Notation tensorielle
\[
\begin{cases}
\nabla\cdot\boldsymbol{u} = 0 \\
\rho \left( \frac{\partial \boldsymbol{u}}{\partial t} + \left[\boldsymbol{u}\cdot\nabla\right]\boldsymbol{u} \right) = -\nabla p + \nabla\cdot\boldsymbol{\tau} + \boldsymbol{f} \\
\rho c\left( \frac{\partial T}{\partial t} + \left[\boldsymbol{u}\cdot\nabla\right] T \right) = -\nabla\cdot\boldsymbol{q} + \boldsymbol{\tau}:\boldsymbol{d} + r
\end{cases}
\]
Notation en indice d'Einstein
\[
\begin{cases}
\frac{\partial u_j}{\partial x_j} = 0 \\
\rho \left( \frac{\partial u_i}{\partial t} + u_j\frac{\partial u_i}{\partial x_j} \right) = -\frac{\partial p}{\partial x_i} + \frac{\partial \tau_{ij}}{\partial x_j} + f_i \\
\rho c\left( \frac{\partial T}{\partial t} + u_j\frac{\partial T}{\partial x_j} \right) = -\frac{\partial q_j}{\partial x_j} + \tau_{ij}d_{ij} + r
\end{cases}
\]
Notation tensorielle
\[
\begin{cases}
\nabla\cdot\boldsymbol{u} = 0 \\
\rho \left( \frac{\partial \boldsymbol{u}}{\partial t} + \left[\boldsymbol{u}\cdot\nabla\right]\boldsymbol{u} \right) = -\nabla p + \nabla\cdot\boldsymbol{\tau} + \boldsymbol{f} \\
\rho c\left( \frac{\partial T}{\partial t} + \left[\boldsymbol{u}\cdot\nabla\right] T \right) = -\nabla\cdot\boldsymbol{q} + \boldsymbol{\tau}:\boldsymbol{d} + r
\end{cases}
\]
Notation en indice d'Einstein
\[
\begin{cases}
\frac{\partial u_j}{\partial x_j} = 0 \\
\rho \left( \frac{\partial u_i}{\partial t} + \color{red}{u_j\frac{\partial u_i}{\partial x_j}} \right) = -\frac{\partial p}{\partial x_i} + \frac{\partial \tau_{ij}}{\partial x_j} + f_i \\
\rho c\left( \frac{\partial T}{\partial t} + \color{red}{u_j\frac{\partial T}{\partial x_j}} \right) = -\frac{\partial q_j}{\partial x_j} + \tau_{ij}d_{ij} + r
\end{cases}
\]
\(
\frac{\partial}{\partial x_j}\left( u_i u_j \right) = u_j\frac{\partial u_i}{\partial x_j} + u_i\underbrace{\frac{\partial u_j}{\partial x_j}}_{0},\hspace{30px}
\frac{\partial}{\partial x_j}\left( T u_j \right) = u_j\frac{\partial T}{\partial x_j} + T\underbrace{\frac{\partial u_j}{\partial x_j}}_{0}
\)
\[
\begin{cases}
\frac{\partial u_j}{\partial x_j} = 0 \\
\rho \left[ \frac{\partial u_i}{\partial t} + \frac{\partial}{\partial x_j}\left( u_i u_j \right) \right] = -\frac{\partial p}{\partial x_i} + \frac{\partial \tau_{ij}}{\partial x_j} + f_i \\
\rho c\left[ \frac{\partial T}{\partial t} + \frac{\partial}{\partial x_j}\left( T u_j \right) \right] = -\frac{\partial q_j}{\partial x_j} + \tau_{ij}d_{ij} + r
\end{cases}
\]
\[
\phi(t) = \underbrace{\overline{\phi}}_{\text{moyenne constante}} + \underbrace{\phi'(t)}_{\text{fluctuations autour de la moyenne}} \\
\]
Où la moyenne est définie comme
\[
\overline{\phi}(t) = \frac{1}{T}\int_{t_0 - \frac{T}{2}}^{t_0 + \frac{T}{2}} \phi(t)~\text{d}t\\
\overline{\phi}(t) = \overline{\phi},\hspace{30px}\overline{\phi'}(t) = 0,\hspace{30px}\overline{a\phi} = a\overline{\phi},\hspace{30px}\overline{\frac{\partial \phi}{\partial t}}\approx\frac{\partial \overline{\phi}}{\partial t},\hspace{30px}\overline{\frac{\partial \phi}{\partial x_j}}\approx\frac{\partial \overline{\phi}}{\partial x_j}
