Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-08T00:32:57.729Z Has data issue: false hasContentIssue false

Computational fluid dynamics application to aerospace science

Published online by Cambridge University Press:  03 February 2016

J. S. Shang*
Affiliation:
[email protected], Wright State University, Dayton, Ohio, USA

Abstract

A brief narration on significant accomplishments in computational fluid dynamics (CFD) for basic research and aerospace application is attempted to highlight the outstanding achievements by scientists and engineers of this discipline. To traverse such a vast domain, numerous and excellent contributions to CFD will be unintentionally overlooked by the author’s limited exposure. Nevertheless it is an ardent hope that the present abridged literature review will aid to reaffirm excellence in research and to identify knowledge shortfalls both in aerodynamics and its modeling and simulation capability. The future modeling and simulation technology needs, as well as potential and fertile research areas, are humbly put forth for consideration.

Type
Research Article
Copyright
Copyright © Royal Aeronautical Society 2009 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Thom, A., The flow past circular cylinders at low speeds, Proceeding of the Royal Society of London, A141, 1933, pp 651666.Google Scholar
2. Von Neumann, J., Proposal and analysis of a numerical method for the treatment of hydrodynamical shock problems, Nat Def and Res Com Rept AM-551, March 1944.Google Scholar
3. Von Neumann, J. and Richtmyer, R.D., A method for the numerical Calculation of hydrodynamic shocks, J App Phys, 1950, 21, pp 232257.Google Scholar
4. Taylor, G.I., The formation of a blast wave by a very intense explosion, Proc Roy soc London, Series A, 1950, 201, pp 159168.Google Scholar
5. Harlow, F.H., The particle-in-cell computing method for fluid dynamics, Method in Computational physics, 1964, 3, p 319.Google Scholar
6. Hirt, C.W., Computer studied of time-dependent turbulent flows, highspeed computing in fluid dynamics, physics of fluid supplement II, American Inst of phys, 1969, New York, NY, USA.Google Scholar
7. Strang, W.Z., Berdahl, C.H., Nutley, E.L. and Murn, A.J., Evaluation of four panel aerodynamic prediction methods (MCAERO, PANAIR, QUANPAN and VSAERO), AIAA-1985-4092, Colorado Springs, CO, USA, 1985.Google Scholar
8. Ferri, A., Application of the method of characteristics to supersonic rotational flow, NACA TN 841, 1946.Google Scholar
9. Fay, J.A. and Riddell, F.R., Theory of stagnation point heat transfer in dissociated air, J Aero Science, 1958, 25, pp 73–85 and 121.Google Scholar
10. Korkegi, R.H., Survey of viscous interaction associated with high Mach number flight, AIAA J, 1971, 9, pp 771784.Google Scholar
11. Lees, L. and Reeves, B.L., Supersonic separated and reattaching flows: I General theory and application to adiabatic boundary layer/shock wave interactions, AIAA J, 1964, 2, pp 19071920.Google Scholar
12. Stewartson, K. and Williams, P.G., Self-induced separation, Proceeding of the Royal Society of London, 1969, A312, pp 181206.Google Scholar
13. Moretti, G. and Abett, M., A time-dependent computational method for blunt body flows, AIAA J, 1966, 4, pp 21362141.Google Scholar
14. Chapman, D.R., Computational aerodynamics development and outlook, AIAA J, 1979, 17, pp 12931313.Google Scholar
15. MacCormack, R.W., The effect of viscosity in hypervelocity impact cratering, AIAA 1969-354, Cincinnati, Oh, USA, 1969.Google Scholar
