Mathematical Analysis of Fluid Flow Dynamics Using Navier-Stokes Equations and Finite Element Method

Abstract

Navier-Stokes Equations (NSEs) and how they have been applied to fluid dynamics will be discussed, along with numerical computations for solving mathematical models based on the equations, as analytical solutions are difficult to obtain for most complex problems. For discretizing problems, the Finite Element Method (FEM) is described for governing equations, selecting shape functions, and deriving the weak form, while assembling a global system and defining boundary conditions. A Newton-Raphson Method (NRM) is applied to solve steady flow problems in 2-D channels by iteratively solving nonlinear systems generated by discretized NSEs. In addition, obstructions such as baffles and blocks affect turbulent flow properties and pressure distribution in a channel. By studying the flow patterns, turbulence intensity, and pressure drop, we aim to find optimal arrangements that maximize mixing with minimal pressure loss, while permitting control over the flow. In addition to the methodologies used in the four case studies, these studies also examine important open questions regarding fluid dynamics. It is possible to compare laminar flow within a flat plate with turbulent flow through a channel with blocks to gain an understanding of how different characteristics of the overlying flow and domain geometry interact. In order to obtain true results, the fluid characteristics, geometry, as well as the boundary conditions (BCs) must be understood. In this paper, the FEM and NRM are combined to solve for various flow regimes. The simulation and computation of fluid flow commonly occurring in engineering applications can be optimized based on the results of this study.