MEC3028: Theoretical and computational exercise for CFD (Total marks: 50)
Answer all questions and provide the answers as a separate pdf file. Upload all the answers to Canvas by the submission deadline.
Problem Specification
[25 marks total]
Laminar fluid flows through a two-dimensional channel of height H = 0.025 m and length L (see Figure 1). The above problem is to be solved using a uniform mesh, with grid spacing of Dx and Δy. The inlet velocity is 0.0065 m/s and the outlet vents to atmospheric pressure. The fluid has a shear viscosity of 𝜇 = 1 mPa.s and a density of 𝜌 = 1000 kg/m3. Use the SIMPLE algorithm and Double Precision solver for all simulations.
Note 1: The default solution algorithm is COUPLED and you need to change that to SIMPLE algorithm.
Note 2: The domain geometry is fixed, but the mesh spacing is changed in subsequent meshes.
(a) Generate a series of computational meshes for the given problem, as specified in the Problem Specification above, solve across the flow domain using Fluent with each mesh and convergence criterion, and complete Table 1. Record numbers to the 5th digit after the decimal point e.g. 0.12345 and do not round up/down the numbers. You will need to use Fluent to determine the maximum velocity, Umax, at the outlet of the channel. For all simulations you should use the Double Precision solver (selected from the Fluent launcher panel).
[10 marks]
Mesh L (m)
Dx (mm) 10
Umax (m/s)
Convergence criterion
12.5 10-6 5 10-3
5 10-6 1 10-3 1 10-6
0.5 10-3 0.5 10-6
(b) Plot Umax at the outlet against 1/Δy (which is proportional to the number of cells along a boundary). You should use a different line style for the results with convergence criterion 10!” and 10!#. Annotate the axes clearly.
(c) Using the tabulated and plotted values, comment on the results with regard to Mesh Independence and Solver Convergence (max 5 lines).
(d) Complete Table 2 by finding the maximum velocity Umax at each of the lengths given in the table below. Analyse the velocity fields and conclude whether the flow is fully developed at these lengths. Use a mesh size of Δ𝑥 = 10 mm and Δ𝑦 = 0.5 mm and convergence criteria of 10-6.
0.01 0.02 0.05 0.25 0.75
Umax (m/s)
Fully developed flow (Y es/No)
Problem Specification [25 marks total]
Laminar fluid flows through a two-dimensional channel of height H = 2h, where h=1 m, and length L=0.2 m in a periodic manner, which implies that the flow is between two infinitely long flat plates (see Figure 2) and is fully developed. Periodic boundary conditions ensure that the fluid which leaves the domain is reimposed at the inflow of the domain (NB. No inlet or outflow boundaries need to be imposed in this case). The fluid has a viscosity of 𝜇 = 1/64 kg/(ms), a density of 𝜌 = 1kg/m3 and a constant pressure gradient 𝜕𝑃/𝜕𝑥 = −1Pa/m is imposed in the axial flow direction. Details on how to apply the periodic boundary conditions as well as imposing the pressure gradient are provided in the Appendix at the end of this document.
Use the SIMPLE algorithm and Double Precision solver for all simulations.
Note1: The domain geometry is fixed, but the mesh spacing is changed in subsequent meshes.
Note2: Note that you should make the geometry for each mesh separately as Fluent causes problems with periodic conditions when duplicating the case setup.
Periodic Conditions
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(a) Generate a series of computational meshes for the given problem, as specified in the Problem Specification above with different mesh specifications given in Table 3 below. Please provide these meshes as figures in the report and clearly label the meshes in the figures.
