DEN403/DENM010 (2020)

Question 1

a)  Suggest an example of implicit integration schemes for ordinary differential

equation        5u . What is the approximation order (order of accuracy) of the

dt

scheme you provided and why? For the full mark, you must provide a detailed answer.

[5 Marks]

b)  For the governing equation from (b), provide an example explicit integration        scheme. With providing step-by-step details of your numerical working, calculate the maximum time step necessary for stability of this scheme.

[5 Marks]

c)  A student wants to solve the linear advection equation            0 (x-space, t-time) t     x

using a finite-difference method. What is numerical dissipation (diffusion) and which adverse effects of numerical diffusion should he/she be aware of? What can be       done to minimise these effects? Provide a detailed answer.

[6 Marks]

d)  Numerical diffusion can also be used to stabilise numerical calculations. For solving the linear advection equation from part c), consider the first-order forward Euler       scheme in time and a central difference scheme in space,

u(i, n1)  u(i, n)     u(i 1, n)  u(i 1, n)

                              2h

where  and h are the time and space steps, i is the space grid index, and n is the time step index.

The scheme is unstable for all time steps.

Provide a step-by-step explanation how the above scheme can be stabilised by adding an artificial Laplacian dissipation term to the right-hand side of the         discretised equation.

[5 Marks]

e)  An engineer wants to simulate a steady flow about a thin aerofoil put at zero           incidence in the free-stream. The flow regime of interest corresponds to a small      subsonic Mach number M=0.1 when the non-linear inertia effects can be ignored    (the flow can be regarded as linear). The engineer checked that the computational scheme and the boundary conditions he/she wants to use for the solution are         consistent. What else does the engineer need to ensure about the scheme in order

to guarantee that the solution converges to the correct result when the CFD grid is refined? Provide a detailed answer with a step-by-step explanation of your            reasoning.

[5 Marks]

f)   Following up with Question 1 e), the engineer now wants to simulate a steady         transonic M=0.9 flow over the same aerofoil when stationary shock waves emerge on the aerofoil surface. In this case the flow is strongly non-linear. Same question   about the solution consistency and convergence as in part f). Can the engineer use the same logic to answer this question as in part f)? For the full mark, you must      provide a detailed explanation.

[7 Marks]

Question 2

Consider a one dimensional linear advection-diffusion equation with the following initial

u     u

t     x

a)  With the use of sketches to outline the solution domain in (x-t) plne, the                   computational scheme stencil, and the update order of solution points (i,n) away     from the boudaries, provide a step-by-step explaination how you would solve the    above initial boundary value problem using the second-order Central Leapfrog (CL) finite-difference scheme

u(i, n1)  u(i, n1)     u(i 1, n)  u(i 1, n)

2                                2h

where h is the computational grid spacing and  is the computational time step.

[3 marks]

b)  Explain the initial time stepping procedure for the scheme. Note that the CL scheme is three-time layer, hence, you must suggest a feasible way to start the calculations from n=0.

[5 marks]

c)  Explain how you would be dealing with the left and the right boundary points where the periodic boundary condition is used.

[3 marks]

d)  Using the Von Neumann method, provide details of analysis how to calculate the maximum time step of the CL scheme for stability.

.                                                                                                                      [5 marks]

e)  Using the Von Neumann method again, provide details of analysis how to evaluate

the numerical dissipation error of the CL scheme. Explain its physical meaning.

.

[5 marks]

f)   Using the Von Neumann method, provide details of dispersion analysis for the CL scheme to derive an expression for the phase error. Explain its physical meaning.

[6 marks]

g)  Following up with Question 2 f), show that for very small grid spacing h  0 (very large number of grid elements for a fixed problem size) the CL scheme is exact for all stable time steps of the scheme. For the full mark, please provide step-by-step details of your numerical working.

[6 marks]

Question 3

a)  Consider the following properties for air and silicone oil:

Air:                  ρ = 1.22 kg m –3            μ = 1.8510–5 kg m – 1 s – 1

Silicone oil:     ρ = 930 kg m –3             μ = 9.310–3 kg m – 1 s – 1

Which one is more viscous considering both forces on other structures as well as transfer of momentum within the fluid itself? Why?

[4 Marks]

b)  The critical Reynolds number for Laminar/Turbulent flow transition in a round pipe is

2300. What is the speed needed in a 10 cm diameter pipe for this transition to happen for the fluids in the previous question (i) air; (ii) silicone oil? (consider that the Reynolds number is based on bulk velocity and diameter)

[4 Marks]

c)  Describe the main principles of the following models of turbulence

(i)  eddy-viscosity

(ii) Reynolds-stress transport

, and give advantages and disadvantages of each type of closure.

[15 Marks]

d)  Indicate which method you would use to simulate an industrial turbulence problem; RANS or DNS? Why?

[4 Marks]

e)  The following  data are measured values of ( u , v ) in an idealised 2-d turbulent flow. Calculate u , v , u 2  , v2  , uv  from this set of numbers.

(2.6,0.3)     (3.4,0.2)     (4.2,–0.4)     (5.1,–0.2)     (3.9,0.1)

[6 Marks]