Treatment of Wall Boundary Conditions in High-Order Compact Gas-Kinetic Schemes
Jiawang Zhang, Xing Ji, Kun Xu
TL;DR
This paper addresses the challenge of achieving high-order accuracy near boundaries for compact gas-kinetic schemes on curved meshes. It develops a one-sided, third-order boundary discretization that leverages updated interior gradients and time-dependent, BGK-based fluxes within a CGKS framework, complemented by a two-stage fourth-order time integration and HWENO reconstruction. The approach is extended to curved geometries and includes a kinetic, isothermal boundary condition to enforce no-penetration. Numerical experiments across laminar to hypersonic flows demonstrate that the proposed boundary treatment outperforms both isothermal/adiabatic second-order methods and third-order ghost-cell schemes, particularly when the boundary layer is under-resolved, highlighting its practical impact for accurate, efficient CFD on complex domains.
Abstract
The boundary layer represents a fundamental structure in fluid dynamics, where accurate boundary discretization significantly enhances computational efficiency. This paper presents a third-order boundary discretization for compact gas-kinetic scheme (GKS). Wide stencils and curved boundaries pose challenges in the boundary treatment for high-order schemes, particularly for temporal accuracy. By utilizing a time-dependent gas distribution function, the GKS simultaneously evaluates fluxes and updates flow variables at cell interfaces, enabling the concurrent update of cell-averaged flow variables and their gradients within the third-order compact scheme. The proposed one-sided discretization achieves third-order spatial accuracy on boundary cells by utilizing updated flow variables and gradients in the discretization for non-slip wall boundary conditions. High-order temporal accuracy on boundary cells is achieved through the GKS time-dependent flux implementation with multi-stage multi-derivative methodology. Additionally, we develop exact no-penetration conditions for both adiabatic and isothermal wall boundaries, with extensions to curved mesh geometries to fully exploit the advantages of high-order schemes. Comparative analysis between the proposed one-sided third-order boundary scheme, third-order boundary scheme with ghost cells, and second-order boundary scheme demonstrates significant performance differences for the third-order compact GKS. Results indicate that lower-order boundary cell treatments yield substantially inferior results, while the proposed third-order treatment demonstrates superior performance, particularly on coarse grid configurations.