High-Order Gas-Kinetic Scheme in Curvilinear Coordinates for the Euler and Navier-Stokes Solutions (Communications in Computational Physics, 2020)
The high-order gas-kinetic scheme (HGKS) has achieved success in simulating compressible flows with Cartesian meshes. To study the flow problems in generalgeometries, such as the flow over a wing-body, the development of HGKS in generalcurvilinear coordinates becomes necessary. In this paper, a two-stage fourth-order gas-kinetic scheme is developed for the Euler and Navier-Stokes solutions in the curvilinear coordinates from one-dimensional to three-dimensional computations. Based onthe coordinate transformation, the kinetic equation is transformed first to the computational space, and the flux function in the gas-kinetic scheme is obtained there andis transformed back to the physical domain for the update of flow variables insideeach control volume. To achieve the expected order of accuracy, the dimension-by-dimension reconstruction based on the WENO scheme is adopted in the computational domain, where the reconstructed variables are the cell averaged Jacobian and theJacobian-weighted conservative variables. In the two-stage fourth-order gas-kineticscheme, the point values as well as the spatial derivatives of conservative variables atGaussian quadrature points have to be used in the evaluation of the time dependentflux function. The point-wise conservative variables are obtained by ratio of the abovereconstructed data, and the spatial derivatives are reconstructed through orthogonalization in physical space and chain rule. A variety of numerical examples from theaccuracy tests to the solutions with strong discontinuities are presented to validate theaccuracy and robustness of the current scheme for both inviscid and viscous flows.The precise satisfaction of the geometrical conservation law in non-orthogonal mesh isalso demonstrated through the numerical example.