16. Hung, C.M. and MacCormack, R.W., Numerical solutions of supersonic and hypersonic laminar flows over two-dimensional compression corner, AIAA 75-2, 1975.Google Scholar
17. Shang, J.S. and Hankey, W.L., Numerical solution of the Navier-Stokes equations for supersonic turbulent flow over a compression ramp, AIAA 75-3, 1975. Also AIAA J, 1975, 13, (10), pp 13681374.Google Scholar
18. Shang, J.S., Hankey, W.L. and Law, C.H., Numerical simulation of shock wave-turbulent boundary-layer interaction, AIAA 76-0095, 1976. Also AIAA J, 14, (10), October 1976, pp 14511457.Google Scholar
19. Horstman, C.C., Kussoy, M.I., Coakley, T.J., Rubesin, M.N. and Marvin, J.G., Shock-wave induced turbulent boundary-layer separation at hypersonic speeds, AIAA 75-4, 1975.Google Scholar
20. Knight, D.D., Numerical simulation of realistic high-speed inlets using the Navier-Stokes equations, AIAA J, 1977, 15, pp 15831589.Google Scholar
21. Dolling, D.S., Fifty years of shock/boundary Interaction: What next, AIAA J, 2001, 39, pp 15171531.Google Scholar
22. Shang, J.S. and Hankey, W.L., Numerical solution of the compressible Navier-Stokes equations for a three-dimensional corner, AIAA J, 1977, 15, pp 15751582.Google Scholar
23. Shang, J.S. and Hankey, W.L., Three-dimensional supersonic interacting turbulent flow along a corner, AIAA J, 17, 1979, pp 706713.Google Scholar
24. Thompson, J.F., Thames, F.C. and Mastin, C.W., Automatic numerical generation of body-fitted curvilinear coordinate system for field containing any number of arbitrary two-dimensional bodies, J Computational Physics, 1974, 15, pp 299319.Google Scholar
25. Murman, E.M. and Cole, J.D., Calculation of plane steady transonic flows, AIAA J, 1971, 9, pp 114121.Google Scholar
26. Jameson, A., Iterative solution of transonic flows over airfoils and wings including flows at Mach 1, Comm Pure Appl Math, 17, 1974, pp 283309.Google Scholar
27. Peaceman, D.W. and Rachford, H.H., The numerical solution of parabolic and elliptic differential equations, J Soc Industrial Applied Math, 1955, 3, pp 2841.Google Scholar
28. Douglas, J. and Gunn, J., A general formulation of alternating direction methods, I, Parabolic and hyperbolic problems, Numerical Math, 1964, 6, pp 428453.Google Scholar
29. Briley, W.R., A Numerical study of laminar separation bubbles using the Navier-Stokes equations, 1971, J Fluid Mech, 47, pp 713736.Google Scholar
30. Briley, W.R. and McDonald, H., Solution of the three-dimensional compressible Navier-Stokes equations by an implicit technique, Lecture Notes, Phys, 35, Spring-Verlag, 1974, New York, USA, pp 105110.Google Scholar
31. Beam, R.M. and Warming, R.F., An implicit factored scheme for the compressible Navier-Stokes equations, AIAA J, 16, 1978, pp 393401.Google Scholar
32. Pulliam, T.H. and Steger, J.L., Implicit finite difference simulation of three-dimensional compressible flow, AIAA J, 18, (2), 1980, pp 159167.Google Scholar
33. MacCormack, R.W. and Paullay, A.J., Computational efficiency achieved by time splitting of finite difference operators, AIAA, pp 72154, 1972.Google Scholar
34. Rizzi, A. and Inouye, M., Time-splitting finite-volume method for three-dimensional blunt-body flow, AIAA J, 11, (11), 1973, pp 14781485.Google Scholar
35. Helliwell, W.S., Dickinson, R.P. and Lubard, S.C., Viscous Flow over Arbitrary geometries at high angle of attack, AIAA J, 19, (2), 1981, pp 191197.Google Scholar
