Comparison of the generalized Riemann solver and the gas-kinetic scheme for inviscid compressible flow simulations
Comparison of the generalized Riemann solver and the gas-kinetic scheme for inviscid compressible flow simulations
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DOI：
10.1016/j.jcp.2011.03.028
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发表时间：
2011-06
期刊：
影响因子：
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通讯作者：
Jiequan Li;Qibing Li;K. Xu
中科院分区：
文献类型：
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作者：
Jiequan Li;Qibing Li;K. Xu
欧拉方程的广义黎曼问题 (GRP) 方案和玻尔兹曼方程的气体动力学方案 (GKS) 是两种用于流体模拟的高分辨率冲击捕获方案。不同之处在于，一种是基于无粘性欧拉方程及其波相互作用的特性，另一种是基于粒子输运和碰撞。它们之间的相似之处在于，两种方法都可以在单元界面周围使用相同的MUSCL型初始重建，并且单元界面两侧的空间斜率参与气体演化过程和依赖于时间的通量函数的构建。尽管这两种方法都已成功应用于无粘可压缩流计算，但它们的性能从未进行过比较。由于两种方法都使用相同的初始重建，因此任何差异仅来自其通量评估中不同的基础机制。因此，这样的比较对于帮助我们理解物理建模和数值性能之间的对应关系很重要。由于GRP如此忠实地求解无粘性欧拉方程，因此该比较也可以用来显示求解欧拉方程本身的有效性。数值比较表明，对于一维情况下的欧拉解，GRP 表现出稍好的计算效率，并且具有与 GKS 相当的精度，但 GKS 比 GRP 更鲁棒。对于二维高马赫数流动模拟，GKS 不存在激波不稳定性，并且比 GRP 更快地收敛到稳态解。 GRP 具有红宝石现象，就像悬在精确黎曼求解器上方的云。 GRP 和 GKS 使用不同的物理过程来描述从不连续性开始的流动运动。一种是基于无限次粒子碰撞的平衡状态假设，另一种是从非平衡自由输运过程开始，通过粒子碰撞演变成平衡状态。通量评估中的不同机制导致其数值表现有所偏差。通过这项研究，我们可以科学地得出结论，使用欧拉方程作为控制方程在单元分辨率有限的离散空间中构造数值通量可能是无效的。改编纳维-斯托克斯 (NS) 方程也是无效的，因为 NS 方程描述了流体动力学尺度上的流动行为，并且没有任何从不连续性开始的相应物理场。这一事实暗示了欧拉和纳维-斯托克斯方程与连续介质假设的一致性，以及在模拟极高马赫数流时在构建数值格式时直接建模离散空间中的物理过程的必要性。数值算法的发展与推导控制方程的建模过程类似，但这里的控制量不能缩小到零。
The generalized Riemann problem (GRP) scheme for the Euler equations and gas-kinetic scheme (GKS) for the Boltzmann equation are two high resolution shock capturing schemes for fluid simulations. The difference is that one is based on the characteristics of the inviscid Euler equations and their wave interactions, and the other is based on the particle transport and collisions. The similarity between them is that both methods can use identical MUSCL-type initial reconstructions around a cell interface, and the spatial slopes on both sides of a cell interface involve in the gas evolution process and the construction of a time-dependent flux function. Although both methods have been applied successfully to the inviscid compressible flow computations, their performances have never been compared. Since both methods use the same initial reconstruction, any difference is solely coming from different underlying mechanism in their flux evaluation. Therefore, such a comparison is important to help us to understand the correspondence between physical modeling and numerical performances. Since GRP is so faithfully solving the inviscid Euler equations, the comparison can be also used to show the validity of solving the Euler equations itself. The numerical comparison shows that the GRP exhibits a slightly better computational efficiency, and has comparable accuracy with GKS for the Euler solutions in 1D case, but the GKS is more robust than GRP. For the 2D high Mach number flow simulations, the GKS is absent from the shock instability and converges to the steady state solutions faster than the GRP. The GRP has carbuncle phenomena, likes a cloud hanging over exact Riemann solvers. The GRP and GKS use different physical processes to describe the flow motion starting from a discontinuity. One is based on the assumption of equilibrium state with infinite number of particle collisions, and the other starts from the non-equilibrium free transport process to evolve into an equilibrium one through particle collisions. The different mechanism in the flux evaluation deviates their numerical performance. Through this study, we may conclude scientifically that it may NOT be valid to use the Euler equations as governing equations to construct numerical fluxes in a discretized space with limited cell resolution. To adapt the Navier–Stokes (NS) equations is NOT valid either because the NS equations describe the flow behavior on the hydrodynamic scale and have no any corresponding physics starting from a discontinuity. This fact alludes to the consistency of the Euler and Navier–Stokes equations with the continuum assumption and the necessity of a direct modeling of the physical process in the discretized space in the construction of numerical scheme when modeling very high Mach number flows. The development of numerical algorithm is similar to the modeling process in deriving the governing equations, but the control volume here cannot be shrunk to zero.