High order finite volume scheme and conservative grid overlapping technique for complex industrial applications. The numerical foundation of the CFD solver FLUSEPA (French trademark N. 13400926) is presented. It is a Godunov’s type unstructured finite volume method suitable for highly compressible turbulent scale-resolving simulations around 3D complex geometries and general non-Cartesian grids. First, a family of k-exact Godunov schemes is developed by recursively correcting the truncation error of the piecewise polynomial representation of the primitive variables. The keystone of the proposed approach is a quasi-Green gradient operator which ensures consistency on general meshes. In addition, a high-order single-point quadrature formula, based on high-order approximations of the successive derivatives of the solution, is developed for flux integration along curved cell faces. Then, a re-centering process is used to reduce as far as possible numerical diffusion. The proposed family of schemes is compact in the algorithmic sense, since it only involves communications between direct neighbors (cells which have common faces) of the mesh cells. To address complex geometries, a conservative grid intersection technique is used. Compressible numerical test cases are investigated to demonstrate the accuracy and the robustness of the presented numerical scheme, then, supersonic RANS/LES computations around the Ariane 5 space launcher are presented to shows the capability of the scheme to predict flows with shocks, vortical structures and complex geometries.