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The failure of convergence was caused by ill-conditioning of the problem. In CFD simulations, numerical values are solved for from large systems of linear equations with the basic fluid variables at the mesh nodes as unknowns. In Case 1, 2a and 2b, the mesh density along the xy plane and the z direction differed by a factor of 2~5. This difference could make some numbers in the intermediate steps of iteration smaller/larger, and could be propagated through the process of matrix operations to into several orders of magnitude in the 3D system of ~10^5 cells. The extremely small number thus appeared could be close to machine precision, and couldn't reduce to any smaller number through iteration, which led to the stagnation of residual plots observed in Figure 1. As was show in Case 3 Figure 1c, after the mesh in xy plane was coarsened to the same order as the mesh in z direction, the residual converged to 10^-9 with 1st order solver and then to 10^-6 with second order solver after enough number of steps.The fluctuation in the tails of the residual plots could either be caused by fluctuation of extremely small numbers, or the fluctuation term defined in the turbulence model. Further experiments should be designed to verify the above hypothesis.
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The results from Case 4 are shown in the following Figures.
a. Contour of energy dissipation rate of the 2D model
b. Contour of energy dissipation rate of the 3D model with periodic boundary condioncondition
Figure 4 Comparison of energy dissipation maps of 2D and 3D models,drawn on the same scale
a. Contour of energy dissipation rate along the plane indicated in Figure 3
b. Contour of energy dissipation rate along the plan indicated in Figure 7
Figure 5 Contours of energy dissipation rate of the 3D model with periodic boundary conditions, drawn on its max/min scale 
a. Velocity profile along the plane and line indicated in Figure 7, drawn on the max/min scale of the z velocity in the indicated line and plane
b. Velocity profile along the plane and line indicated in Figure 7, drawn on the max/min scale of the x,y,z velocity in the whole domain.
Figure 6 z velocity profile
Figure 7 The plane and line referred to in Figure 5 and Figure 6, in red
Observations:
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The difference of the prediction from 3D and 2D models indicates the importance of z component in the flow field, which made the assumption of the equivalence of periodic 3D model and 2D model invalid. When we used 2D model to approximate 3D flows, other than only uniform condition, along z direction, we were actually assuming the all the fluid variables only have x,y components and no z components. When applying the periodic boundary conditions to the walls in xy plane, although the assumption that periodic repetition in z direction were equivalent to "uniform" could be valid, uniformity in z direction alone wasn't equivalent to "no components" in z direction. Thus these two models could not be good approximation to each other, and could generate significantly different predictions. As shown in Figure 6, there were still non-zero components of velocity in z direction, though insignificant in magnitude, and not uniform along the z direction.
Furthermore, the importance of z components could put the validity of 2D model as an approximation in question: in Case 4, even small components in z direction could make significant difference in results from 2D model, let alone in the real flow.
However, the above hypothesis must be further investigated, ruling out all other possible causes of differences. Particularly, the effect of the length of the period must be investigated by vary the width of the flocculator. Ideally, periodic repetition with infinitely small period length is equivalent to "uniform".
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Possible future simulation experiments and research topics are:
- To
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- To create refined mesh with proper boundary conditionslayers, and run simulations on SGI server, check grid convergence;
- To investigate the fluctuating tail of residual plots;
- Design numerical experiments to observe the fluctuation of extremely small numbers;
- Use difference turbulence models and compare the shape of the tails;
- Simulation with periodic boundary conditions with various period length
- Use backstep experimental data to validate 3D models with periodic boundary conditions;
- Find and compare with experimental data of free/confine jets .