Computational Fluid Dynamics: a Practical Approach
Table of Contents
Introduction 3 Development of numerical schemes 5 Partial Differential Equations 5 Initial and Boundary condition 5 Modelling Approaches 6 Numerical Methods 6 Explicit method 8 Implicit method 8 Numerical Coding 10 Explicit method 10 Final code 11 Implicit Method 15 Final Code 16 Numerical results 18 Analysis of the Numerical results 23 Conclusion 24 References 25
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).
Introduction 3 Development of numerical schemes 5 Partial Differential Equations 5 Initial and Boundary condition 5 Modelling Approaches 6 Numerical Methods 6 Explicit method 8 Implicit method 8 Numerical Coding 10 Explicit method 10 Final code 11 Implicit Method 15 Final Code 16 Numerical results 18 Analysis of the Numerical results 23 Conclusion 24 References 25
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).
References: * J. M. McDonough. (2009). LECTURES IN ELEMENTARY FLUID DYNAMICS. * Jhon D. Anderson, Jr. (1995). Computational Fluid Dynamics. The Basics with Application. University of Maryland. * Kuzmin, D. Introduction to Computational Fluid Dynamics. Available: http://www.mathematik.uni-dortmund.de/~kuzmin/cfdintro/lecture1.pdf. * Zuo, W. (2005). Introduction of Computational Fluid Dynamics. FAU Erlangen-Nürnberg * CHEM 520. Navier-Stokes Equations. Available: http://depts.washington.edu/chemcrs/bulkdisk/chem520A_aut05/notes_Week_05_Lecture_07.pdf. * André Bakker. (2002). Applied Computational Fluid Dynamics. Lecture 5 - Solution Methods * Gilberto E. Urroz. (2004). Convergence, Stability, and Consistency of Finite Difference Schemes in the Solution of Partial Differential Equations. * Anderson, j. Dick, E. Degrez, G. Grundmann, R. (2009). In: John F. Wendt Computational Fluid Dynamics: An Introduction. 3rd ed: Technology & Engineering * Randall J. LeVeque. (2005). Finite Difference Mehtods for Differential Equations.