Eddy plot

The following pictures highlight the behaviour of the DFSEM equation WITHOUT the shape function. Basically they only illustrate the cross product $ \mathbf{r}^k \times \mathbf{\alpha}^k $. The different behaviour of the X direction and the other two comes from the fact that only the $ \mathbf{x}^k $ eddy coordinate is affected by time (and convected then), while the Y and Z component are not. As a result, the plane is fixed for the U component, instead moves for the others. These pictures, instead, show the velocity components, respectively, U, V and W. The simulation performed has only one single eddy, which is always generated in the middle of the grid.
plane_x.gif
$ u'=\sqrt{\frac{V_{b}}{N\sigma^{3}}}\sum_{k=1}^{N}((y-y^{k})\alpha_{3}^{k}-(z-z^{k})\alpha_{2}^{k}) $

single_eddy_sinus_x.gif
U velocity component
plane_z.gif
$ v'=\sqrt{\frac{V_{b}}{N\sigma^{3}}}\sum_{k=1}^{N}((z-z^{k})\alpha_{1}^{k}-(x-x^{k})\alpha_{3}^{k}) $

single_eddy_sinus_x.gif
V velocity component
plane_z.gif
$ w'=\sqrt{\frac{V_{b}}{N\sigma^{3}}}\sum_{k=1}^{N}((x-x^{k})\alpha_{2}^{k}-(y-y^{k})\alpha_{1}^{k}) $

single_eddy_sinus_x.gif
W velocity component
$ \mathbf{u}'=\sqrt{\frac{V_{b}}{N\sigma^{3}}}\sum_{k=1}^{N}\mathbf{r}^{k}\times\boldsymbol{\alpha}^{k} $
$ \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\boldsymbol{\alpha}^{k} $

U eddy shape

u_eddy.jpg

In U direction the eddy is basically has always the same shape

(basically divided into two parts, a upper one and a lower one.)

In is just streched and compressed as time goes by.

This because in the plane equation for this eddy there is no time

dependance.

Further the eddy orientation depends only on the random

intensities $ \alpha_i $.

V-W eddy shape

z1_eddy.jpg z1_eddy.jpg z1_eddy.jpg
z1_eddy.jpg z1_eddy.jpg In V and W direction the eddy shape is totally different. It evolves during time, starting from a upper (or lower) part, which later on depevolps a lower (upper) one. This evolution comes from the time dependence of the plane pictured at the top of the page, in V and W direction.

Q-criterion

$ Q = \cfrac{1}{2} [ |\mathbf{\Omega}|^2 + |\mathbf{S}|^2 ] = \cfrac{1}{2} \cfrac{\partial u_i}{\partial x_j} \cfrac{\partial u_j}{\partial x_i}$

q_criterio_single_eddy.gif
q_criterio_multiple_eddy.gif

ciao

qcriterion_multipleddy.gif

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Topic revision: r9 - 2010-06-03 - 16:33:57 - RuggeroPoletto
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