## Computational Programming

24 Mar

Homework 3: 2-D Convection (Computational Programming)

Problem 1 (100 points)

Consider two-dimensional steady-state convection without diffusion (Γ = 0) over a square domain define by  and governed by the partial differential equation (PDE):

and with the boundary conditions such that along the left edge (x = 0) , along the bottom edge (y = 0) , and along the other edges, the gradient of  in the stream-wise direction is zero. The velocity field is defined as .

Problem setup and analytical solution:

 Problem setup Contour plot of  from analytical solution Line plot of at X–X axis from analytical solution

Find:

• Derive the discretization equations for the above governing PDE by using(a) first-order up-winding method, and (b) central differencing method.
• Contour plots of by using (a) first-order up-winding method, and (b) deferred-correction method.
• Line plots of across the domain (along the XX axis) by using (a) first-order up-winding method, and (b) deferred-correction method.
• Compare the numerical simulation results by thefirst-order up-winding method andthe deferred-correction method with the analytical solution shown in the figuresabove.

(Note: Blending factor  for the deferred correction method.)