CFD technology for hypersonic vehicle design

CFD technology for hypersonic vehicle design

2 CFD Technology and Reacting Flows CM 30:1/7-D 0045.7919198 u.al + o.at 6 1988 ibgimm Rar pk. CFD TECHNOLOGY FOR HYPERSONIC VEHICLE DESIGN GE...

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2 CFD Technology and Reacting Flows

CM

30:1/7-D

0045.7919198 u.al + o.at 6 1988 ibgimm Rar pk.

CFD TECHNOLOGY

FOR HYPERSONIC

VEHICLE DESIGN

GERALD C. PAYNTER CFD Anaiysis Group, Boeing Advanced Systems, Seattle, WA 98124 U.S.A. Abstract-Because of tbe recent national interest in hypersonicaircraff CFD technologywith hypersonic applicationhas been and is under intensivedevelopmentat Boeing. A substantial CFD capabilitynow existsthat can be appliedto a wide range of hypersonic Rows, including those with finite-rate chemistry etkck The paper describes this hypersonic CFD development progress. It inctudcs a review of the code features required for bypcrsonic application, the development approach, progress and a discussion of the limitations of tbe current physical modeling.

INTRODUCI’ION

stage-to-orbit launch system, and a hypersonic transport. Propulsion systems include various combinations of ramjets, scramjets, turbojets and rockets. Propulsion installations for hypersonic aircraft are unique in that a large portion of the airframe is typically integrated into the propulsion system. Design of these vehicles requires accurate knowledge of the vehicle flow environment at critical Right conditions. Some of the important design issues are the aerodynamic heating and thermal protection system, the lift and drag, control system effectiveness and the inlet, combustor and nozzle performance. Hypersonic flow phenomena have a significant influence on these design issues. These include real gas effects, merged shock-shear layers, high wall

As noted by Hearth and Preyss [l], manned aircraft have been limited by aerodynamic heating and turbojet engine performance to speeds below about Mach 3 (except for re-entry vehicles and the X-15). Interest in new aircraft operating in the hypersonic speed regime has resurged. One notable example of thii is the ~jet-twerp, airb~athing-t~orbi~ aerospace plane [2]. As shown in Fig. 1, future hypersonic aircraft are expected to operate at Mach numbers above three and at altitudes below the space vehicle reentry corridor. Military aircraft under consideration include advanced interceptors, long-range reconnaissance and strike/reconnaissance vehicles. Civil aircraft include an airb~athing launch vehicle, a single-

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Fig. 1. Hypersonic vehicle flight regimes. 39

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GERALD C. PAYNTER

cooling, radiation, catalytic wall effects, laminar and turbulent shock-boundary layer interactions, transition and finite-rate air and fuel-air chemistry. A significant portion of the hypersonic flight envelope cannot be simulated experimentally. At low densities and high velocities, the molecular reaction rates are ‘slow’ in comparison to the time required for a fluid element to travel a given distance over or through the vehicle. Thii suggests that the internal molecular energy modes, chemical dissociation, species ionization, etc., are out of chemical equilibrium [3]. Experimental simulation of a flow in which nonequilibrium chemistry is important requires that the density, velocity and geometric scale all be reproduced in the experiment and this is not possible within current test facilities. Furthermore, the small model scales, short run times, low total enthalpies and unsteady test conditions of the existing test facilities make accurate measurement of flow and performance properties very difficult. Computational Fluid Dynamics (CFD) is an avenue around the constraints imposed by the existing test facilities. CFD can provide solutions to the governing flow equations (and therefore the vehicle flow environment and performance) in the hypersonic speed regime. Existing experimental data can be used to evaluate the accuracy of the CFD simulations for specific phenomena CFD is a way to extrapolate the existing experimental data base to support vehicle design outside of this data base. As pointed out in [3] and above, hypersonic CFD methods must address a number of problems not encountered within the flight envelopes of conventional aircraft. These include the following (1) The equations describing the flow are more complex. Linear or Euler plus boundary layer formulations are inadequate. Even extensions of the Navier-Stokes quations are necessary in some flight regimes (2) A noncontinuum flow model is needed when the molecular mean-free-path becomes ‘large’ relative to the scale of vehicle geometry features (such as the nose or leading edge radius). (3) Nonequilibrium chemistry effects arc important. In the air flow about the vehicle and in the fuel-air reactions of the propulsion system, nonequilibrium chemistry cannot be neglected. The overall goal of the hypersonic CFD technology development at Boeing is to provide aeroheating, control, aerodynamic and propulsion design information. Devefopment of the hypersonic CFD technology requires that each of the elements of the CFD technology be addressed to provide the needed increase in capability. The elements of the CFD technology are generally identified [4] as the computer system, CFD codes and algorithms, geometry representation, mesh generation, physical modeling, validation, and pre- and post-processing The purpose of this paper is to describe progress at Boeing toward developing the CFD technology

