In recent years, high-order discontinuous Galerkin (DG) methods have emerged as an attractive approach for numerical simulations of compressible flows. This paper presents an overview of the recent development of DG methods for compressible flows with particular focus on hypersonic flows. First, we survey state-of-the-art DG methods for computational fluid dynamics. Next, we discuss both matrix-based and matrix-free iterative methods for the solution of discrete systems stemming from the spatial DG discretizations of the compressible Navier–Stokes equations. We then describe various shock capturing methods to deal with strong shock waves in hypersonic flows. We discuss adaptivity techniques to refine high-order meshes, and synthetic boundary conditions to simulate free-stream disturbances in hypersonic boundary layers. We present a few examples to demonstrate the ability of high-order DG methods to provide accurate solutions of hypersonic laminar flows. Furthermore, we present direct numerical simulations of hypersonic transitional flow past a flared cone at Reynolds number , and hypersonic transitional shock wave boundary layer interaction flow over a flat plate at Reynolds number . These simulations run entirely on hundreds of graphics processing units (GPUs) and demonstrate the ability of DG methods to directly resolve hypersonic transitional flows, even at high Reynolds numbers, without relying on transition or turbulence models. We end the paper by offering our perspectives on error estimation, turbulence modeling, and real gas effects in hypersonic flows.

中文翻译：

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高超声速流动的不连续伽辽金方法


近年来，高阶间断伽辽金（DG）方法已成为可压缩流数值模拟的一种有吸引力的方法。本文概述了可压缩流的 DG 方法的最新发展，特别关注高超声速流。首先，我们调查了最先进的计算流体动力学 DG 方法。接下来，我们讨论基于矩阵和无矩阵迭代方法来求解源于可压缩纳维-斯托克斯方程的空间 DG 离散化的离散系统。然后，我们描述了处理高超音速流中强冲击波的各种冲击捕获方法。我们讨论了细化高阶网格的自适应技术，以及模拟高超声速边界层中自由流扰动的合成边界条件。我们举了几个例子来证明高阶 DG 方法提供高超声速层流精确解的能力。此外，我们提出了在雷诺数下通过喇叭形锥体的高超声速过渡流和在雷诺数下在平板上的高超声速过渡激波边界层相互作用流的直接数值模拟。这些模拟完全在数百个图形处理单元 (GPU) 上运行，并展示了 DG 方法直接解析高超音速过渡流的能力，即使在高雷诺数下也是如此，而无需依赖过渡或湍流模型。我们通过提出我们对高超音速流中的误差估计、湍流建模和真实气体效应的观点来结束本文。