一种改进的HLLEM格式及其激波稳定性分析

A modified HLLEM scheme and shock stability analysis

使用低耗散激波捕捉格式对高超声速流动问题进行数值模拟时经常会遭受不同形式的激波不稳定性.本文基于二维无黏可压缩Euler方程,对低耗散HLLEM格式进行激波稳定性分析.结果表明:激波面横向通量中切向速度的扰动增长诱发了格式的不稳定性.通过增加耗散来治愈HLLEM格式的激波不稳定性.为了避免引入过多的耗散进而影响剪切层的分辨率,定义激波探测函数和亚声速区探测函数,使得只有在计算激波层亚声速区的横向数值通量时才增加耗散,其余地方的数值通量依然采用低耗散的HLLEM格式来计算.稳定性分析和数值模拟的结果表明,改进的HLLEM格式不仅保留了原格式高分辨率的优点,还大大提高了格式的鲁棒性,在计算强激波问题时能够有效地抑制不稳定现象的发生.

Reliable numerical simulations for hypersonic flows require an accurate,robust and efficient numerical scheme.The low-dissipation shock-capturing methods often suffer various forms of shock wave instabilities when used to simulate hypersonic flow problems numerically.For the two-dimensional(2D)inviscid compressible Euler equations,the stability analysis of the low-dissipation HLLEM scheme is conducted.The odd and even perturbations are added to the initial state in the streamwise direction and the transverse direction respectively,and the evolution equations of perturbations are deduced to explore the mechanism of instability inherent in the HLLEM scheme.The results of stability analysis show that the perturbations of density and shear velocity in the flux transverse to the shock wave front are undamped.Due to the symmetry,the 2D Sedov blast wave problem is computed to prove the multidimensionality of the shock instability.In the one-dimensional case which is free from the instability,the undamped property of density perturbation is also existent but no shear velocity is found.The conclusion can be drawn as follows:the shock instability of HLLEM scheme is triggered by the perturbation growth of shear velocity in the flux transverse to the shock wave front.Based on the conclusion of stability analysis,the instability of HLLEM scheme is cured by adding the shear viscosity to the transverse flux.In order to avoid affecting the resolution of the shear layer due to the introduction of too high shear viscosity,two functions to detect the shock wave and the subsonic regimes are defined,so that the shear viscosity is only added to the transverse flux in the subsonic regime of the shock layer,while the rest of numerical fluxes are still computed by the original HLLEM scheme.The results of stability analysis and some challenging numerical test problems show that the modified HLLEM scheme not only retains the merits of the original HLLEM,such as,resolving contact discontinuity and shear wave accurately,but also has greatly improved its robustness,inhibiting the unstable phenomena from occurring effectively when computing the strong shock wave problems.