Turbulence model validation

Validation of Turbulence Models:

 Turbulence Modeling Resolution Problem

Computational Fluid Dynamics works by iteratively changing the values of the variables to reduces the residuals, or errors of the governing equation. The governing equations in our model is conservation of mass and conservation of momentum as shown below in Fig. 1 and Fig. 2:

 Figure 1. Conservation of Mass

 Figure 2. Conservation of Momentum

The conservation of Momentum equation can be simplified to Reynolds Averaged Navier Stokes Equation in Fig. 3 by estimating velocity in terms of mean velocity (u) and velocity fluctuation (u') as shown below:

 
With the conservation of mass equation, and conservation of momentum equations in each direction, there are four governing equations. However with the introduction of the velocity fluctuation variable (u'), there are six variables (u,v,w,u',v',w'). Thus, the problem becomes unresolvable unless additional equations are formulated to relate the variables. This is the Turbulence Modeling Resolution Problem.
 
The k-eps models overcome this problem by 

 
 
We have turbulence flow in our flocculation tank. Since we have unresolvable term in turbulence flow, different turbulence models or "estimation" were created to resolve turbulence flow with different characteristic. To decide on the turbulence model to use, a flow over backstep was compared with the literature experimental data. Figure 1 shows the flow of Re = 48000 over the channel. In the middle of the channel, the flow separate due to the small step size of height h. The flow reattaches at about 7 times the step height further downstream. This flow properties is similar to the 180 degree bend in the flocculation tank where we have flow separation and reattachment downstream (Figure 2).

Figure 1: Flow over backstep in a open channel (Re = 48000, Reattachment length = 7h)

Figure 2: Flow over 180 degree turn in flocculation tank

We compared the back step flow using K-e, K-W SST, K-e realizable, K-e RNG, RSM turbulence models. We need to pinpoint the reattachment points of all the turbulence models so that we can compare the reattachment ration with the experimental data.

Plotting the derivative du/dy, the change in direction of velocity in x direction with respect to y at the wall, we will be able to accurately pinpoint the reattachment point. At the wall, separation flow will give negative du/dy and and the flow reattaches, the du/dy will reach zero and becomes positive. Figure 3 shows the derivative of du/dy vs x direction for different turbulence model.
 Figure 3: dU/dy for Different Turbulence Models

Table 1 shows the comparison results of different turbulence models.

Table 1: Reattachment ratio with different turbulence models

From table 1, we see that the K-e under-predict the reattachment length, as known by most literature. KW SST and K-e realizable gives the most accurate representation of the back step flow with reattachment length of 6.37 and 6.18. However, from literature review, it has been known that K-e realizable is more appropriate for this type of flow, K-e realizable was chosen as the model for flow in the flocculation tank.

 
Figure 4: Flow over backstep using K-e realizable model
 
Now that we have finish validating all the appropriate step, we are fairly confident with the accuracy of our model. We can start optimizing the geometry to get the desired flow properties that will promote particles agglomeration. For analyzing different geometry, meshes with different dimension parameters will have to be created. It is time consuming to create a new mesh from scratch just to change some small parameters. Therefore a journal script was written to automate the mesh creation process.