Numerical Analysis and Fluid Flow Modeling of Incompressible Navier-Stokes Equations

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

A Review of Time Relaxation Methods

The time relaxation model has proven to be effective in regularization of Navier–Stokes Equations. This article reviews several published works discussing the development and implementations of time relaxation and time relaxation models (TRMs), and how such techniques are used to improve the accuracy and stability of fluid flow problems with higher Reynolds numbers. Several analyses and computational settings of TRMs are surveyed, along with parameter sensitivity studies and hybrid implementations of time relaxation operators with different regularization techniques