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

本项目的目标是研究在外加磁场作用下液态金属中二维(2D)和三维(3D)湍流之间的转换机制。虽然这个问题的某些方面已经被研究过了，但一些基本的问题，比如知道转变是否有滞后，找出用相关的无量纲数表示的明确的转变标准，或者理解墙的影响，直到今天仍然没有答案。这些应该是我们想要解决的问题，使用原始的数值模拟和低维模型。这些模型将利用这种湍流呈现出大的各向异性结构的趋势来建立。我们将分两个主要步骤进行：首先，我们研究3D周期盒更简单情况下的跃迁。其次，我们解决了与外加磁场垂直的两壁之间的湍流问题，在空间的其他方向上有周期性的边界条件。两种情况下得到的结果之间的差异将指出墙所起的作用，并给出周期性边界条件与墙边界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.