\]
\[
\begin{align}
\overline{\rho \left[ \frac{\partial u_i}{\partial t} + \frac{\partial}{\partial x_j}\left( u_i u_j \right) \right]} &= \overline{-\frac{\partial p}{\partial x_i} + \frac{\partial \tau_{ij}}{\partial x_j} + f_i} \\
\rho \left[ \frac{\partial \overline{u}_i}{\partial t} + \frac{\partial}{\partial x_j}\left( \overline{u_i u_j} \right) \right] &= -\frac{\partial \overline{p}}{\partial x_i} + \frac{\partial \overline{\tau}_{ij}}{\partial x_j} + \overline{f}_i \\
\end{align}
\]
\[
\begin{align}
\overline{\rho \left[ \frac{\partial u_i}{\partial t} + \frac{\partial}{\partial x_j}\left( u_i u_j \right) \right]} &= \overline{-\frac{\partial p}{\partial x_i} + \frac{\partial \tau_{ij}}{\partial x_j} + f_i} \\
\rho \left[ \frac{\partial \overline{u}_i}{\partial t} + \color{red}{\frac{\partial}{\partial x_j}\left( \overline{u_i u_j} \right)} \right] &= -\frac{\partial \overline{p}}{\partial x_i} + \frac{\partial \overline{\tau}_{ij}}{\partial x_j} + \overline{f}_i \\
\end{align}
\]
\[
\begin{align}
\frac{\partial}{\partial x_j}\left( \overline{u_i u_j} \right) &= \frac{\partial}{\partial x_j}\left( \overline{\left[\overline{u}_i + u_i' \right]\left[\overline{u}_j + u_j'\right]} \right)\\
&=\frac{\partial}{\partial x_j}\left( \overline{ \overline{u}_i\overline{u}_j + \overline{u}_iu_j' + u_i'\overline{u}_j + u_i'u_j'} \right)\\
&=\frac{\partial}{\partial x_j}\left( \overline{\overline{u}_i\overline{u}_j} + \overline{\overline{u}_iu_j'} + \overline{u_i'\overline{u}_j} + \overline{u_i'u_j'} \right)\\
&=\frac{\partial}{\partial x_j}\left( \overline{u}_i\overline{u}_j + \overline{u}_i\underbrace{\overline{u_j'}}_{0} + \underbrace{\overline{u_i'}}_0\overline{u}_j + \overline{u_i'u_j'} \right)\\
\Rightarrow \frac{\partial}{\partial x_j}\left( \overline{u_i u_j} \right) &= \frac{\partial}{\partial x_j}\left( \overline{u}_i\overline{u}_j + \overline{u_i'u_j'} \right)
\end{align}
\]
\[
\begin{align}
\overline{\rho c \left[ \frac{\partial T}{\partial t} + \frac{\partial}{\partial x_j}\left( T u_j \right) \right]} &= \overline{-\frac{\partial q_j}{\partial x_j} + \tau_{ij}d_{ij} + r }\\
\rho c \left[ \frac{\partial \overline{T}}{\partial t} + \frac{\partial}{\partial x_j}\left( \overline{Tu_j} \right) \right] &= -\frac{\partial \overline{q}_j}{\partial x_j} + \overline{\tau_{ij}d_{ij}} + \overline{r}\\
\end{align}
\]
\[
\begin{align}
\overline{\rho c \left[ \frac{\partial T}{\partial t} + \frac{\partial}{\partial x_j}\left( T u_j \right) \right]} &= \overline{-\frac{\partial q_j}{\partial x_j} + \tau_{ij}d_{ij} + r }\\
\rho c \left[ \frac{\partial \overline{T}}{\partial t} + \color{red}{\frac{\partial}{\partial x_j}\left( \overline{Tu_j} \right)} \right] &= -\frac{\partial \overline{q}_j}{\partial x_j} + \color{red}{\overline{\tau_{ij}d_{ij}}} + \overline{r}\\
