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In this study, large eddy simulations (LES) of supersonic combustion using a compressible flamelet/progress variable (CFPV) approach are performed for the HIFiRE scramjet at Mach 8 flight conditions. The LES results show good agreement with the ground test experimental measurements. The combustion model is based on an efficient flamelet approach, where compressibility corrections are devised based on assumed functional forms of important thermo-chemical quantities. Specifically, the source term of the progress variable is rescaled with the local density and temperature in the LES, leading to improved predictions relative to existing flamelet models. A modified equilibrium wall-model, capable of predicting the viscous heating, is used in the viscous near-wall region. Temperature near the wall increases significantly due to viscous heating, enhancing the reaction rate and heat-release. This is shown to be a crucial step for accurately predicting the pressure rise in the combustor.