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Modelling and Simulation 2 (Dr M Jabbari): CFD CW 2a Due Thursday, April 30, 2020 – 17:00

CFD CW 2A – MATLAB

Background

Practical problems of heat transfer by conduction are often quite intricate and cannot be solved by ana-lytical methods. Their mathematical models may include non-linear differential equations with complexboundary conditions. In such cases, the only recourse is to obtain approximate solutions by employingdiscrete numerical techniques like the Finite Difference Method (FDM).

A description of the physical problem is shown in Figure 1. A suitable model for this physical problemcould be based on Fourier’s law of heat transfer. Conduction through a thin layer of fluid (L << W and

boundary

Figure 1: A 3D solid wall and its 1D FD discretisation along thickness.

Now consider the energy balance for a small element “i” of fluid, shown in Figure 1, to be

rate of heat conduction rate of heat conductioninto control volume out of control volume+ = +rate of heat generation rate of energy storageinside control volume inside control volume

which can be expressed algebraically as

Modelling and Simulation 2 (Dr M Jabbari): CFD CW 2a Due Thursday, April 30, 2020 – 17:00

Question 1 continued on next page. . . Page 4 of 5

Modelling and Simulation 2 (Dr M Jabbari): CFD CW 2a Due Thursday, April 30, 2020 – 17:00

Task 2: Use MATLAB to plot the temperature along x-direction (T − x diagram) at the end of 100seconds — (tagged: Fig. 1)

Task 3: Use MATLAB to plot the temperature at each time step at grid points “2”, “4” and “6”,

T (x2, tn), T (x4, tn) and T (x6, tn), over time (T − t diagram) — (all in one graph and tagged: Fig.2)

Question 2

Question Weighting: 40%

Use the program you developed in Question 1 to run two different tasks:

Task 1: Run the simulations with different grid resolution, P = 9, 11, 29, and compare the “T − x”at t = 100s (all in one graph and tagged: Fig. 3) and “T − t” diagram for the centre grid point(all in one graph and tagged: Fig. 4). Discuss your results briefly in terms of numerical modellingaccuracy. Specifically, how does changing the resolution of the grid change the solution?

Task 2: Run the simulations with grid resolution of P = 29, but different final time of tf =

100, 400, 700, 1000s and compare “T − x” diagrams as well as “T − t” diagram (all in one graphand tagged: Fig. 5) for the centre grid point (all in one graph and tagged: Fig. 6). Discuss yourresults briefly in terms of steady/unsteady phenomena. Is this problem unsteady or steady? Whathappens when you run your simulation for longer?

Assessment

You will need to submit yourm-file and a single PDF document in the blackboard that includes Figs. 1–6

and all necessary discussions. This assessment is worth 10% of the whole unit. The marking scheme

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