Computational Fluid Dynamics (CHE314)
Mid-Sem Exam (20 marks)
Date: 21/02/2022
1. Derive a 2nd order accurate one-sided difference expression for
∂𝑇
at a wall. [2 marks]
∂𝑥
2. Laplace’s equation in polar coordinates is:
Write the discretized form of it using central differencing. Use indices i,j in radial (s) and
azimuthal (𝜙) directions respectively. [2 marks]
3. Derive the stability criterion for the numerical solution of the two-dimensional heat
equation (only conduction) using the explicit FTCS discretization on a square grid of size
∆𝑥. [3 marks]
4. Determine a second order accurate in space and first order accurate in time
discretization for the following equation:
2
∂𝑢
∂𝑡
+
∂𝑢
2
∂𝑥
2
+
∂𝑢
∂𝑥∂𝑦
= 𝑠𝑖𝑛(𝑥)
[3 marks]
5. The DuFort-Frankel method for solving the heat equation involves the solution of the
following difference equation:
𝑛+1
𝑢𝑗
𝑛−1
− 𝑢𝑗
2 ∆𝑡
=
α
𝑛
2
𝑛+1
(𝑢𝑗 + 1 − 𝑢𝑗
𝑛−1
− 𝑢𝑗
𝑛
+ 𝑢𝑗 − 1)
(∆𝑥)
Find out the amplification factor for this difference equation. Is this scheme numerically
dissipative or dispersive? [2 marks]
Derive the stability conditions necessary for solving this difference equation. [1 mark]
6. Write one example each of hyperbolic, parabolic and elliptic equations. [2 marks]
7. Assume that we have a two dimensional matrix with the values of concentration of
oxygen stored in it for a grid of size 20x20. The corners of this domain are located at
(0,0) and (2,2). Write a code (specify the programming language in which you write it) to
do the following tasks:
a. Loop through all the grid points to store a value of 0.1 at all points [1]
b. An oxygen source with concentration 0.5 is located in the form of a square with
width 0.2 and centered at (1,1). Moreover, the concentration is zero at all the
domain boundaries. Write a code to modify the concentration matrix such that
this oxygen source and other boundary conditions are accounted for. [2]
c. Write a code (hint: use loops) to solve the diffusion equation for the concentration
(using an explicit scheme, FTCS). [2]