36. Levy, L.L. Jr, Experimental and computational steady and unsteady transonic flows about a thick airfoil, AIAA J, 16, (6), 1978, pp 564572.Google Scholar
37. Steger, J.L. and Bailey, H.E., Calculation of transonic aileron buzz, AIAA J, 1980, 18, (3), pp 249255.Google Scholar
38. Fasel, H., Investigation of the Instability of Boundary Layers by a finite-difference model of the Navier-Stokes equations, J Fluid Mech, 1976, 72, (2), pp 355383.Google Scholar
39. Lomax, H., Recent progress in numerical techniques for flow simulation, AIAA J, 1976, 14, (4), pp 512528.Google Scholar
40. Redhed, D.D., Chen, A.W. and Hotovy, S.G., New approach to the 3D transonic flow analysis using the STAR 100 computer, AIAA J, 1979, 17, pp 9899.Google Scholar
41. Pulliam, T.H. and Lomax, H., Simulation of three-dimensional compressible viscous flow on the ILLIAC IV computer, AIAA Paper 79-0206, 1979.Google Scholar
42. Shang, J.S., Buning, P.G., Hankey, W.L. and Wirth, M.C., Performance of vectorized three-dimensional Navier-Stokes code on the Cray-1 computer, AIAA J, 18, 1980, pp 10731079.Google Scholar
43. Roe, P.L., Approximate Riemann solvers, parameter vectors and difference scheme, J Comp. Physics, 43, 1981, pp 351372.Google Scholar
44. Shang, J.S., An assessment of the solutions of the Navier-Stokes equations, J Aircr, 22, 1985, pp 354370.Google Scholar
45. Viviand, H. and Ghazzi, W., Numerical solution of the compressible Navier-Stokes equations at high Reynolds numbers with application to the blunt body problem, Lecture Notes in Physics, 59, 1976, pp 434439.Google Scholar
46. Peake, D.J., Fisher, D.F. and McRae, D.S., Flight, wind tunnel and numerical experiments with a slender cone at incidence, AIAA J, 20, (10), October 1982, pp 13381345.Google Scholar
47. Agarwal, R. and Rakich, J.V., Supersonic laminar viscous flow past a cone at angle of attack in spinning and coning motion, AIAA J, June 1982, 20, pp 761768.Google Scholar
48. Gnoffo, P.A., A finite-volume, adaptive grid algorithm applied to planetary entry flow fields, AIAA J, 21, 1983, pp 12491254.Google Scholar
49. Tassa, Y. and Sankar, N.L., Dynamic stall of an oscillating airfoil in turbulent flow using time dependent Navier-Stokes solver, Unsteady Turbulent Shear Flow, Springer-Verlag, New York, USA, 1981, pp 185196.Google Scholar
50. Tannehill, J.C., Venkatapathy, E. and Rakich, J. V., Numerical solution of supersonic viscous flow over blunt delta wings, AIAA J, 20, 1982, pp 203210.Google Scholar
51. Fujii, K. and Kutler, P., Numerical simulation of the leading edge separation vortex for a wing and strake-wing configuration, AIAA Paper 83-1908, July 1983.Google Scholar
52. Hollanders, H., Lerat, A. and Peyret, R., 3D calculation of transonic viscous flows by an implicit method, AIAA Paper 1983-1953, July 1983.Google Scholar
53. Anderson, O.L., Calculation of internal viscous flows in axisymmetric ducts at moderate to high Reynolds numbers, Int J Computers and Fluids, 8, 1980, pp 391411.Google Scholar
54. Kumar, A., Two-dimensional analysis of a scramjet inlet flowfield, AIAA J, 20, (1), January 1982, pp 9697.Google Scholar
55. Patankar, S.V., Pratap, V.S. and Spalding, D.B., Prediction of laminar flow and heat transfer in helically wield Pipes, J Fluid Mechanics, 62, 1974, pp 539551.Google Scholar
56. Markatos, N.C., Spalding, D.B., Tatchell, D.G. and Mace, A.C.H., Flow and combustion in the base-wall region of a rocket exhaust plume, Combustion Science and Technology, 1982, 28, pp 1529.Google Scholar
