Page 215 - Process Modelling and Simulation With Finite Element Methods

202 Process Modelling and Simulation with Finite Element Methods

the same constant, this is legitimately a periodic quantity. The streamfunction-
vorticity approach eliminates pressure, which is a non-periodic quantity, in
favour of v and O, which can be modelled as periodic in the flowwise direction.

Exercise 5.1: Computing generalized eigenvalues
Use eig ( ) to compute the solution to the generalized matrix problem, see after
(5.15), for
A=[l 0 0 0 0; -2 1 0 0 0; 1 -2 1 0 0; 0 1 -2 1 0; 0 0 1 -2 11;
B=[l 0000;01000;00100;00000;00000];
Then compute eig (A) and compare. Why do you think you get these answers?
Now use

B=[l0000;01000;00100;1 111 1;12345];
What changes? Why?
Finally, try
B=[l 0 0 0 0; 0 1 0 0 0; 0 0 10 0; 0 0 0 1 0; 0 0 0 0 I];
What can you conclude about the generalized eigenvalue problem (5.14) from
this exercise? Why do you think we have always asked for the smallest
magnitude eigenvalues from eigs ( ) for FEM augmented eigensystems? What
if we asked for the largest eigenvalues?

5.3.1 Streamfunction-vorticity model with periodic BCs

We have previously seen the streamfunction-vorticity Poisson equation in (2.7)
and (3.3):
v21y = -ci)

where the streamfunction v is defined by its differential relationships:

(5.16)

Additionally, by making a tranformation to the moving frame, x’=x-Ut, and then
dropping the prime, we can write the momentum equation (5.10) and the
convective-diffusion equation (5.12) in terms of the streamfunction v:

(5.17)