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JDM questions for DEN331 Main sit CFD (January 2023)

Question 3

The unsteady diffusion equation in one-dimension,

with a concentration C, time t, the constant diffusion coefficient k = 0.2 m2 s and spatial dimension x describes the diffusion of a property C. It also represents effects of physical or artificial viscosity in a fluid.

You are asked to model the concentration in a long channel discretised into equally spaced nodes i = 1, 7.

The data for the initial solution at n = 0. is given as

Table 1: Initial data for C, x is given in [m].

a) Calculate the residual (i.e. the rate of change) for node i = 3 of the initial solution given in Tab. 1.

Use a central difference for the second derivative. [9 marks]

b) Use a forward difference in time with a timestep of ∆t = 1.25 s and the residual from question part a) to calculate the new value for C of node i = 3 after the first timestep. [9 marks]

c) Your supervisor wants you to calculate the concentration C at node 2 at timelevel n = 2. Tick the correct response[s] to your supervisor. Every incorrect answer is two negative marks.

i) C = 0.8,

ii) C = 0.85,

iii) can be computed after the solution at timelevel n = 1 has been computed,

iv) can’t be computed with the given data,

v) The solution of the central scheme is incorrect, the central scheme is not conservative. [4 marks]

d) For a different case, note the changes in x, a solution at timelevel n = 3 is given as shown in Tab. 2

Table 2: Data for C 3 at n = 3, x is given in [m].

Calculate the mesh width h, the timestep ∆t and the solution for nodes i = 3, 5 for timelevel n = 4, using the same discretisation as in a), b), but with a timestep of ∆t = h 2/k. [12 marks]

e) Analyse the results and tick the correct statements. Every incorrect answer is two negative marks.

i) The solution is monotonic.

ii) The solution has overshoots.

iii) The timestep is too small, the solution is inaccurate.

iv) The timestep is too large, the solution is unstable.

v) The solution of the central scheme is incorrect, the central scheme is not conservative.

vi) The solution is guaranteed to be stable. [4 marks]

Question 4

You are asked to set up a CFD simulation for incompressible laminar flow in a blood vessel geometry. Blood enters at vessels 1,2, blood exits at vessels 3,4. Vessel 5 connects between 1-4. You are asked to produce accurate results for flow rate and velocity profile at the cross section indicated with a dashed line in vessel 5, see Fig. 1.

Figure 1: Blood vessel geometry.

Your colleague proposes the list of options from a typical commercial CFD solver in the questions below, select from that list the boundary conditions that would be the best choice for the particular boundary in the question. If two choices are equally appropriate, then tick both.

Every incorrect answer is two negative marks.

a) Boundary 3:

i) velocity outlet

ii) velocity inlet

iii) slip wall

iv) non-slip wall

v) pressure outlet

vi) symmetry

vii) no boundary condition to specified here  [4 marks]

b) Boundary 4:

i) velocity outlet

ii) velocity inlet

iii) slip wall

iv) non-slip wall

v) pressure outlet

vi) symmetry

vii) no boundary condition to specified here    [4 marks]

c) Boundary 5:

i) velocity outlet

ii) velocity inlet

iii) slip wall

iv) non-slip wall

v) pressure outlet

vi) symmetry

vii) no boundary condition to specified here    [4 marks]