where ZP= 104/Tb,r and ZG = ln(104/Tb,r). Then, partial derivatives of the tempera-ture, for example, the translational temperature Ttr, are given by
∂Ttr
∂ρ = 1 2q2 ρCbv
, (A.23)
∂Ttr
∂l =− u
ρCbv, (A.24)
∂Ttr
∂m =− v
ρCbv, (A.25)
∂Ttr
∂n =− w
ρCbv, (A.26)
∂Ttr
∂E = 1
ρCbv, (A.27)
∂Ttr
∂ρs =
−Cv,sTtr+ ∆hs
ρCbv ,(s̸= e)
0. (s = e)
(A.28)
References
[1] Y. Suzuki, T. Imada, “Concept and technology of HTV-R: an advanced type of H-II transfer vehicle,”Transactions of the Japan Society for Aeronautical and Space Sciences, Aerospace Technology Japan, 10(ists28), Tg9-Tg18, 2012.
[2] W. M. McCabe and C. F. Stolwyk, “Electromagnetic propagation through shock ionized air surrounding glide re-entry spacecraft,” IRE Transactions on Space Electronics and Telemetry, Vol. SET-8, No. 4, pp. 257-266, 1962.
[3] P. W. Huber and T. E. Sims, “The entry-communications problem,”Astronautics and Aeronautics, Vol. 2, No. 10, pp. 30-40, 1964.
[4] J. P. Rybak and R. J. Churchill, “Progress in reentry communications,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 7, No. 5, pp. 879-894, 1971.
[5] G. E. Wood, S. W. Asmar, T. A. Rebold, and R. A. Lee, “Mars Pathfinder en-try, descent, and landing communications,” The Telecommunications and Data Acquisition Progress Report 42-131, pp. 1-19, 1997.
[6] T. E. Horton, “The JPL thermochemistry and normal shock computer program,”
Jet Propulsion Laboratory Technical Report 32-660, 1964.
[7] P. W. Huber, J. S. Evans, and C. J. Jr. Schexnayder, “Comparison of theoretical and flight-measured ionization in a blunt body re-entry flowfield,”AIAA Journal, Vol. 9, No. 6, pp. 1154-1162, 1971.
[8] W. L. Weaver and J. T. Bowen, “Entry trajectory, entry environment, and analysis of spacecraft motion for the RAM C-III flight experiment,”, NASA Technical Memorandum vol X-2562, pp. 1-95, 1972.
[9] Y. Wada, Y. Watanabe, T. Akimoto, and H. Yasui, “Data analysis of electrostatic-probe,” Special Publication of the National Aerospace Laboratory SP-24, pp. 227-236, 1994.
[10] D. Isakeit, A. Wilson, P. Watillon, T. Leveugle, C. Cazaux, and G. Br¨eard, “The atmospheric reentry demonstrator,” No. BR-138, 1998.
[11] H. Hodara, “The use of magnetic fields in the elimination of the re-entry radio blackout,”Proceedings of the IRE, Vol. 49, No. 12, pp. 1825-1830, 1961.
[12] R. M. Manning, “Analysis of electromagnetic wave propagation in a magnetized re-entry plasma sheath via the Kinetic equation,”NASA/TM-2009-216096, 2009.
[13] C. Thoma, D. V. Rose, C. L. Miller, R. E. Clark, and T. P. Hughes, “Electro-magnetic wave propagation through an overdense magnetized collisional plasma layer,” Journal of Applied Physics, Vol. 106, No. 4, pp. 043301, 2009.
[14] R. E. Good and J. J. Rossi, “An injection system for alleviation of radio blackout during re-entry,” MITHRAS CAMBRIDGE MA No. MC-63-78-R2, 1966.
[15] J. W. Meyer, “System and method for reducing plasma induced communication disruption utilizing electrophilic injectant and sharp reentry vehicle nose shaping,”
U.S. Patent No. 7237752, 2007.
[16] E. D. Gillman, J. E. Foster, and I. M. Blankson, “Review of leading approaches for mitigating hyper-sonic vehicle communications blackout and a method of ceramic particulate injection via cathode spot arcs for blackout mitigation,” NASA/TM-2010-216220, 2010.
[17] M. Keidar, M. Kim, and I. Boyd, “Electromagnetic reduction of plasma density during atmospheric reentry and hypersonic flights,” Journal of Spacecraft and Rockets, Vol. 45, No. 3, pp. 445-453, 2008.
[18] M. Kim, M. Keidar, and I. Boyd, “Analysis of an electromagnetic mitigation scheme for reetnry telemetry through plasma,”Journal of Spacecraft and Rockets, Vol. 45, No. 6, pp. 1223-1229, 2008.