57. Weidner, E.H. and Drummond, J.P., Numerical study of a staged fuel injection for supersonic combustion, AIAA J, 20, (10), October 1982, pp 14261427.Google Scholar
58. Griffin, M.D., Anderson, J.D. JR. and Diwaker, R., Navier-Stokes solutions of the flowfield in an internal combustion engine, AIAA J, December 1976, 14, (12), pp 16651666.Google Scholar
59. Shang, J.S. and Scherr, S.J., Navier-Stokes solution for a complete reentry configuration, J Aircr, 1986, 23, pp 881888.Google Scholar
60. Barthelemy, R.R., The National Aero-Space Plane program, AIAA 1989-5053, July 1989.Google Scholar
61. Gomez, R., Vicker, D., Rogers, S.E., Aftosmis, M.J., Chan, W.M., Meakin, R. and Murman, S.M, STS-107 Investigation ascent CFD support, AIAA 2004-2226, 2004.Google Scholar
62. Reznick, S.G. and Flores, J., Strake-generated vortex interactions for a fighter-like configuration, AIAA 87-0589; also, J Aircr, 26, 1989, pp 289294.Google Scholar
63. Huband, G.W., Rizzetta, D.P. and Shang, J.S., Numerical solution of the Navier-Stokes equation for an F-16A configuration, AIAA 1988-2057; also J Aircr, 26, 1989, pp 634640.Google Scholar
64. Thomas, J.L., Walters, R.W., Reu, T., Ghaffari, F., Weston, R.P. and Luckring, J.M., Application of a patched-grid algorithm to the F/A-18 forebody-leading-edge extension configuration, J Aircr, 27, 1990, pp 749756.Google Scholar
65. Schiff, L.B., Cummings, R.M, Sorenson, R.L. and Rizk, Y.M., Numerical simulation of high-incidence flow over the F-18 fuselage forebody, AIAA 1989-0339, 1989; also Numerical simulation of high- incidence flow over the isolated F-18 fuselage forebody, J Aircr, 28, 1991, pp 609617.Google Scholar
66. Rizk, Y.M. and Gee, K., Unsteady simulation of viscous flowfield around F-18 Aircraft at large incidence, J Aircr, 29, 1992, pp 986992.Google Scholar
67. Harten, A., Egquist, B., Osher, S. and Chakravarthy, S., Uniformly high-order essentially non-oscillatory schemes III, J Comp Phys, 71, 1987, pp 231257.Google Scholar
68. Shu, C.W., Total–variation-diminishing time discretizations, SIAM J Sci Stat, Comput. 9, 1988, pp 10731084.Google Scholar
69. Candler, G.V. and MacCormack, R.W., Computation of weakly ionized hypersonic flows in thermochemical nonequilibrium, J Thermophysics, 5, 1991, pp 266273.Google Scholar
70. Shankar, V., Hall, W.F. and Mohammadian, A.H., A time-domain differential solver for electromagnetic scattering problems, Proceedings of Institute of Electrical and Electronics Engineers, 1989, 77, p 709.Google Scholar
71. Shang, J.S., Characteristic-based algorithms for solving the Maxwell equations in the time domain, IEEE Antennas and Propagation Magazine, 37, June 1995, pp 1525.Google Scholar
72. MccLinton, C., Bittner, R. and Kamath, P., CFD support to NASP design, AIAA 90-5249, October 1990.Google Scholar
73. Curran, E.T., Heiser, W.H. and Pratt, D.T., Fluid phenomena in scramjet combustion systems, Annul Rev Fluid Mech, 1966, pp 323360.Google Scholar
74. Curran, E.T., Scramjet Engines: The first forty years, J Propulsion and Power, 17, 2001, pp 11381148.Google Scholar
75. Walters, R.W., Cinnella, P., Slack, D.C. and Halt, D., Characteristic based algorithms for Flows in thermochemical nonequi-librium supersonic combustion fields, A1AA J, 1992, 30, pp 13041313.Google Scholar
76. Tannehill, J. C., Buelow, P. E., Levalts, J. O. and Lawrence, S. L., three-dimensional upwind parabolized Navier-Stokes code for real gas flows, J Spacecraft and Rockets, 27, 1990, pp 150159.Google Scholar
