This website is a platform on which visitors can discuss Computational Fluid Dynamics (CFD) and get some feedback from CFD experts or other visitors.

One major advantage of this blog is that it works in parallel with different courses taught in fluid mechanics and fundamental books in numerical methods.

The structure of the blog is based on different levels of CFD (elementary, basic, intermediate and advanced) which will be mentioned in the topic of each post. Many common problems of fluid mechanics are approached by proposing discussions about their main aspects and some codes for solving them.

All visitors are welcomed to participate in discussions and suggest solutions. Furthermore, the questions or problems, which are not fully solved during the discussion, will be answered frequently (e.g. every 10 days) by experts.

# CFD Education Center

## Tuesday, November 24, 2015

## Tuesday, January 20, 2015

### Computational Fluid Dynamics - Elementary

In this post, the following topics are presented:

(a) Linear Algebraic Systems

(b) 1-D discrete data analysis

(b-1) Lagrange interpolation

(b-2) Spline interpolation

(b-3) Least squares method

(c) Curvilinear Coordinates

(d) Higher Order Ordinary Differential Equations (ODEs)

(d-1) Turning a higher order ODE to a system of 1st-order ODEs

(d-2) Finding solutions for homogeneous equations

(e) Partial Differential Equations - Classification

(e-1) Hyperbolic PDEs

(e-2) Parabolic PDEs

(e-3) Elliptic PDEs

(f) Finite Difference Approximations

(g) Finite Differences - Higher Order Approximations

(h) Stability of the Finite Difference Schemes

(i) Integration: quadrature formulae

(j) Runge-Kutta Method

(k) Newton's Method (1-D, 2-D, and 3-D)

(a) Linear Algebraic Systems

(b) 1-D discrete data analysis

(b-1) Lagrange interpolation

(b-2) Spline interpolation

(b-3) Least squares method

(c) Curvilinear Coordinates

(d) Higher Order Ordinary Differential Equations (ODEs)

(d-1) Turning a higher order ODE to a system of 1st-order ODEs

(d-2) Finding solutions for homogeneous equations

(e) Partial Differential Equations - Classification

(e-1) Hyperbolic PDEs

(e-2) Parabolic PDEs

(e-3) Elliptic PDEs

(f) Finite Difference Approximations

(g) Finite Differences - Higher Order Approximations

(h) Stability of the Finite Difference Schemes

(i) Integration: quadrature formulae

(j) Runge-Kutta Method

(k) Newton's Method (1-D, 2-D, and 3-D)

## Monday, January 19, 2015

### Computational Fluid Dynamics - Basic

The following topics are going to be discussed in this post:

(a) 1-D Linear Ordinary Differential Equations of 1st and 2nd order

(a-1) First order derivatives discretization

(a-2) Second order derivatives discretization considering different boundary conditions

(b) 1-D time dependent Parabolic differential equations

(b-1) Diffusion equation: finite difference discretization

(b-2) Diffusion equation: application of different boundary conditions

(b-3) Example: transient heat conduction

(c) 1-D time dependent Hyperbolic differential equations

(d) 1-D non-linear partial differential equations (Newton's and explicit methods)

(d-1) Blasius boundary layer

(d-2) Steady Burgers' equation

(d-3) Unsteady, viscous Burgers' equation

(d-4) Unsteady, inviscid Burgers' equation

(d-5) 2-D, steady, Kovasznay flow$ ^{New}$

(e) 1-D flow model in nozzles

(e-1) Incompressible fluid through a converging nozzle

(e-2) Compressible fluid through a converging nozzle

(e-3) Multiphase flow through a converging nozzle

(e-4) Normal shock waves

(e-5) Compressible fluid through a converging-diverging (de Laval) nozzle

Each topic would be thoroughly discussed in a time period of approximately one to three weeks.

