Multiresolution Reconstruction of the Hypersonic Heat Flux

Abstract

A multiresolution formulation for the direct and inverse reconstruction of the heat flux from temperature sensors distributed over a multidimensional solid in an hypersonic flow is presented. The thermal response is determined by approximating the system Green’s function with the Galerkin method and optimizing the heat flux distribution by fitting the distributed surface temperature data. Coating and glue layers are treated as separated domains for which the Green’s function is obtained independently. Connection conditions for the system Green’s function are derived by imposing continuity of heat flux and temperature concurrently at all interfaces. The heat flux is decomposed in a space-time basis where the temporal functions are multiresolution wavelet with arbitrary scaling. We develop quadrature formulas for the convolution product between wavelets and Green’s function, a reconstruction approach based on isoparametric mapping of three-dimensional geometries, and boundary wavelets for inverse problems. We validate the approach against turbulent conjugate heat transfer simulations and wind tunnel experiments at Mach 0.8 and 6. The main findings are that multidimensional effects are important near the wedge shoulder in the short time scale, that the L-curve regularization must be locally corrected to analyze transitional flows, and that proper regularization leads to sub-cell resolution of the inverse problem.