\end{align}
\]
\[
\begin{align}
\frac{\partial}{\partial x_j}\left( \overline{T u_j} \right) &= \frac{\partial}{\partial x_j}\left( \overline{\left[\overline{T} + T' \right]\left[\overline{u}_j + u_j'\right]} \right)\\
&=\frac{\partial}{\partial x_j}\left( \overline{ \overline{T}\overline{u}_j + \overline{T}u_j' + T'\overline{u}_j + T'u_j'} \right)\\
&=\frac{\partial}{\partial x_j}\left( \overline{\overline{T}\overline{u}_j} + \overline{\overline{T}u_j'} + \overline{T'\overline{u}_j} + \overline{T'u_j'} \right)\\
&=\frac{\partial}{\partial x_j}\left( \overline{T}\overline{u}_j + \overline{T}\underbrace{\overline{u_j'}}_{0} + \underbrace{\overline{T'}}_0\overline{u}_j + \overline{T'u_j'} \right)\\
\Rightarrow \frac{\partial}{\partial x_j}\left( \overline{T u_j} \right) &= \frac{\partial}{\partial x_j}\left( \overline{T}\overline{u}_j + \overline{T'u_j'} \right)
\end{align}
\]
\[
\begin{align}
\Rightarrow \overline{\tau_{ij}d_{ij}} &= \overline{ \left[ \overline{\tau}_{ij} + \tau'_{ij} \right]\left[ \overline{d}_{ij} + d'_{ij} \right] } \\
&=\overline{ \overline{\tau}_{ij}\overline{d}_{ij} + \tau'_{ij}\overline{d}_{ij} + \overline{\tau}_{ij}d'_{ij} + \tau'_{ij} d'_{ij}} \\
&=\overline{\tau}_{ij}\overline{d}_{ij} + \underbrace{\overline{\tau'_{ij}}}_0\overline{d}_{ij} + \overline{\tau}_{ij}\underbrace{\overline{d'_{ij}}}_0 + \overline{\tau'_{ij} d'_{ij}} \\
\Rightarrow \overline{\tau_{ij}d_{ij}} &= \overline{\tau}_{ij}\overline{d}_{ij} + \overline{\tau'_{ij} d'_{ij}}
\end{align}
\]
\[
\begin{cases}
\frac{\partial \overline{u}_i}{\partial x_i} = 0 \\
\rho \left[ \frac{\partial \overline{u}_i}{\partial t} + \frac{\partial}{\partial x_j}\left( \overline{u}_i \overline{u}_j \right) \right] = -\frac{\partial \overline{p}}{\partial x_i} + \frac{\partial }{\partial x_j}\left(\overline{\tau}_{ij} \color{blue}{- \rho\overline{u_i'u_j'}} \right) + \overline{f}_i \\
\rho c \left[ \frac{\partial \overline{T}}{\partial t} + \frac{\partial}{\partial x_j}\left( \overline{T}\overline{u}_j \right) \right] = -\frac{\partial }{\partial x_j}\left( \overline{q}_j \color{red}{- \rho c \overline{T'u'_j}} \right) + \overline{\tau}_{ij}\overline{d}_{ij} + \color{darkviolet}{\overline{\tau'_{ij}d'_{ij}}} + \overline{r}\\
\end{cases}
\]
\[
\begin{cases}
\frac{\partial \overline{u}_i}{\partial x_i} = 0 \\
\rho \left[ \frac{\partial \overline{u}_i}{\partial t} + \frac{\partial}{\partial x_j}\left( \overline{u}_i \overline{u}_j \right) \right] = -\frac{\partial \overline{p}}{\partial x_i} + \frac{\partial }{\partial x_j}\left(\overline{\tau}_{ij} + \color{blue}{\overline{\sigma}^t_{ij}} \right) + \overline{f}_i \\
\rho c \left[ \frac{\partial \overline{T}}{\partial t} + \frac{\partial}{\partial x_j}\left( \overline{T}\overline{u}_j \right) \right] = -\frac{\partial }{\partial x_j}\left( \overline{q}_j +\color{red}{\overline{q}^t_j} \right) + \overline{\tau}_{ij}\overline{d}_{ij} + \color{darkviolet}{\rho \overline{\epsilon}} + \overline{r}\\
