1. FALL 2015 Esra Sorgüven Öner
ME 333 Fluid Mech. Lab.
FALL 2015
Esra Sorgüven Öner

2. Contents Summary of what we did so far:
You have learned how to generate a CFD mesh and execute a CFD analysis with a CFD code.
What are we going to do now:
Learn what is actually going on inside the code.
Learn how to choose the models, discretisation schemes…
Estimate the errors involved in your CFD analyses.

3. Steps in CFD The actual physical phenomena Modeling Discretising
Solving
Modeling error
Discretisation error
Numerical error
Does the CFD solution really represent the actual phenomena?

4. Contents 1. Modeling 2. Discretising 3. Solving
What is happening in the actual flow?
Is the flow compressible?
Is the flow turbulent? If so how “turbulent”?
Is there only one phase or more? (for example, flow in a condenser)
How many chemical species are there? Is there a chemical reaction (like combustion?)
2. Discretising
What discretisation method should I choose? FDM, FEM, FVM?
Discretisation scheme? UDS, CDS?
Order of discretisation? 1st, 2nd, …
How should I prepare mesh? Structured, unstructured? How can one judge about the “quality” of a CFD mesh?
3. Solving
What kind of a solver should I choose? If iterative, than what is the under relaxation factor?

5. 1. Modeling

6. The governing equations for a Newtonian fluid with constant r and m
in cartesian coordinate system:
in cylindrical coordinate system:

7. Different forms of NS equations
Compressible
Incompressible
Reynolds-Averaged NS (RANS) (for turbulence modeling)
With Chemical Reactions
Multiphase flows

8. Compressible NS in integral form

9. Compressible NS in differential form

10. Incompressible NS in differential form

11. Reynolds averaged NS (RANS)

12. RANS with chemical reactions

13. RANS for multiphase flows

14. General form of Governing Equations
Mass:
Momentum:
Energy:
Accumulation
Convection
Diffusion
Source

15. 2. Discretization

16. How can we solve these equations numerically?
We have to solve a set of partial differential equations.
Equations involve:
DISCRETISATION
Accumulation
Convection
Diffusion
Source

17. Example of discretization

18. Discretization methods
Finite Difference
Finite Element
Finite Volume
Nodes
Differential form of NS is solved.
Derivatives are approximated as differences
FEM results in ODE’s
Volumes
Integral form of NS
Fluxes are calculated
Inherently conservative
Results in algebraic equations

19. Discretization 1D 2D

20. Finite Difference
1st order:
Backward
Forward
Central

21. Backward difference 1st order accuracy
Higher order approximations:
Backward difference 1st order accuracy
Forward difference 1st order accuracy
Central difference 2nd ord.
Backward diff. 3rd ord.
Forward diff. 3rd ord.
Central diff. 4th ord.

22. Discretising 2nd order Derivatives
For equidistant grids (ie. Dx remains constant through the whole flow domain)
Central 2nd ord.
Central 4th ord.
Discretising Diffusionterms with central differences:

23. Example 1D Diffusion – convection problem:
Discretize the 1D diffusion-convection equation. In this equation only variable is f. r, u, G are real numbers.
Make a grid of 8 points and write the discretized equation for each grid point.
How many equations do you have? How many unknowns do you have? How many solutions can you have for this equation set?

24. Finite Volume Method
Steady NS equations in integral form:

25. Fluxes in FVM

26. Solver: FVM Mimics the underlying physical conservation principle
Rate of change of  in the CV = Convection Flux of  into CV + Diffusion Flux of  into the CV + Rate of Source Creation of  in the CV

28. Algebraic Equations

29. CFD Mesh

30. CFD Mesh Components Components are defined in preprocessor
Cell = control volume into which domain is broken up
computational domain is defined by mesh that represents the fluid and solid regions of interest.
Face = boundary of a cell
Edge = boundary of a face
Node = grid point
Zone = group of nodes, faces, and/or cells
Boundary data assigned to face zones.
Material data and source terms assigned to cell zones.
2D mesh
node
face
cell
cell center
face
cell
node
edge
3D mesh

33. Grid Types Unstructured adaptive mesh
Small control volume where equations are applied
Structured H-mesh
inlet
periodic
Unstructured adaptive mesh
exit
exit
inlet

51. 3. Iterative Solver