As I stated in this previous page, the SEM method that was implemented showed some convergence problems, mainly with 3rd and 4th order averaged terms. So comes the need to do many other simulations to check how SEM works with differents parameters.
We focused some main parameters and we investigate over them. These parameters are:
I believe that, even if might not be as important as time and eddies number, the position needs to be investigated, as well as the box volume, because of completeness.
Another important task that has been developed this week was the rearrangement of the SEM code. In fact, the hight number of simulation that had to be done required lot of CPU time. So I modified my fortran code, deleting some useless functions and modifying the code with the purpouse to reach a faster computational speed. For this task I also used the N. Jarrin's fortran codes as idea to improve my own code. As a result of this work, the simulations speed increased of 2025% from the previous version of my SEM fortran code.


First we proceed, we need to take a look at the simulation time interval. In fact we must be sure that the simulation reaches its convergence, otherwise any further
discussion would be useless. To check this particular simulation feature I run several long time interval simulation, and check how the main variables behave.
At first sight, the highest averaged order have the lowest convergence speed. This can be easily checked by having a look at the last graphs of my previous result page. So I focused my attenction to these parameters and their output graphs. A 10,000 timesteps simulation long (every time step is 0.005 s, so overall is 50 s) showed that a kind of convergence is reached after 6,000  7,000 timesteps (3035 s). There results came from a 2000 eddies simulations, as the one described in the previous page. Taking a look to another simulation (1000 eddies), to check the results validity, we can confirm (see Fig. 2) the previous hypothesis.
As conclusion of our first discussion we can say that the minimun time step number to be sure that we reach a convergence situation is around 6.0007.000, even if is firmly suggested to use a highter number.
The first obvious analysis is the comparison between 2 or more points results in the same grid. In fact we expect to have very similar results moving around the same grid. This because of the random nature of the SEM method that should not be spacedependent then.
In the graph you can see plotted the %$<u^2>$% in different points all over the grid (you can see the points coordinates in the legend). As we can see, the central point is the one that best fit the theoretical results, instead the lateral points are inside a 20% range around the theoretical result, even if they are not right at the grid boundary.
Some similar results can be seen also taking in consideration other parameters, such as the %$<S_i>$% and the %$<F_i>$% (as you can see below).
In both cases, in fact, the results in the central point is the one that better fits with the theoretical result, and it is in between all other results.
All these results lead us to two different conclusions:


Another very important parameter is the eddies number, or better, the eddies density inside the box. For our simulation, since the volume is %$ ( 2 \pi )^2 * 4 = 25.13 $%, the densiti in so defined as:
%$ \rho_{eddy} = \frac{N_{eddies}}{Vol} = \frac{N_{eddies}}{25.13} $% 
In the following picture can be compared some results as the density increases. The value considered are:
%$ \rho_1 = \frac{500}{25.13} = 19.90 m^{3} $% 
%$ \rho_2 = \frac{1000}{25.13} = 39.79 m^{3} $% 
%$ \rho_3 = \frac{2000}{25.13} = 79.59 m^{3} $% 
%$ \rho_4 = \frac{3000}{25.13} = 119.38 m^{3} $% 
%$ \rho_5 = \frac{4000}{25.13} = 159.17 m^{3} $% 
Ando here you are some graphs.
FIG 1 
FIG 2 
FIG 3 
FIG 4 
From these grapsh, and mainly from the fig. 2 and fig. 4, but also from the others, we can see the strong relationship with the eddies density. In this study, until we reach a density of about 80 eddies per cubed meter. In few words, all parameters shows a better behaviour as the eddies increases. This must be taken in consideration during the simulation. We have to assure a minimum density to be sure a convergence situation has been reached.
By the way, we were expecting such a behaviour for the %$ <F_u> $% as from the theory we demonstrated that it has a strange relationship with the eddies number and the box volume, and so with the eddies density.
All the simulation in this section were run with 2000 eddies, that is defined the standard case.
FIG 1 
FIG 2 
FIG 3 
FIG 4 
Here I plotted the usually 4 main variable that has to be checked: %$ <u^2>, <uv>, <S_u>, <F_u> $%. Here it is more difficult to find a relationship, since the variable seemsto have a kind of random behaviour. In fact,for example, the %$ <F_u> $% in this case varies because the eddies density varies with the volume, and the same for all the others paramethers. To get some usefull results is then necessary to complete some simulation with a fixed eddies density, as the one that are now running.
Understand better why the results "jump", particularly and why a non uniform convergence has reached (Ex. the %$<u^2>$% never shows a common behaviour as it should)
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