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12.3.4 Modeling the Turbulent Production

The production term, $G_\nu$, is modeled as

 G_\nu = C_{b1} \rho \tilde{S} \tilde{\nu} (12.3-5)

 \tilde{S} \equiv S + \frac{\tilde{\nu}}{\kappa^2 d^2} f_{v2} (12.3-6)

 f_{v2} = 1 - \frac{\chi}{1 + \chi f_{v1}} (12.3-7)

$C_{b1}$ and $\kappa$ are constants, $d$ is the distance from the wall, and $S$ is a scalar measure of the deformation tensor. By default in FLUENT, as in the original model proposed by Spalart and Allmaras, $S$ is based on the magnitude of the vorticity:
 S \equiv \sqrt{2 \Omega_{ij} \Omega_{ij}} (12.3-8)

where $\Omega_{ij}$ is the mean rate-of-rotation tensor and is defined by
 \Omega_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right) (12.3-9)

The justification for the default expression for $S$ is that, for the wall-bounded flows that were of most interest when the model was formulated, turbulence is found only where vorticity is generated near walls. However, it has since been acknowledged that one should also take into account the effect of mean strain on the turbulence production, and a modification to the model has been proposed [ 73] and incorporated into FLUENT.

This modification combines measures of both rotation and strain tensors in the definition of $S$:
 S \equiv \vert\Omega_{ij}\vert + C_{\rm prod} \; \min \left(0, \vert S_{ij}\vert - \vert\Omega_{ij}\vert \right) (12.3-10)


C_{\rm prod} = 2.0, \;\; \vert\Omega_{ij}\vert \equiv \sqrt{2... ...ega_{ij}}, \;\; \vert S_{ij}\vert \equiv \sqrt{2 S_{ij} S_{ij}}

with the mean strain rate, $S_{ij}$, defined as
 S_{ij} = \frac{1}{2}\left(\frac{\partial u_j}{\partial x_i} + \frac{\partial u_i}{\partial x_j} \right) (12.3-11)

Including both the rotation and strain tensors reduces the production of eddy viscosity and consequently reduces the eddy viscosity itself in regions where the measure of vorticity exceeds that of strain rate. One such example can be found in vortical flows, i.e., flow near the core of a vortex subjected to a pure rotation where turbulence is known to be suppressed. Including both the rotation and strain tensors more correctly accounts for the effects of rotation on turbulence. The default option (including the rotation tensor only) tends to overpredict the production of eddy viscosity and hence overpredicts the eddy viscosity itself in certain circumstances.

You can select the modified form for calculating production in the Viscous Model panel.

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