Synthetic Eddy Method


The first code is the semequationsv1.1. This is a simple 2D code that solve the SEM equations as explained in Charpter 4 of Nicolas Jarrain Thesis (available here).It is a simple 2D grid (128 x 128 grid points) inside a eddy box. Using the SEM method, I tried to get the same results Nicolas had got. Here you are what i did.

The first simulation that I show has the following features:

  • eddy box dimension 4 x 2 $\pi$ x 2 $\pi$ m
  • 2000 eddies
  • 5000 total time step, $\Delta$t = 0.005 s \x{2192} 25 s

The first image shows 4 main graphics. In three of them you can see the instant velocity in a given point inside a grid (in this casethe point with y=64 and z=64), in which you can easily see the velocity fluctuation aroud a average constant value.The lastgraphics instead shows the U-Velocity distribution allover the grid in the last time step.


The following pictures instead shows the behaviour of time averaged speed. The first graphs shows the first order time averaged <u>, <v> and <z>. As we expected from the theory, the <u> is, after an initial transitory zone, is exactly the same as the U inlet velocity (10 m/s), while both <v> and <w> have a zero-average. A much more interesting result comes from the second graph. It shows the second order speed combinations <uv>, <uw> and <vw>. This totally agree with the theory, since I imposed as initial condition a zero-mixed components in the Reynolds stress tensor. I stress again the initial transitory zone, that needs to be reduced in time and space dimension to improve the SEM method.


The last graph shows our first problem (that will be dealt in a wider manner later) is that the second order average speed does not converge completely. In fact it fluctuates in the range [1.1 - 1.5], that does not include 1, the value we were expecting for the convergence.

The last graphs coming from the simulations show the third and fourth order averaged speed. These order are very usefull for the sistem Fourier-Analysis. From theory we know that:

   \begin{equation} S_{u_i} = \frac{&amp;lt;u_{i}^{'3}&amp;gt;}{{&amp;lt;u_{i}^{'2}&amp;gt;}^{3/2}} = \frac {1}{{(N R_{ii})}^{3/2}}&amp;lt;{(\sum_{k=1}^N X_{i}^{(k)})}^3&amp;gt; = 0 \label{eq:thirdterm} \end{equation}  (1)

   \begin{equation} F_{u_i} = \frac{&amp;lt;u_{i}^{'4}&amp;gt;}{{&amp;lt;u_{i}^{'2}&amp;gt;}^{2}} = \frac {1}{N^2 R_{ii}^2}&amp;lt;{(\sum_{k=1}^N X_{i}^{(k)})}^4&amp;gt; = 3 + \frac{1}{N} (4 F_{f}^3 F_\epsilon \frac {V_B}{\sigma^3} - 3) \label{eq:fourthterm}\end{equation}  (2)

As we can see from the graphs, the zero-value is almoust reached by the S term, as the F one reach a stationary value. As we can see from the (2) , the stationary value for F → 3 as N (eddies number) → ∞. In our case, N = 2000 and, we can see, the steady value is yet higher than 3. In any case, we reached a convergence situation.


Also in these two last graphs we can see some mismatches with our theoretical results. Mainly both the 3rd and the 4th averaged order seem to approach a value that is not really the calculated one. Further there is another problem: it seems that the three components converge to three different value.

Future issues

Understand how SEM parameters, as different position in the same grid, time, eddies number and box volume, affect the results.

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I Attachment Action Size Date Who Comment
elsef90 semequations_v1.1.f90 manage 13.9 K 2009-11-17 - 13:18 RuggeroPoletto Source file of the v1.1 of SEM equation solver. This version is a 2D version, that allow to see velocity fluctuations in a y-z 128x128 grid.
Topic revision: r11 - 2009-11-17 - 13:18:09 - RuggeroPoletto
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17 Dec 2017


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