Statistics

For scale-resolving simulations, statistics can be computed using the command ComputeStatistics. They are computed as accumulating averages starting at the specified time step and stored in a separate CGNS file. For a generic fluctuating quantities \(\phi\) and \(\psi\), the moments are computed as follows:

• First moment Reynolds average

  \begin{equation} \overline \phi = \frac 1 N \sum_n^N \phi_n, \quad \phi = \overline \phi + \phi' \end{equation}

  and Favre average

  \begin{equation} \tilde \phi = \frac{\overline{\rho \phi}}{\rhobar}, \quad \phi = \tilde \phi + \phi'' \end{equation}

• Second moment

  \begin{eqnarray} \overline{\phi' \psi'} &=& \overline{\phi \psi} - \overline \phi \overline \psi \\ \widetilde{\phi'' \psi''} &=& \widetilde{\phi \psi} - \tilde \phi \tilde \psi \end{eqnarray}

The sub-grid stresses are computed based on the Boussinesq assumption (see Original model) and represent the modeled amount of Reynolds stresses. For LES simulations the expression

\begin{equation} \frac{2}{3} \rho^* k^* \delta_{ij} \end{equation}

disappears, because there is no modeled turbulent kinetic energy available (k=0).

Currently the following statistics can be computed.

Moment	CGNS variable name	Symbol	Description
First	Density	\(\rhobar\)	Reynolds-averaged density
First	Pressure	\(\overline p\)	Reynolds-averaged pressure
First	VelocityX/Y/Z	\(\tilde u_i\)	Favre-averaged velocity vector
Second	ReynoldsStressXX/XY/XZ/YY/YZ/ZZ	\(\uu ij\)	Favre-averaged Reynolds stress tensor
Second	ReynoldsStressSgsXX/XY/XZ/YY/YZ/ZZ	\(\uu ij_{SGS}\)	Favre-averaged Reynolds stress (SGS) tensor
Second	PressureStrainXX/XY/XZ/YY/YZ/ZZ	\(\Pi_{ij} = \frac{1}{\rhobar} \overline{p' \left(\parfrac{u_i''}{x_j} + \parfrac{u_j''}{x_i}\right)}\)	Pressure-strain tensor
Second	TurbulentDissipationXX/XY/XZ/YY/YZ/ZZ	\(\epsilon_{ij} = \frac{2}{\rhobar} \overline{\mu s^*_{kj} \parfrac{u_i''}{x_k} + \mu s^*_{ki} \parfrac{u_j''}{x_k}}\)	Turbulent dissipation tensor
Third	SkewnessX/Y/Z	\(\frac{\widetilde{u_i''^3}}{\widetilde{u_i''^2}^{3/2}}\)	Normalized Skewness vector
Fourth	FlatnessX/Y/Z	\(\frac{\widetilde{u_i''^4}}{\widetilde{u_i''^2}^{2}}\)	Normalized Flatness vector

With the trace-free strain rate tensor

\[ s^*_{ij} = s_{ij} - \frac 1 3 s_{qq}, \quad s_{ij} = \frac 1 2 \left(\parfrac{u_i}{x_j} + \parfrac{u_j}{x_i}\right) \]