Math 270E: Assignment 6

Assigned Wednesday, Oct. 28, 2008, due Wednesday, Nov. 5, 2008

In this assignment you will modify the programs from Assignment 4 (and possibly Assignment 5) to create a program that solves the vorticity/stream-function formulation of the Euler equations.

[1] Modify the programs in Assignment 4 to solve the Euler equations in the vorticity/stream function formulation,

\[\begin{array}{c} \dfrac{\partial \omega}{\partial t} \, + \, \vec u \cdot \nabla \omega = 0 \\ \, \\ \Delta \Psi = - \omega \\ \, \\ u = \dfrac{\partial \Psi}{\partial y} \qquad v = - \dfrac{\partial \Psi}{\partial x} \end{array}\]

in the rectangular region $[-4,4] \times [-4,4]$ with an initial vorticity distribution

\[ \omega(x,y) = \left\{ \begin{array}{c|c} ( 1 - ((x-2)^2 + y^2))^3 & (x-2)^2 + y^2 \leq 1 \\ ( 1 - ((x+2)^2 + y^2))^3 &(x+2)^2 + y^2 \leq 1 \\ 0 & elsewhere \end{array} \right. \]

and homogeneous (no-flow) normal velocity boundary conditions.

To help you plan your implementation strategy, I've made a list of some of the tasks that you will need to perform in order to extend the program in Assignment 4 to a program that solves the Euler equations.

• Create a routine that computes the velocity field from a given vorticity distribution. One can modify the routine initializeVelocity(...) from Assignment 4 for this purpose. Remember to compute the velocities at boundary grid points.
  
• Create a routine that initializes the vorticity.
  
  
• Modify the convection/diffusion operator routine so that an approximation to the convection term at the boundary is computed.
  
  
• Modify advanceTimeStep(...) so that the velocity is computed before the convection/diffusion operator routine is called.
  
  
• Create routines for computing the discrete L² norm of the velocity components and the vorticity.

[2] For this problem, use the program you created for [1] with a centered/non-conservative difference approximation (method (c)).

1. Compute the solution to time t = 200. Create plots of the vorticity distribution. Does your solution appear qualitatively correct?
   
   
2. Using the discrete ${\rm L}^2$ norm of the velocity and vorticity distribution at time t= 5, estimate the rate of convergence of the numerical method. (A reasonable choice of N for this calculation is N = 40, 80 and 160 panels). Comment on your observed rates of convergence. Is it what you expect?

What You Should Turn In

• Your answers to the questions 2(a) and 2(b).