Magnetohydrodynamic Turbulence: numerical simulation and study of the transition between two-dimensional and three-dimensional turbulence, with and without walls

该项目的目标是研究在外部施加的磁场下液态金属中二维（2d）和三维（3d）湍流过渡的机制。虽然这一问题的某些方面已经得到了研究，但一些基本问题，如知道过渡是否有滞后，找到用相关的非量纲数字表示的明确过渡标准，或理解壁的影响，至今仍未得到解答。这些将是我们想要解决的问题，使用原始的数值模拟和低维模型。利用这种湍流表现出大的各向异性结构的趋势来建立模型。我们将分两个主要步骤进行：首先，我们研究三维周期盒的简单情况下的跃迁。其次，我们解决了与施加磁场正交的两个壁之间的湍流问题，并在空间的其他方向上具有周期性边界条件。两种情况下得到的结果之间的差异将指出壁面所起的作用，并给出周期边界条件与壁面约束问题的相关性的度量。

The goal of this project is to study the mechanisms which govern the transition between twodimensional (2d) and three-dimensional (3d) turbulence in liquid metals under an externally imposed magnetic field. Although some aspects of this problem have already been studied, basic questions such as knowing whether the transition has an hysteresis, finding clear transition criteria expressed in terms of the relevant non-dimensional numbers, or understanding the influence of walls remain to this day unanswered. Those shall be the questions we want to address, using original numerical simulations and low-dimensional models. The models will be built by taking advantage of the tendency of this kind of turbulence to exhibit big anisotropic structures. We shall proceed in two main steps: firstly, we study the transition in the simpler case of a 3d periodic box. Secondly we tackle the problem of turbulence confined between two walls orthogonal to the imposed magnetic field, with periodic boundary conditions in the other directions of space. The difference between the results obtained in both case will point out the role played by the walls, and give a measure of the relevance of periodic boundary conditions to walls bound problems.