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

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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.