Date of Graduation

5-2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics (PhD)

Degree Level

Graduate

Department

Mathematical Sciences

Advisor/Mentor

Kaman, Tulin

Committee Member

Bradshaw, Zachary

Second Committee Member

Walters, Keith

Third Committee Member

Liu, Chen

Keywords

Front-tracking method; ghost-fluid method; Richtmyer-Meshkov Instability; weighted essentially non-oscillatory schemes

Abstract

Turbulent mixing due to hydrodynamic instabilities occurs in a broad spectrum of engineering, astrophysical and geophysical applications. Theory, experiment, and numerical simulation help us to understand the dynamics of interface instabilities between two fluids. This thesis presents an increasingly accurate and robust front tracking method for the numerical simulations of shock-induced turbulent mixing known as Richtmyer-Meshkov Instability (RMI). Front tracking is an adaptive computational method, where the interface instability is explicitly represented as lower dimensional manifolds moving through a rectangular grid. All the cell-center states (density, velocity and pressure) are updated using higher order weighted essentially non-oscillatory (WENO) scheme. Performance of fifth- and ninth-order WENO schemes, with and without monotonicity-preserving bounds, as well as the WENOZ method, for the one-dimensional Sod shock tube, shock-entropy wave interaction problem and a scalar advection test problem are presented. Then the Richtmyer--Meshkov instability (RMI) between air and SF6 simulations are performed in order to show the improvements achieved using the new method. The fifth- and ninth-order WENO schemes with and without monotonicity preserving bounds are explored in the numerical solution of the shock-driven interface problem.

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