necessary to support design of hypersonic vehicles. This is addressed through a review of the Design Process in the next section followed by sections on Hypersonic CFD Requirements and Development, Applications and Conclusions. THE DESIGN PROCESS Flow related info~ation is needed for design of the aerodynamic, propulsion, flight control and thermal control systems. In a typical design cycle, initial concepts are chosen on the basis of preliminary design studies. Critical flight conditions are then selected to characterize the operating envelope for the various vehicle systems. Parametric variation of the baseline design at the critical flight conditions is used to obtain design information on the performance of the various aircraft systems for configuration refinement. This process is illustrated in Figs 2 and 3. For ~nv~tional aircraft, either wind tunnel testing and/or CFD are sources of the performance information needed for configuration selection. As noted above, wind tunnel testing is unavailable as a source for this information for a significant portion of the hypersonic flight envelope. CFD is thus used much like a numerical wind tunnel (or flight test) to acquire the necessary information. Because the total number of cases analyzed in a design study is the product of the number of configurations and the number of critical flight conditions, it is important to constrain both of these factors (as well as the sophisti~tion of the CFD analysis) to keep the overall CFD computing task within the bounds of the available computing resources. HYPERSONIC CFD REQUIREMEh’TS AND DEVELOPMENT

Work toward achieving a viable hypersonic CFD capability was accelerated in 1985 in response to an increased national interest in this flight regime. The initial objective was to extend the existing capability to continuum flow describable by the Reynoldsaveraged Navier-Stokes equations (or a subset thereof) augmented by specie conservation equations for air and hydrogen-air reacting flows. As noted by Bradley [3], this extended capability would be applicable to hypersonic flight in the lower atmosphere. The argument for initially not addressing the CFD requirements for hypersonic flight in the upper atmosphere (noncontinuum flow and the required extensions of the Navier-Stokes equations) was that the peak aerodynamic loads and heating rates were expected to occur in the lower atmosphere below 150,oooft). The overall approach to the development, acquisition and application of the CFD Technology is that described in [S]. Use of CFD to provide hypersonic design information for configuration selection requires that the elements of CFD

CFD technologyfor hypersonicvehiclerksiga

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Fig. 2. The design cyctc. technology be addressed as shown in Fig. 4. These include the computer hardware and system, codes and algorithms, geometry representation and mesh generation, physical modeling, validation and preand post-processing. Two observations guide the CFD technology preparation. These observations are that: (1) The computer storage and speed currently available severely constrains the level of analysis and the number of cases computed. (2) Rapid acquisition and development of the technology needed to achieve the initial objective could provide a competitive advantage. Two strategies were and are being used to address the computer limitations and the desire for a rapid development of the capability. The first strategy, zonal modeling, is a way to achieve accurate flow simulation within the available computer capacity. The primary zonal strategy for aircraft analysis in the hypersonic regime interacts parabolized Navier-Stokes (PNS) and full NavierStokes (FNS) codes. The FNS codes require storage of up to 30 quantities at each point in the flow domain and time relaxation to achieve a solution. FNS codes have the widest range of application but are the most expensive in the computer CPU time and storage required to achieve a solution. PNS codes march plane-by-plane through the flow domain and require storage of about 30 quantities per point in the plane immediately upstream of the plane to be computed. PNS codes require much less storage

