View-Factor Approach as a Radiation Model for the Reentry Flowfield Daniil A. Andrienko * University of Michigan, Ann Arbor, Michigan 48109 Sergey Surzhikov † Institute for Problems in Mechanics, 119526, Moscow, Russia and Joseph Shang ‡ Wright State University, Dayton, Ohio 45435 DOI: 10.2514/1.A33305 An accurate and efficient radiation transfer model, based on the view-factor approach, is proposed for the reentry flowfield. The radiation transfer equation is coupled to the gasdynamic system of equations in order to describe the energy transfer in the absorbing and emitting air plasma in two-dimensional axisymmetric and three-dimensional geometries. A newly derived semi-analytical expression for the radiative flux density dramatically simplifies the calculation of the spectral and integral characteristics of the radiation flowfield in the axisymmetric geometry. The comparison with the ray-tracing method and tangent slab approximation revealed an asymptotic accuracy of radiation flux density. Nomenclature A = surface of element F, E = incomplete elliptic integral of first and second kind J = spectral intensity of radiation J b = spectral intensity of the black body T = attenuation factor V = volume of element W = radiation flux density κ ν = absorption coefficient τ = optical thickness Subscripts i, j = indices of gasdynamic grid ν = spectral index I. Introduction S UPERORBITAL reentry experiments [1–3] have revealed a significant contribution of radiation transfer into the total heat flux on the surface of a space vehicle. At the same time, a large number of studies that deal with the problem of an accurate and efficient simulation of radiation transfer have been published over the past decades. However, the problem of radiation transfer in the nonequilibrium gas, heated by a shock wave, is not yet completely solved. One of the possible reasons for this is a strong dependence of radiation properties from chemical composition and thermodynamic state of high-temperature gas. Another factor that complicates the numerical simulation of radiation transfer is the mathematical com- plexity of the radiation energy conservation equation. The integral radiation intensity depends on the spatial and angular coordinates as well as on the wavelength. These problems make the numerical simulation of radiation transfer very prohibitive in multidimensional geometries. An efficient and accurate model of radiation transfer is currently a very desirable item in aerothermodynamics. Until recently, the tangent slab (TS) approximation [4] was generally used to evaluate the radiation flux incident on the heat shield of a space vehicle, while the radiation heating of the leeward surface was often neglected due to its relatively small contribution and the inaccuracy of the TS approximation for nonshock flows. With increasing computational resources, the ray-tracing method (RTM) became available [5]. One of the important advantages of the RTM is the accurate description of radiation flux for the entire flowfield [6,7]. However, the RTM is computationally expensive, especially in multidimensional geom- etries. Another disadvantage of the RTM is a necessary approxi- mation of the computed flowfield on the tracing ray, involving an algorithm of the nearest neighbor search (NNS). This approximation is required even when no estimation of optical thickness of the tracing ray is required. While a significant effort must be spent to obtain a high-quality, artifact-free flowfield that captures a shock wave [8], inaccurate approximation may compromise the overall accuracy of the radiation transfer model. On the other hand, the spherical harmonics method [9] offers a relatively simple and efficient way to account for the multi- dimensional radiation transfer utilizing the same mesh, used for the solution of gasdynamic equations. Recently, the P 1 approximation was successfully applied to develop a two- and three-dimensional model of radiation transfer for the entry of a Martian probe [10,11]. While the reasonable agreement with the RTM was achieved, the P 1 approximation was found to give inaccurate results in the optically transparent gas. Among other radiation transfer models, the view-factor (VF) ap- proach can be used to access the heat flux on the surface of a space vehicle. This approach, also known as the zonal method, was origi- nally proposed in [12] to account for the radiation energy exchange between isothermal elements separated by an absorbing and emitting media. The key idea of the VF approach is to establish the parameter of visibility between the elements participating in the radiative energy exchange. In the case of nonparticipating media, the visibility parameter simply equals to the fraction of the solid angle at which the emitting element is observed by the receiving element. Hence, the visibility parameter is often referred to as a VF. When the media between exchanging elements has its own radiative properties, the attenuation of radiation is often included in the definition of the VF. The latter can be established for the surface-to-surface, Presented as Paper 2014-2488 at the 45th AIAA Plasmadynamics and Lasers Conference, AIAA Aviation and Aeronautics Forum and Exposition 2014, Atlanta, GA, 16–20 June 2014; received 19 March 2015; revision received 6 August 2015; accepted for publication 11 September 2015; published online 25 November 2015. Copyright © 2015 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 1533-6794/15 and $10.00 in correspondence with the CCC. *Postdoctoral Research Fellow, Department of Aerospace Engineering, 1320 Beal Ave. † Professor, Director, pr. Vernadskogo 101-1. ‡ Research Professor, 3640 Col Glenn Hwy. Article in Advance / 1 JOURNAL OF SPACECRAFT AND ROCKETS Downloaded by UNIVERSITY OF MICHIGAN LIBRARY on December 10, 2015 | http://arc.aiaa.org | DOI: 10.2514/1.A33305