Estimation Methods for Two-Dimensional Conduction Effects of Shock-Shock Heat Fluxes

Highly localized heating rates occur in extreme thermal environments, such as shock–shock interaction at hy-personic speeds. Experimental estimation of heat ux usually involves measurement of discrete unsteady surface temperatures on low conductivity materials that are chosen to reduce conduction effects. Typically, temperature measurementsarereduced toheat uxesusingone-dimensionalconductiontechniques.However, lateral conduction from localized high heat ux regions into low heat ux regions is signi cant and in uences the one-dimensional solution. The one-dimensional solution also suffers mathematical instabilities. To evaluate the nonuniform, un-steady surface ux from measured temperatures, an inverse technique was devised that damps instabilities in the temporal direction and resolves large and sudden ux changes in the spatial direction. Based on previous work, a simple inverse method was used in time, and a function speci cation method was used in space. Furthermore, a technique was devised to expedite the solution by marching in space as well as in time. The new multidimensional inversemethodwas found to resolve steep spatial gradientsmore accurately in ux thana one-dimensionalmethod. Furthermore, the inverse procedure exhibits better stability than a multidimensional forward technique. Nomenclature Ch = normalized heat ux; Stanton number H = order of regularizationmatrix, dimensionless q = heat ux, W/cm2 R = regularization term S = objective function, K2 T = calculated temperature, K t = time, s X = sensitivitymatrix Y = measured temperature, K a = regularizationparameter, K2cm4/W2 W ¡ 1 = matrix of measurement variances, 1/K2 Subscripts i, j = temporal index n = normalized s = space t = time 0 = current iteratio