77. Wadawadigi, G., Tannehill, J.C., Buelow, P.E. and Lawrence, S., Three-dimensional upwind parabolized Navier-Stokes code for supersonic combustion fields, J Thermophysics and Heat Transfer, 7, 1993, pp 661667.Google Scholar
78. Drummond, J.P. and Bouche, M., Overview of NATO Background on Scramjet Technologies, 2, Technologies for Propelled Hypersonic Flight, Chapter 1, NATAO-RTO-TR-AVT-007, 2006.Google Scholar
79. Buning, P.G., Parks, S.J., Chan, W.M. and Renze, K.J., Application of the Chimera overlapped grid scheme to simulation of space shuttle accent flows, Proceedings of the 4th international Symposium on CFD, Davis, CA, USA, 1991.Google Scholar
80. Sotnick, J.P., Kandula, M. and Buning, P., Navier-Stokes simulation of the space shuttle launch vehicle flight transonic flowfield using a large scale chimera grid system, AIAA 1994-1860, June 1994.Google Scholar
81. Chakravarthy, S.R., Szema, K.Y. and Chen, C.L., An universe-series code for inviscid CFD with Space Shuttle applications using unstructured grids, AIAA 1991-3340, 1991.Google Scholar
82. Yang, R.J., Chang, J.L.C. and Kwak, D., Navier-Stokes flow simulation of the space shuttle main engine hot gas manifold, J Spacecraft and Rocket, 1992, 29, pp 253262.Google Scholar
83. Friedlander, S.K. and Topper, L. (Eds) Classic Paper on Statistical Theory, Interscience, 1962, New York, USA, p 186.Google Scholar
84. Shang, J.S., Assessment of technology for aircraft development, J Aircr, 32, 1995, pp 611617.Google Scholar
85. Bradshaw, P., Turbulence: The chief outstanding difficulty of our subject, Experimental in Fluids, 1994, 16 Nos ¾, pp 203216.Google Scholar
86. Clark, R.A., Ferziger, J.H. and Reynolds, W.C., Evaluation of subgrid scale models using an accurately simulated turbulent flow, J Fluid Mechanics, 1979, 91, (1).Google Scholar
87. Piomelli, U., Large-eddy simulation: achievement and challenges, Progress in Aerospace Sciences, 35, 1999, pp 335362.Google Scholar
88. Gaitonde, D.V. and Shang, J.S., Optimized compact-difference-based finite-volume solvers for linear wave phenomenon, JCP, 138, 1997, pp 617643.Google Scholar
89. Lele, S.K., Finite difference schemes with spectral-like resolution, J Comp. Phys, 1992, 103, pp 16–14.Google Scholar
90. Rizzetta, D.P. and Visbal, M.R., Application of large eddy simulation to supersonic compression ramps, AIAA J, 40, 2002, pp 15741581.Google Scholar
91. Rizzetta, D.P. and Visbal, M.R., Large eddy simulation of supersonic cavity flowfield including flow control, AIAA J, 2003, 41, pp 14521462.Google Scholar
92. Shang, J.S., Wagner, M., Pan, Y. and Blake, D.C., Strategies for adopting FVTD on mutlicomputers, Computing in Science and Engineering, 2000, 2, pp 1021.Google Scholar
93. Strang, W.Z., Tomaro, R.F. and Grismer, M.J., The defining methods of Cobalt: A parallel, implicit, unstructured Euler/Navier-Stokes flow solver, AIAA 99-0786, January 1999.Google Scholar
94. Cao, H.V. and Su, T.Y., Navier–Stokes analyses of a 747 high-lift configuration. AIAA Paper, 98-2623, 1998.Google Scholar
95. Rogers, S.E., Roth, K., Cao, H.V., Slotnick, J.P, Whitlock, M., Nash, S.M. and Baker, M.D., Computation of viscous flow for a Boeing 777 aircraft in landing configuration, 2001, J Aircr, 48, pp 10601068.Google Scholar
96. Vos, S.B., Rizzi, A., Darracq, D. and Hirshel, E.H., Navier-Stokes solvers in european aircraft design, Progress in Aerospace Sciences, 38, 2003, pp 601697.Google Scholar