and require an order-of-ma~tude less CPU time to achieve a solution (relative to FNS) on a given grid. PNS codes are, however, applicable only to those flows where upstream influence is not important. In the hypersonic zonal strategy, FNS is used only in the localized leading edge, stagnation, or separated flow regions where PNS analysis is inappropriate. PNS analysis is used over the remainder of the vehicle. Five or six million grid points may be required to adequately resolve local Sow fatures for a full vehicle analysis. Since 90% (or more) of a vehicle flow domain is attached predominantly supersonic flow, the tonal strategy can be used to reduce the CPU cost from weeks to hours without a significant loss in solution accuracy. The second strategy is to base the hypersonic CFD technology on the existing CFD capability to minimize development cost, time and risk. Because a substantial CFD capability had been developed and demonstrated for the subsonic, transonic, and supersonic speed regimes, this capability could be extended to the hypersonic regime through the addition of the technology features unique to this speed regime while minimizing the development time and cost. Hypersonic CFD technology progress is addressed herein by reviewing, in subsections below, the status at the beginning of development, additional hypersonic requirements, current status, and planned development for the computer system, codes and algorithms, geometry and mesh, physical modeling and vaiidation.

GERALD

42

C. PAYMER

Fofa givenaircraft component

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Windtunnel

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Fit. cond. 2

Alt.

A Configuration

M Select critical uwrating_wintt

Select confiir~tion & pmdii performnce

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Generate design information at aitcal operating points

Fig. 3. Generation of design information. The computer system A CRAY XMP was available at the initiation of the hypersonic CFD development effort. While the XMP was and is representative of the state-of-theart in scientific computing, it (and the associated data transmission system) did not satisfy two requirements for hypersonic CFD support: (1) The need for a classified computing mode. (2) The need for high-speed data links with major government computing centers. A classified computing mode is essential because most of the hypersonic aircraft under consideration would satisfy a national defense requirement. Highspeed data links to major government computing centers are necessary because these resources are often made available to support major government contracts. The ability to quickly access and use government computing resources could be a factor in a design competition. A VAX 83OO/PPS 264 mini-supercomputer was acquired to satisfy the need for a stand-alone classified computing capability. The FPS 264 has 4.5 (106) words of memory and the effective CPU speed is about 25% that of one processor of the CRAY XMP. A 56-Kb link has been established with the

NASA PSCN network. This makes work on a number of government CRAY XMP computers at various NASA and Air Force centers feasible. It also allows access to the CRAY II at NASA Ames (for unclassified computing). Further development of the computer system (for hypersonic CFD support) is dependent on the development of the market in this area.

With reference to the primary zonal solution strategy described above, a variety of 2- and 3-D parabolized and full Navier-Stokes codes were available at the initiation of the development. Most of these codes featured ideal gas modeling and either constant wall temperature or adiabatic wall boundary conditions. Most of the available codes were Machnum~r-limits - if the freestream Mach number was above about M = 15, the code would fail. In 3-D. no practical codes with either air or Hz-air finite-rate chemistry modeling were available. Within the context of the initial CFD objective, the following code features were required for hypersonic application. - High Mach number capability.

CFD technology

for

hypersonic

43

vehicle design

VALIDATION

?

GEOMETRY REPRESENTATION

MESH GENERATION

ZONAL SOLUTION FOR COMPUTATIONAL EFFICIENCY

a NUMERICAL ALGORITHMS

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COORDINATED DEVELOPMENT/ ACOUISITION OF CFD ELEMENTS

PRE & POST. PROCESSING

l

CONSTRUCT ZONAL SOLUTION METHODS FOR SPECIFIC APPLICATIONS

GENERATE DESIGN INFORMATION

Fig. 4. The technology elements of CFD.

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Ideal gas, equilibrium air, finite-rate air and finite-rate Hz-air chemistry modeling. - General wall energy balance boundary conditions. - An efficient algorithm for 3-D FNS simulations. (Note: FNS is used herein to denote the class of codes requiring storage of the dependent variables, Jacobian matrices, etc., for the full solution domain.) The following 3-D codes were prepared for hypersonic application: Code/Ret ARC3D/ [6] SPEAR3D/ [7J Ames PNS/ [8]

Description/Sources Thin-layer

Navier-Stokes/ NASA Ames Parabolixed Navier-Stokes/ AMTEK Parabolii Navier-Stokes/ NASA Ames

These codes share the following features: - High Mach number capability. - Ideal gas and equilibrium air real gas modeling. wall -A conduction-convection-radiation energy balance boundary condition. - Baldwin-Lomax algebraic turbulence modeling (SPEAR3D also features K-E turbulence modeling). All of the above codes solve for the flow properties over the solution domain on body-fitted meshes. ARC3D ‘captures’ all shocks. SPEAR3D and Ames

