Question 4\ a) Consider the discretisations given below to approximate

(delu)/(delx)

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)\

(delu)/(delx)|_(i-1)~~(u_(i-1)-u_(i-2))/(h)

\ ii)\

(delu)/(delx)|_(i-1)~~(u_(i+1)-u_(i-1))/(2h)

\ iii)\

(delu)/(delx)|_(i-1)~~(u_(i+1)-u_(i)+u_(i-1))/(4h)

\ [18 marks]\ 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\ vii) Improve the mesh quality in areas of uniform flow\ ix) Improve the mesh quality in areas of high gradients

a) Consider the discretisations given below to approximate xu in the point i1. 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 i1 happen to be as follows: and the mesh width is h=0.5. i) xui1hui1ui2 ii) xui12hui+1ui1 iii) xui14hui+1ui+ui1 [18 marks] 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