## First simulation considerations

Here you are the first results of the SEM method in a simple channel flow. The simulation run, even if was just a try (so was run with just a coarse grid with the followings dimensions 20 x 2 x 3.5, and this cell division 40 x 20 x 20), is very usefull regardless. We are able to understand in a better way how to improve the standard SEM method (improvements that might be usefull for the DFSEM too).

Here I plot the SEM inlet (upper image) and the LES outlet (lower one). Even from just these two images we can understand that the SEM method needs to have a kind of "boundary condition implementation".

The SEM I used to generate this inlet has the following form for %$\sigma$% %reflatex{eq:Sigma}%

Since the SEM version I implemented is a slightly modified version that consider a direction and velocity dependence of %$\sigma$% as showed in %reflatex{eq:Sigmaij}%

 %BEGINLATEX{label="eq:Sigma"}%\begin{equation*} \sigma = MAX(min(\frac{k^{\frac{3}{2}}}{\epsilon},\kappa \delta),\Delta),~~~~\Delta = MAX(\Delta x, \Delta y, \Delta z) ~~~~~~~~~ \end{equation*}%ENDLATEX% %BEGINLATEX{label="eq:Sigmaij"}%\begin{equation*} \sigma = \sigma_{ij} ~~~~ used~ for~ each~ velocity~ component~ i~ and~ each~ direction~ j~ \end{equation*}%ENDLATEX%

The main difference in the method implemented are the "shapes" of eddies.It is easier to show in a pseudo-code how I implemented this feature, presents in Jarrain thesis.

 DO j = 1,3 IF ( %$\mathbf{X}point_i - \mathbf{X}eddy_i < \sigma_{ij}$%) %$V_j = A (1 - \frac{\mathbf{X}point_i - \mathbf{X}eddy_i}{\sigma_{ij} }$%) END IF END DO

Nicolas, in his thesis, run several channel flow and many others different geometry. Actually, the expression used for %$\sigma$% in %reflatex{eq:Sigma}% is the one that Nicolas used in a "general RANS-LES coupling", and is based on the hypothesis that %$\sigma_{ij} = \sigma$%, that is constant in all directions and velocity components.

Actually Nicolas used once a different specification for each component of %$\sigma$% tensor in one channel flow. He defined it in the following way:

 %$\sigma_{i2} = \sigma_{i3}$% %$\sigma_{i3} =$% defined from a previous LES periodic channel flow, as the distance at which the two-point correlations drop to 0.1 %$\sigma_{i1} = U_c T_i$% where %$U_c$% is the average velocity and %$T_i$% is a timescale, calculated again from a previous LES periodic channel flow

### SEM Boundary condition

The implementation of a boundary condition in the SEM method should not be so difficult. In fact, considering a simple channel flow, the only condition to be satisfied is the following: %$v' = 0$%. Since, considering the %reflatex{eq:Sigmaij}% the v' component is:

%BEGINLATEX{label="eq:vprimo"}%\begin{equation*} v' = K \frac{(\sigma_{12}-\alpha)(\sigma_{22}-\beta)(\sigma_{32}-\gamma)}{\sqrt{\sigma_{12} \sigma_{22} \sigma_{32}} (\sigma_{12} \sigma_{22} \sigma_{32}) } \end{equation*}%ENDLATEX%

%BEGINLATEX{label="eq:vprimoDEF"}%\begin{equation*} \alpha = \mathbf{X}point_1 - \mathbf{X}eddy_1, \beta = \mathbf{X}point_2 - \mathbf{X}eddy_2, \gamma = \mathbf{X}point_2 - \mathbf{X}eddy_2, \end{equation*}%ENDLATEX%

A logical way to implement a kind of boundary condition in the SEM method could be a modification in the eddy box. So far this box is defined by the following relations %$x_{i,min} = \underset{min}{x \in S}(x_i - \sigma(\mathbf{x})) ~~~~~~~~ x_{i,max} = \underset{max}{x \in S}(x_i + \sigma(\mathbf{x}))$%.

In a real case the turbulence applies only in a region which does not include the viscous sublayer. Could be interesting then to limit the SEM method only to a zone with %$y^+ > 5 \div 30$% (I think that a lower limit is better because of the random eddies positions, which need a kind of buffer region to be fully developed). This allow us to be in the log-law area, where turbulence is fully developed.

## Problem in parallelization

 Serial in MAN2E Parallel (2 processors) in MAN2E Parallel (8 processors) in MAN2E Serial in my own PC
At the moment I am experiencing some computational problem during the parallelization of the subroutine I created. In fact, as we can easily compare from the two pictures above, the parallel inlet has a kind lower turbulence length scale (the simulation I run to create these images are exactly corresponding but for the number of processors involved).

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Topic revision: r4 - 2010-02-16 - 09:22:28 - RuggeroPoletto
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19 Sep 2019

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