A Robust and Low-Dissipation Flux Splitting Scheme
-
摘要: 随着计算流体力学的快速发展,设计精确、高效并且健壮的数值格式变得尤为重要.通过对3种流行的通量分裂方法(AUSM、Zha-Bilgen和Toro-Vázquez)的对流通量和压力通量进行特征分析,构造了一种简单、低耗散并且健壮的通量分裂格式(命名为R-ZB格式).采用Zha-Bilgen分裂方法将Euler方程的通量分裂成对流通量和压力通量,其中对流通量采用迎风方法来计算,压力通量采用低耗散的HLL格式来计算,从而克服了原始的HLL格式不能精确分辨接触间断的缺点.数值实验表明,该文给出的R-ZB格式不仅保留了原始Zha-Bilgen格式简单高效、能够精确分辨接触间断等优点,而且具有更好的健壮性,在计算二维问题时不会出现数值激波不稳定现象.Abstract: With the rapid development of computational fluid dynamics, it is particularly important to design accurate, efficient and robust numerical schemes. Through the characteristics analyses of 3 popular flux splitting methods (AUSM, Zha-Bilgen and Toro-Vázquez), a simple, low-dissipation and robust flux splitting scheme (named as R-ZB) was constructed. The flux of Euler equations was split into a convection flux and a pressure flux with the Zha-Bilgen splitting procedure. The convection flux was computed with a simple upwinding scheme, and the pressure flux was evaluated with a low-dissipation HLL scheme to overcome the flaw of failing to capture contact discontinuities. Numerical experiments show that, the proposed R-ZB scheme not only retains the merits of the original Zha-Bilgen scheme, such as simpleness, efficiency and capturing contact discontinuities accurately, etc., but also has better robustness, which eliminates the numerical shock instabilities in the calculation of 2D problems.
-
Key words:
- Euler equations /
- Zha-Bilgen splitting /
- HLL /
- R-ZB /
- low dissipation /
- numerical shock instability
-
[1] LIOU M S. Open issues in numerical fluxes: proposed resolutions[C]// 〖STBX〗20th AIAA Computational Fluid Dynamics Conference, Fluid Dynamics and Co-located Conferences,2011. DOI: 10.2514/6.2011-3055. [2] QU F, YAN C, YU J, et al. A new flux splitting scheme for the Euler equations[J]. Computers & Fluids,2014,102: 203-214. [3] ROE P L. Approximate Riemann solvers, parameter vectors, and difference schemes[J]. Journal of Computational Physics,1981,43(2): 357-372. [4] HARTEN A, LAX P D, VAN LEER B. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws[J]. SIAM Review,1983,25(1): 35-61. [5] TORO E F, SPRUCE M, SPEARES W. Restoration of the contact surface in the HLL-Riemann solver[J]. Shock Waves,1994,4(1): 25-34. [6] TORO E F. Riemann Solvers and Numerical Methods for Fluid Dynamics [M]. 3rd ed. Berlin: Springer, 1999. [7] KIM S D, LEE B J, LEE H J, et al. Robust HLLC Riemann solver with weighted average flux scheme for strong shock[J]. Journal of Computational Physics,2009,228(20): 7634-7642. [8] STEGER J L, WARMING R F. Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods[J]. Journal of Computational Physics,1981,40(2): 263-293. [9] VAN LEER B. Flux vector splitting for the Euler equations[C]// 〖STBX〗8th International Conference on Numerical Methods in Fluid Dynamics . Berlin, Heidelberg: Springer-Verlag, 1982: 507-512. [10] ANDERSON W K, THOMAS J L, VAN LEER B. Comparison of finite volume flux vector splittings for the Euler equations[J]. AIAA Journal,1986,24(9): 1453-1460. [11] TORO E F, VZQUEZ-CENDN M E. Flux splitting schemes for the Euler equations[J]. Computers & Fluids,2012,70: 1-12. [12] LIOU M S, STEFFEN JR C J. A new flux splitting scheme[J]. Journal of Computational Physics,1993,107(1): 23-39. [13] LIOU M S. A sequel to AUSM: AUSM+[J]. Journal of Computational Physics,1996,129(2): 364-382. [14] LIOU M