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

壁面流动的随机涡法和蒙特卡罗模拟

• 批准号: 2592790
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2592790
• 关键词: Random; vortex; method; Monte; Carlo

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.

湍流流动的动力学由著名的Navier-Stokes方程描述。这些方程的非线性结构解释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。湍流的特征在于速度的可感知的不规则变化，这引起了这种流动的数值模拟的计算困难。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找到有限差分方法中的解，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，这可能是有利的，因为蒙特-卡罗方案在处理多变量动态时更好。然而，要实现这种方法，不可压缩的Navier-Stokes方程的解需要显式表示的一些分布。由于随机涡旋方法，这是可能的。使用这种方法，我们可以用布朗粒子的分布集合来表示流体的速度，这些布朗粒子遵循相关的泰勒扩散。这样，就可以将原来的不可压Navier-Stokes方程等价地表示为求解McKean-Vlasov型随机微分方程的问题。因此，解决泰勒扩散的封闭问题，人们推导出泰勒扩散的速度场的积分表示，这些泰勒扩散很容易模拟，因为它们满足一些（普通）随机微分方程。在这种情况下，可以使用蒙特-卡罗方法来数值计算流动速度的积分表示。这种方法最近由Z.特别是对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，对于湍流的研究也很重要。实际上，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观察到，然而，对于壁边界区域，即使在二维情况下，也可在边界附近观察到湍流运动。该项目的目的是使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对具有边界的特定区域进行湍流模拟的计算实验。