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

本计画的目的是研究外加磁场下液态金属中二维与三维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，如知道过渡是否有滞后，找到用相关的无量纲数表示的明确的过渡准则，或理解墙壁的影响，至今仍未得到解答。这些都是我们想要解决的问题，使用原始的数值模拟和低维模型。模型将利用这种湍流呈现大的各向异性结构的趋势来建立。我们将进行两个主要步骤：首先，我们研究的过渡在一个简单的情况下的三维周期盒。其次，我们解决的问题，湍流之间的两个墙壁正交施加的磁场，在空间的其他方向的周期性边界条件。在这两种情况下得到的结果之间的差异将指出的作用所发挥的墙壁，并给出了一个衡量的相关性的周期性边界条件的墙壁约束问题。

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.