S. Recent progress and applications of AUSM+[C]//16th International Conference on Numerical Methods in Fluid Dynamics . Berlin, Germany, 1998. [15] LIOU M S. A sequel to AUSM, part II: AUSM+-up for all speeds[J]. Journal of Computational Physics,2006,214(1): 137-170. [16] ZHA G C, BILGEN E. Numerical solution of Euler equations by using a new flux vector splitting scheme[J]. International Journal for Numerical Methods in Fluids,1993,17(2): 115-144. [17] ZHA G C, SHEN Y, WANG B. An improved low diffusion E-CUSP upwind scheme[J]. Computers & Fluids,2011,48(1): 214-220. [18] KAPEN P T, TCHUEN G. An extension of the TV-HLL scheme for multi-dimensional compressible flows[J]. International Journal of Computational Fluid Dynamics,2015,29(3/5): 303-312. [19] TORO E F, CASTRO C E, LEE B J. A novel numerical flux for the 3D Euler equations with general equation of state[J]. Journal of Computational Physics,2015,303: 80-94. [20] XIE W, LI H, TIAN Z, et al. A low diffusion flux splitting method for inviscid compressible flows[J]. Computers & Fluids,2015,112: 83-93. [21] SHIMA E, KITAMURA K. Parameter-free simple low-dissipation AUSM-family scheme for all speeds[J]. AIAA Journal,2011,49(8): 1693-1709. [22] MANDAL J C, PANWAR V. Robust HLL-type Riemann solver capable of resolving contact discontinuity[J]. Computers & Fluids,2012,63: 148-164. [23] QUIRK J J. A contribution to the great Riemann solver debate[J]. International Journal for Numerical Methods in Fluids,1994,18(6): 555-574. [24] HU L J, YUAN L. A robust hybrid HLLC-FORCE scheme for curing numerical shock instability[J]. Applied Mechanics and Materials,2014,577: 749-753. [25] WU H, SHEN L J, SHEN Z J. A hybrid numerical method to cure numerical shock instability[J]. Communications in Computational Physics,2010,8: 1264-1271. [26] REN Y X. A robust shock-capturing scheme based on rotated Riemann solvers[J]. Computers & Fluids,2003,32(10): 1379-1403. [27] 胡立军, 袁礼. 一种治愈强激波数值不稳定性的混合方法[J]. 应用数学和力学, 2015,36(5): 482-493.(HU Lijun, YUAN Li. Analysis of numerical shock instability and a hybrid curing method[J]. Applied Mathematics and Mechanics,2015,36(5): 482-493.(in Chinese)) [28] SUN M, TAKAYAMA K. An artificially upstream flux vector splitting scheme for the Euler equations[J]. Journal of Computational Physics,2003,189(1): 305-329. [29] GOTTLIEB S. On high order strong stability preserving Runge-Kutta and multi step time discretizations[J]. Journal of Scientific Computing,2005,25(1/2): 105-128. [30] LI B, YUAN L. Convergence issues in using high-resolution schemes and lower-upper symmetric Gauss-Seidel method for steady shock-induced combustion problems[J]. International Journal for Numerical Methods in Fluid,2013,71(11): 1422-1437. [31] JIANG G S, SHU W C. Efficient implementation of weighted ENO schemes[J]. Journal of Computational Physics,1996,126(1): 202-228. [32] WOODWARD P, COLELLA P. The numerical simulation of two-dimensional fluid flow with strong shocks[J]. Journal of Computational Physics,1984,54(1): 115-173. [33] KITAMURA K, ROE P L, ISMAIL F. Evaluation of Euler fluxes for hypersonic flow computations[J]. AIAA Journal,2009,47(1): 44-53. [34] LAX P D, LIU X D. Solution of two-dimensional Riemann problems of gas dynamics by positive schemes[J]. SIAM Journal on Scientific Computing,1998,19(2): 319-340.
点击查看大图
计量
- 文章访问数: 883
- HTML全文浏览量: 132
- PDF下载量: 486
- 被引次数: 0