Number of divisions in x-direction
Number of divisions in y-direction
4 equidistant points
6 equidistant points
20 equidistant points
100 equidistant points
40 points with a bias factor of 8 towards both walls
(b) Solve across the flow
SIMPLE algorithm. For
order upwind and 2nd order upwind) for the momentum equation and record the values of 𝜕𝑢/𝜕𝑦 at the bottom wall (𝐻=0) for all the meshes and both discretisation schemes in table 4 below. Use convergence criterion of 10!# and a minimum of 35,000 iterations for each simulation. The values for 𝜕𝑢/𝜕𝑦 can be obtained by the derivative function under Plots, x-y plot option.
Please comment (max 5 lines) on the values obtained and reason behind the differences observed.
Note 1: Export the data and use exact numbers rather than reading them from the plot Note 2: Report the numbers to 4th digit after the decimal point for $%
[5 marks for values + 5 mark for discussion =10 marks] Values of 𝜕𝑢/𝜕𝑦 from the simulations
domain using Fluent with different meshes given in Table 3 using each case run the simulation for two different discretisation (1st
4 equidistant points
6 equidistant points 20 equidistant points 100 equidistant points 40 points with bias 8
1st order upwind
2nd order upwind
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(c) In the case of wall bounded flows wall shear stress 𝜏’ is an important quantity which is defined as: 𝜏’ = 𝜇(𝜕𝑢/𝜕𝑦) at the wall (i.e. 𝐻 = 0 and 𝐻 = 2 in this case). From the solution obtained in Fluent previously record the values of 𝜏’, in table 5, on the bottom wall.
Note 1: Export the data and use exact numbers rather than reading them from the plot Note 2: Report the numbers to 6th digit after the decimal point
Values of 𝜏’ from the simulations
1st order upwind
4 equidistant points
6 equidistant points 20 equidistant points 100 equidistant points 40 points with bias 8
2nd order upwind
(d)Wall friction velocity 𝑢( is one of the most important quantity to evaluate the shear stress in wall bounded flows and it is defined as: 𝑢( = (|𝜏’| /𝜌))/+. Evaluate 𝑢( for all the meshes and all the numerical schemes used and evaluate the non-dimensional velocity 𝑢, = 𝑢/𝑢( (where 𝑢 is the velocity of the flow in the x-direction) against the non- dimensional distance from the wall defined as 𝑦, = (𝜌𝑢(𝑦/𝜇) (where 𝑦 is the dimensional distance away from the wall). An analytical solution to this problem can be obtained by using 𝑢, = 𝑢/𝑢( = [𝑦/h − 0.5(𝑦/h)+]/𝜇 and the result is shown in Figure 3 (NB. The x-axis in Figure 3 is on log scale). Plot the analytical solution and compare it with the data obtained from Fluent simulations for all the meshes and 2nd order scheme used. Please use different line colours for each line and label them appropriately.
Appendix: Imposing periodic boundary conditions and pressure drop
Assume that the domain faces in question 2 at along the 𝑦-axis at 𝑥 = 0𝑚 and 𝑥 = 0.2𝑚 are x-left and x-right respectively. In the case of question 2 both x-left and x-right should be assigned interface boundary conditions under the boundary conditions tab as shown in Figure A1 below. This can be done by double clicking the Boundary Conditions tab and then selecting the appropriate options.
Once both x-left and x-right have been assigned as interface boundaries then expand the Boundary Conditions tab on the left window in Fluent and then expand the Interface tab. Select both x-left and x-right while pressing the Ctrl button on the keyboard. Then right click and select Periodic as shown in Figure A2 below. A new window will appear as shown in Figure A3 where the Auto Compute Offset box needs to be unchecked and the value for Offset needs to be entered for X [m], which is 0.2m in this case. Once all the appropriate options have been selected then click the Create button. This will create a new boundary condition which will have the name as specified in the Zone name box as shown in Figure A3.
Once the new zone has been created, select the new zone, and then click on the Periodic conditions button and a new window will appear as shown in Figure A5. Under the Pressure Gradient option specify the pressure gradient as -1Pa/m, and this will specify the constant pressure drop condition. Once all of these steps have been followed, please perform the simulation as done previously in the tutorials.
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