- THESIS - MAGNETOHYDRODYNAMICS
Codes will be uploaded once the Journal of Scientific Computing approves the manuscript!
- Motivation -
Magnetohydrodynamics (MHD) explores the dynamics of electrically conductive fluids under the influence of magnetic fields. A quintessential example is our own sun, which operates as an MHD system. Enhancing our comprehension of such systems necessitates the continual refinement of algorithms. Traditionally, improving a code involves a trade-off between accuracy and computational speed. One approach prioritizes refining the model for heightened accuracy, albeit at the expense of longer simulation times. Alternatively, adjusting the model to expedite flow simulation sacrifices accuracy. However, employing a stabilized penalty projection finite element method enables the preservation of competitive accuracy levels while significantly reducing simulation time. Under the guidance of Dr. Muhammad Mohebujjaman, I began research in employing this method.
With the improvement of this algorithm comes numerous applications. One of the major benefits of utilizing MHD flow is that it requires no moving parts to produce a flow in the electrically conducting fluid. Such precision is necessary for many applications such as weather forecasting, liquid metal cooling of nuclear reactors such as the Tokamak, process metallurgy, artificial heart, and MHD propulsion. One of the many applications of MHD can be used to simulate plasma propulsion. Electromagnetic plasma propulsion systems offer high exhaust velocities and produce higher thrust densities than space-charge limited electric propulsion systems. These systems use magnetic fields to accelerate and expel plasma, enabling efficient propulsion within a vacuum.
- Application -
We propose, analyze, and test two efficient and accurate, fully-discrete algorithms for computing the solution of stochastic magnetohydrodynamic (MHD) flow problems having random noises within the initial conditions, boundary conditions, forcing functions, and viscosity parameters. The stable decoupled algorithms use the Elsässer variable formulations and a decoupling of the ensemble MHD system into two Oseen-type sub-problems for each realization. For each of the sub-problems, the first algorithm uses a grad-div stabilization and ensemble eddy-viscosity terms in conjunction with penalty-projection based splitting method and provides unconditional stability, and first order accuracy in time. The second algorithm is second order accurate in time in practice, and optimally accurate in space. Both algorithms are designed in a way that at each time step the finite element assembly of the system provides the same coefficient matrix for each of the J realizations but with different right-hand-side vectors. The stability and convergence theorems of both algorithms are ascertained rigorously. The numerical tests are given to support the predicted convergence rates using some manufactured analytical solutions. Finally, we test the penalty-projection scheme on benchmark channel flow over a step, and a regularized lid-driven problems, and found it works well.
We demonstrate the flow by considering a 2D benchmark regularized lid-driven cavity problem. No slip boundary conditions are applied to all sides excluding the lid of the cavity located at the top of the simulation. The simulation to the left, exhibiting the white ripples and streamlines over speed contours displays the velocity strength of the simulation. The simulation to the right with streamlines over speed contours displays the magnetic field of the simulation.