 fligt. Pressure coefficient along vehicle's surface atvarious instants . L = 12 ft, q_ = 1000 Ibf / ft2 , PMAX = 2594 Ibf / ft2 , PMIN = 33 Ibf / ft2 .

94


AIAA 98-2412 CP

0.464

0.462

0.460

0.458

0.456

0.454

0.452

0.450(

0.024

0.023

0.022

0.021

O.02O

0.019

O.018

O.017

normal force

s e

axial force

pitching moment( Xc = - 0.466, Yc = 0.0891 )

e -0.35-0.30-0.25-0.20-0.1S-0.10-0.050.00 0.05 0.10 0.15

X/L

Fig. 8 Force and moment histories on hypersonic vehicle during M_ = 5 flight asit separates from the arm and the booster. Motion : 8 = to t, AX = Vx t.

0.315normal force axial force

0.032

0.031

0.030

0.029

0.028

0.027

0.026

0.025

pitching moment( X,. = -0.466, Yc = 0.0891 )

°-022o———\———£———f

e" -0.35-0.30-0.2S-0.20-0.15-0.10-O.050.00 0.05 0.10 0.15

X/L

Fig. 9 Force and moment histories on hypersonic vehicle during M_=10 flight asit separates from the arm and the booster. Motion :9 = oo t , A X = Vx t.

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