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\section{Discr\'etisation}
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L'intégration des termes de gradient transposé 
 $\dive\,(\mu_{\,tot}\,^t\,\ggrad(\underline{v}))$ et de viscosité
secondaire \\
$-\displaystyle \frac{2}{3}\,\grad (\mu_{\,tot}\,\dive(\underline{v}))$ est la suivante\footnote{la viscosité de volume
$\kappa$ est supposée nulle, cf. \fort{navsto}} : 
\begin{equation}\notag
\begin{array}{llll}
&\displaystyle \int_{\Omega_i}\dive (\mu_{\,tot}
\,^t\,\ggrad(\underline{v})\,)\,d\Omega \\
&= \sum\limits_{l=x,y,z}\left[ \sum\limits_{j \in 
Vois(i)} \mu_{\,tot,ij} ((\displaystyle \frac {\partial v_x}{\partial
l})_{\,moy,ij}\,n_{\,ij}^{\,x} +(\frac {\partial v_y}
 {\partial l})_{\,moy,ij}\,n_{\,ij}^{\,y}+ \displaystyle(\frac {\partial v_z}
{\partial l})_{\,moy,ij}\,n_{\,ij}^{\,z}) S_{\,ij}\right.\\
&+\left.\displaystyle \sum\limits_{k \in \gamma_b(i)} \mu_{\,tot,\,b_{ik}}
((\frac {\partial v_x}{\partial l})_{\,moy,\,b_{ik}}\,n_{\,b_{ik}}^{\,x} +(\frac
{\partial v_y}{\partial l})_{\,moy,\,b_{ik}}\,n_{\,b_{ik}}^{\,y}+
\displaystyle(\frac {\partial v_z}{\partial
l})_{\,moy,\,b_{ik}}\,n_{\,b_{ik}}^{\,z}) S_{\,b_{ik}}\right] \underline
{e}_{\,l} \\\\
&-\displaystyle \frac{2}{3}\,\int_{\Omega_i}\grad (\mu_{\,tot} \dive\,(\underline{v})) 
\,d\Omega \\ 
&= -\displaystyle \frac{2}{3}\,\sum\limits_{l=x,y,z}\left[\sum\limits_{j \in Vois(i)}(\mu_{\,tot}\,\dive(\underline{v}))_{\,ij} S_{\,ij}^{\,l} + \displaystyle \sum_{k \in \gamma_b(i)}(\mu_{\,tot}\,\dive(\underline{v}))_{\,b_{ik}} S_{\,b_{ik}}^{\,l}\right]\,\underline {e}_{\,l}
\end{array}
\end{equation}
Le terme de viscosité $\mu_{\,tot}$ est calculé en fonction du modèle de turbulence utilisé :
\begin{equation}\notag
\mu_{\,tot}=
\begin{cases}
\mu + \mu_t & \text{pour les modèles à viscosité turbulente ou en LES}, \\
\mu& \text{pour les modèles au second ordre ou en laminaire}.
\end{cases}
\end{equation}
où $\mu$ et $\mu_t$ repr\'esentent respectivement la viscosit\'e dynamique moléculaire et la viscosité dynamique turbulente.
