ME 540

Numerical Solutions Applied To Heat Transfer And Fluid Mechanics Problems

Selection of Grid Pattern

Each equation to be solved needs a grid.

Consider the required accuracy of results

Higher Accuracy  Finer Grid spacing  Larger Number of Nodes  $ Select Grid Type Orthogonal Grid Cartesian Usually depends on boundary
Cylindrical configuration of region Non-Orthogonal Grid Irregular geometries How many grid patterns do we need?
Example: Primitive variable set of equations for flows Vector: x & y momentum
Scalar: pressure, temperature

You may wish to have 3 grids, one for scalars, and one for each flow direction.

Formation of finite difference equation

Covered in future lectures. Considerations include: accuracy in space and (were appropriate) time; stability; robustness; and applicability to target computer architectures. Interior and Boundary nodes require different difference forms.  

Solution Algorithm

Select a method for solving sets of algebraic equations simultaneously. This may become a fairly complicated iterative procedure.   For flow calculations – must handle the coupling of the various equations and the non-linear terms in the momentum equation.  

Perform Auxiliary Calculations

For example: Invert transformed variables if necessary
Calculate shear stresses and heat flux at boundaries
For transients audit mass and energy conservation


Summary of steps in problem solution

    1. Determine appropriate mathematical model
    2. Classification of partial differential equation
    3. Transformation of mathematical model
    4. Select grid pattern
    5. Formation of finite difference equations
    6. Solution algorithm
    7. Perform auxiliary calculations

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Created by  Frank Schmidt
Maintained by John Mahaffy : jhm@psu.edu