(a) 1-D Linear Ordinary Differential Equations of 1st and 2nd order

(a-1) First order derivatives discretization

(a-2) Second order derivatives discretization considering different boundary conditions

(b) 1-D time dependent Parabolic differential equations

(b-1) Diffusion equation: finite difference discretization

(b-2) Diffusion equation: application of different boundary conditions

(b-3) Example: transient heat conduction

(c) 1-D time dependent Hyperbolic differential equations

(d) 1-D non-linear partial differential equations (Newton's and explicit methods)

(d-1) Blasius boundary layer

(d-2) Steady Burgers' equation

(d-3) Unsteady, viscous Burgers' equation

(d-4) Unsteady, inviscid Burgers' equation

(d-5) 2-D, steady, Kovasznay flow$ ^{New}$

(e) 1-D flow model in nozzles

(e-1) Incompressible fluid through a converging nozzle

(e-2) Compressible fluid through a converging nozzle

(e-3) Multiphase flow through a converging nozzle

(e-4) Normal shock waves

(e-5) Compressible fluid through a converging-diverging (de Laval) nozzle

Each topic would be thoroughly discussed in a time period of approximately one to three weeks.

## Sunday, January 18, 2015

### Computational Fluid Dynamics - Intermediate

The following topics are going to be discussed in this post:

(a) 2-D viscous, steady Burgers' equation

(a-1) Exact solution

(a-2) Numerical solution (Newton's method) - Dirichlet boundary conditions

(a-3) Numerical solution - Dirichlet+Neumann boundary conditions$ ^{New}$

(b) 2-D Poisson's equation

(c) 2-D Helmholz equation

(d) Lid-driven cavity flow

(d-1) Lid-driven cavity unsteady solution - stream function-vorticity formulation

(d-2) Lid-driven cavity solution - staggered grid

(a) 2-D viscous, steady Burgers' equation

(a-1) Exact solution

(a-2) Numerical solution (Newton's method) - Dirichlet boundary conditions

(a-3) Numerical solution - Dirichlet+Neumann boundary conditions$ ^{New}$

(b) 2-D Poisson's equation

(c) 2-D Helmholz equation

(d) Lid-driven cavity flow

(d-1) Lid-driven cavity unsteady solution - stream function-vorticity formulation

(d-2) Lid-driven cavity solution - staggered grid

## Saturday, January 17, 2015

### Computational Fluid Dynamics - Advanced

More advanced topics will be presented in this post:

(a) de Laval Nozzle; 1-D model, Fluent and OpenFOAM simulation

(a-1) Introduction to Laval Nozzle (Converging-Diverging Nozzle)

(a-2) Subsonic laminar/turbulent flow in Laval Nozzle ($\texttt{rhoSimpleFoam}$)

(a-3) Supersonic laminar/turbulent flow in Laval Nozzle

(b) Natural Convection in a Rectangular Channel using OpenFOAM

(a) de Laval Nozzle; 1-D model, Fluent and OpenFOAM simulation

(a-1) Introduction to Laval Nozzle (Converging-Diverging Nozzle)

(a-2) Subsonic laminar/turbulent flow in Laval Nozzle ($\texttt{rhoSimpleFoam}$)

(a-3) Supersonic laminar/turbulent flow in Laval Nozzle

(b) Natural Convection in a Rectangular Channel using OpenFOAM

## Friday, January 16, 2015

### Fluid Dynamics - Learning Materials Directory

In this post, different materials related to fluid dynamics will be published.

Experts are invited to contribute in publishing learning materials in this section agreeing the terms and conditions of the "CFD Education Center". Contact us using the comments tool below or via social networks for more information.

