Introduction3

Development of numerical schemes5

Partial Differential Equations5

Initial and Boundary condition5

Modelling Approaches6

Numerical Methods6

Explicit method8

Implicit method8

Numerical Coding10

Explicit method10

Final code11

Implicit Method15

Final Code16

Numerical results18

Analysis of the Numerical results23

Conclusion24

References25

Introduction

Over the years the importance of fluid dynamics has grown exponentially. It represents the theoretical and physical aspects of the fluid in motion, as it flows naturally or when effected by a force. This application can be applied to liquids and gases providing a deeper understanding of pure sciences such as atmospheric, geophysics and oceanography. However the main application of this study is in industries such as turbo machinery, aerospace, civil engineering, automotive, water industry and more as these are realising the importance of this field. Over the last century there were two different approaches used in this field. The theoretical-analytical approach requires the uses of partial differential equations which consist of continuity, Euler and Navier-Stokes equations. These helped to study and understand the behaviour of the fluid as it flows and predict and changes which may occur. However there are some issues which occur in mathematical analysis, calculating the data can be extremely difficult complicated therefore assumptions are made in order to simplify these equations. Due to this the results produced are not completely accurate and contain errors. On the other hand experimental approaches were used to physically model the flow in a test or a lab to understand the behaviour. It can show different aspects of the flow during separation or the transition of laminar to turbulent, but as the data is only qualitative it doesn’t provide any information on velocity or pressure of the fluid (J. M. McDonough, 2009).

(Anderson. Jr, 1995)

As the technology advanced it allowed the theoretical and experimental approach to be combined together though the use of computer simulation techniques, this is known as computational fluid dynamics. This uses mathematical equations, numerical methods and software to produce quantitative and qualitative data for the flow. The advantages of CFD compared to the previous approach are shown in the table blow (Kuzmin, D ).

Experiment| CFD|

* Calculations made for a signal quantity * Model size limited to test or laboratory size * Range of problems are limited * Expensive to carry out experiments * Slow and extremely time consuming | * For all the quantities at the same time * Can represent the actual flow area * Can be applied to any problem and conditions * Relatively cheap to model flows * Fast and can complete many iterations |

The report shows a detailed study of the different methods used in CFD, the process or creating a program and analysis of the results gained. This will help to gain deeper understanding of how fluids work and how CFD can be applied to different scenarios to predict the flow outcome. Aim

The purpose of this report is to create a model of a couette flow and analyse the results. This consists of a vicious fluid between two parallel plates. As the upper plate moves and the lower plate remain stationary and the fluid in the middle flows due to the shear stress created by the upper plate. This creates a linear velocity across the flow, which can be calculated by using the Navier-Stokes equation and Finite Difference Method (FDM). Objectives

* A detail research on physical and theoretical aspects of the CFD * To develop a numerical schemes

* Numerical coding using a programming language such as MatLab * Graphic output of the numerical results

* Analysis of numerical results

Development of numerical schemes

Partial Differential Equations

Fluid dynamics is bases on...