Random vortex method and Monte-Carlo simulations for wall-bounded flows

紊流的动力学是由著名的Navier-Stokes方程描述的。这些方程的非线性结构说明了它们的数学复杂性，并产生了这些在自然界中可观察到的湍流解。紊流的特点是速度的变化具有明显的不规则性，这给紊流数值模拟带来了计算困难。例如，这意味着在直接数值模拟的情况下，需要使用小尺寸的网格来寻找有限差分方法的解，从而导致计算成本高。为了克服上述困难，我们开发了基于蒙特卡罗模拟的数值方法。事实上，它可能是有利的，因为蒙特卡罗方案在处理多变量动态时更好。然而，为了实现这种方法，不可压缩的Navier-Stokes方程的解需要用一些分布来显式表示。由于随机涡旋方法，这是可能的。使用这种方法，人们可以根据布朗粒子在相关的泰勒扩散下的一系列分布来写出流速。因此，可以将原始不可压缩的Navier-Stokes方程表述为求解一些McKean-Vlasov型随机微分方程的等价问题。因此，在求解泰勒扩散的闭包问题时，可以推导出速度场的泰勒扩散的积分表示，这种扩散很容易模拟，因为它满足一些（普通）随机微分方程。在这种情况下，可以使用蒙特卡罗方法来计算流速度的数值积分表示。这种方法是由Z. Qian最近开发的，特别适用于占据壁面区域的流动。这种情况在流体动力学中特别有趣，对湍流的研究也很重要。的确，对于占据整个无边界空间的自由流体流动，湍流现象仅在三维情况下可见，而对于有壁面的区域，即使在二维情况下也可以看到靠近边界的湍流运动。该项目的目的是利用我们上面概述的方法开发数值方案，并在二维和三维情况下进行具有边界的特定区域湍流模拟的计算实验。

The dynamics of turbulent fluid flows is described by the prominent Navier-Stokes equations. The non-linear structure of these equations accounts for their mathematical complexity and gives rise to these turbulent solutions observable in nature. Turbulent flows are characterised by perceivably irregular changes in velocity which induces computational difficulty of numerical simulations of such flows. For instance, this implies that in the case of direct numerical simulations, one is required to use a mesh of small size to find the solution in finite difference methods leading to high cost in computations. To overcome the aforementioned difficulty, we develop numerical methods based on Monte-Carlo simulations. Indeed, it might be advantageous as Monte-Carlo schemes are better when dealing with multivariate dynamics. However, to implement this approach, the solution to the incompressible Navier-Stokes equations is required to be explicitly represented in terms of some distributions. This turns out to be possible due to the random vortex method. Using this method, one writes the velocity of the flow in terms of a collection of distributions of Brownian particles following the associated Taylor's diffusion. Thus, one is able to formulate the original incompressible Navier-Stokes equations as an equivalent problem of solving some McKean-Vlasov type stochastic differential equations. Therefore, solving the closure problem for Taylor's diffusion, one derives an integral representation for the velocity field in terms of Taylor's diffusions which are easily simulated as they satisfy some (ordinary) stochastic differential equations. In this case, one can use Monte-Carlo methods to compute the integral representations for the velocity of the flow numerically. This approach has been recently developed by Z. Qian particularly for flows occupying wall-bounded regions. This case is especially interesting in fluid dynamics and additionally important for the study of turbulence. Indeed, for free fluid flows occupying the whole space without boundary, the phenomenon of turbulence is observable only in the three-dimensional case, however, for wall-bounded regions turbulent motion is seen close to the boundary even in the two-dimensional case. The aim of the project is to develop numerical schemes using the approach we outlined above and conduct computational experiments for simulation of turbulent flows for particular regions with boundary in the two- and three-dimensional cases.