The shape functions that Nicholas used are the followings:
%BEGINLATEX{label="eq:DFSEM4"}%
\begin{equation*} q_\sigma(y) =
\begin{cases} \frac{1}{\sqrt{4 \pi}} \sqrt{\frac{8}{3}} (sin(\pi y))^2 y, & if~|y| < 1 \\ 0, & otherwise
\end{cases}
\end{equation*}
%ENDLATEX%
%BEGINLATEX{label="eq:DFSEM5"}%
\begin{equation*} q_\sigma(y) =
\begin{cases} \frac{1}{\sqrt{4 \pi}} \sqrt{\frac{35}{87}} \frac{27}{2} (-27x^5+5x^3), & if ~ |y| < \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), & if~|y| < 1 ~ and ~ |y| \ge \frac{1}{3}\\ 0, & otherwise
\end{cases}
\end{equation*}
%ENDLATEX%
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:
%BEGINLATEX{label="eq:shapefunctionrelation"}%
\begin{equation*} \frac{1}{r^2}\frac{d}{dr}q(r) = g(r) \end{equation*}
%ENDLATEX%
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) $%, the shape function in the vorticity field.
Plot of the shape function |
Plot of the derivative of the shape function |
The last picture represents the %$ g(r) $% shape function calculated using %reflatex{eq:shapefunctionrelation}%.
Some question rise: how did N.J. choose his functions? The book which I found suggests every shape function to satisfy the condition %$ \int g(r) dr = 1 $%.
To have a better view of the situation I decided to run a parametrical simulation. Basically I modified the DFSEM method and now the velocity field is calculated as:
%BEGINLATEX{label="eq:schemeparam"}%
\begin{equation*} u = K (C_1 f(r) + C_2 \frac{df(r)}{dr}) \end{equation*}
%ENDLATEX%
This equality is valid even in the following exaples, mutatis mutandis. In this first one I chose the function %reflatex{eq:DFSEM4}%, and of course its derivative.
FIG1 - <uu> with %reflatex{eq:DFSEM4}% and its derivative |
FIG2 - <uv> with %reflatex{eq:DFSEM4}% and its derivative |
FIG3 - <uu> with %reflatex{eq:DFSEM5}% and its derivative |
FIG4 - <uv> with %reflatex{eq:DFSEM5}% and its derivative |
Equation %reflatex{eq:schemeparam}% suggests us another interesting idea: use two (or more) shape function for each eddy, in order to get the reynolds stresses we need.
FIG3 - <uu> with %reflatex{eq:DFSEM4}% and %reflatex{eq:DFSEM5}% |
FIG3 - <uv> with %reflatex{eq:DFSEM4}% and %reflatex{eq:DFSEM5}% |
This is a personal consideration of mine: shouldn't these %$ C_i $% coefficient in equation %reflatex{eq:schemeparam}% satisfy some conditions?
I was thinking the following one: %$ \sum_{k=1}^N C_i=1 $%, which is, for me, a kind of consistency condition.
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:
%BEGINLATEX{label="eq:reynolds"}%
\end{equation*} |
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.
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I | Attachment | Action | Size | Date | Who | Comment |
---|---|---|---|---|---|---|
m | funzioni_dfsem.m | manage | 1.1 K | 2010-03-02 - 14:57 | UnknownUser | MATLAB script used to create a study these shape function |