[19] N. Sternberg and A. I. Smolyakov, “Resonant transmission of electromagnetic waves in multilayer dense-plasma structures,” IEEE Transactions on Plasma Sci-ence, Vol. 37, No. 7, pp. 1251-1260, 2009.
[20] R. L. Stenzel and J. M. Urrutia, “A new method for removing the blackout problem on reentry vehicles,” Journal of Applied Physics, Vol. 113, No. 10, pp. 103303, 2013.
[21] S. Mudaliar and V. I. Sotnikov, “Radiation characteristics of antennas embedded in a medium with a two-temperature electron population,” IEEE Transactions on Antennas and Propagation, Vol. 60, No. 10, pp. 4545-4555, 2012.
[22] L. C. Schroeder and F. P. Russo, “Flight investigation and analysis of alleviation of communications blackout by water injection during gemini 3 reentry,” NASA TM X-1521, pp. 1-56, 1968.
[23] I. F. Belov, V. Y. Borovoy, V. A. Gorelov, A. Y. Kireev, A. S. Korolev, and E.
A. Stepanov, “Investigation of remote antenna assembly for radio communication with reentry vehicle,” Journal of Spacecraft and Rockets, Vol. 38, No. 2, pp. 249-256, 2001.
[24] Y. Takahashi, K. Yamada, and T. Abe, “Radio frequency blackout possibility for an inflatable vehicle,” AIAA Paper 2012-3110, 2012.
[25] Y. Takahashi, K. Yamada, and T. Abe, “Examination of radio frequency black-out possibility for an inflatable vehicle during atmospheric reentry,” Journal of Spacecraft and Rockets, Vol. 51, No. 2, pp. 430-441, 2014.
[26] P. Tran, J. C. Paulat, P. Boukhobza, “Re-entry flight experiments lessons learned - the atmospheric reentry demonstrator ARD,” Education Notes RTOENAVT130 Flight Experiments for Hypersonic Vehicle Development, Vol. 10, pp. 101 -10-46, 2007.
[27] H. Usui, H. Matsumoto, F. Yamashita, M. Yamane, and S. Takenaka, “Computer experiments on radio blackout of a reentry vehicle,” Proceedings of 6th Spacecraft Charging Technology Conference, pp. 107-110, 2000.
[28] H. Matsumoto and Y. Omura, “Particle simulation of electromagnetic waves and its application to space plasmas,” Computer simulations of space plasmas, Terra Scientific Publishing Company, 1984.
[29] U. Shumlak and J. Loverich, “Approximate Riemann solver for the two-fluid plasma model,” Journal of Computational Physics, Vol. 187, No. 2, pp. 620-638, 2003.
[30] A. Hakim, J. Loverich, and U. Shumlak, “A high resolution wave propagation scheme for ideal Two-Fluid plasma equations,”Journal of Computational Physics, Vol. 219, No. 1, pp. 418-442, 2006.
[31] J. Loverich, A. Hakim, and U. Shumlak, “A discontinuous Galerkin method for ideal two-fluid plasma equations,” Communications in Computational Physics, Vol. 9, No. 2, pp. 240-268, 2011.
[32] B. Srinivasan, A. Hakim, and U. Shumlak, “Numerical methods for two-fluid dis-persive fast MHD phenomena,” Communications in Computational Physics, Vol.
10, No. 1, pp. 183-215, 2011.
[33] H. Kumar and S. Mishra, “Entropy stable numerical schemes for two-fluid plasma equations,” Journal of Scientific Computing, Vol. 52, No. 2, pp. 401-425, 2012.
[34] M. R. Visbal, S. E. Sherer, and M. D. Michael, “High-order methods for wave propagation,” DTIC Document, 2008.
[35] M. Kundrapu, J. Loverich, K. Beckwith, and P. Stoltz, “Modeling radio com-munication blackout and blackout mitigation in hypersonic vehicles,” Journal of Spacecraft and Rockets, Vol. 52, No. 3, pp. 853-862, 2015.
[36] J. Loverich, S. C. D. Zhou, K. Beckwith, M. Kundrapu, M. Loh, S. Mahalingam, P.
Stoltz, and A. Hakim, “Nautilus: a tool for modeling fluid plasmas,” 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, 2013.
[37] M. Kundrapu, J. Loverich, K. Beckwith, P. Stoltz, M. Keidar, T. Zhuang, and A.
Shashurin, “Modeling and simulation of weakly ionized plasmas using nautilus,”
51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, 2013.
[38] A. Shashurin, T. Zhuang, G. Teel, M. Keidar, M. Kundrapu, J. Loverich, I. I.
