Nichola's shape function

The shape functions that Nicholas used are the followings:


 <br /><br />\begin{equation*} q_\sigma(y) =<br /><br />\begin{cases} \frac{1}{\sqrt{4 \pi}} \sqrt{\frac{8}{3}} (sin(\pi y))^2 y, &amp; if~|y| &amp;lt; 1 \\ 0, &amp; otherwise<br /><br />\end{cases}<br /><br />\end{equation*}<br /><br />(1)

 <br /><br />\begin{equation*} q_\sigma(y) =<br /><br />\begin{cases} \frac{1}{\sqrt{4 \pi}} \sqrt{\frac{35}{87}} \frac{27}{2} (-27x^5+5x^3), &amp; if ~ |y| &amp;lt; \frac{1}{3} \\ \frac{1}{\sqrt{4 \pi}} \sqrt{\frac{35}{87}} \frac{27}{2} (\frac{1}{2}x^3-x^2+\frac{1}{2}x), &amp; if~|y| &amp;lt; 1 ~ and ~ |y| \ge \frac{1}{3}\\ 0, &amp; otherwise<br /><br />\end{cases}<br /><br />\end{equation*}<br /><br />(2)

These shape function (which are referred to the velocity field) are obtained from a eddy shape function in the vorticity field. The relation between these two kind of shape function is the following:

 <br /><br />\begin{equation*} \frac{1}{r^2}\frac{d}{dr}q(r) = g(r) \end{equation*}<br /><br />(3)

Here I plot these function to show their dependency with the distance R. I plot the derivative of each shape function as well because of its relation with the $ g(r) $, the shape function in the vorticity field.

shapefunction.jpg

Plot of the shape function

shapefunction_derivative.jpg

Plot of the derivative of the shape function

gfunction.jpg

The last picture represents the $ g(r) $ shape function calculated using (3).

Some question rise: how did N.J. choose his functions? The book which I found suggests every shape function to satisfy the condition $ \int g(r) dr = 1 $.

shapefunctionsuggested.jpg

Implementation of f(r) and df(r)/dr in a DFSEM

To have a better view of the situation I decided to run a parametrical simulation. Basically I modified the DFSEM method and now the velocity field is calculated as:

 <br /><br />\begin{equation*} u = K (C_1 f(r) + C_2 \frac{df(r)}{dr}) \end{equation*}<br /><br />(4)

This equality is valid even in the following exaples, mutatis mutandis. In this first one I chose the function (1), and of course its derivative.

uu_func_deriv.jpg

FIG1 - <uu> with (1) and its derivative

uv_func_deriv.jpg

FIG2 - <uv> with (1) and its derivative

uu_func2_deriv2.jpg

FIG3 - <uu> with (2) and its derivative

uv_func2_deriv2.jpg

FIG4 - <uv> with (2) and its derivative

Linear combination of different eddy shapes

Equation (4) suggests us another interesting idea: use two (or more) shape function for each eddy, in order to get the reynolds stresses we need.

uu_func_func2.jpg

FIG3 - <uu> with (1) and (2)

uv_func_func2.jpg

FIG3 - <uv> with (1) and (2)

Consideration

This is a personal consideration of mine: shouldn't these $ C_i $ coefficient in equation (4) satisfy some conditions?

I was thinking the following one: $ \sum_{k=1}^N C_i=1 $, which is, for me, a kind of consistency condition.

Reynolds stresses influences

njreynolds.jpg

But is it really so important to have such a reynolds stress approximation? To answer this question I looked back to N. Jarrin job, and in ch. 7 I found this reynolds stresses influence study, of course valid for the SEM method and not for the divergence free one. Anyway from this study we stress that some controll over the output reynold stresses it's important. In fact from this study we can see that controlling the reynolds stresses is very important: even if a fully and completly reconstructed reynolds stress tensor is somehow useless, a schematic one define:

 <br /><br />\begin{equation*} R = \begin{bmatrix} \frac{2}{3}k &amp; 0 &amp; 0 \\ 0 &amp; \frac{2}{3}k &amp; 0\\ 0 &amp; 0 &amp; \frac{2}{3}k \end{bmatrix}  \end{equation*}<br /><br />(5)

This schematic rapresentation of the Reynolds tensor gives us the lowest convergence to the LES fully developed average value. Anyway we need to keep in hand the situation, because with the Reynolds stress coming directly from the DFSEM we are not able to set even this simplification of the tensor.

~

~

#

#

#

#

#

#

#

#

#

#

#

#

#

#


Current Tags:
create new tag
, view all tags
Topic attachments
I Attachment Action Size Date Who Comment
elsem funzioni_dfsem.m manage 1.1 K 2010-03-02 - 14:57 RuggeroPoletto MATLAB script used to create a study these shape function
Topic revision: r6 - 2010-03-09 - 15:25:58 - RuggeroPoletto
Main Web
17 Dec 2017

Site

Manchester CfdTm
Code_Saturne

Ongoing Projects

ATAAC
KNOO

Previous Projects

DESider
FLOMANIA

Useful Links:

User Directory
Photo Wall
Upcoming Events
Add Event
 

Computational Fluid Dynamics and Turbulence Mechanics
@ the University of Manchester
Copyright © by the contributing authors. Unless noted otherwise, all material on this web site is the property of the contributing authors.