Lab 7 - Duct Problem, TWS Algorithm


Problem Statement

TWS Formulation for Laminar Flow

This lab exercise picks up where the previous lab left off. It was observed in Lab 6 that 2dX oscillatory waves were present and that artificial diffusion might be needed. Per the course lectures and problem experiences, we have the full Taylor Series modification of the Galerkin Weak Statement.

This lab seeks to use only the beta term to apply artificial diffusion. Thus


As discussed in the previous lab, a full Newton Jacobian formulation is used. This Jacobian will resemble the previously discussed Jacobian with the addition of the proper artificial diffusion terms created above.

TWS Formulation for Turbulent Flow

For the case of turbulent flow, we recall the formulations developed and used in Lab 3 for the Turbulent Kinetic Energy closure model.

The potential term, phi, satisfies our pressure projection needs.

Now we are faced with a computationally intense Jacobian formulation. A quasi-Newton scenario reduces the coupling and allows for a more tractable solution with the trade-off of a slower convergence rate. It should be noted that the final pressure calculations are totally post-processing.

Discussion of Results

Here is the non-uniform mesh used for the following laminar and turbulent entrance duct flow problems.

2-D Laminar Duct Flow

Re = 100, Beta = 0

Little spikes of instability are clearly seen in the velocity, phi, and sum phi plots for the GWS problem solution.

2-D Laminar Duct Flow

Re = 100, Beta = 0.2

Now we add a little artificial diffusion via the TWS. Observe how the little spike seen in the velocity plot has now been reduced. Also the oscillations in the phi and sum phi plots has been reduced.

2-D Laminar Duct Flow

Re = 100, Beta = 0.3

Now, with a little more artificial diffusion, the velocity plot is completely smooth. In addition, the oscillations seen in the phi and sum phi plots has been reduced by an order of magnitude.

2-D Laminar Duct Flow

Re = 1000, Beta = 0

Moving on to Re = 1000 we see even more instability in the velocity profile.

2-D Laminar Duct Flow

Re = 1000, Beta = 0.2

I'm not quite sure what has happened to the phi plot. It looks rather strange and does not jive with the previous plots.

2-D Laminar Duct Flow

Re = 1000, Beta = 0.3

Now all is well, once again, and artificial diffusion has done a great job in cleaning the solution.

2-D Turbulent Duct Flow

Re = 10,000; Beta = 0.2

For the turbulent duct flow, it is interesting to note the dramatic changes seen in the solutions with such a small change in beta. Observe, for instance the large reduction in flow velocity when Beta = 0.3. Also, observe the significant change in the turbulent Reynolds numbers. Phi has changed dramatically, as well.

2-D Turbulent Duct Flow

Re = 10,000; Beta = 0.3

2-D Turbulent Duct Flow

Conjugate Heat Transfer

Finally, we look at the case of conjugate heat transfer in turbulent duct flow. The non-uniform meshing has been changed to suit this problem geometry as show. I am not sure what has caused the spike seen on the temperature field. Perhaps more artificial diffusion is needed?

You can watch the time elapsed temperature field development by clicking HERE.


In conclusion, we have clearly seen how the beta term from the TWS can stabilize solutions via the addition of artificial diffusion. For the laminar flow cases, the addition of the beta term is seen to smooth out any spurrious peaks. The Phi and SumPhi functions, however, will always show some "jagged" behavior as they apply their corrections after every step through time.

For the turbulent flow case, it is evident that one must exercise caution when choosing the strength of the artificial diffusion. In going from Beta = 0.2 to 0.3, it is observed that the maximum flow velocity reduces by nearly 15%! I would also note that I found it interesting that the flow velocities were on the order of 500 ft/s! Clearly, if this were the case, flow compressibility would have to be considered.

Finally, the turbulent duct flow with conjugate heat transfer clearly shows the turbulent velocity profile seen from Lab 3. The temperature field solution also clearly shows the diffusion of heat through the fluid via the heated wall.