Beilis, and Y. Raitses, “Laboratory modeling of the plasma layer at hypersonic flight,” Journal of Spacecraft and Rockets, Vol. 51, No. 3, pp. 838-846, 2014.
[39] Y. Takahashi, K. Yamada, and T. Abe, “Prediction performance of blackout and plasma attenuation in atmospheric reentry demonstrator mission,” Journal of Spacecraft and Rockets, Vol. 51, No. 6, pp. 1954-1964, 2014.
[40] R. J. Luebbers., “Frequency-dependent finite-difference time-domain formulation for transient propagation in plasma,” IEEE Transactions on Antennas and Prop-agation, Vol. 39, No. 1, pp. 29-34, 1991.
[41] K. S. Kunz and R. J. Luebbers, Finite Difference Time Domain Method for Elec-tromagnetics, Boca Raton, FL: CRC, 1993.
[42] K. Kinefuchi, I. Funaki, and T. Abe, “Frequency-dependent FDTD simulation of the interaction of microwaves with rocket-plume,” IEEE Transactions on Anten-nas and Propagation, Vol. 58, No. 10, pp. 3282-3288, 2010.
[43] S. A. Cummer, “An analysis of new and existing FDTD methods for isotropic cold plasma and a method for improving their accuracy,” IEEE Transactions on Antennas and Propagation, Vol. 45, No. 3, pp. 392-400, 1997.
[44] C. Park,Nonequilibrium Hypersonic Aerothermodynamics, New York, Wiley, 1990.
[45] C. Park, “Review of chemical-kinetic problems of future NASA missions, I: Earth entries,” Journal of Thermophysics and Heat Transfer, Vol. 7, No. 3, pp. 385-398, 1993.
[46] C. Park, R. L. Jaffe, and H. Partridge, “Chemical-kinetic parameters of hyperbolic earth entry,” Journal of Thermophysics and Heat Transfer, Vol. 15, No. 1, pp. 76-90, 2001.
[47] R. N. Gupta, J. M. Yos, R. A. Thompson, and K.-P. Lee, “A review of reaction rates and thermodynamic and transport properties for an 11-species air model for chemical and thermal nonequilibrium calculations to 30000 K,” NASA RP-1232, 1990.
[48] P. A. Gnoffo, R. N. Gupta, and J. L. Shinn, “Conservation equations and physical models for hypersonic air flows in thermal and chemical nonequilibrium,” NASA TP-2867, 1989.
[49] D. Bose and G. V. Candler, “Thermal rate constants of the N2+O → NO+N reaction using ab initio 3A′′ and 3A′ potential energy surfaces,” The Journal of chemical physics, Vol. 104, No. 8, pp. 2825-2833, 1996.
[50] D. Bose and G. V. Candler, “Thermal rate constants of the O2+N → NO+O reaction based on the A2′ and A4′ potential-energy surfaces,” The Journal of chemical physics, Vol. 107, No. 16, pp. 6136-6145, 1997.
[51] D. S. Liechty, “Extension of a kinetic approach to chemical reactions to electronic energy levels and reactions involving charged species with application to DSMC simulations,” NASA/TP-2014-218254, 2014.
[52] J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, New York, Wiley, 1954.
[53] M. Capitelli, C. M. Ferreira, B. F. Gordiets, and A. I. Osipov,Plasma Kinetics in Atmospheric Gases, Springer, 2000.
[54] M. Nagulapally, G. V. Candler, C. O. Laux, and C. H. Kruger, “Numerical sim-ulation of a constant current density discharge in a flowing air plasma,” AIAA Paper 99-3477, 1999.
[55] J. M. Yos, “Transport properties of nitrogen, hydrogen, oxygen, and air to 30,000 K,” TRAD-TM-63-7, Research and Advanced Development Division, AVCO Cor-poration, 1963.
[56] T. Sakai and J. Olejniczak, “Improvements in a Navier-Stokes Code for Arc Heater Flows,” AIAA Paper 2003-3782, 2003.
[57] M. Fertig, A. Dohr and H. H. Fr¨uhauf, “Transport coefficients for high-temperature nonequilibrium air flows,” AIAA Paper 98–2937, 1998.
[58] M. Fertig, A. Dohr and H. H. Fr¨uhauf, “Transport coefficients for high-temperature nonequilibrium air flows,”Journal of Thermophysics and Heat Trans-fer, Vol. 15, No. 2, pp. 148-156, 2001.
[59] C. F. Curtiss and J. O. Hirschfelder, “Transport properties of multicomponent gas mixture,” The Journal of chemical physics, Vol. 17, No. 6, pp. 550-555, 1949.
[60] C. Park, “Rotational relaxation of N2 behind a strong shock wave,” Journal of Thermophysics and Heat Transfer, Vol. 18, No. 4, pp. 527-533, 2004.
