  The high-order gas-kinetic scheme (HGKS) has achieved success in simulating
compressible flow in Cartesian mesh. To study the flow problem in general
geometry, such as the flow over a wing-body configuration, the development of a
three-dimensional HGKS in general curvilinear 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. Based on the
coordinate transformation, the kinetic equation is transformed first to the
computational space, and the flux function in the gas-kinetic scheme is
obtained there and is transformed back to the physical domain for the update of
conservative flow variables inside each 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 the Jacobian-weighted conservative
variables, and the conservative variables are obtained by ratio of the above
reconstructed data at Gaussian quadrature points of each cell interface. In the
two-stage fourth-order gas kinetic scheme (GKS), similar to the generalized
Riemann solver (GRP), the initial spatial derivatives of conservative variables
have to be used in the evaluation of the time dependent flux function in GKS,
which are reconstructed as well through orthogonalization in physical space and
chain rule. A variety of numerical examples from the order tests to the
solutions with strong discontinuities are presented to validate the accuracy
and robustness of the current scheme. The precise satisfaction of the
geometrical conservation law in non-orthogonal mesh is also demonstrated
through the numerical example.
