Assigned Wednesday, Nov. 5, 2008, due Wednesday, Nov. 12, 2008
[1] Modify the programs in Assignment 5 to solve the Navier-Stokes equations in the vorticity/stream function formulation,
| $\qquad \qquad \dfrac{\partial \omega}{\partial t} \, + \, \vec u \cdot \nabla \omega = \nu \Delta \omega$ | $\begin{array}{c}\Delta \Psi = - \omega \\ \, \\ u = \dfrac{\partial \Psi}{\partial y} \qquad v = - \dfrac{\partial \Psi}{\partial x} \end{array}$ |
in the rectangular region $[0,1] \times [0,1]$ with velocity boundary conditions as specified in the following figure
$\qquad \qquad \qquad \qquad$
and homogeneous initial data $\omega(x,y,0)=0$. Use the method (6.5.10) on page 185 of Peyret and Taylor to determine the boundary
vorticity.
(2) Use your program to compute the steady state solution at Reynold number 400 (n = .0025) of the problem described in section 6.6.1 of Peyret and Taylor.
(a) Select one method of spatial discretization and compute the results using a 20 X 20 grid. Compare your results
with those presented in the table 6.6.1 on page 202 of the text by evaluating the functionals listed in Table
6.6.1.
(b) Compute the solutions using a larger grid (one with more grid panels in each direction). How do your results depend upon grid size?