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
NonOrthogonal 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 nonlinear 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

Determine appropriate mathematical model

Classification of partial differential equation

Transformation of mathematical model

Select grid pattern

Formation of finite difference equations

Solution algorithm

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