[61] J. G. Parker, “Rotational and vibrational relaxation in diatomic gases,” The Physics of Fluids, Vol. 2, No. 4, pp. 449-462, 1959.
[62] R. C. Millikan and D. R. White, “Systematics of vibrational relaxation,” The Journal of chemical physics, Vol. 39, No. 12, pp. 3209-3213, 1963.
[63] C. Park, “Problems of rate chemistry in the flight regimes of aeroassisted orbital transfer vehicles,” AIAA Paper 84-1730, 1984.
[64] J. P. Appleton and K. N. C. Bray, “The conservation equations for a nonequilib-rium plasma,” Journal of Fluid Mechanics, Vol. 20, No. 4, pp. 659-672, 1964.
[65] M. Mitchner and C. H. Jr. Kruger, Partially Ionized Gases, New York, Wiley, 1973.
[66] M. Nishida and M. Matsumoto, “Thermochemical nonequilibrium in rapidly ex-panding flows of high-temperature air,”Zeitschrift f¨ur Naturforcuhung A, Vol. 52, No. 4, pp. 358368, 1997.
[67] S. S. Lazdinis and S. L. Petrie, “Free electron and vibrational temperature nonequi-librium in high temperature nitrogen,”Physics of Fluids, Vol. 17, No. 8, pp. 1539-1546, 1974.
[68] J. H. Lee, “Electron-impact vibrational relaxation in high-temperature nitrogen,”
Journal of Thermophysics and Heat Transfer, Vol. 7, No. 3, pp. 399-405, 1993.
[69] C. Park and S. H. Lee, “Validation of multitemperature nozzle flow code,”Journal of Thermophysics and Heat Transfer, Vol. 9, No. 1, pp. 9-16, 1995.
[70] M. Vinokur, “An analysis of finite-difference and finite-volume formulations of conservation laws,”NASA CR-177416, 1986.
[71] S. K. Godunov, “A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics,” Matematicheskii Sbornik, Vol. 89, No. 3, pp. 271-306, 1959.
[72] P. L. Roe, “Approximate Riemann solvers, parameter vectors, and difference schemes,” Journal of Computational Physics, Vol. 43, No. 2, pp. 357-372, 1981.
[73] P. L. Roe, “High resolution schemes for hyperbolic conservation laws,” Journal of Computational Physics, Vol. 49, No. 3, pp. 357-393, 1983.
[74] H. C. Yee, “A class of high-resolution explicit and implicit shock-capturing meth-ods,” NASA TM-101088, 1989.
[75] J. J. Quirk, “A contribution to the great Riemann solver debate,” International Journal for Numerical Methods in Fluids, Vol. 18, No. 6, pp. 555-574, 1994.
[76] M. Pandolfi and D. D. Ambrosio, “Numerical instabilities in upwind methods:
analysis and cures for the carbuncle phenomenon,” Journal of Computational Physics, Vol. 166, No. 2, pp. 271-301, 2001.
[77] J. L. Steger and R. F. Warming, “Flux vector splitting of the inviscid gasdynamics equations with application to finite-difference methods,”Journal of Computational Physics, Vol. 40, No. 2, pp. 263-293, 1981.
[78] B. Van Leer, “Flux-vector splitting for the Euler equations,” Lecture Notes in Physics, Vol. 170, pp. 507-512, 1982.
[79] D. Hanel, R. Schwane, and G. Seider, “On the accuracy of upwind schemes for the solution of the Navier-Stokes equations,” AIAA Paper 87-1105-CP, 1987.
[80] M. S. Liou and C. J. Steffen, “A new flux splitting scheme,” Journal of Compu-tational Physics, Vol. 107, No. 1, pp. 23-39, 1993.
[81] E. Shima and K. Kitamura, “Parameter-free simple low-dissipation AUSM-family scheme for all speeds,” AIAA Journal, Vol. 49, No. 8, pp. 1693-1709, 2011.
[82] K. Kitamura and E. Shima, “Towards shock-stable and accurate hypersonic heat-ing computations: a new pressure flux for AUSM-family schemes,” Journal of Computational Physics, Vol. 245, pp. 62-83, 2013.
[83] B. Van Leer, “Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection,” Journal of Computational Physics, Vol. 23, No. 3, pp. 276-299, 1977.
[84] B. Van Leer, “Towards the ultimate conservative difference scheme. V. A second-order sequel to godunov s method,”Journal of Computational Physics, Vol. 32, No. 1, pp. 101-136, 1979.
[85] Y. Wada and M. S. Liou, “A flux splitting scheme with high-resolution and ro-bustness for discontinuities,” AIAA Paper 94-0083, 1994.