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.
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
Laminar or Turbulent - if turbulent \rightarrow +turbulence model + near-wall treatment
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?
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:
- L-VEL & yPlus
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 you’ve had beforehand (CFD results, measurement data).
Add constitutive relations:
- Shear Stress
- Dynamic Viscosity
- Kinematic Viscosity
Add Boundary Conditions and Initial Conditions.
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 “critical” regions and set size:
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
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), 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.
Once the problem is well defined with the boundary conditions, and if necessary with initial conditions, the problem is solved with a software. Open∇FOAM is a popular option for a solver which is used by several companies that provide CFD software. SimScale is among them.
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
- Statistical analysis
According to AIAA (1998) & Oberkampf and Trucano (2002) the following terminology is widely used and accepted:
Verification (“Are we solving the equations right?”) :
\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:
Iterative Convergence Error
Validation (“Are we solving the right equations?”) :
\rightarrow Quantification of input & physical model uncertainty
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
Read more about CFD in our related article in the SimWiki.
Also see our SimWiki for more about other interesting simulation related questions.
- Laurien & Oertel: Numerische Strömungsmechanik - Grundgleichungen und Modelle - Lösungsmethoden - Qualität und Genauigkeit
- Versteeg & Malalasekera: An Introduction to Computational Fluid Dynamics - The Finite Volume Method - 2nd Edition