高阶紧致气体动理学格式中壁面边界条件的处理
Treatment of Wall Boundary Conditions in High-Order Compact Gas-Kinetic Schemes
AI总结：
针对高阶紧致气体动理学格式（GKS）的边界处理难题，提出一种基于随时间变化分布函数的单侧三阶边界离散方法，实现了空间与时间高精度，在粗网格上表现优越。
摘要：
边界层是流体力学中的基本结构，准确的边界离散能显著提升计算效率。本文提出了一种针对紧致气体动理学格式（GKS）的三阶边界离散方法。宽模板和弯曲边界给高阶格式的边界处理带来了挑战，尤其是在时间精度方面。通过利用随时间变化的气体分布函数，GKS同时计算通量并更新网格界面处的流动变量，从而在三阶紧致格式中实现网格平均流动变量及其梯度的同步更新。所提出的单侧离散方法通过在无滑移壁面边界条件的离散中利用更新后的流动变量和梯度，在边界单元上实现了三阶空间精度。通过采用多阶段多导数方法的GKS随时间变化通量实现，边界单元上的高阶时间精度得以实现。此外，我们针对绝热和等温壁面边界开发了精确的无穿透条件，并扩展至弯曲网格几何结构，以充分发挥高阶格式的优势。对所提出的单侧三阶边界格式、使用幽灵单元的三阶边界格式以及二阶边界格式进行比较分析，表明它们对三阶紧致GKS的性能存在显著差异。结果表明，低阶边界单元处理产生明显较差的结果，而所提出的三阶处理方法展现出优越的性能，特别是在粗网格配置上。
英文摘要：
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.
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