# Computational Fluid Dynamics: An Introduction pdf lecture notes and ebook

Neeraj Yadav | 12-Jun-2016 |

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INDEX

1 Introduction:................................................................................................................ 2
2 Mathematical Formulation..........................................................................................3
2.1 Governing equations.............................................................................................. 3
2.2 Boundary Conditions............................................................................................. 5
2.2.1 Example .......................................................................................................7
3 Techniques for Numerical Discretization ................................................................... 9
3.1 The Finite Difference Method ............................................................................... 9
3.2 The Finite Element Method................................................................................. 11
3.3 The Finite Volume Method ................................................................................. 14
3.4 Spectral Methods ................................................................................................. 15
3.5 Comparison of the Discretization Techniques..................................................... 16
4 Solving The Fluid Dynamic Equations..................................................................... 17
4.1 Transient-Diffusive Terms .................................................................................. 17
4.1.1 Finite Difference Approach ....................................................................... 17
4.1.2 Finite Element Approach........................................................................... 21
4.2 Transient-Convective Terms ............................................................................... 24
4.3 Shock Capturing Methods ................................................................................... 26
4.4 Convective-Diffusive Terms ............................................................................... 27
4.5 Incompressible Navier-Stokes Equations............................................................ 30
4.5.1 Pressure-Based Methods............................................................................30
5 Basic Solution Techniques........................................................................................ 34
5.1 Direct Method...................................................................................................... 34
5.2 Iterative Methods................................................................................................. 34
5.2.1 Jacobi and Gauss-Seidel methods.............................................................. 35
5.2.2 Relaxation methods. ..................................................................................37
5.2.3 ADI Method: .............................................................................................38
5.3 Convergence and Stability................................................................................... 39
5.4 Von Neuman Stability Analysis .......................................................................... 39
5.5 Convergence of Jacobi and Gauss-Seidel Methods (iterative methods): ............ 41
6 Building a Mesh........................................................................................................ 44
6.1 Element Form ...................................................................................................... 44
6.2 Structured Grid .................................................................................................... 46
6.2.1 Conformal mapping method......................................................................47
6.2.2 Algebraic method ......................................................................................47
6.2.3 Differential equation method.....................................................................47
6.2.4 Block-structured method ...........................................................................47
6.3 Unstructured grid................................................................................................. 47
7 References................................................................................................................. 49

This File is intended as an introductory guide for Computational Fluid Dynamics
CFD. Due to its introductory nature, only the basic principals of CFD are introduced
here. For more detailed description, readers are referred to other textbooks, which are
devoted to this topic.1,2,3,4,5 CFD provides numerical approximation to the equations that
govern fluid motion. Application of the CFD to analyze a fluid problem requires the
following steps. First, the mathematical equations describing the fluid flow are written.
These are usually a set of partial differential equations. These equations are then
discretized to produce a numerical analogue of the equations. The domain is then divided
into small grids or elements. Finally, the initial conditions and the boundary conditions of
the specific problem are used to solve these equations. The solution method can be direct
or iterative. In addition, certain control parameters are used to control the convergence,
stability, and accuracy of the method.