与显式LES一致的SPH

摘要：

介绍了光滑粒子流体动力学（SPH）的一致速度光滑方法。首先，局部平均的纳维-斯托克斯方程推导出一个严格的数学方法来证明“失踪”的湍流应力在标准SPH公式。然后，它表明，与适当的选择速度平滑，SPH和大涡模拟（LES）等效地解决了同一组过滤方程，需要关闭近似。SPH和LES之间的唯一区别，本文所证明的是控制方程的表示;前者的方程是积分微分形式，而后者是微分形式。SPH和LES之间的这种等价性的一个直接结果是，最初为LES开发的湍流建模技术可以很容易地被采用到这个版本的SPH中。我们的亚网格应力张量的积分形式的表示将提供洞察处理湍流建模的替代方法。虽然使用Smagorinsky模型是LES中湍流建模的常见做法，但对于SPH，最自然的选择是使用近似反卷积方法。我们选择速度平滑的另一个特别重要的后果是，由此产生的滤波方程在性质上是非保守的，并且不能容易地构造正确的拉格朗日量。

英文摘要：

The aim of this paper is to introduce a consistent velocity smoothing method for smoothed particle hydrodynamics (SPH). First the locally averaged Navier-Stokes equations are derived in a mathematically rigorous way to demonstrate the "missing" turbulent stress in standard SPH formulations. It is then shown that with a proper choice of velocity smoothing, SPH and large eddy simulation (LES) equivalently solve the same set of filtered equations that require closure approximations. The only difference between SPH and LES, as demonstrated in this paper is the representation of governing equations; for the former the equations are in integro-differential form whereas for the latter they are in differential form. One direct consequence of this equivalence between SPH and LES is that turbulence modeling techniques originally developed for LES can easily be adopted into this version of SPH. Our representation of the sub-grid stress tensor in integral form will provide insight into alternative approaches for dealing with turbulence modeling. Although the use of the Smagorinsky model is the common practice for turbulence modeling in LES, it will be shown that for SPH the most natural choice is to use approximate deconvolution methods. The other particularly fundamental consequence of our choice of velocity smoothing is that the resulting filtered equations are nonconservative in nature and a correct Lagrangian cannot be easily constructed.

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