Implementation of High Temperature Modelling in Two Dimensional Navier-Stokes Solvers. G.U. Aero Report 9330

The study reported here is concerned with hypersonic viscous flows and involves the development and implementation of an equilibrium air model within Computational Fluid Dynamics (CFD) codes to account for high temperature effects. This work is included in a research program to develop accurate and efficient computational tools for hypersonic re-entry problems. The flow regimes encountered by spacecraft vehicles during atmospheric reentry highlight the need to develop new efficient computational techniques to study chemistry effects. Hypersonic flows are characterized by very strong shock waves, large pressure and heat flux variations and involve chemical processes such as vibrational excitation, dissociation and ionization, which cause air to depart from perfect gas behaviour. As a consequence, the application of CFD techniques to hypersonic flight vehicles requires methods that compute efficiently the composition of air over a wide range of temperatures and densities. An accurate and efficient equilibrium air model has been developed by J.M. Anderson and is based on the solution of the law of mass action to compute the chemical properties of the gas mixture. The air model currently implemented consists of six chemical species (O2, N2, 0, NO, N, Ar) and involves three chemical reactions. The model does not take account of ionization. For the temperature range being considered, the assumption of thermal and chemical equilibrium remains valid and air is assumed to be a reacting mixture of thermally perfect gases. The equilibrium chemical composition of the mixture is obtained from the solution of the law of mass action. The thermodynamic properties of the pure species are obtained by using the least-squares method of curve fitting. The thermodynamic properties of the gas mixture are calculated from the thermodynamic species properties and the equilibrium species concentrations. This physically based model was preferred to more general curve fit techniques since it provides a more flexible and more suitable model for CFD applications. An important aspect for solving the equilibrium equations is the choice of the independent variables. Because the mixture properties are computed on the basis of a uniform equilibrium temperature for all the component species, one needs to retain the equilibrium temperature as an independent variable, the second one usually being density. However, within the time dependent solution of the flow equations, it is the conserved total energy density which is computed as an independent variable and input into the state equations. It is therefore necessary to develop efficient techniques for inverting the general equations of state to give temperature as a function of energy and density. Due to the complex formulation of the equation of state for air at high temperatures, there is no alternative to resorting to an iterative procedure. Since this iterative process can be quite time consuming, the success of employing such physical model for CFD applications will reside in the efficiency of this procedure. Different techniques have been investigated for inverting the equations of state, and the most efficient is an algorithm based on a Newton-Raphton iteration. The number of iterations is minimized by computing an accurate guess for the equilibrium temperature. The present model has already been implemented in an implicit central difference scheme for application in one-dimensional nozzle flow problem. Here, the numerical solution of the Navier-Stokes equations is performed by using an upwind scheme together with a finite volume discretization method. The main difficulty which is encountered when extending upwind methods to equilibrium air models is in the formulation of the flux Jacobian matrix, which must take into account the temperature variations of the ratio of specific heats. The formulation used here is a generalization of Roe average scheme for equilibrium air and was first proposed by Vinokur & Montague. A simple explicit time stepping scheme is used and the boundary conditions treatment is based on the method ofn characteristics. In order to illustrate the application of the equilibrium air model within the CFD environment, a solution of the quasi-one-dimensional Euler equations is given for a hypersonic expansion nozzle flow. Application is then extended to two-dimensional hypersonic inviscid and viscous flows over blunt body shapes. Calculations are performed for both the perfect gas flow and the equilibrium flow for comparison. The effects of the chemistry are discussed