The following steps are going to explain the mathematical approach behind a CFD simulation. For you to understand it more easily, they are categorized into 7 steps.
First step:
Problem Statement:
The first step of the simulation is to gather information about the simulation process in general.
 What is the most convenient way of solving this problem in an economic way:
 Cheap solution: No high computational costs
 Fast solution: Fast solution possible without giving up much information of the solution
 Uncomplicated solution: Simplify the problem as much as possible without restating a new problem
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Modelling:

Laminar or Turbulent  if turbulent \rightarrow +turbulence model + nearwall treatment

Combustion

Other Physical Models

Is the flow steady or unsteady?

Are there any problems about the flow simulation that others have dealt with in the past?

Will physical phenomena influence the simulation?

What is the goal of the CFD simulation?

Second step:
Mathematical Fundamental:
The Initial Boundary Value Problem consists of the Partial Differential Equation the Initial Conditions as well as the Boundary Conditions:

Choose flow model that fits your simulation:
 SpalartAllmaras
 kepsilon
 komega
 LVEL & yPlus
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Identify the forces which cause and influence the motion of the fluid.

Define the Computational Domain of the problem.

Formulate conservation laws for mass, momentum and energy.
\rightarrow Governing Equations
~ 
If possible, simplify the equations:
 Check for Symmetry
 Check for dominant flow directions (1D/2D).
 Terms that have no influence on the solution can be neglected.
 Incorporate knowledge that youâve had beforehand (CFD results, measurement data).
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Add constitutive relations:
 Shear Stress
 Viscosity
 Dynamic Viscosity
 Kinematic Viscosity
~

Add Boundary Conditions and Initial Conditions.
Third step:
Discretization:
The system of Partial Differential Equations is transformed into algebraic equations. The discretion process is divided into three parts.
1. Mesh generation  Nodes and Cells
 Structured Mesh / Unstructured Mesh / Hybrid Mesh.
 Mesh adaption in âcriticalâ regions and set size:
 rRefinement
 hRefinement
 pRefinement
2. Space discretization  Coupled Ordinary Differential Equation/ Differential algebraic equation systems
 FiniteDifferenceMethod / FiniteVolumeMethod / FiniteElementMethod.
 HighOrderApproximation / LowOrderApproximation.
3. Time discretization  Algebraic System (Ax = b).
 Explicit Schemes / Implicit Schemes
Fourth step:
Iterative solution of the algebraic equation:

Solving systems of linear equations:
 Direct Methods: Gaussian elimination, LU decomposition.
 Iterative Methods: Strongly Implicit Procedure (SIP) , Alternating Direction Implicit (ADI) , Tridiagonal Matrix Algorithm (TDMA), RungeKutta method, Multigrid method.
 Coupled systems of equations.
 Nonlinear Equations
 Methods for transient problems: Linear multistep method etc.
Convergence: Check if the iterations converge.
 Residuals (Decrease by three orders of magnitude indicate at least qualitative convergence).
 Mass, Momentum, Energy, and Scalar balances are achieved.
Fifth step:
Simulation Run:
Once the problem is well defined with the boundary conditions, and if necessary with initial conditions, the problem is solved with a software. OpenâFOAM is a popular option for a solver which is used by several companies that provide CFD software. SimScale is among them.
Sixth step:
PostProcessing:
Looking at the solutions from the the computed flow.
 PostProcessing of integral parameters (Drag, Lift etc.)
~  Visualization in different dimensions:
 1D: Straight lines
 2D: Contour plots, Streamlines
 3D: Isosurfaces, Isovolumes, Streamtracer
 Animation of the flow
~
 Statistical analysis
Seventh step:
According to AIAA (1998) & Oberkampf and Trucano (2002) the following terminology is widely used and accepted:
Verification (âAre we solving the equations right?â) :
\rightarrow Quantification of errors
 Compare results with analytical solutions if possible.
If we ignore the fact that there might be coding errors and user errors, we can examine the following:

Roundoff Error

Iterative Convergence Error

Discretization Error
Validation (âAre we solving the right equations?â) :
\rightarrow Quantification of input & physical model uncertainty

Input uncertainty

Physical uncertainty
General tips
Influencing parameters for computation times in CFD

Code used in order to solve the flow (\rightarrow MPI, Vectorization)

Hardware (CPU, RAM, etc.)

Mesh size / Mesh Quality

Algorithms

Solvers
Read more about CFD in our related article in the SimWiki.
Also see our âSimWiki for more about other interesting simulation related questions.
Literature References:
 Laurien & Oertel: Numerische StrĂ¶mungsmechanik  Grundgleichungen und Modelle  LĂ¶sungsmethoden  QualitĂ€t und Genauigkeit
 Versteeg & Malalasekera: An Introduction to Computational Fluid Dynamics  The Finite Volume Method  2nd Edition
 http://www.mathematik.unidortmund.de/~kuzmin/Transport.pdf
 http://www.mathematik.unidortmund.de/~kuzmin/cfdintro/lecture1.pdf