Reference no: EM133109882

SEM400 Computational Fluid Dynamics - Deakin University

Assignment: Programming Exercise

In your first assignment, you will have to complete a programming task. You can use any programming language you prefer, but we recommend MatLab, as it allows for relatively easy implementation and visualization. You can use programming libraries only for the solution of linear algebraic problems (e.g. linsolve(A,b) or inv(A)) and for plotting data. All other routines need to be programmed in primitive statements (i.e, if statements, loops and assignments). We can offer support for MatLab, C and Fortran.

Your task is to write a code using the Finite-Volume-Method (FVM) to solve the following 1D equations.

Question 1): Solve the 1D heat conduction equation without a source term.

The 1D heat conduction equation without a source term can be written as:

d/dx (k.dT/dx) = 0

Where k is the thermal conductivity, T the local temperature and x the spatial coordinate.

Using the Finite Volume Method, use this equation to solve for the temperature T in a rod. The rod has a length of L = 2.0 m, a cross-section area of A = 10·10^-2m³, the thermal conductivity is k = 1500 W/Km, and the temperatures at the ends are held constant at 200°C and 600°C, respectively

Question 2): Solve the 1D heat conduction equation with a source term.

The 1D heat conduction equation with a source term can be written as:

d/dx (k.dT/dx) + q = 0

with k being the thermal conductivity, T the local temperature, x the spatial coordinate and q the source term.

Using the Finite Volume Method, use this equation to solve for the temperature T across the thickness of a flat plate of thickness L = 3 cm. The thermal conductivity is k = 1.25 W/Km, and the temperatures at the two ends are held constant at 150°C and 300°C, respectively. An electric current creates a constant heat source of q = 1100 kW/m³.

Question 3): Solve the 1D convection-diffusion problem

The 1D convection-diffusion problem can be written as:

d/dx (ρuΦ) = d/dx(ΓdΦ/dx)

With Φ the property that is being transported, u the convection speed, Γ the diffusivity and ρ the density.

The length of the domain is L = 1.0 m, the density is ρ = 1.0 kg/m3, the diffusivity Γ = 0.1 kg/ms. Determine the distribution of Φ for the following cases.

i. u = 0.1 m/s using 5 cells.
ii. u = 2.5 m/s using 5 cells.
iii. u = 2.5 m/s using 20 cells

For Assignment 1, you need to submit your results for the programming exercise:

1) The MatLab code(s) (eg. myFVM.m)
2) A report

This part of the assignment will contribute 20% to your final grade. You will be graded on

• The code (working, easy readable, adaptable, efficient) (5 percentage points)
• Verification (compare to analytical results) (5 percentage points)
• Project report (10 percentage points)
a. Explain approach and code (!).
b. Show verification, results and
c. Interpretation, especially regarding your results from the convection-diffusion problem.
d. Show how the code can be used for other problems (i.e. set your own boundary conditions and solve, 1 example).