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ME 709/809 Final [20 points]
You will have 72 hours to complete this assignment. You will be simulating a 2D, fully developed
channel flow using the k – model.
• TASK 1 [2 points]: Without using index notation, show the governing equations for a 2D,
time averaged k – model.
– Show the time averaged continuity equation.
– Show each of the components of the averaged Naveir-Stokes equations. Use the Boussinesq assumption as listed in section 6 of the notes. You can ignore the term in blue
from the assumption.
– Start from the modeled k and equations (equations 11.97 and 11.98 in
turbulence-modelling.pdf) and show the modeled k and equations for a 2D, timeaveraged flow.
• TASK 2 [2 points]: Simplify the equations from TASK1 for a 2D fully developed channel
flow. Clearly indicate what terms you are setting to 0 and why. Prove that the governing
equations reduceh to:
0 = − ρ1 ∂P
+ Pk − ;
0 = ∂y
ν + σνt ∂y
Pk = νt ∂U
(c1 Pk − c2 );
νt = cµ k .
• TASK 3 [5 points]: Discretize the equations using central difference and a non-equidistant
mesh. Note that you need to keep k and positive to avoid divergence of the solution. To
achieve that, put all negative source terms in SP where S = SP ΦP + SU . For example, the
dissipation term in the k equations should be in the SP and look similar to: SP = − k ∆y or
SP = − k ∆V (depending on how you integrate the equations).
• TASK 4 [3 points]: Write down the steps of the algorithm for solving this problem in Matlab.
For the boundary conditions use U = k = ∂y
Use channel height of 2 m, ρ = 1 kg/m , uτ = 1 m/s and use − ∂P
∂x = τw .
Use a Gauss-Seidel solver.
• TASK 5 [8 points] Simulate this flow using Matlab.
Plot the velocity profile.
Please submit the following on Canvas under FINAL exam submission:
– A pdf file containing the work for TASK 1, 2, 3, and 4; The plot from TASK 5, and a
printout of the code from TASK 5. The work from TASKS 1 – 4 can be scanned papers,
an exported one-note document or have any other form, as long as everything is legible
and the submission is a single pdf file.
– A zipped folder with all your work on the assignment, including the final .m file.