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

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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.