Large Eddy Simulations in Magnetohydrodynamics Flows

磁流体动力学描述了在磁场存在的情况下导电流体流动的行为。导电流体的应用领域包括天体物理、地球物理、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、舰船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，星系磁场影响星际云形成恒星。这些应用需要比目前存在的更好的建模和仿真能力。没有明确的尺度分离的问题，如湍流，仍然处于多尺度建模和模拟的前沿。当流体导电时，流体的湍流运动伴随着磁涨落。对于磁流体力学(MHD)湍流，数值模拟发挥了比流体力学湍流更大的作用，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)太复杂，无法与理论结果进行比较。本研究项目将为这些重要问题开发改进的计算方法。本研究项目研究数学严谨和计算高效的方法来分析受MHD模型约束的正反问题。这包括计算算法的数值分析，使用Elsasser变量的隐式显式时间推进格式，通过时间过滤器进行的后处理，空间线性和非线性过滤器，频谱过滤，针对MHD湍流的空间过滤器的开发，最优控制，以及参数估计。该项目的另一个目标是研究几种用于模拟湍流、粘性、不可压缩、导电流动中的大涡的模型的数学性质，以及允许通过时间分裂进行长期模拟的新的数值模型。该项目对培养本科生和研究生在磁流体力学、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.