High-order gas-kinetic scheme in curvilinear coordinates for the Euler and Navier-Stokes solutions

TL;DR

The paper develops a two-stage fourth-order gas-kinetic scheme in general curvilinear coordinates to solve Euler and Navier–Stokes equations. By transforming the BGK equation to a computational space and reconstructing Jacobian-weighted variables with a WENO scheme, the method achieves high-order accuracy on non-Cartesian grids. A two-stage temporal discretization coupled with Gazing quadrature-based flux evaluation provides fourth-order time accuracy, while orthogonalization and chain-rule mapping yield necessary physical-space derivatives for the GKS fluxes. The approach is validated through extensive accuracy tests, geometric conservation law verification, and challenging flow problems, demonstrating high accuracy, robustness, and exact GCL satisfaction on non-orthogonal meshes. This enables precise, high-order simulations of complex geometries such as wing-body configurations and turbulent flows in curvilinear domains.

Abstract

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.

Figures (11)

• Figure 1: Schematic of physical domain $(x,y)$ and computational domain $(\xi,\eta)$ in two-dimensional case.
• Figure 2: Accuracy test: 2D nonuniform orthogonal mesh (a) and nonorthogonal mesh (b).
• Figure 3: One dimensional Riemann problem: the density, velocity, pressure distributions for Sod problem at $t=0.2$, and for blast wave problem at $t=0.038$.
• Figure 4: 2D Riemann problems: the nonuniform orthogonal (a) and nonorthogonal (b) mesh $40\times40$ cells.
• Figure 5: 2D Riemann problems: the density distributions on the orthogonal mesh with $500\times500$ cells.