Development of Versatile Mixed Finite Element Methods for the Non-Isothermal Incompressible and Compressible Navier-Stokes Equations
Open Access
Author:
Miller, Edward
Graduate Program:
Mechanical Engineering
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
December 12, 2023
Committee Members:
Robert Kunz, Professor in Charge/Director of Graduate Studies Robert Kunz, Major Field Member James Coder, Outside Unit & Field Member Stephen Lynch, Major Field Member Xiang Yang, Major Field Member David Williams, Chair & Dissertation Advisor
Keywords:
Finite Element Methods Computational Fluid Dynamics Compressible Flow Incompressible Flow
Abstract:
The ever increasing demand for accurate numerical methods has led to the development of more and more sophisticated methods for simulating fluid flow. These methods are often designed to handle a specific flow regime or be valid under specific circumstances. What is needed in the field is a method that is accurate and robust over a wide range of conditions. Here, we propose a finite element method designed to work over a broad range of flow regimes and remain consistent and accurate in each regime. This is accomplished utilizing a mixed finite element method whose properties are rigorously analyzed to demonstrate the method’s effectiveness at handling these different flow regimes. We first use standard mathematical techniques to prove that the method is stable and obtains optimal error estimates for the non-isothermal incompressible Navier-Stokes equations. We then demonstrate on a series of test cases that the method accurately captures the physics of the non-isothermal incompressible Navier-Stokes equations. Next, we extend our method to the compressible Navier-Stokes equations where again the order of accuracy is demonstrated, this time using a series of numerical experiments. Finally, we present a series of compressible flow test cases to prove that the method can capture the physics of this regime
