Two different turbulence models, the standard k-ε Model and Reynolds Stress Model are compared to see which model that could predict the flow most accurately. The idea of this is to form a basis for the comparison with the GENUS code.
As a starting point, the simulations were performed for the 2D axisym-metric case with the finer grid. This because, the results that FLUENT pro-duced should be able to be repropro-duced in the same way by a casual user who has created the grid, unfortunately this was not the case. The finer grid re-sulted in a really bad convergence and reversed flow on the outlet. The sec-ond approach was to use the same input values as FLUENT used. Change the turbulence intensity to 22% and the hydraulic diameter to 0.055 m (strange! This is not the correct value of the hydraulic diameter for this co-axial cylinder) at the swirling inlet. This change of the boundary condition did not improve the solution.
The following solution strategy was applied to reproduce the results:
1. Change the under-relaxation factor for momentum (radial and axial ve-locities) from 0.7 to 0.4.
2. Use the PRESTO! scheme for pressure which is well suited for swirling flows.
3. Solve the problem without swirl to get a defined initial field.
4. Start with a low tangential velocity, perhaps 10% of the final tangential velocity.
5. Turn off the momentum equations for radial and axial velocities and solve the momentum equation for the tangential velocity together with the turbulence equations.
6. Turn off the momentum equation for the tangential velocity and turn on the momentum equations for radial and axial velocities and solve them together with the turbulence equations.
7. Turn on all equations and solve them simultaneously to obtain a fully coupled solution.
8. Increase the tangential velocity, perhaps doubling it, and use the former fully coupled solution as an initial field.
9. Repeat step 5 to 8 until you reach the desired tangential velocity.
Even though the "recipe" above looks circumstantial, it is a simplification of the true methodology. Because it seems that, not every fully coupled so-lution for a certain tangential velocity can serve as an initial field for the next tangential velocity. So it is a little bit of trial and error. This is not the way the casual user wants to solve the problem. Not surprisingly, it was much easier to get a converged solution with the Standard k-ε Model than it was with Reynolds Stress Model, for whom it takes considerably longer to converge. With the Standard k-ε Model, one can use the recipe above.
To make the comparison "valid", the case has to be computed in 3D. It is really exciting to see how a 3D model behaves when a 2D calculation is so complicated to perform. Instinctively, one thinks that a 3D calculation is even worse, but this is not the case! For the k-ε model it is just to set the case and "push the button". The strange thing is that the 3D calculation con-verged much faster than the 2D. The 3D calculation needed 1500 iterations to reach the desired convergence criterion, whereas the 2D calculation needed 6600 iterations. For the Reynolds Stress Model a simplification of the recipe above is used. Start the calculation with a tangential velocity of 4 m/s to get a defined initial field. Increase the tangential velocity to 7 m/s and use the former solution as an initial field and when this solution is con-verged, set the tangential velocity to the final value (the profile). But with this solution, reversed flow on the outlet is present. However, this will be used as a solution anyway despite of the error. The 3D sector needed 20000 iterations to reach convergence, whereas the 2D axisymmetric case needed 30000 iterations.
During the simulation in FLUENT the following interpolation schemes are used: PRESTO! (PREssure STaggering Option) for the pressure, SIM-PLE for the pressure-velocity coupling and Second Order Upwind (CDS) for the rest. For the 3D sector with Reynolds Stress Model it was necessary to use First Order Upwind interpolation to reach convergence. A conver-gence criterion of 10-4 are used for all residuals and they are: continuity, momentum (velocity components), the turbulence kinetic energy (k) and the turbulence dissipation rate (ε). When Reynolds Stress Model is used, equa-tions are also solved for each of the six Reynolds stress components with the consequence that there will be six more residuals to take into account. Due to false convergence with the 3D sector using Reynolds Stress Model, a tougher convergence criterion was applied.
Available measurements are axial velocity, swirl velocity, RMS axial velocity and RMS swirl velocity at three different positions [16]. The RMS axial and swirl velocities are not automatically calculated and therefore user-defined functions have to be defined. In the Standard k-ε Model, the reynolds stresses are assumed to be isotropic
w
The turbulent kinetic energy is defined (Eq.(6-15)) as
w
This implies that the RMS axial and swirl velocities are given by
k w
uRMS RMS 3
= 2
= (8-4)
When Reynolds Stress Model is used, equations for each of the six com-ponents in the reynolds stress tensor are solved. Therefore, the RMS axial velocity is given by
u u
uRMS = ′ ′ (8-5)
and the RMS swirl velocity is given by w
w
wRMS = ′ ′ (8-6)
The reference that FLUENT used (reference [15]) did not contain the ex-perimental data that they have used. One explanation might be that they have contacted IFRF to get more experimental data. Anyhow, the experi-mental data from IFRF [15] is also shown in appendix 2. Due to lack of raw data, the curves shown in [15] are reproduced in Excel. Available measure-ments are axial and swirl velocities at five different positions, unfortunately there are no measurements of the RMS axial and swirl velocities.
8.4.1 Results
A comparison between the two grids for the 2D axisymmetric case shows that there is no major difference between the results. The reverse ax-ial velocity is about 3% higher for the finer grid. The coarser grid produces zones with reverse axial velocity farther downstream from the inlet. These zones are not present with the finer grid. All results shown are with the finer grid.
Experimental data show that the central air injection should penetrate fully through the reverse flow created by the swirling flow. The Standard k-ε Model does not predict penetration, see figure 8.2, that occurs in the ex-periment, while Reynolds Stress Model shows full penetration, see figure 8.3.
Figure 8.2. 2D axisymmetric. Reverse axial velocity (m/s) for the Standard k-ε Model that shows no penetration
Figure 8.3. 2D axisymmetric. Reverse axial velocity (m/s) for Reynolds Stress Model that shows full penetration.
The Reynolds Stress Model prediction agrees well close to the inlet showing full penetration and reverse flow. Swirling flows result in anistropy in the turbulence field, which is more accurately predicted by the Reynolds Stress Model. Farther from the inlet, the prediction accuracy decreases, and the Standard k-ε Model is actually more accurate. The axial and swirl ve-locities are compared with experimental data [16] at three different posi-tions. Measurements of RMS (root mean square) axial and swirl velocities are also available at the three positions. There is also a comparison with experimental data [15] for axial and swirl velocities at five different
posi-tions. See appendix 2 for a comparison between experimental data and pre-dicted values.
From a qualitative point of view, the Reynolds Stress Model predicts the flow correct where the jet fully penetrates the reverse flow. However, quan-titatively there are discrepancies between measured and predicted values.
For the 3D sector with the Standard k-ε Model compared with the 2D axisymmetric case, there is no major difference, but the maximum reverse axial velocity is 28 % higher for the 3D case. Generally the results are pretty much the same. For the Reynolds Stress Model, the reverse flow is 19 % higher for the 3D sector compared to the 2D axisymmetric case. Great im-provements are made with the 3D sector compared to the axisymmetric case, but it does not predict full penetration of the jet and there is reversed flow on the outlet. The anistropy is more carefully taken into account when a 3D model is used compared to a 2D axisymmetric model, see appendix 3.