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:
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Laminar or Turbulent - if turbulent \rightarrow +turbulence model + near-wall treatment
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Combustion
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Other Physical Models
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Is the flow steady or unsteady?
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Are there any problems about the flow simulation that others have dealt with in the past?
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Will physical phenomena influence the simulation?
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What is the goal of the CFD simulation?
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Second step:
Mathematical Fundamental:
The Initial Boundary Value Problem consists of the Partial Differential Equation the Initial Conditions as well as the Boundary Conditions:
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Choose flow model that fits your simulation:
- Spalart-Allmaras
- k-epsilon
- k-omega
- L-VEL & yPlus
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Identify the forces which cause and influence the motion of the fluid.
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Define the Computational Domain of the problem.
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Formulate conservation laws for mass, momentum and energy.
\rightarrow Governing Equations
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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
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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:
- r-Refinement
- h-Refinement
- p-Refinement
2. Space discretization - Coupled Ordinary Differential Equation/ Differential algebraic equation systems
- Finite-Difference-Method / Finite-Volume-Method / Finite-Element-Method.
- High-Order-Approximation / Low-Order-Approximation.
3. Time discretization - Algebraic System (Ax = b).
- Explicit Schemes / Implicit Schemes
Fourth step:
Iterative solution of the algebraic equation:
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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), Runge-Kutta 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:
Post-Processing:
Looking at the solutions from the the computed flow.
- Post-Processing of integral parameters (Drag, Lift etc.)
~ - Visualization in different dimensions:
- 1-D: Straight lines
- 2-D: Contour plots, Streamlines
- 3-D: Isosurfaces, Isovolumes, Streamtracer
- Animation of the flow
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- 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:
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Roundoff Error
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Iterative Convergence Error
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Discretization Error
Validation (âAre we solving the right equations?â) :
\rightarrow Quantification of input & physical model uncertainty
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Input uncertainty
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Physical uncertainty
General tips
Influencing parameters for computation times in CFD
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Code used in order to solve the flow (\rightarrow MPI, Vectorization)
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Hardware (CPU, RAM, etc.)
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Mesh size / Mesh Quality
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Algorithms
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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 Strömungsmechanik - Grundgleichungen und Modelle - Lösungsmethoden - QualitÀt und Genauigkeit
- Versteeg & Malalasekera: An Introduction to Computational Fluid Dynamics - The Finite Volume Method - 2nd Edition
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