## Eddies length-scale %$\sigma$%

A parameter with a fundamental importance is the length scale. Untill now we investigate the influences of several parameters on the results, but, in all our cases, we used a fixed-constant-spacial independent length-scale %$\sigma$%.This is, obviously, a non phisical approximation. In fact %$\sigma$% is strongly anisotropic and space-dependent. This week then I modified the SEM script see the behaviour of the model with different kind of sigma degree of freedom and I checked the results.

### Space dependence

Sigma is not constant all over our domain. It varies continuously (mainly it decreases as we approach the wall). I improved then this dependence.

I improved in this model also the possibility to set a user velocity profile as initial condition.

The simulation I ran has the followings features:

• %$N_{eddy} = 1000$%
• %$X_{length} = 2 \sigma_{max}$%
• %$T_{step} = 50000$%
• parabolic U-velocity profile
• %$\sigma = MAX(MIN(0.02 * y,0.02 * (2 \pi - y )),0.5)$%

The first plot is the modified U-velocity profile after the application of the SEM method. The velocity profile is modified with the synthetic turbulence created by the SEM method.

Another interesting plot is the instantaneus V-W velocity component in the grid.

Again, in this simulation, all the averaged variable have the same behaviour as the one with constant %$\sigma$%. The only different behaviour is the %$<F_u_i>$%, justifiable because of the space variance of %$\sigma$% that does not allow to achieve a full convergence.

### Anisotropy

These first two graphs show the different length scale between the U and the W component. The simulation ran ( %$N_{eddies} = 1000, X_{lenght} = 2 * \sigma_{max}, T_{step} = 50000$%) has a different length scale over the three components.Basicly the implemented %$\sigma$% model is the following:

 %$\sigma = \begin{array}{ccc} \sigma_{xu} & \sigma_{xv} & \sigma_{xw} \\ \sigma_{yu} & \sigma_{yv} & \sigma_{yw} \\ \sigma_{zu} & \sigma_{zv} & \sigma_{zw} \end{array}$%

The implemented length-scale is then a tensor in the space-velocity plane. With this we have "deformed" eddies that allow us to have a general behaviour closer to the phisical one.Again the VW components in the YZ plane is plotted.

The main difference between this case and the "standard" (= sigma consant) case, is in the fourth order averaged speed. In fact as this order depends on the ratio %$Vol \over \sigma$% that is no more constant over all the grid, then every velocity component converges towards a different value, as the following graph shows.

## Conclusions

Some features for sigma have now been implemented. Two more tasks have now to be done:

1. keep on studying some different features that require to be implemented (like the Divergence Free method)
2. study an "automatic" procedure that allow to calculate all these imput parameters

Both these task are very important to reach a solution as close as possible to the LES one and that does not require many backstream points to reach its developed value.

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Topic revision: r3 - 2009-11-23 - 17:43:40 - RuggeroPoletto
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23 Aug 2019

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