The sub-sections are ordered by author's name:

**Bigarella, E. D. V.**

(a) Introduction to Computational Fluid Dynamics

**Chilvers, J.**

(a) CFD and Turbulence: Method in the Madness

**Demirel, E.**

(a) CFD Applications in Civil Engineering

**Romano', F.**

(a) Numerical Methods in Fluid Dynamics

## Thursday, January 15, 2015

### MATLAB Codes Bank

Many topics of this blog have a complementary Matlab code which helps the reader to understand the concepts better. In this post, quick access to all Matlab codes which are presented in this blog is possible via the following links:

ID | Topic | Code Link |
---|---|---|

Elem. a | Linear Algebraic Systems | LinearAlgebraicSystems.m |

Elem. b | 1D Discrete Data analysis | 1D_Discrete_Data.tar.gz |

Elem. i | Integration: quadrature formulae | 1D_Quadrature_Formulae.tar.gz |

Elem. j | Runge-Kutta method | 2ndto4thOrd_Runge-Kutta_Method.tar.gz |

Elem. k | Newton's method (1-D, 2-D, and 3-D) | 1to3D_Newton-Raphson_Method.tar.gz |

Basic. a | 1-D Linear ODEs of 1st order | 1st_Order_Eq.tar.gz |

Basic. a | 1-D Linear ODEs of 2nd order | 2nd_Order_Eq.tar.gz |

Basic. b | 1-D time dependent Parabolic differential equations | 1D_Diffusion_Equation.tar.gz |

Basic. b-3 | 1-Dimensional, transient heat conduction | FTCS.m |

Basic. d-1 | Blasius boundary layer | BlasiusBoundaryLayer.m |

Basic. d-2 | 1-Dimensional, steady Burgers' equation | Burgers1D_SteadyViscous.m |

Basic. d-3 | 1-D, unsteady, viscous Burgers' equation | 1D_Burgers_Unsteady_Viscous.tar.gz |

Basic. d-4 | 1-D, unsteady, inviscid Burgers' equation | 1D_Burgers_Unsteady_Inviscid.tar.gz |

Basic. d-5 | 2-D, steady, Kovasznay flow | Kovasznay_Cartesian.m |

Basic. e-1 | 1-Dimensional, incompressible fluid through a nozzle | 1D_Nozzle_Incomp.rar |

Basic. e-2 | 1-Dimensional, compressible fluid through a converging nozzle | 1D_Nozzle_ConvCompr.tar.gz |

Basic. e-3 | Multiphase flow through a converging nozzle | 1D_Nozzle_ConvMultiphase.tar.gz |

Basic. e-4 | Normal shock waves | 1D_NormalShockWaves.tar.gz |

Interm. a-1 | Steady Burgers' equation exact solution, 2-Dimensional | Cartesian_2D_BURGER_Exact.m |

Interm. a-2 | Burgers' equation: numerical solution - Dirichlet boundary conditions | Cartesian_2D_BURGER_Exact_Numeric.m |

Interm. a-3 | Burgers' equation: Neumann + Dirichlet boundary conditions | Cartesian_BURGER_Neumann_right.m |

Interm. b | 2-D Poisson's equation | Poisson_2D.m |

Interm. c | 2-D Helmholtz equation | 2D_Helmholtz_Equation.tar.gz |

Interm. d-1 | Lid-driven cavity unsteady solution - stream function-vorticity formulation | Unsteady_2DLidDrivenCavity.m |

Adv. a | Laval Nozzle, Matlab codes and OpenFOAM setup | Appendix_B.pdf |

## Wednesday, January 14, 2015

### Introduction to MATLAB

What is MATLAB?

MATLAB (

**mat**rix

**lab**oratory) is a multi-paradigm numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages, including C, C++, Java, and Fortran.

Although MATLAB is intended primarily for numerical computing, an optional toolbox uses the MuPAD symbolic engine, allowing access to symbolic computing capabilities. An additional package, Simulink, adds graphical multi-domain simulation and Model-Based Design for dynamic and embedded systems.

In 2004, MATLAB had around one million users across industry and academia. MATLAB users come from various backgrounds of engineering, science, and economics. MATLAB is widely used in academic and research institutions as well as industrial enterprises.

Since MATLAB is a useful tool for calculation of the fluid dynamics problems, it is recommended to be familiar with it. Through the following link, many useful resources could be found about MATLAB for beginners and also professionals. Moreover, many tutorials can be downloaded online for free.

Useful MATLAB tutorials

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