光滑粒子流体动力学中的混合问题
Resolving mixing in Smoothed Particle Hydrodynamics
摘要：
光滑粒子流体动力学（SPH）的标准公式无法解决流体边界处的混合。我们使用的广义SPH运动方程的误差和稳定性分析证明，这是由于两个不同的问题。第一个是动量方程中的领先阶误差。这应该随着相邻数的增加而减少，但不会因为数值不稳定性导致内核被不规则地采样。我们确定了两个重要的不稳定性：结块不稳定性和带状不稳定性，我们表明，这两个都是治愈一个合适的选择内核。第二个问题是局部混合不稳定性（LMI）。这发生在粒子试图在内核尺度上混合，但由于熵守恒而无法混合时。其结果是在边界处的压力不连续性，将不同熵的流体推开。我们使用加权密度估计，确保压力是单值的整个流动的LMI固化。这也为粒子提供了更好的体积估计，减少了连续性和动量方程中的误差。我们证明混合在我们新的优化光滑粒子流体动力学（OSPH）计划使用开尔文亥姆霍兹不稳定性（KHI）测试密度对比度1：2，和“斑点测试”-1：10密度比气体球在风洞中-发现OSPH和欧拉代码之间的良好协议。
英文摘要：
Standard formulations of smoothed particle hydrodynamics (SPH) are unable to resolve mixing at fluid boundaries. We use an error and stability analysis of the generalised SPH equations of motion to prove that this is due to two distinct problems. The first is a leading order error in the momentum equation. This should decrease with increasing neighbour number, but does not because numerical instabilities cause the kernel to be irregularly sampled. We identify two important instabilities: the clumping instability and the banding instability, and we show that both are cured by a suitable choice of kernel. The second problem is the local mixing instability (LMI). This occurs as particles attempt to mix on the kernel scale, but are unable to due to entropy conservation. The result is a pressure discontinuity at boundaries that pushes fluids of different entropy apart. We cure the LMI by using a weighted density estimate that ensures that pressures are single valued throughout the flow. This also gives a better volume estimate for the particles, reducing errors in the continuity and momentum equations. We demonstrate mixing in our new Optimised Smoothed Particle Hydrodynamics (OSPH) scheme using a Kelvin Helmholtz instability (KHI) test with density contrast 1:2, and the 'blob test' - a 1:10 density ratio gas sphere in a wind tunnel - finding excellent agreement between OSPH and Eulerian codes.
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