A numerically stable and low dissipation shock-capturing scheme

HU Lijun; LI Long

doi:10.7638/kqdlxxb-2023.0062

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A numerically stable and low dissipation shock-capturing scheme

更新时间：2026-07-27

- A numerically stable and low dissipation shock-capturing scheme
- Vol. 42, Issue 7, Pages: 35-47(2024)
- 作者机构：
  
  衡阳师范学院数学与统计学院,衡阳,421002
- 作者简介：
- 基金信息：
- DOI：10.7638/kqdlxxb-2023.0062
  CLC： O354.4
- Published：2024
- 稿件说明：
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胡立军, 李龙. 一种数值稳定的低耗散激波捕捉格式[J]. 空气动力学学报, 2024, 42(7): 35−47.

HU Lijun, LI Long. A numerically stable and low dissipation shock-capturing scheme[J]. 2024, 42(7): 35-47.
胡立军, 李龙. 一种数值稳定的低耗散激波捕捉格式[J]. 空气动力学学报, 2024, 42(7): 35−47. DOI： 10.7638/kqdlxxb-2023.0062

HU Lijun, LI Long. A numerically stable and low dissipation shock-capturing scheme[J]. 2024, 42(7): 35-47. DOI： 10.7638/kqdlxxb-2023.0062.

摘要

HLLC格式应用广泛且可以准确捕捉激波间断，其满足熵条件和保正性，易于推广到其他双曲系统。但是，在计算多维强激波时出现的数值不稳定现象大大限制了它在高马赫数流动问题中的应用。本文基于三维无黏可压缩欧拉方程组构造一种激波数值稳定的HLLC格式（SS-HLLC）：将传统的HLLC通量显式地改写成耗散的HLL通量和反扩散项的形式，利用充分光滑的控制函数调节激波附近横向通量中的反扩散项来改善格式的激波数值稳定性，分析证明了反扩散控制技术的有效性。此外，采用简单的THINC重构方法和BVD算法来减少数值耗散项中的密度差，从而构造了一种低耗散的激波数值稳定格式（LD-SS-HLLC），可进一步提高格式对接触间断的分辨率。经过一系列标准算例测试，结果证明了LD-SS-HLLC格式的低耗散性和激波数值稳定性。

Abstract

It is well known that the Harten-Lax-van Leer with contact (HLLC) scheme suffers from unstable performance in capturing strong multi-dimensional shock waves

which impedes its application in high-Mach-number flows. This paper presents a simple strategy to develop a stable HLLC-type scheme for three-dimensional inviscid Euler equations. The conventional HLLC flux is split into a dissipative HLL flux and an anti-diffusion term. The improvement of the numerical stability

justified by stability analyses and numerical experiments

is achieved by using a sufficiently smooth control function to adjust the numerical dissipation term of the transverse flux adjacent to the shock wave. In addition

the simple THINC (tangent of hyperbola for interface capturing) reconstruction method and the BVD (boundary variation diminishing) algorithm are employed to reduce the density difference in the numerical dissipation term

thus further improving the resolution for contact discontinuities. The numerical results of one- to three-dimensional benchmark examples demonstrate the proposed scheme’s high resolution for contact discontinuities and stability for multi-dimensional strong shock waves. The numerical results of a shock/boundary-layer interaction problem promise the future application of the proposed local anti-diffusion control method in simulating viscous flows. The modularization of the proposed scheme and its integration into Computational Fluid Dynamics software to simulate

for instance

hypersonic flows

multi-component chemical reaction flows

and turbulent combustion are worthy of further research in the future.

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references

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