PNS feature bow shock fitting or capturing as a user-specified option. In addition, finite-rate air [9] and Hz-air chemistry models [lo] have been added to the ARC3D flow solver. A 2-D zonal flow analysis capability was acquired from SAIC [ll]. These codes include SCRAMP, SCRINT, SCORCH and SCHNOZ, and provide an end-to-end vehicle flow simulation for scramjetpowered hydrogen-fueled vehicles. These codes feature ideal or equilibrium air gas modeling, general finite-rate chemistry modeling, a wall energy balance boundary condition and algebraic or K-E turbulence modeling. Planned further development includes improvement of the PNS code robustness, testing of the Euler option of ARC3D and the PNS codes for application in the low hypersonic regime, implementation of K-E turbulence modeling in ARC3D, and the addition of air and Hz-air finite-rate chemistry modeling in a PNS code. In addition, all of the present codes are in a research stage of development. User features will be implemented so that the codes can be used in a ‘production’ mode. Geometry and mesh Geometry programs provide an analytic representation of the surfaces (wing, nacelle, etc.) that bound the flow domain of interest. The CFD codes solve discretized partial differential equations on a mesh fit to the boundary surfaces that extends through the domain to be solved. The body surface is a

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GERALDC.PAYMER

coordinate surface in the transformed coordinate system on which the solution is obtained. The vehicle surface geometry along with the geometry of the outer boundary is needed to define the transformation to the body fitted mesh system. Since the geometry description of a surface is Mach number independent, a substantial capability was available at the beginning of development from prior work in other speed regimes. No additional requirements were identified for application of this existing capability to the hypersonic speed regime. At present, surface geometry is described in one of two ways for 3-D flow simulations: (1) Surface mesh points are obtained directly from the GRAFTEK CAD system. (2) An AGPS description [12] of the surface is developed. AGPS represents a surface as parametric bicubics, biquintics or rational bsplines. Each way has advantages and disadvantages. Use of GRAFTEK to obtain surface points is faster for the flow analyst but this method has less flexibility in the modification of the surface point distribution. Use of AGPS offers good flexibility in surface point distribution modification but requires a trained user and l-2 weeks of time to prepare a surface geometry description from which ‘cuts’ and points can be obtained. GRAFTEK will generate an IGES (International Geometry Exchange System) description of a surface (as will AGPS), but at present, no computer interface exists between GRAFTEK and AGPS surface descriptions. Planned development includes implementation of an IGES input feature with AGPS so that a CAD description can be used to develop an AGPS description of a geometry. Since mesh generation is also Mach number independent, a substantial capability was available from prior work in other speed regimes. No additional mesh generation requirement (unique to the hypersonic speed regime) was identified. Elliptic [ 133 and algebraic mesh generation procedures are used at present for hypersonic analysis. The primary advantages of the elliptic procedure are grid smoothness and control of orthogonality at the grid boundaries; the disadvantages are the cost and complexity. The primary advantages of the algebraic procedures are low computational cost and explicit control of the grid point distribution. A disadvantage is that algebraic grids are less ‘smooth’ than those of the elliptic procedure. Planned development includes an investigation of solution adaptive grid generation methods for improved accuracy and efficiency, and a newly developed interactive hyperbolic grid generation technique. Physical modeling

The physical modeling is defined herein to consist of the gas models, chemistry models, transition

models and turbulence models of the CFD codes. (One could argue that the selection of a solution strategy and the assumption of a continuum are also important elements of the physical modeling. These issues are discussed in the paragraphs above on the overall solution strategy.) At the initiation of development, ideal gas modeling, no chemistry modeling, Re-theta/M transition modeling, and algebraic Baldwin-Lomax and/or K-E turbulence models were available in the codes selected for hypersonic development. It was necessary to satisfy at least three additional requirements to achieve the initial hypersonic CFD objective-CFD support for hypersonic vehicle design in the lower atmosphere: - Equilibrium chemistry real gas modeling for air. - Finite-rate air chemistry modeling. - Finite-rate H,-air chemistry modeling. Even with the implementation of the available air and Hi-air chemistry modeling, the accuracy of this modeling still must be established. As noted by Bradley [3], the reaction rate constants used in the chemistry modeling are based on measurements at temperatures at the lower end of the range of interest. The existing transition and turbulence modeling may be very deficient for hypersonic CFD applications in the lower atmosphere. Morkovin [14] and Reshotko et al. [lS] summarize the present understanding of hypersonic transition modeling and raise a number of questions that can only be addressed with a long-term fundamental research effort. Bradley [3] points out that the existing turbulence modeling is either too complex for practical implementation or is based on 2-D incompressible data. With all of these reservations in mind, however, no specific additional requirements were defined that would improve the accuracy of the hypersonic transition and turbulent flow simulations. At present, all of the codes have the following physical modeling features: - Equilibrium air real gas modeling. - Re-Theta/M transition modeling. - Algebraic turbulence modeling. In addition, SPEAR3D and the 2-D SAIC codes both have K-E turbulence modeling. Boeingmodified ARC3D has air [9] and Hz-air [IO] finiterate chemistry modeling and the SAIC codes SCORCH and SCHNOZ both have general reaction finite-rate chemistry modeling. Planned improvements include the following: - Addition of K-E turbulence modeling in ARC3D. - The addition of Hz-air finite-rate chemistry modeling in a 3-D PNS code.

