# Solution of 2-D Incompressible Navier Stokes Equations with Artificial Compressibility Method Using Ftcs Scheme

**Topics:**Fluid dynamics, Computational fluid dynamics, Navier–Stokes equations

**Pages:**20 (4419 words)

**Published:**October 23, 2011

IMRAN AZIZ

Department of Mechanical Engineering College of EME

National University of Science and Technology

Islamabad, Pakistan

Imran_9697@hotmail.com

Abstract— The paper deals with the 2-D lid-driven cavity flow governed by the non dimensional incompressible Navier-Stokes theorem in the rectangular domain. Specific boundary conditions for this case study have been defined and the flow characteristics pertaining to the scenario have been coded in MATLAB using artificial compressibility method and FTCS scheme. The results are compared successfully with an authentic research paper by Ghia, Ghia & Shin.

Keywords: Navier stokes equations, Artificial Compressibility, FTCS scheme. Introduction

The Navier-Stokes equations describe the motion of fluid substances and are used to solve wide range of problems in Fluid Dynamics. These equations include conservations of mass, momentum and energy. In this documentation we present the solution of 2-d navier stokes equations in a flow driven lid cavity used as a model for subsonic bombers weapons bay. The weapons bay is a compartment on fighter aircrafts to carry bombs [1].It is a highly critical area for any bomber design in the aeronautical industry. The design optimization of weapon bay involves various considerations namely weapon load capacity, extent of vibration and severe aerodynamic drag. In order to approximate the problem, a MATLAB code is developed to solve two dimensional incompressible navier stokes equations. Incompressible flows are those in which density variation is not linked to the pressure. The mass conservation is a constraint on velocity field. The continuity equation can be combined with momentum equation to derive the equation for pressure or it can be solved independently by applying artificial compressibility. We have used the later approach by employing FTCS scheme. The results have been validated on a 81 × 81 uniform grid and at Re = 1000 by comparing the u-velocity distribution along vertical centerline and v-velocity distribution along hoizontal centerline with the results obtained in various numerical studies (Ghia, Ghia & Shin in particular). A tolerance criteria of 1e-6 is used in this regard. The results are then verified using Grid independent studies for the above mentioned parameters. After the validation and the consequent verification of the data obtained through the coded program, the horizontal and the vertical velocity contours span wise velocity contours and the velocity vector plots have been created to graphically ascertain the results. In the ensuing passage, the behavior of convergence with time and the analysis of the convergence on various grid sizes have been performed. The analysis has been complemented with a CPU time vs. grid sizes plot. The tolerance criteria for convergence have then further been expedited and by using a semilog plot, the CPU time vs. tolerance graph is shown. Finally, the variation in the qualitative change in the flow structure; especially on the location of primary and secondary vortices, has been determined by giving a Re variation from 100-1000.

Numerical mehodology

The vector q = (u (x, y), v(x, y)) and pressure p(x,y) are used to describe completely the two dimensional flow field of an incompressible fluid. These functions are the solutions of the following mass and momentum conservation equations [2]. * Mass conservation

∇.q=0

Or it can be written in explicit form as

∂u∂x+∂v∂y=0

In order to solve the system, an artificial compressibility is introduced in the above continuity equation as follows:

∂p∂τ+1β2∂uj∂xj=0

Where τ=1/β2 is taken as the “artificial compressibility” of the fluid. The compressibility is a pseudo term and can be related to the pseudo-speed of sound and to an artificial density by using the following relations [3]...

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