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HLLC+: Low-Mach Shock-Stable HLLC-Type Riemann Solver for All-Speed Flows
SIAM Journal on Scientific Computing ( IF 3.0 ) Pub Date : 2020-07-20 , DOI: 10.1137/18m119032x Shusheng Chen , Boxi Lin , Yansu Li , Chao Yan
SIAM Journal on Scientific Computing ( IF 3.0 ) Pub Date : 2020-07-20 , DOI: 10.1137/18m119032x Shusheng Chen , Boxi Lin , Yansu Li , Chao Yan
SIAM Journal on Scientific Computing, Volume 42, Issue 4, Page B921-B950, January 2020.
The approximate Riemann solver of Harten--Lax--van Leer (HLL) and its variant HLLC (HLL with Contact restoration) solver are widely used as flux functions of finite volume Godunov-type methods for the solution of the gas dynamic Euler equations. However, the HLLC solver suffers from two significant difficulties: an accuracy problem at low-speed flows and shock instability at high-speed flows. To remedy such drawbacks, a novel low-Mach shock-stable HLLC-type scheme called HLLC+ is developed for all speeds. The antidissipation pressure fix is introduced first to overcome the accuracy problem in low Mach number limits. Then, shear viscosity is identified and scaled into the original HLLC scheme to overcome shock instability. A new pressure-based factor function without switching coefficients is devised to prevent shear viscosity from smearing the boundary layer. The new HLLC+ scheme involves no empirical parameters and is easy to implement. Asymptotic analysis and low Mach number test cases show the excellent behaviors of HLLC+ in low Mach number limits: no global cut-off problem, damping pressure checkerboard modes, having expected ${Ma^2}$ scaling of pressure and density fluctuations, and satisfaction of divergence constraint. Furthermore, this work manifests that the accuracy problem is associated with the normal velocity jumps of the flux interface, while shock instability is related to the transverse velocity jumps. Numerical test cases across a wide range of Mach numbers demonstrate the superior performance and potentiality of HLLC+ to simulate all Mach number flows.
中文翻译：
HLLC +：适用于全速流动的低马赫冲击稳定HLLC型黎曼求解器
SIAM科学计算杂志，第42卷，第4期，第B921-B950页，2020年1月。
Harten-Lax-van Leer（HLL）的近似Riemann求解器及其变体HLLC（带接触恢复的HLL）求解器被广泛用作有限体积Godunov型方法的通量函数，用于求解气体动力学Euler方程。但是，HLLC求解器面临两个重大困难：低速流动时的精度问题和高速流动时的冲击不稳定性。为了弥补这些缺点，开发了适用于所有速度的新型低马赫冲击稳定HLLC型方案HLLC +。首先介绍了抗耗散压力修正方法，以克服低马赫数限制下的精度问题。然后，确定剪切粘度并将其缩放到原始HLLC方案中，以克服冲击不稳定性。设计了一种新的无转换系数的基于压力的因数函数，以防止剪切粘度涂抹边界层。新的HLLC +方案不包含任何经验参数，并且易于实现。渐近分析和低马赫数测试案例显示了HLLC +在低马赫数限制下的出色表现：无全局截止问题，阻尼压力棋盘模式，预期压力和密度波动的缩放比例为{{Ma ^ 2} $}和满意程度散度约束。此外，这项工作表明，精度问题与通量界面的正常速度跳跃有关，而冲击不稳定性与横向速度跳跃有关。
更新日期：2020-07-20
The approximate Riemann solver of Harten--Lax--van Leer (HLL) and its variant HLLC (HLL with Contact restoration) solver are widely used as flux functions of finite volume Godunov-type methods for the solution of the gas dynamic Euler equations. However, the HLLC solver suffers from two significant difficulties: an accuracy problem at low-speed flows and shock instability at high-speed flows. To remedy such drawbacks, a novel low-Mach shock-stable HLLC-type scheme called HLLC+ is developed for all speeds. The antidissipation pressure fix is introduced first to overcome the accuracy problem in low Mach number limits. Then, shear viscosity is identified and scaled into the original HLLC scheme to overcome shock instability. A new pressure-based factor function without switching coefficients is devised to prevent shear viscosity from smearing the boundary layer. The new HLLC+ scheme involves no empirical parameters and is easy to implement. Asymptotic analysis and low Mach number test cases show the excellent behaviors of HLLC+ in low Mach number limits: no global cut-off problem, damping pressure checkerboard modes, having expected ${Ma^2}$ scaling of pressure and density fluctuations, and satisfaction of divergence constraint. Furthermore, this work manifests that the accuracy problem is associated with the normal velocity jumps of the flux interface, while shock instability is related to the transverse velocity jumps. Numerical test cases across a wide range of Mach numbers demonstrate the superior performance and potentiality of HLLC+ to simulate all Mach number flows.
中文翻译：
HLLC +：适用于全速流动的低马赫冲击稳定HLLC型黎曼求解器
SIAM科学计算杂志，第42卷，第4期，第B921-B950页，2020年1月。
Harten-Lax-van Leer（HLL）的近似Riemann求解器及其变体HLLC（带接触恢复的HLL）求解器被广泛用作有限体积Godunov型方法的通量函数，用于求解气体动力学Euler方程。但是，HLLC求解器面临两个重大困难：低速流动时的精度问题和高速流动时的冲击不稳定性。为了弥补这些缺点，开发了适用于所有速度的新型低马赫冲击稳定HLLC型方案HLLC +。首先介绍了抗耗散压力修正方法，以克服低马赫数限制下的精度问题。然后，确定剪切粘度并将其缩放到原始HLLC方案中，以克服冲击不稳定性。设计了一种新的无转换系数的基于压力的因数函数，以防止剪切粘度涂抹边界层。新的HLLC +方案不包含任何经验参数，并且易于实现。渐近分析和低马赫数测试案例显示了HLLC +在低马赫数限制下的出色表现：无全局截止问题，阻尼压力棋盘模式，预期压力和密度波动的缩放比例为{{Ma ^ 2} $}和满意程度散度约束。此外，这项工作表明，精度问题与通量界面的正常速度跳跃有关，而冲击不稳定性与横向速度跳跃有关。
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