\end{cases}\\
\]
Par définition avec une décomposition sphérique - déviatoire. \[ \overline{\sigma}_{ij} = \color{red}{\frac{1}{3}\overline{\sigma}_{kk}\delta_{ij}} + \color{blue}{\overline{\tau}_{ij}} \]
De manière analogue pour le terme turbulent \[ \overline{\sigma}^t_{ij} = \color{red}{\frac{1}{3}\overline{\sigma}^t_{kk}\delta_{ij}} + \color{blue}{\overline{\tau}^t_{ij}} = \color{red}{-\frac{2}{3}\rho\overline{k}\delta_{ij}} + \color{blue}{\overline{\tau}^t_{ij}}\]
On peut donc réécrire la conservation de la qdm
\[
\rho \left[ \frac{\partial \overline{u}_i}{\partial t} + \frac{\partial}{\partial x_j}\left( \overline{u}_i \overline{u}_j \right) \right] = -\frac{\partial}{\partial x_i}\left( \overline{p} + \color{red}{\frac{2}{3}\rho\overline{k}\delta_{ij}} \right) + \frac{\partial }{\partial x_j}\left(\overline{\tau}_{ij} + \color{blue}{\overline{\tau}^t_{ij}} \right) + \overline{f}_i \\
\]
On modélise les vitesses de déformations turbulentes comme celles moléculaires
\[
\overline{\tau}_{ij} = 2\mu\overline{d}_{ij} \sim \overline{\tau}^t_{ij} = 2\mu_t\overline{d}_{ij}
\]
Pareil entre le flux de chaleur turbulent et moléculaire
\[
\overline{q}_j = -k\frac{\partial T}{\partial x_j} \sim \overline{q}^t_j = -k_t\frac{\partial T}{\partial x_j}
\]
\[
\mu_t = \frac{-\rho\overline{u'_iu'_j}}{\frac{\partial u_i}{\partial x_j}},\hspace{30px} k_t = \frac{-\rho c \overline{T'u'_j}}{\frac{\partial T}{\partial x_j}}\
\]
\[
\begin{cases}
\frac{\partial \overline{u}_i}{\partial x_i} = 0 \\
\rho \left[ \frac{\partial \overline{u}_i}{\partial t} + \frac{\partial}{\partial x_j}\left( \overline{u}_i \overline{u}_j \right) \right] = -\frac{\partial }{\partial x_i}\left( \overline{p} + \frac{2}{3}\rho\overline{k}\delta_{ij} \right) + \left[ \mu + \mu_t \right]\frac{\partial^2 \overline{u}_i}{\partial x^2_j} + \overline{f}_i \\
\rho c \left[ \frac{\partial \overline{T}}{\partial t} + \frac{\partial}{\partial x_j}\left( \overline{T}\overline{u}_j \right) \right] = -\left[k + k_t\right]\frac{\partial^2 T}{\partial^2 x_j} + \mu\left( \frac{\partial \overline{u}_j}{\partial x_j} \right)^2 + \rho \overline{\epsilon} + \overline{r}
\end{cases}
\]
\[
\begin{cases}
\frac{\partial \overline{u}_i}{\partial x_i} = 0 \\
\rho \left[ \frac{\partial \overline{u}_i}{\partial t} + \frac{\partial}{\partial x_j}\left( \overline{u}_i \overline{u}_j \right) \right] = -\frac{\partial \overline{p} }{\partial x_i} + \frac{\partial}{\partial x_j}\left( \overline{\tau}_{ij} + \overline{\sigma}^t_{ij} \right) + \overline{f}_i \\
\rho c \left[ \frac{\partial \overline{T}}{\partial t} + \frac{\partial}{\partial x_j}\left( \overline{T}\overline{u}_j \right) \right] = -\frac{\partial }{\partial x_j}\left( \overline{q}_j + \overline{q}^t_j \right) + \overline{\tau}_{ij}\overline{d}_{ij} + \rho \overline{\epsilon} + \overline{r}
\end{cases}
\]