97. Fraishtadt, V.L., Kuranov, A.L. and Sheikin, E.G., Use of MHD systems in hypersonic aircraft, Technical Physics, 11, 1998, p 1309.Google Scholar
98. Shang, J.S., Recent research in magneto-aerodynamics, Progress in Aerospace Science, 2001, 37, pp 120.Google Scholar
99. Ganiev, Y., Gordeev, V., Krasilnikov, A., Lagutin, V., Otmennikov, V. and Panasenko, , Aerodynamic drag reduction by plasma and hot-gas injection, J Thermophysics and Heat Transfer, 14, 2000, pp 1017.Google Scholar
100. Shang, J.S., Plasma injection for hypersonic blunt body drag reduction, AIAA J, 2002, 40, (6), pp 11781186.Google Scholar
101. Macheret, S.O., Shneider, M.N. and Miles, R.B., Magneto-hydrody-namic and electro-hydrodynamic control of hypersonic flows of weakly ionized plasma, AIAA J, 42, 2004, pp 13781387.Google Scholar
102. Kimmel, R., Hayes, J., Menart, J. and Shang, J.S., Effect of magnetic fields on surface plasma discharges at Mach 5, J Spacecraft and Rocket, 2006, 43, pp 13401346.Google Scholar
103. Shang, J.S., Surzhikov, S.T., Kimmel, R.L., Gaitonde, D.V., Menart, J.A. and Hayes, J.R., Mechanisms of plasma actuator for hypersonic flow control, Progress in Aerospace Sciences, 2005, 41, (8), pp 642668.Google Scholar
104. Leger, L., Moreau, E. and Touchard, G., Effect of a DC corona electrical discharge on the airflow along a flat plate, 2002, IEEE Trans Indust Appl, 38, pp 1478–85.Google Scholar
105. Post, M.L. and Corke, T.C., Separation control on high angle-of-attack airfoil using plasma actuator, AIAA J, 2004, 42, (2), pp 21772184.Google Scholar
106. Moreau, E., Airflow Control by non-thermal plasma actuators, J Physics D: App Physics, 2007, 40, pp 605636.Google Scholar
107. Jacobsen, L.S., Carter, C.D. and Jackson, T.A., Toward plasma-assisted ignition in scramjets, AIAA 2003-0871, Reno, NV, USA, January 2003.Google Scholar
108. Anikin, N., Kukaev, E., Starikovskaia, S. and Starikovskii, A., Ignition of hydrogen-air and methane-air mixtures at low temperatures by nanosecond high voltage discharge, AIAA 2004-0833, Reno, NV, USA, January 2004.Google Scholar
109. Kolesnichenko, Y., Basics in beams mw energy deposition for flow/flight control, AIAA 2004-0669, Reno NV, January 2004.Google Scholar
110. Adelgren, R.G., Yan, H., Elliots, G.S., Knight, D.D., Beutner, T.J. and Zhetovodov, A.Z., Control of Edney IV interaction by pulsed energy deposition, AIAA J, 2005, 43, pp 256269.Google Scholar
111. Miles, R.B., Brown, G.L., Lempert, W.R., Yetter, R., Williams, G.L., Bogdonoff, S.M., Natelson, D. and Guest, J.R., Radiatively driven hypersonic wind tunnel, AIAA J, 33, (8), 1995, pp 14631470.Google Scholar
112. Park, C., Mehta, U.B. and Bogdanoff, D.W., MHD energy bypass scramjet performance with real gas effects, J Propulsion and power, 19, (5), 2001, pp 10491057.Google Scholar
113. Gaitonde, D.V., Three-dimensional flow-through scramjet simulation with MGD energy-bypass, AIAA 2003-0172, January 2003.Google Scholar
114. Mitchner, M. and Kruger, C.H., Partially Ionized Gas, John Wiley & Sons, 1973, New York, USA.Google Scholar
115. Surzhikov, S.T. and Shang, J.S., Two-component plasma model for two-dimensional glow discharge in magnetic field, Comp Physics, 199, September 2004, pp 437464.Google Scholar
116. Solov’ev, V.R., Konchakov, A.M., Krivtsov, V.M. and Aleksandrov, N.L., Numerical simulation of a surface barrier discharge in air, Plasma Phy Rept, 7, (7), 2008, pp 594608.Google Scholar
