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A finite difference scheme for inviscid flows with non-equilibrium chemistry and internal energy
Maihl. Comput. Modelling Vol. 16, No. 2, pp. 21-27, Printed in Great Britain. All rights reserved
1992 Copyright@
@x95-7177/92 $5.00 + 0.00 1992 Pergamon Press plc
A FINITE DIFFERENCE SCHEME FOR INVISCID FLOWS WITH NON-EQUILIBRIUM CHEMISTRY AND INTERNAL ENERGY P. GLAISTER Department
of Mathematics,
P.O. Box 220, University of Reading, Whiteknights
Reading, RG6 2AX, United Kingdom (Received
July 1990)
Abstract-A finite difference scheme is presented for the inviscid terms of the equations of compressible fluid dynamics with general non-equilibrium chemistry and internal energy.
1. INTRODUCTION Recently [l], an approximate Riemann solver was presented for treating one-dimensional, inviscid, chemically reacting flows, and uses numerical characteristic decomposition for the general flux terms arising in non-equilibrium flow. In addition to this, a stiff solver was suggested as a means of treating the source terms arising in the decoupled, scalar equations. An integral part of this scheme is the special averages of flow variables required to make shock-capturing automatic. In particular, approximations are required for the derivatives of the equation of state in each computational cell. In this respect, a significant improvement in the efficiency of this scheme was made in [2], where the number of function evaluations of the equation of state is an optimum. We note, however, that although [l] (and [2]) treats flows with non-equilibrium chemistry via a production equation for each species’ mass density (or mass fraction), the resulting scheme only applies to the case of thermodynamic equilibrium. That is, although an allowance has been made for imperfect gas effects due to chemical changes in the amount of mass of each species, no account has been made for the activation of internal energy modes which behave non-linearly with temperature. The purpose of the present paper is to extend the scheme in [l] to treat this more general case of thermodynamic non-equilibrium. When the pressure is sufficiently low, away from the gas triple point, it has been determined that each species of the gas mixture will behave as a thermally perfect gas, i.e., a state of thermodynamic equilibrium. An assumption of this kind was made in [3] and suggested as a means of constructing the equation of state required in [l]. At very high temperatures, however, portions of the internal energy may be out of equilibrium. Then, the internal energy of each species will generally depend on the translational temperature, the electron temperature, and various internal energy modes, which must be computed as functions of space and time [4]. The scheme presented here assumes that a portion of the internal energy of each species is in thermodynamic equilibrium, and that the remaining portion is in a non-equilibrium state. The non-equilibrium part of the energy is assumed to be modelled by appropriate production rates. Finally, we assume that the translational temperature of all species is the same, i.e., the scheme is not applicable to flows with free electrons. As in [l], we concentrate only on the modelling of the flux terms, since the source terms can vary according to the chemical model of the production of species. In particular, the number of reaction steps chosen together with the production of the non-equilibrium internal energy and the corresponding numerical treatment will, generally, be via an appropriate stiff-solver. In addition, we assume a general form for the equation of state; particular forms, together with their appropriate construction, depend on the model chosen. Typeset by A,&%‘&$ 21
1992 Copyright@
@x95-7177/92 $5.00 + 0.00 1992 Pergamon Press plc
A FINITE DIFFERENCE SCHEME FOR INVISCID FLOWS WITH NON-EQUILIBRIUM CHEMISTRY AND INTERNAL ENERGY P. GLAISTER Department
of Mathematics,
P.O. Box 220, University of Reading, Whiteknights
Reading, RG6 2AX, United Kingdom (Received
July 1990)
Abstract-A finite difference scheme is presented for the inviscid terms of the equations of compressible fluid dynamics with general non-equilibrium chemistry and internal energy.
1. INTRODUCTION Recently [l], an approximate Riemann solver was presented for treating one-dimensional, inviscid, chemically reacting flows, and uses numerical characteristic decomposition for the general flux terms arising in non-equilibrium flow. In addition to this, a stiff solver was suggested as a means of treating the source terms arising in the decoupled, scalar equations. An integral part of this scheme is the special averages of flow variables required to make shock-capturing automatic. In particular, approximations are required for the derivatives of the equation of state in each computational cell. In this respect, a significant improvement in the efficiency of this scheme was made in [2], where the number of function evaluations of the equation of state is an optimum. We note, however, that although [l] (and [2]) treats flows with non-equilibrium chemistry via a production equation for each species’ mass density (or mass fraction), the resulting scheme only applies to the case of thermodynamic equilibrium. That is, although an allowance has been made for imperfect gas effects due to chemical changes in the amount of mass of each species, no account has been made for the activation of internal energy modes which behave non-linearly with temperature. The purpose of the present paper is to extend the scheme in [l] to treat this more general case of thermodynamic non-equilibrium. When the pressure is sufficiently low, away from the gas triple point, it has been determined that each species of the gas mixture will behave as a thermally perfect gas, i.e., a state of thermodynamic equilibrium. An assumption of this kind was made in [3] and suggested as a means of constructing the equation of state required in [l]. At very high temperatures, however, portions of the internal energy may be out of equilibrium. Then, the internal energy of each species will generally depend on the translational temperature, the electron temperature, and various internal energy modes, which must be computed as functions of space and time [4]. The scheme presented here assumes that a portion of the internal energy of each species is in thermodynamic equilibrium, and that the remaining portion is in a non-equilibrium state. The non-equilibrium part of the energy is assumed to be modelled by appropriate production rates. Finally, we assume that the translational temperature of all species is the same, i.e., the scheme is not applicable to flows with free electrons. As in [l], we concentrate only on the modelling of the flux terms, since the source terms can vary according to the chemical model of the production of species. In particular, the number of reaction steps chosen together with the production of the non-equilibrium internal energy and the corresponding numerical treatment will, generally, be via an appropriate stiff-solver. In addition, we assume a general form for the equation of state; particular forms, together with their appropriate construction, depend on the model chosen. Typeset by A,&%‘&$ 21