Analysis of Numerical Methods for Incompressible Multiphase Flows with Thermal Convection

This dissertation is dedicated to the development and analysis of numerical methods to approximate the solutions of incompressible multiphase flows with thermal convection whose dynamics is governed by the Navier-Stokes equations and the temperature equation under the Boussinesq approximation. Unlike traditional schemes that rely on using the temperature and velocity as primary variables, our schemes use the momentum and internal energy. In addition, our schemes involve a semi-implicit projection method, which effectively decouples momentum and pressure. This choice of primary variables and decoupling leads to time independent stiffness matrices which enhance the computational performance of our numerical methods. We establish the stability and convergence properties of the proposed time marching algorithms. These analyses are based on assumptions regarding the gradient of the density approximation sequence, which we assume is either bounded by a constant or controlled by the time step. Under the Courant–Friedrichs–Lewy condition, these assumptions lead to an order of convergence of one or one-half, respectively. To validate our conclusions and assess the performance of our hydrodynamic and thermal solvers, we conduct a series of tests in both two-dimensional and three-dimensional domains. These tests are designed to challenge the schemes under various conditions, including setups with sharp density gradients and discontinuous density functions. Through these tests, we aim to demonstrate the effectiveness and versatility of our numerical schemes in accurately simulating multiphase incompressible flows. Our numerical illustrations are performed using the FreeFEM++ and SFEMaNS codes. Additionally, we contribute to the advancement of the SFEMaNS code by integrating our thermal solver and use this solver to study of turbulent thermal convection.

Name thesis:degree_name

Doctor of Philosophy

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Houston

Year dc:date.issued

2024

Author dc:creator

• Vu, An Thuy

Advisor dc:contributor.advisor

• Cappanera, Loic

Committee members dc:contributor.committeemember

• Guermond, Jean-Luc
• Timofeyev, Ilya
• Olshanskii, Maxim

• Mathematics

Language dc:language.iso

en

Handle dc:identifier.uri

https://hdl.handle.net/10657/19865

OAI identifier oai:identifier

oai:uh-ir.tdl.org:10657/19865