Collaborative Research: Effective Numerical Schemes for Fundamental Problems Related to Incompressible Fluids

该项目旨在开发一种新的计算方法来研究由浮力驱动的流体和由电导引起的湍流的稳定性，即磁流体动力学（MHD）湍流。通过精确模拟这些现象，这项研究将为改进极端天气事件的建模和预测提供见解，如龙卷风、天文现象、北极光和太阳耀斑等现象，以及等离子体和液态金属的导电流体。这种新的计算方法将成为科学计算界的一个有价值的工具。研究生，包括那些来自代表性不足群体的研究生，将在理论和计算机领域接受培训。这项研究还将吸引本科生和K-12学生，使当地学校和社区受益。该项目旨在研究求解不可压缩Navier-Stokes方程的数值方法，提供发散自由速度，可证明的稳定性，高雷诺数下的鲁棒性，以及长时间计算的高效率。本研究将着重改进经典投影法的不足，目标如下：首先，该方法将达到无发散的条件。这一特性对于浮力驱动流体的精确模拟至关重要。这些流体中的温度将被速度场传输、重新排列和分层，并且执行无散度条件将对演化和最终状态提供更准确的说明。其次，该算法将能够以高精度处理任意大的雷诺数。高雷诺数的三维Navier-Stokes流的模拟可以得到最优收敛，避免非物理振荡。第三，将建立该方案的稳定性和误差估计，其中结果将独立于雷诺数。研究人员将应用该数值方法模拟流体静力平衡附近的浮力驱动流体和背景磁场附近的导电流体。在第一个问题中，将求解一个控制流体静力平衡附近扰动的Boussinesq-Navier-Stokes系统。Boussinesq系统将受浮力影响的Navier-Stokes方程与温度输运方程耦合在一起。数值方法也将扩展到模拟垂直黏度远小于水平黏度的各向异性流动。这个问题出现在模拟大气和海洋中的埃克曼层湍流时。在第二个问题中，将计算和分析导电流体在引导磁场下的稳定性和长期行为。当与严格的分析相辅相成时，这些模拟将有助于加速解决关于Boussinesq和MHD方程的几个开放稳定性问题。该项目由计算数学项目和促进竞争研究的既定项目（EPSCoR）共同资助。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project aims to develop a novel computational method to investigate the stability of buoyancy-driven fluids and turbulent flows due to electrical conduction, known as magnetohydrodynamic (MHD) turbulence. By accurately simulating these phenomena, the research will provide insights into improving modeling and prediction of extreme weather events such as tornados, astronomical occurrences, phenomena like Northern lights and solar flares, and electrically conducting fluid of plasma and liquid metals. The new computational method will be a valuable tool for the scientific computing community. Graduate students, including those from underrepresented groups, will be trained in both theoretical and computer fields. The research will also engage undergraduates and K-12 students, benefiting local schools and communities.The project aims to investigate numerical methods to solve the incompressible Navier-Stokes equations, delivering the divergence free velocity, provable stability, robustness in high Reynolds numbers, and high efficiency in long-time computations. The research will focus on improving the shortcomings of the classical projection method with the following goals. First, the method will achieve the divergence-free condition. This feature is critical in the accurate simulation of buoyancy-driven fluids. The temperature in these fluids will be transported, rearranged, and stratified by the velocity field, and the enforcement of the divergence-free condition will give a more accurate account of the evolution and the eventual states. Second, this algorithm will be able to handle arbitrarily large Reynolds numbers with high accuracy. The simulations of the 3D Navier-Stokes flows with a high Reynolds number will result in optimal convergence and avoid nonphysical oscillations. Third, stability and error estimates will be established for this scheme, where the results will be independent of the Reynolds numbers. The investigators will apply this numerical method to simulate the buoyancy-driven fluids near the hydrostatic equilibrium and the electrically conducting fluids near a background magnetic field. In the first problem, a Boussinesq-Navier-Stokes system governing the perturbations near the hydrostatic equilibrium will be solved. The Boussinesq system couples the Navier-Stokes equations forced by buoyancy with the temperature transport equation. The numerical method will also be extended to simulate anisotropic flows for which the vertical viscosity is much smaller than the horizontal one. This problem arises in modeling turbulent flows in Ekman layers occurring in the atmosphere and the ocean. In the second problem, the stability and long-time behavior of electrically conducting fluids under a guiding magnetic field will be computed and analyzed. When complemented with rigorous analysis, these simulations will help accelerate the resolution of several open stability problems on the Boussinesq and the MHD equations.This project is jointly funded by the Computational Mathematics program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.