Alternative Shape Functions

From the final version of the DF-SEM it is possible to obtain the equations the Shape function has to satisfy in order to produce a DF method which is able to reproduce all the States of turbulence!!

 <br /><br />\begin{equation*}  \mathbf{u}'=\sqrt{\frac{V_{b}}{N\sigma^{3}}}\sum_{k=1}^{N}\frac{q_{\sigma}(r^{k})}{(r^{k})^{3}}\mathbf{r}^{k}\times [ R_L^M \{ \boldsymbol{ C \alpha }^{k} \} ]  \end{equation*}<br /><br />(1)

Assuming the method uses several shape functions, a different one in each direction ($ q_x $, $ q_y $ and $ q_z $), it is possible, defining $ A_i = \cfrac{q_i}{(\frac{r^k}{\sigma})^3} $ and $ \boldsymbol{\beta} = \mathbf{r}^{k}\times [ R_L^M \{ \boldsymbol{ C \alpha }^{k} \} ] $, to write down the system which defines the shape functions:

 <br /><br />\begin{equation*}  \begin{cases}<br />(y-y^{k})\cfrac{\partial A_{1}}{\partial x}=(x-x^{k})\cfrac{\partial A_{2}}{\partial y}\\<br />(z-z^{k})\cfrac{\partial A_{1}}{\partial x}=(x-x^{k})\cfrac{\partial A_{3}}{\partial z}\end{cases}  \end{equation*}<br /><br />(2)

In case $ q_x = q_y = q_z = q_{\sigma} $ and $ q_{\sigma}(r^k) $ is function of the only distance $ r^k $, since $ \frac{\partial r}{\partial x_i} - \frac{x_i - x_i^k}{r^k} $, the system (2) is verified for every $ q_{\sigma} $!!

Process:

  • define q_x
  • use system (2) to find q_y and q_z

It is important to stress out that the shape functions $ q_x, q_y, q_z $ must always respect these conditions:

  1. approach zero at leas as fast as $ (r^k)^3 $
  2. possibly $ q(r/ \sigma)=0 $ if $ (r/ \sigma)&amp;gt;1 $

##################

Equation from system:

$ (x-x^k) (\cfrac{\partial q_y}{\partial y})(\cfrac{r^k}{\sigma})^3 - 3 \cfrac{(x-x^k)(y-y^k)}{r^k \sigma}(r^k)^2 (q_y - q_x) - (y-y^k)(\cfrac{\partial q_x}{\partial x})(\cfrac{r^k}{\sigma})^3 = 0 $

qy_shape_function_if_qxsinpixx2.png

TO BE DONE:

verify the new Reynolds stresses since they are dependent on $ q_i^2 $, the new shape functions with different components in x, y and z may positively affect them and let us reproduce every kind of turbulence!


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pngpng q_sigma.png manage 121.4 K 2010-09-20 - 11:14 RuggeroPoletto  
Topic revision: r5 - 2010-09-20 - 11:14:35 - RuggeroPoletto
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