## Reynolds/Vorticity relationship

### Direct relationship

After a text research, I found the articles in the reference. These articles contain a relationship formula which links somehow the Reynolds stresses and the vorticity.

%BEGINLATEX{label="eq:VortRey"}%\begin{equation*} \frac{\partial}{\partial x_j}(\overline{u'_j u'_i}) = - \epsilon_{ijk} \overline{u'_j \omega_k} + \frac{1}{2} \frac{\partial}{\partial x_i} (\overline{u'_j u'_j}) \end{equation*}%ENDLATEX%

That developed in the x-direction becomes:

%BEGINLATEX{label="eq:VortReyCompx"}%\begin{equation*} \frac{\partial}{\partial x}(\overline{u'^2}) + \frac{\partial}{\partial y}(\overline{u' v'}) + \frac{\partial}{\partial z}(\overline{u' w'}) = \overline{u'_3 \omega_2} - \overline{u'_2 \omega_3} + \frac{1}{2} \frac{\partial}{\partial x} (\overline{u'^2} + \overline{v'^2} + \overline{w'^2} ) \end{equation*}%ENDLATEX%

In Reference 2 you will be able to find the demonstration of the %reflatex{eq:VortRey}%.

This is not a direct relation between the reynold stress and the vorticity. In fact, basically the %reflatex{eq:VortRey}% and %reflatex{eq:VortReyCompx}% could be easier used directly in the Navier-Stokes equations rather than for our purpose.

### 'Vorticity Reynolds stresses'

The second possibility is to use a kind of vorticity Reynolds stresses defined as %$R_{i,j} = \overline{\omega_i \omega_j}$%. This solution simplify a lot the numerical scheme showed in this page, because it allow us to use the same SEM method used for the vorticity field for the vorticity, in particulare with the LUND coefficient, and calculate then the velocity field. I started then a research about this parameter but two problems came out:

1. I found some DNS data (examples: data1, data2), but there is no information about a istantaneous vorticity, only of the mean vorticity field, that is not useful in our scope
2. Even if we found some information, how could these be useful? In fact, from the RANS solution we do not have any information about the vorticity reynolds stresses and I think these should be then modelled somehow.

### Towards a model

First of all let's calculate some important relations and correlations in the theoretical model. For our purpuose we use the Jarrin results in the SEM method and we compare these with the corrispondent results for the DFSEM.

 ORIGINAL SEM DFSEM METHOD %$\mathbf{u}' = \frac{1}{\sqrt{N}} \sum_{k=0}^N a_{ij} \epsilon_j f_{\sigma (\mathbf{x})}(\mathbf{x} - \mathbf{x}^k)$% %$\mathbf{\omega}' = \frac{1}{\sqrt{N}} \sum_{k=0}^N \epsilon_j g_{\sigma (\mathbf{x})}(\mathbf{x} - \mathbf{x}^k)$% auto-correlation %$= a_{im}a_{jm} = R_{ij}$% %$<\omega'_i \omega'_j> = 1$% 2 point correlation %$R_{ij}(\mathbf{u},\mathbf{r}) = R_{ij} \prod_{l=1}^3[f_{\sigma(\mathbf{x})} * f_{\sigma(\mathbf{x+r})}](r_l)$% %$R_{ij}(\mathbf{\omega},\mathbf{r}) = \prod_{l=1}^3[g_{\sigma(\mathbf{x})} * g_{\sigma(\mathbf{x+r})}](r_l)$%

These results are totally equals (except for the Lund coefficients).

#### Vorticity SEM equation from the velocity one

%BEGINLATEX{label="eq:SemDependencies"}%\begin{equation*} u'_i = \frac{1}{\sqrt{N}} \sum_{k=0}^N \left (\sum_{j=1}^3 a_{ij}(\mathbf{x}) ~ \epsilon_j \right ) f_{\sigma (\mathbf{x})}(\mathbf{x} - \mathbf{x}^k) \end{equation*}%ENDLATEX%

%BEGINLATEX{label="eq:Semderivative"}%\begin{equation*} \frac{\partial u'_i}{\partial x_l} = \frac{1}{\sqrt{N}} \sum_{k=0}^N \left [ \left (\sum_{j=1}^3 \frac{\partial a_{ij}(\mathbf{x})}{\partial x_l} ~ \epsilon_j \right ) f_{\sigma (\mathbf{x})}(\mathbf{x} - \mathbf{x}^k) + \left ( \sum_{j=1}^3 a_{ij}(\mathbf{x}) ~ \epsilon_j \right ) \frac{\partial f_{\sigma (\mathbf{x})}(\mathbf{x} - \mathbf{x}^k)}{\partial x_l} \right ] \end{equation*}%ENDLATEX%

Since the general formulation of the vorticity %$\omega_i = \epsilon_{ijk} \frac{\partial u_j}{x_k}$% better write at least one component!!