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CFD technology for hypersonic vehicle design

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Validation Experimental validation of a CFD analysis for design occurs at two levels, as noted in [S]. At the first level, it must be shown that the modeling inherent in the codes and in the analysis strategy are capable of predicting the specific flow phenomena expected for a given application. At the second level, it must be demonstrated that the CFD analysis strategy and the way in which it is applied are capable of providing performance data of sufficient accuracy for configuration selection. In addition, code accuracy can be evaluated through comparisons between the CFD code and analytic or well-tested correlation methods and through global conservation checks. Specific hypersonic validation requirements are application-dependent. One potential source of validation data for trans-atmospheric vehicles is the National Aerospace Plane CFD validation option [16]. Although not unique to the hypersonic speed regime, global conservation checks are useful for evaluating the solution accuracy and are, therefore, listed as a requirement. One should note

that the dearth of hvrnzrsonic data suitable for either level I or level II C?D validation makes validation for a wide range of hypersonic flows impractical in the near term. The following work toward code validation has been completed. -

-

-

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SPEAR3D results have been compared with data for a laminar shock/boundary layer interaction at M = 14 [17J Ames PNS results have been compared with data Cl83 for laminar and turbulent flow over a biconic body at M = 8 and 16. ARC3D (with Hz-air finite-rate chemistry modeling) results have been compared with data from the Burrows and Kurkov [19] tangential H, injection experiment. Global conservation checks on mass, momentum and energy have been added to all of the hypersonic codes undergoing development.

Further validation work is dependent on the development of the aircraft market and the availability of suitable data.

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GERALDC. PAYNTER

46

5. G. C. Paynter and E. Tjonneland, CFD - a user’s technology assessment. In Progress in Astronautics and

APPLICATIONS

Application

and

development

aspects

of the

hypersonic CFD capability were reported in [ll] and [20]425]. The xonal analysis of a generic wing-fuselage at M = 16 from [22] b an example of the current capability. This full-scale geometry (Fig 5) is an axisymmetric ogive between the nose and the wingfuselage junction at 600 in. Swept inlet compression ramps emerge from the undersurfaceof the fuselage

at about !I00in. The end of the analysis domain is at 1292 in., the leading edge of the inlet cowl lip. The geometry was assumed to be at a 2.5” angleof-attack to the freestream flow, the flow over the vehicle was assumed to be laminar, and the wall temperature was assumed to be 22WR. The flow over the first 800 in. was computed with a PNS code [8], and the flow over the remainder of the geometry was computed with ARC3D [6]. Contour plots of the computed static pressure and Mach number distributions at the cowl lip axial station are also given in Fig. 5. The details of this analysis and more completed results are reported in c221. More recently, modified ARC3D with air and H,air finite-rate chemistry modeling has been applied to nose region analysis and to the flow through a scramjet combustor. The scramjet combustor analysis will be reported in [26]. CONCLUSIONS

Substantial progress has been made toward CFD technology supporting design of hypersonic vehicles flying in the lower atmosphere. The primary limitations of this are our limited understanding of key features of the physical modeling - specifically transition, turbulence and finite-rate chemistry. Experiments to evaluate the accuracy of the current CFD capability and to improve the current physical modeling are needed. Acknowledgement-The

authors wish to acknowledge that

the work reported here-in was accomplished by the members of the CFD Group, Hypersonic Technology Staff, Boeing Advanced Systems, Seattle.

