Chemical equilibrium and non-equilibrium inviscid flow computations using a centered scheme

Abstract

Within the framework of the collaboration between IMHEF and CERFACS a 2D Inviscid Flow solver for Hypersonic flows has been developed. The Euler equations are discretized in space on a structured mesh using the Finite Volume method with centered differences. The resulting system of ordinary differential equations is integrated in time using the explicit Runge Kutta scheme. Artificial dissipation terms are added to damp odd/even oscillations allowed for by centered space differences, and to damp spurious oscillations near discontinuities. External shock waves in the flow field are treated by a shock fitting procedure, while (weaker) internal shock waves are captured by the numerical scheme. A complete description of the numerical method can be found in [1]. The strong shock waves present in hypersonic flows give rise to high temperatures directly behind the shock wave, which may result into the dissociation of air. This is a process which costs energy, hence temperatures in the flow field will be reduced. Air dissociation can be modelled on different levels, which depend on the ratio of the characteristic time scales of the flow and the chemistry. If the characteristic time scale of the chemistry is much smaller than that of the flow, it can be assumed that the flow is in chemical equilibrium, i.e. chemical reactions are taking place, but the production of a chemical species is balanced by its destruction. The other limit is that the chemistry time scale is much smaller than that of the flow, hence no chemical reactions are taking place. The chemistry is frozen, and the air is treated as a thermally perfect gas. If the time scales are of the same order of magnitude the flow is in chemical non-equilibrium. These three levels of modelling have been included in the Euler solver. Incorporation of equilibrium and frozen chemistry is straightforward for the centered scheme described above, since only the relation which connects the pressure to the density and total energy had to be changed. This has been done using the effective Y approach, where γ is the effective ratio of specific heats. For explicit schemes, this y needs to be calculated only once per 100 time steps. Non-equilibrium chemistry has been incorporated by solving three partial differential equations for the partial densities of the species N, O and NO using the Runge Kutta scheme described above, together with two algebraic equations for the species N2 and 02. The time step used in the Runge Kutta time stepping is in this case the minimum of the chemical time step and the fluid dynamic time step. Calculations showed that only in the initial phase of the time integration process the chemical time step was the smallest of the two. Figure 1 shows the calculated temperatures for the flow around a sphere. Owing to the shock fitting procedure the external bow shock is sharp and oscillation free. The highest temperatures are found when the gas is treated as frozen since in this case no dissociation is taking place. The lowest temperatures along the stagnation line are obtained when it is assumed that the flow is in equilibrium. For the non-equilibrium calculation, the flow is frozen across the shock wave, and is in chemical equilibrium at the stagnation point. This explains the strong temperature gradient between shock and body.