N.Jarrin solution

So far, the DF-SEM is capable of reproducing a iso-tropic turbulence. This because all the eddies created are "alligned" with the reference system (I imposed their intensities to be $ \alpha_i = \{-1;+1\} $. Substantially we had only 8 different types of eddies introduced in our domain!!!This fact led us to a small possibiliy in controlling the Reynolds tensor. Why not increasing our possibilities?

The intensities limits rose in the old SEM, and they were defined in order to get the Reynolds stress: obviously the new DF-SEM may have a totally different implementation.

First of all, I tried to set $ \{ -\sqrt{3} < \alpha_i < +\sqrt{3} \} $. This substantially lead to the same result we had had previously, since it is substantially the same distribution ( $ <\alpha_i> = 0 $ and $ <\alpha_i^2> = 1 $) but with a totally news: now we are inserting several different types of eddies!

Further then, after a look at some old N. Jarrin SEM method, I noticed he used a coefficient which increased the intensities of the eddies alligned with the Reynolds Stress tensor main component defined as follow:

 \begin{equation*}  C = 1 + \cfrac{\mathbf{\alpha} \mathbf{S} \mathbf{\alpha}^T}{||\mathbf{\alpha}||\:||\mathbf{S}||\:||\mathbf{\alpha}||}  \end{equation*}(1)

The new DF-SEM equation then becomes:

 \begin{equation*}  \mathbf{u'}(\mathbf{x})=\sqrt{\cfrac{V_{b}k}{N\sigma^{3}}}\sum_{k=1}^{N}\mathbf{K}_{\sigma}\left(\cfrac{\mathbf{x}-\mathbf{x}^{k}}{\sigma}\right)\times C \boldsymbol{\alpha}^{k}  \end{equation*}(2)


Fig 1 - Main components of the Reynolds stress tensor


Fig 2 - <uv> <uw> and <vw> components of the Reynolds stress tensor

These two pictures are taken from a simulation where I used (2). They clearly show that is actually possible to get a different composition of Reynolds stresses (even if I have to underline that I have not properly understood their mechanis).

I will increase my efforts in developing a theory about it which could allow us to better understand it!

Final Scheme implementation

A totally new approach to the problem came in my mind seeing N. Jarrin solution and L. Davidson work. Here it is explained.

As I illustrated earlier on, with the DFSEM is very simple to get an isotropic turbulent fluctuation (where only the main diagonal components of the Reynolds tensor appear).

An elegant solution is then to generate an eddy in the local Reynolds stresses principal coordinate system. In this system basically we will have to generate only the main diagonal components of the Reynolds tensor, which is definitely an easier job. Further, since obviously the three main components are not equals, we may choose a different amplitude in the intensities generation, which should guarantee us the generation of different components!

So step by step scheme:

  1. Calculate the eighenvalues and the eighenvectors for the local Reynolds tensor (QR algorithm? what else?)
  2. Generate all the intensities in the local Reynolds tensor main system ( $ \alpha_i $ must be referenced to the eighenvalues )
  3. Rotate the intensities from the eighenvectors reference system to the main one
  4. Calculate all the velocities as usually

This method is very elegant and it will be potentially able to generate any Reynolds stresses. Further it does not increase the total computational time considerably because computing the eighenvalue and the eighenvectors is necessary only the first time the DF-SEM method is called, since neither Reynolds stresses nor their eighenvalues change as the simulation goes on.

The problem which I have not solved yet are rather numerical (algorithm to calculate eighenvalues and eighenvectors) or scaling-problems (I have not found yet the explicit relation between the intensities amplitude and the output reynolds stresses, because the cross product makes it a bit complicated ), but they all seem easy to be resolved, and this makes me quite confident the method will bring us positive results!


Fig 3 - Reynolds stresses. The $ &amp;lt;\alpha_2^2 &amp;gt; = 4 $ let us have higher values for both

<uu> (blue line) and <ww> (green line), instead <vv> in untouched. This because even if we

modified the y-component of the intensity, it affects all but the <vv> component of the tensor

because of the cross product presence in (2)

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Topic revision: r5 - 2010-06-16 - 14:31:00 - RuggeroPoletto
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22 Jul 2018


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