Award Date

5-1-2019

Degree Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematical Sciences

First Committee Member

Monika Neda

Second Committee Member

Zhonghai Ding

Third Committee Member

Xin Li

Fourth Committee Member

Dong-Chan Lee

Number of Pages

60

Abstract

The Navier-Stokes equations (NSE) are an essential set of partial differential equations for governing the motion of fluids. In this paper, we will study the NSE for an incompressible flow, one which density ρ = ρ0 is constant.

First, we will present the derivation of the NSE and discuss solutions and boundary conditions for the equations. We will then discuss the Reynolds number, a dimensionless number that is important in the observations of fluid flow patterns. We will study the NSE at various Reynolds numbers, and use the Reynolds number to write the NSE in a nondimensional form.

We will then derive energy and enstrophy balances for the NSE. At high Reynolds numbers, a fluid’s velocity u has many small spatial scales, which become difficult to account for, especially in three-dimensional flow. We discuss the time relaxation model (TRM), which aims to truncate these small scales while allowing the large scales to be accurately resolved, [25]. We will derive the energy and enstrophy balances for the TRM and show that the energy and enstrophy are the same as the NSE, but with enhanced dissipation terms.

Finally, we will derive a continuous finite element variational formulation for the TRM. Using FreeFEM++, we will run numerical results for the TRM for a specific benchmark problem.

Keywords

Finite element method; Navier-Stokes equations; Numerical analysis; Reynolds number; Scientific computing; Taylor-Green vortex

Disciplines

Aerodynamics and Fluid Mechanics | Applied Mathematics | Mathematics

Language

English


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