117. Shang, J.S., Huang, P.G., Yan, H. and Surzhikov, S.T., Computational simulation of direct current discharge, J Applied Physics, 105, 2009, pp 023303–1-14.Google Scholar
118. Landau, L. and Teller, E., Zur theore der schalldispersion, Physik Z. Sowjetunion, b. 10, h. 1, 1936, p 34.Google Scholar
119. Treanor, C.E. and Marrone, P.V., Effect of dissociation on the rate of vibrational relaxation, Physics of Fluids, 5, (9), 1962, pp 10221026.Google Scholar
120. Park, C., Chemical-kinetic parameters of hypersonic earth entry, J Thermophysics and heat Transfer, 2001, 15, (1), pp 7690.Google Scholar
121. Park, C., Stagnation-point ablation of carbonaceous flat disk, Part 1: Theory, AIAA J, 1983, 21, (1), pp 15881594.Google Scholar
122. Zhluktov, S. and Abe, T., Viscous shock-layer simulation of air flow past ablating blunt body with carbon surface, J Thermo-Physics and Heat Transfer, 13, (1), 1999, pp 5059.Google Scholar
123. Chen, Y.K. and Milo, F.S., Navier-Stokes Solutions with finite rate ablation for planetary mission earth mission earth reentries, J Spacecraft and Rockets, 2005, 42, (6), pp 961970.Google Scholar
124. Josyula, E. and Bailey, W.F., Governing equations for weakly ionized plasma flow field of aerospace vehicles, J Spacecraft and Rockets, 40, (6), 2003, pp 845857.Google Scholar
125. Shang, J.S., Surzhikov, S.T. and Yan, H., Simulate Hypersonic Nonequlibrium Flow Using Kinetic Models, AIAA 2009-0386, Orlando, FL, USA, January 2009.Google Scholar
126. Capitelli, M., Gorse, C. and Longo, S. and Giordano, D., Collision integrals of high-temperature air species, J Theromphysics and Heat Transfer, 14, 2000, pp 259268.Google Scholar
127. Levin, E. and Wright, M.J., Collision integrals for ion-neutral interactions of nitrogen and oxygen, J Thermophysics and Heat Transfer, 18, (1), 2004, pp 143147.Google Scholar
128. Chaban, G., Jaffe, R., Schwenke, D.W. and Huo, W., Dissociation cross sections and rate coefficients for nitrogen from accurate theoretical calculations, AIAA 2008-1209, Reno, NV, USA, January 2008.Google Scholar
129. Mueller, T.J., Kellogg, J.C., Ifju, G.I. and Shkarayev, S.V., Introduction to the design of fixed-wing micro air vehicles including three cases studies, AIAA Education Series, Reston, VA, USA, 2006.Google Scholar
130. Berman, G.J. and Wang, Z.J., Energy-minimizing kinematics in hovering insect flight, J Fluid Mech, 2007, 582, pp 153168.Google Scholar
131. Cockburn, B. and Shu, C.W., TVB Runge-Kutta local projection discontinuous Galerkin Finite element method for conservation laws 2: General frame work, Mathematics of Comp, 1989, 52, pp 411435.Google Scholar
132. Wang, Z.J., Spectral finite Volume method for conservative laws on unstructured grids: basic formulation, J Comp physics, 2002, 178, pp 210251.Google Scholar
133. Liu, Y., Vinokur, M. and Wang, Z.J., Spectral difference method for unstructured grids I: basic formulation, J Comp physics, 2006, 216, pp 780801.Google Scholar
134. Korpriva, D. and Kolias, J., A conservative Staggered-grid Chebyshev multidomain method for compressible flows, J Comp. Physics, 1996, 125, pp 244261.Google Scholar
135. Van Den Abeele, K., Lacor, C. and Wang, Z.J., On the stability and accuracy of the spectral Difference Method, J Scientific Computing, 2008, 37, pp 162188.Google Scholar
136. Gautsche, W., Orthogonal polynomials, computational and approximation, Oxford Science Publications, Oxford University Press, Oxford, UK, 2004.Google Scholar