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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.
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Keywords:
- hypersonic flow /
- low dissipation scheme /
- tangential dissipation /
- shock instability
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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图 1 超声速情形下扰动量的发展曲线 (a)初始扰动量和初值(I); (b)初始扰动量和初值(II)
Figure 1. Evolutionary curves of perturbation under supersonic condition: (a) Initial perturbation values and initial conditions (I); (b) initial perturbation values and initial conditions (II).
图 2 亚声速情形下扰动量的发展曲线 (a)初始扰动量和初值(III); (b)初始扰动量和初值(IV)
Figure 2. Evolutionary curves of perturbation under subsonic condition: (a) Initial perturbation values and initial conditions (III); (b) initial perturbation values and initial conditions (IV).
图 3 横向扰动下扰动量的发展曲线 (a)初始扰动量(V); (b)初始扰动量(VI); (c)初始扰动量(VII); (d)初始扰动量(VIII)
Figure 3. Evolutionary curves of perturbation in transverse direction: (a) Initial perturbation values (V); (b) initial perturbation values (VI); (c) initial perturbation values (VII); (d) initial perturbation values (VIII).
图 4 二维Sedov爆轰波问题的密度等值线 (a) HLLEM; (b) x-HLLE+y-HLLEM (c) x-HLLEM+y-HLLE
Figure 4. Density contours for 2D Sedov blast wave problem: (a) HLLEM; (b) x-HLLE+y-HLLEM (c) x-HLLEM+y-HLLE.
图 5 三种激波探测函数的函数曲线
Figure 5. Curves for three different shock-detecting functions.
图 6 随机数值噪声问题的密度等值线 (a) HLLEM; (b) M-HLLEM-g; (c) M-HLLEM-g1; (d) M-HLLEM-g2
Figure 6. Density contours of random numerical noise problem: (a) HLLEM; (b) M-HLLEM-g; (c) M-HLLEM-g1; (d) M-HLLEM-g2.
图 7 随机数值噪声问题
$y$ 方向速度的最大量值Figure 7. Maximum magnitude of velocity in y-direction of random numerical noise problem.
图 8 高超声速绕柱流问题的密度等值线 (a) HLLEM; (b) M-HLLEM
Figure 8. Density contours of hypersonic flow over a cylinder: (a) HLLEM; (b) M-HLLEM.
图 9 双马赫反射问题的密度等值线 (a) HLLEM; (b) M-HLLEM
Figure 9. Density contours of double Mach reflection problem: (a) HLLEM; (b) M-HLLEM.
图 10 二维Sedov爆轰波问题的压力等值线 (a) HLLEM; (b) M-HLLEM
Figure 10. Pressure contours of 2D Sedov blast wave problem: (a) HLLEM; (b) M-HLLEM.
图 11 二维无黏接触间断问题的密度分布
Figure 11. Density distribution of 2D inviscid contact discontinuity problem.
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[1] Tchuen G, Fogang F, Burtschell Y, Woafo P 2014 Comput. Phys. Commun. 185 479
Google Scholar
[2] 全鹏程, 易仕和, 武宇, 朱杨柱, 陈植 2014 63 084703
Google Scholar
Quan P C, Yi S H, Wu Y, Zhu Y Z, Chen Z 2014 Acta Phys. Sin. 63 084703
Google Scholar
[3] Quirk J J 1994 Int. J. Numer. Methods Fluids 18 555
Google Scholar
[4] Gressier J, Moschetta J M 2000 Int. J. Numer. Methods Fluids 33 313
Google Scholar
[5] Dumbser M, Moschetta J M, Gressier J 2004 J. Comput. Phys. 197 647
Google Scholar
[6] 谢文佳, 李桦, 潘沙, 田正雨 2015 64 024702
Google Scholar
Xie W J, Li H, Pan S, Tian Z Y 2015 Acta Phys. Sin. 64 024702
Google Scholar
[7] Shen Z J, Yan W, Yuan G W 2016 J. Comput. Phys. 309 185
Google Scholar
[8] Kim S D, Lee B J, Lee H J, Jeung I S 2009 J. Comput. Phys. 228 7634
Google Scholar
[9] Wu H, Shen L J, Shen Z J 2010 Commun. Comput. Phys. 8 1264
Google Scholar
[10] Shen Z J, Yan W, Yuan G W 2014 Commun. Comput. Phys. 15 1320
Google Scholar
[11] Ren Y X 2003 Comput. Fluids 32 1379
Google Scholar
[12] Rodionov A 2017 J. Comput. Phys. 345 308
Google Scholar
[13] Rodionov A 2019 Comput. Fluids 190 77
Google Scholar
[14] Rodionov A 2018 J. Comput. Phys. 361 50
Google Scholar
[15] Chen S S, Yan C, Lin B X, Liu L Y, Yu J 2018 J. Comput. Phys. 373 662
Google Scholar
[16] Simon S, Mandal J C 2018 Comput. Fluids 174 144
Google Scholar
[17] Simon S, Mandal J C 2019 J. Comput. Phys. 378 477
Google Scholar
[18] Xie W J, Li W, Li H, Tian Z Y, Pan S 2017 J. Comput. Phys. 350 607
Google Scholar
[19] Fleischmann N, Adami S, Hu X Y, Adams N 2020 J. Comput. Phys. 401 109004
Google Scholar
[20] Einfeldt B, Munz C D, Roe P L, Sjögreen B 1991 J. Comput. Phys. 92 273
Google Scholar
[21] Chauvat Y, Moschetta J M, Gressier J 2005 Int. J. Numer. Methods Fluids 47 903
Google Scholar
[22] Kitamura K, Roe P L, Ismail F 2009 AIAA J. 47 44
Google Scholar
[23] Xu K, Li Z W 2001 Int. J. Numer. Methods Fluids 371
[24] Huang K, Wu H, Yu H, Yan D 2011 Int. J. Numer. Methods Fluids 65 1026
Google Scholar
[25] Woodward P, Colella P 1984 J. Comput. Phys. 54 115
Google Scholar
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