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|
- 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 X–X 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.)