This study focuses on developing a fourth-order finite-volume method to solve the compressible Navier-Stokes equations with combustion on a generalized curvilinear coordinate system. The complex geometry is treated by transforming the physical space to a computational space through a generalized curvilinear coordinate transformation. Such a transformation enables the straightforward application of our existing fourth-order finite-volume method for Cartesian grids to body-fitted meshes. The fourth-order solution accuracy of the algorithm is verified using a vortex convection and diffusion problem. Results of the multi-species mixing flow in a lid-driven cavity are obtained. These results are in good agreement with literature data. The combustion capability of the algorithm is demonstrated using a detonation wave generated from the premixed stoichiometric methane combustion.
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