Large Eddy Simulations in Magnetohydrodynamics Flows

磁流体力学描述了存在磁场时导电流体流动的行为。导电流体的应用包括天体物理学、地球物理学、等离子体约束、受控热核聚变、核反应堆的液态金属冷却、金属的电磁铸造、低轨道卫星的离子推进器、船舶和潜艇的磁流体动力驱动、微流体装置和分子生物学。地核的流体运动维持着地球磁场，太阳磁场产生太阳黑子和太阳耀斑，银河磁场影响着星际云中恒星的形成。这些应用程序需要比目前更好的建模和仿真能力。湍流等没有明确尺度分离的问题，仍处于多尺度建模与仿真的前沿。当流体具有导电性时，紊流运动伴随着磁波动。对于磁流体动力学（MHD）湍流，数值模拟比流体动力学湍流发挥更大的作用，因为实验室实验实际上是不可能的，而且天体物理系统（太阳风湍流，最重要的高雷诺数MHD系统，可在现场测量）太复杂，无法与理论结果相比较。这个研究项目将为这些重要问题开发改进的计算方法。本课题研究数学严谨、计算高效的方法来分析MHD模型约束下的正、逆问题。这包括计算算法的数值分析、使用Elsasser变量的隐式显式时间步进方案、通过时间滤波器的后处理、空间线性和非线性滤波器、光谱滤波、针对MHD湍流的空间滤波器的开发、最优控制和参数估计。该项目的另一个目标是研究几种模型的数学性质，用于模拟湍流粘性、不可压缩、导电流中的大涡流，以及允许通过时间分裂进行长时间模拟的新数值模型。该项目对培养本科生和研究生在磁流体动力学、湍流和反问题的分析和数值方面具有广泛的影响。

Magnetohydrodynamics describes the behavior of an electrically conducting fluid flow in the presence of magnetic fields. Electrically conducting fluids arise in applications including astrophysics, geophysics, plasma confinement, controlled thermonuclear fusion, liquid-metal cooling of nuclear reactors, electromagnetic casting of metals, ion thrusters for low orbiting satellites, magnetohydrodynamic drive for ships and submarines, microfluidic devices, and molecular biology. The fluid motion of the Earth's core maintains the terrestrial magnetic field, the solar magnetic field generates sunspots and solar flares, and the galactic magnetic field influences the formation of stars from interstellar clouds. These applications require substantially better modeling and simulation capabilities than presently exist. Problems without a clear scale separation, such as turbulence, are still at the frontier of multiscale modeling and simulation. When the fluid is electrically conducting, the turbulent fluid motions are accompanied by magnetic fluctuations. For Magnetohydrodynamic (MHD) turbulence, numerical simulations play a greater role than they play for hydrodynamic turbulence, since laboratory experiments are practically impossible and astrophysical systems (solar-wind turbulence, the most important system of high-Reynolds-number MHD accessible to in situ measurements) are too complex to be comparable with theoretical results. This research project will develop improved computational methods for these important problems.This research project studies mathematically rigorous and computationally efficient methods to analyze direct and inverse problems constrained by MHD models. This includes the numerical analysis of computational algorithms, implicit explicit time-stepping schemes using the Elsasser variables, post processing via time-filters, spatial linear and nonlinear filters, spectral filtering, development of spatial filters specific to MHD turbulence, optimal control, and parameter estimation. Another objective of this project is to investigate the mathematical properties of several models for the simulation of the large eddies in turbulent viscous, incompressible, electrically conducting flows and new numerical models that permit long-time simulations, by time-splitting. The project has a broad impact for training undergraduate and graduate students in analytical and numerical aspects of magnetohydrodynamics, turbulence, and inverse problems.