%BEGINLATEX{label="eq:SemOmega"}%\begin{equation*} \omega'_1 = \frac{1}{\sqrt{N}} \sum_{k=0}^N \left \{ \left [ \left (\sum_{j=1}^3 \frac{\partial a_{2j}(\mathbf{x})}{\partial x_3} ~ \epsilon_j \right ) f_{\sigma (\mathbf{x})}(\mathbf{x} - \mathbf{x}^k) + \left ( \sum_{j=1}^3 a_{2j}(\mathbf{x}) ~ \epsilon_j \right ) \frac{\partial f_{\sigma (\mathbf{x})}(\mathbf{x} - \mathbf{x}^k)}{\partial x_3} \right ] - \left [ \left (\sum_{j=1}^3 \frac{\partial a_{3j}(\mathbf{x})}{\partial x_2} ~ \epsilon_j \right ) f_{\sigma (\mathbf{x})}(\mathbf{x} - \mathbf{x}^k) + \left ( \sum_{j=1}^3 a_{3j}(\mathbf{x}) ~ \epsilon_j \right ) \frac{\partial f_{\sigma (\mathbf{x})}(\mathbf{x} - \mathbf{x}^k)}{\partial x_2} \right ] \right \} \end{equation*}%ENDLATEX%

Considering that the %$f_{\sigma}$% is a symmetric function, all its derivative with respect every direction have exactly the same results, then we are allowed to state that:

%BEGINLATEX{label="eq:SemOmegaDerivative"}%\begin{equation*} \frac{\partial f_{\sigma (\mathbf{x})}(\mathbf{x} - \mathbf{x}^k)}{\partial x_1} = \frac{\partial f_{\sigma (\mathbf{x})}(\mathbf{x} - \mathbf{x}^k)}{\partial x_2} = \frac{\partial f_{\sigma (\mathbf{x})}(\mathbf{x} - \mathbf{x}^k)}{\partial x_3} \end{equation*}%ENDLATEX%

Then we can replace and simpify the formulation as following:

%BEGINLATEX{label="eq:SemOmegaBis"}%\begin{equation*} \omega'_1 = \frac{1}{\sqrt{N}} \sum_{k=0}^N \left \{ \left [ \sum_{j=1}^3 \left ( \frac{\partial a_{2j}(\mathbf{x})}{\partial x_3} - \frac{\partial a_{3j}(\mathbf{x})}{\partial x_2} \right ) \epsilon_j \right ]f_{\sigma (\mathbf{x})}(\mathbf{x} - \mathbf{x}^k) + \left [ \sum_{j=1}^3 \left ( a_{2j}(\mathbf{x}) - a_{3j}(\mathbf{x}) \right ) ~ \epsilon_j \right ] \frac{\partial f_{\sigma (\mathbf{x})}(\mathbf{x} - \mathbf{x}^k)}{\partial x_2} \right \} \end{equation*}%ENDLATEX%

### Structured-based model (by Kassinos & Reynolds)

Some more interesting results came from these model. Basically in chapter 2 we can find this useful model:

%BEGINLATEX{label="eq:Psi"}%\begin{equation*} \frac{\partial \Psi'_i}{\partial x_i} = 0~~~~~~~~~~~~\frac{\partial^2 \Psi'_i}{\partial x_n^2} = - \omega'_i~~~~~~~~~~~~ u'_i = \epsilon_{its} \frac{\partial \Psi'_s}{\partial x_t} \end{equation*}%ENDLATEX%

%BEGINLATEX{label="eq:DiEffeCi"}%\begin{equation*} D_{i,j} = \overline{\frac{\partial \Psi'_n}{\partial x_i}\frac{\partial \Psi'_n}{\partial x_j}}~~~~~~~~~~~~F_{i,j} = \overline{\frac{\partial \Psi'_i}{\partial x_n}\frac{\partial \Psi'_j}{\partial x_n}}~~~~~~~~~~~~C_{i,j} = \overline{\frac{\partial \Psi'_i}{\partial x_n}\frac{\partial \Psi'_n}{\partial x_j}}\end{equation*}%ENDLATEX%

%BEGINLATEX{label="eq:ConstitutiveTurbulence"}%\begin{equation*} R_{i,j} + D_{i,j} + F_{i,j} - (C_{i,j} + C_{j,i}) = q^2 \delta_{j,i} \end{equation*}%ENDLATEX%

where in %reflatex{eq:ConstitutiveTurbulence}%: %$q^2 = R_{i,i}$%

Basically %reflatex{eq:Psi}% is a Posson equation for each component, which has to be solved (suggestion: Green Function).

#### Modelling

All the variables showed in %reflatex{eq:DiEffeCi}% are defined from this tensor:

%BEGINLATEX{label="eq:Qtensor"}%\begin{equation*} Q_{ijk} = - \overline{u'_j \frac{\partial \Psi'_i}{\partial x_k}} \end{equation*}%ENDLATEX%

This tensor is then modelled (notice: it's a 27 equation model!!!) and from this are calculate alle the variable in %reflatex{eq:DiEffeCi}%.

Something more interesting comes from "A simplified structure-based model using standard turbulence scale equations ..." by Kassinos et alt.,where they modelled directly the %$D_{ij}$% tensor as a function of:

%BEGINLATEX{label="eq:Dmodel"}%\begin{equation*} d_{ij} = d_{ij}(S,\tau,\Omega) \end{equation*}%ENDLATEX%

but then we will have still to face with a way to get %$\overline{\omega_i \omega_j}$% from the calculated tensor.

## References

1. Velocity--vorticity correlations related to the gradients of the Reynolds stresses in parallel turbulent wall flows (Download)

J. C. Klewicki, Phys. Fluids A 1, 1285 (1989), DOI:10.1063/1.857354

Nuovo Cimento della Societa Italiana di Fisica C 31 (5-6), pp. 791-811

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pdf GetPDFServlet.pdf manage 404.5 K 2010-01-21 - 15:38 RuggeroPoletto Velocity-vorticity correlations related to the gradients of the Reynolds stresses in parallel turbulent wall flows
pdf ncc9367.pdf manage 4394.0 K 2010-01-25 - 17:28 RuggeroPoletto Three-dimensional turbulence vorticity: Numerical and experimental modeling
Topic revision: r13 - 2010-02-16 - 15:00:59 - RuggeroPoletto
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