Axisymmetric radiative heat transfer calculations for flows in chemical non-equilibrium

In the present study, numerical techniques for the calculation of radiative heat transfer in w'symmetric flowfields are developed and applied to the simulation of hypersonic reactive flows. The present approach removes the need for simplified models such as the OneDimensional Slab theory, and can be easily extended to three-dimensional flow problems. Preliminary results are presented for the Mach 47 flow past a sphere, and the predictions obtained from the axisymmetric algorithm are compared with those given by simplified radiative models. A non-negligible difference is found in the predicted values of heat flux a t the stagnation point. 1. I N T R O D U C T I O N In recent years, the need for accurate predictions of aerothermodynamic heating loads, and of radiative heat transfer in particular, has been spurred by interest in aerospace applications, such as the manned mission to Mars, and the re-establishment of human presence on the Moon'. Moreover, the design of thermal protection systems for the Aeroassisted Orbital Transfer Vehicle (AOTV), and for planetary probes such as Pioneer (for Venus) and Galileo (for Jupiter), is almost entirely driven by predictions of radiative heat transfer'. There are two major techniques utilized for the simulation of gasdynamic problems that include radiative heat transfer. Several investigators use an uncoupled approach, whereby the thermodynamic state of the flow is determined from a solver without radiative heat transfer, followed by a post-processing phase where the radiative field is e ~ a l u a t e d ' ~ ~ ~ ~ . On the other hand, the fully coupled approach is advocated by because the decoupling can significantly overestimate both radiative and conductive heat fluxes. Other factors that can play a significant role in the accuracy of radiative heat transfer predictions include: the need for detailed emission and absorption models, the importance of thermo-chemical non-equilibrium phenomena, t Research Assistant, Student Member AIAA Assistant Professor of Aerospace Engineering, Member AIAA Copyright @ 1993 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. 1 the possibility of ablation, and the presence of threedimensional effects away from the stagnation point5. The numerical simulation of hypersonic flowfields in thermo-chemical non-equilibrium without radiative transfer has been successfully attempted by Candler and MacCormack', and uncoupled calculations have been presented by Candler and Park*. Moss and Simmonds', and Gupta et aL9 have used the viscous shock layer equations to solve axisymmetric reentry problems with fully coupled radiative heat transfer. However, the previous analyses employ the OneDimensional Slab approximation, which is valid in the stagnation region only. Hartung3 compares the 1D Slab model to a Modified Differential Approach, and shows that the former is a potential liability if used away from the stagnation point. Several researchers have developed fully coupled hypersonic/radiation simulations, but in all cases the 1D Slab approximation is In a related paper6, the writers have successfully attempted truly two-dimensional numerical solutions to the radiative heat transfer equations, fully coupled with an existing thermo-chemical non-equilibrium flow solver'', co-developed by the second author. The a p proach is compatible with existing spectrographic information and/or radiative models, which can be used as input in order to predict coupled radiative, conductive and diffusive heat transfers over relatively complex geometries. Preliminary results obtained for the inviscid flow over a cylinder at a freestream Mach number of M , = 47 indicated that there is a significant difference between predictions obtained by means of the 1D Slab theory and fully two-dimensional radiative heat transfer calculations. Moreover, the inadequacy and inaccuracy of uncoupled approaches at that high Mach numher was graphically demonstrated. In the present study, the techniques developed for two-dimensional flowfields are extended to axisymmetric problems. The approach chosen has the distinctive advantage of being readily portable to threedimensional problems as well, with a reasonable increase in computational requirements for the radiative source term at a point. Due to the computational costs associated with the development and testing of the new model, two simplifying assumptions are employed: a grey gas approximation has been used for the absorptivity coefficient, although provisions for multi band models are included; and thermodynamic equilibrium b d has been assumed. Although it would be necessary to lift the above simplifications in order to obtain quantitatively accurate predictions of radiative heat transfer for geometries of practical interest, it is necessary to stress at this point that the objective of the present study is to develop the numerical tools for the simulation of axisymmetric flowfields via the inclusion of a truly multi-dimensional radiative source term. In the following, the mathematical and physical foundations for the study of flows in chemical nonequilibrium, including radiative heat transfer effects, will be briefly outlined. A discussion of the numerical techniques employed for the discretization of the flow equations and the determination of the source term due to the presence of a radiation field will follow. A few preliminary results, both viscous and inviscid, will be presented in order to assess the performance of the present algorithm when compared with one-dimensional approximations. w' 2. GOVERNING EQUATIONS 2.1. Thermodynamic Model In the following, a gas mixture composed of N thermally perfect species is considered. The thermal equation of slate will be given by Dalton's Law, whereby the mixture pressure, p, is the summation of u' partial pressures