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DEN331 Computer Aided Engineering

Question 3

Your team leader asks you to model the transport of a chemical reactant in a pipe with varying cross section. She suggests you ignore the velocity profile and assume an average velocity that is constant for each cross section, but varies along the pipe as the cross-section varies.

Consider the linear advection equation for a concentration c with flux f = ac,

where a varies spatially as a = ai , which does not depend on time.

Measurements at time-level n = 0 at stations i produce the following data for ai in and

The stations are equally spaced at xi − xi−1 = 0.6 m.

a) Use a first-order accurate upwind approximation and compute the flux either side of node 2.

c) Employ a finite-volume scheme with explicit time-stepping and first order upwind flux discreti-sation for the advection equation.The concentration of the flow upstream of the test section is c = 0.35, its velocity is 0.75 . The concentration of the flow downstream of the test section is c = 0.55, its velocity is 0.9 .

Using a timestep of ∆t = 0.4 s, compute the concentrations at n = 1, n = 2 and the value of (You don’t need to compute the other concentrations at time-level 3). Work to 4 digits of accuracy.

d) Your colleague complains about the accuracy of your simulation and suggests to use a central discretisation for the flux based on the averages of the fluxes either side,

as it is second order accurate.

Using this flux, and the same data and timestep as in the upwind case c), compute the concentra-tions at n = 1, n = 2 and the value of . Work to 4 digits of accuracy.

e) Which of the following statements is correct, tick any correct answer, each incorrect answer pro-duces 2 negative marks.

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

ii) The upwind solution is incorrect, the scheme is inconsistent.

iii) The central solution is incorrect, the timestep is too large.

iv) The upwind solution is too dissipative, the timestep is too small.

v) The central solution is not dissipative enough, the scheme is unstable.

Question 4

a) Consider the discretisations given below to approximate in the point i − 1.

Derive their truncation error up to and including second-order termsin the truncation error (Note: do not include third-order terms in your calculation). Use that to calculate its value if the the function value u and its first to fourth derivatives at i − 1 happen to be as follows:

and the mesh width is h = 0.5.

i)

ii)

iii)

b) Your colleague presents you with results from a CFD calculation for a small wind-turbine for a developing country modelled in turbulent and steady flow. You are asked to reduce the truncation error, without increasing the computational cost.

Select all the correct answers in the following list. Wrong answers incur 2 negative marks.

[7 marks]

i) Reduce the mesh width overall

ii) Reduce the time-step

iii) Use an implicit method

iv) Refine the mesh where velocity gradients are large, coarsen where they are small

v) Refine the mesh at stagnation points

vi) Increase the order of accuracy of the space discretisation

vii) Use a better turbulence model

viii) Improve the mesh quality in areas of uniform flow

ix) Improve the mesh quality in areas of high gradients