2D lid diven cavity final report.pdf

1 MAE 561 Computational Fluid Dynamics Final Project Simulation of Lid Driven Cavity Problem using Incompressible Navier-Strokes Equation AKSHAY BATRA 1205089388

2 TABLE OF CONTENTS 1. Abstract ………………………………………………………………………………………………… .... 3 2. Acknowledgement ………………………………………………………………………………………...4 3. Introduction ……………………………………………………………………………………………… .5 4. Problem Statement ……………………………………………………………………………………… .5 5. Governing Equations ……………………………………………….. ........................................................ 5 5.1. Stream Function…………………………………………………………………………………… ...5 5.2 Vorticity ………………………………………………….…….. ........................................................ 6 5.2.1 Boundary conditions for the vorticity ……………………………………………………… 6 5.3 Stream function equation ………………………………………. ......................................................... 7 5.4 Successive Over relaxation………………………………………………………………………… 7 5.5 Pressure Calculation…………………………………………… ......................................................... 7 6. Development of the coding algorithm…………………………………………………………………… .8 7. Results…………………………………………………………………………………………………… .9 7.1. Numerical s imulation results ………………………………… .......................................................... 9 7.2. Ansys Fluen t results………………………………………………………………………………… 18 8. Conclusions………………… ... ………………………………………………………………………… .20 9. References ………… ... ………………………………………………………………………………… .. 21 10. Appendix (matlab code) ………………………………………… ............................................................. 22

3 1. ABSTRACT Develop an understanding of the steps involved in solving the Navier-Stokes equations using a numerical method . Write a simple code to solve the “driven cavity” problem using the Incompressible - Navier- Stokes equations in Vorticity form. This project requires that the Vorticity streamline function, u and v velocity profiles, pressure contours for the lid driven rectangular cavity for Reynolds number 100 and 1000. The lid driven cavity is a classical problem and closely resembles actual engineering problems that exist in research and industry areas. The vorticity equation will be solved utilizing a forward time central space (FTCS) explicit method. The streamline equation is solved using the successive over relaxation method. The obtained results follows and are illustrated in the report.

4 2. ACKNOWLEDGEMENTS I would take this opportunity to thank Dr. H.P Huang, my instructor for this course, Computational Fluid Dynamics (MAE 561). He has been a great help for me during this course. Without his support I wouldn’t have been able to achieve what I have achieved in this course. He has been very instrumental in my understanding of numerical methods for computational fluid dynamics. Also I would like to thank Donley Hurd for his constant technical support during this course.

5 3. INTRODUCTION In previous homework assignments an analysis of a how to solve partial differential equations (PDEs) using point Gauss-Seidel (PGS) iterative method and using forward time center space (FTCS) explicit method has been explored. In this project an analysis will be conducted that will utilize these two methods in one problem. But successive over relaxation (SOR) method would be used as the iteration method. This project will consider a rectangular cavity with a moving top wall. This moving wall will slowly cause the fluid to move within the cavity. It is the final steady state solution that this project seeks to acquire (Re 100 and 1000). Finally the similar problem is computed in ANSYS FLUENT, commercial fluid simulation software and results are compared. 4. PROBLEM STATEMENT The upper plate of a rectangular cavity shown in Figure 1 moves to the rights with a velocity of u o . The rectangular cavity has dimensions of L by L . Use the FTCS explicit scheme and the SOR formation to solve for the vorticity and the stream function equations, respectfully. The cavity flow problem is to be solved for the vorticity, streamline, pressure contours and u-v profiles for Re=100 and Re 1000. Later, a case has to be solved where the rectangular cavity as dimensions 2L and L to obtain same contours. 5. GOVERNING EQUATIONS 5.1 Stream Function The derivation for the FTCS starts with the vorticity equation seen in Equation 1. It is important to notice how similar this equation is to the 2-D Navier-Stokes momentum equation. (1) Equation 1 then has a forward difference Taylor Series expansion for first derivatives applied to the first term, a central difference Taylor Series expansion for first derivatives applied to the second term and third term, and Figure 1: (Hoffmann) Figure P8-1 page 357

6 central difference Taylor Series for second derivative applied to the fourth and fifth term. The result is shown in Equation 2. (2) Also if, in equation 1, ? = 𝜕? 𝜕? , ? = − 𝜕? 𝜕? (3) Then, the values of u and v are substituted in equation 1 to obtain equation 3, 𝜕? 𝜕? = - 𝜕? 𝜕? 𝜕? 𝜕? + 𝜕? 𝜕? 𝜕? 𝜕? + 1 𝑅? { 𝜕 𝜕? ( 𝜕? 𝜕? ) + 𝜕 𝜕? ( 𝜕? 𝜕? ) } (4) For this problem we are considering, dx=dy=ds, thus the final discretized equation becomes, ? i,j,n+1 = ? i,j,n - dt[( ? i,j+1,n - ? i,j-1,n ) ( ? i+1,j,n - ? i-1,j,n )] 4ds*ds - dt[( ? i+1,j,n - ? i-1,j,n ) ( ? i,j+1,n - ? i,j-1,n )] 4ds*ds + dt[ ( ? i+1,j,n - ? i-1,j,n - ? i,j+1,n - ? i,j-1,n - 4 ? i,j,n )] (5) Re* ds*ds Where dt is the time step and ds is the space step. 5.2 Vorticity 5.2.1 Boundary conditions for the Vorticity The boundary conditions for the vorticity stream line approach is quite complicated. The boundary conditions were formulated using the lecture notes (scan set 20) and equations 8-111 to 8-117 in the book. For our problem the boundary conditions are:- At the bottom wall ( j =1): ? wall = { ? i,1 – ? i,2 } 2 ??^2 + U wall 2 ?? (6) At the top wall ( j =ny): ? wall = { ? i,ny – ? i,ny-1 } 2 ??^2 - U wall 2 ?? (7) At the left wall (i=1) ? wall = { ? 1,j – ? 2,j } 2 ??^2 + U wall 2 ?? (8)

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