REFERENCES

1. D. P. Hearth and A. E. Preyss. Hypersonic technology -

approach

to an expanded

program.

Astronaut.

Aeronaut. 14, 12 (1976). 2. R. M. Williams, National aerospace plane: technology for America’s future. Aerospace Am. 24, 11 (1986).

3. NRC Committee, R. G. Bradley (Chairman), Current Capabilities and Future Directions in Computational Fluid Dynamics. National Academy Press, Washington,

D.C. (1986). A perspective of theoretical and applied computational fluid dynamics. AIAA Paper 83-0037 (1983).

4. P. Kutler,

Aeronau~cs. Vol. 12. ALU, New York (1986). 6. T. H. Pulliam and J. L. Stener. _ Recent imorovements r in efficiency, accuracy and convergence for implicit approximate factorization algorithms. AIAA paper 86 0360 (1985). 7. S. Yaghmae-e and D. W. Roberts, Modeling shock/ boundary layer interactions with a partially parabolic Navier-Stokes analvsis. AIAA naner 87-2178 f1987). Code lith 8. G. Molvik, A Pa*;lbolired N&e-Stokes Real Gas Effects. NASA Ames. 9. F. G. Blottner, M. Johnson, and M. Ellis, Chemically reacting viscous Row program for multicomponent gas mixtures. Report No. SC-RR-70-754. Sandia Labs (1971). 10. J. P. Drummond, R. C. Rogers and M. Y. Hussaini, A detailed numerical model of a supersonic reacting mixing layer. AIAA paper 86-1427 (1986). 11. T. R. A. Bussing and G. L. Lidstone, An improved computational model for a Scramjet propulsion system. AIAA paper 87-2078 (1987). 12. D. K. Snepp and R. C. Pomeroy, A geometry system for aerodynamic design. AIAA paper 87-2902 (1987). 13. R. L. Sorenson and J. L. Steger. Grid generation in three dimensions by Poisson equations with control of cell size and skewness at boundary surfaces. Advances in Grid Generation, ASME Fluid Engineering Conference, Houston, Texas (1983). 14. M. V. Morkovin, Transition at hypersonic speeds. NASA CR178315 (1987). 15. E. Reshotko, D. M. Bushnell and M. D. Cassidy, Report of the task force for boundary layer transition. NASP TM 1007 (1987). 16. S. 0. Schmitt (Chairman), Fourth National Aero-Space Plane Technology Symposium, Vol. II - Computational Fluid Dynamics. NASP Conference Publication 4023 (1988). 17. M. S. Holden, Theoretical and experimental studies of shock wave/boundary layer interaction on curved compression surfaces. ARL Symposium on Viscous Interaction Phenomena Flow, Wright-Patterson

in Supersonic and Hypersonic

Air Force Base, Ohio (1969). 18. M. D. Ryder, Skin friction, heat transfer and pressure measurements on hypersonic inlet compression surfaces in the Mach number range 7.5 to 16. Technical Report AFFDL-TR-65-199 (1965). 19. M. C. Burrows and A. P. Kurkov, Analytical and experimental study of supersonic combustion of hydrogen in a vitiated ah-stream. NASA TM X-2828 (1978). 20. T. R. A. Bussing and S. Eberhardt, Chemistry associated with hypersonic vehicles. AIAA paper 87-1292 (1987). 21. J. C. Wai, S. C. Dao and D. Chaussee, Navier-Stokes simulation of a hypersonic generic wing/fuselage. AIAA paper 87-1192 (1987). 22. D. Sommerlield, S. C. Dao, M. Schwartz and J. C. Wai, Hypersonic CFD analysis of generic forebodies. AIAA paper 88-0372 (1988). 23. S. C. Dao, Numerical simulation of high-speed inlet flows using a Navier-Stokes solver. (Submitted for presentation.) 24. M. Schwartz, A two-dimensional Navier-Stokes analysis of a Scramjet inlet. AIAA Young Engineers Technical Conference (1987). 25. D. W. Maver and J. Y. Baltar. PNS oredicted shock

location and jump conditions at &ersonic and hvnersonic sneeds. AIAA oaoer 85-1407 (1985). 26. J.-C. Wai and D. Sommet%eld, Scramjet analysis using three-dimensional approximate factorization with chemical reaction. (Submitted to 1989 AIAA Aerospace Sciences Meeting.)