# Latex dans WP

Posted on mai 22, 2008

Je viens de trouver un truc bien pratique: on peut mettre du $\LaTeX$ dans les posts. Ben tiens je viens de le faire pour écrire latex Comme quoi, des fois ça sert de lire les FAQ.

Alors je vais un petit test. Si ça vous intéresse lisez la suite.

Now we shall have a look to the variance budget equation. We will start from the conservation equation for inert scalar that reads

$\label{eq1}\frac{\partial {C}}{\partial t}+u_j\frac{\partial {C}}{\partial {x_j}}=+\nu_c\frac{\partial^2C}{\partial{x^2_j}} .$

Splitting the variables into mean and turbulent parts: $C=\overline{c}+c^{\prime}$ and $u_j=\overline{u_j}+u_j^{\prime}$ gives

$\label{eq2}\frac{\partial {\overline{c}}}{\partial t} +\frac{\partial {c^{\prime}}}{\partial t}+\overline{u_j}\frac{\partial {\overline{c}}}{\partial {x_j}}+{u^{\prime}_j}\frac{\partial {\overline{c}}}{\partial {x_j}}+\overline{u_j}\frac{\partial {c^{\prime}}}{\partial {x_j}}+{u^{\prime}_j}\frac{\partial{c^{\prime}}}{\partial {x_j}}=+\nu_c\frac{\partial^2 \overline{c}}{\partial {x^2_j}}+\nu_c\frac{\partial^2 c^{\prime}}{\partial {x^2_j}}$

and the equation for the mean reads

$\label{eq2bis}\frac{\partial {\overline{c}}}{\partial t} +\frac{\partial {c^{\prime}}}{\partial t}+\overline{u_j}\frac{\partial {\overline{c}}}{\partial {x_j}} =-\frac{\partial{\overline{{u^{\prime}_j}c^{\prime}}}}{\partial {x_j}}+\nu_c\frac{\partial^2\overline{c}}{\partial {x^2_j}}$

Then we subtract the mean part from the second equation leaving us for the prognostic equation for the turbulent concentration $c^{\prime}$

$\label{eq3}\frac{\partial {c^{\prime}}}{\partial t}+{u^{\prime}_j}\frac{\partial {\overline{c}}}{\partial {x_j}}+\overline{u_j}\frac{\partial {c^{\prime}}}{\partial {x_j}}+{u^{\prime}_j}\frac{\partial{c^{\prime}}}{\partial {x_j}}=+\frac{\partial {\overline{{u^{\prime}_j}c^{\prime}}}}{\partial {x_j}}+\nu_c\frac{\partial^2c^{\prime}}{\partial {x^2_j}}$

We multiply it by $2c^{\prime}$

$\label{eq4}2c^{\prime}\frac{\partial {c^{\prime}}}{\partial t}+2{u^{\prime}_j}c^{\prime}\frac{\partial {\overline{c}}}{\partial {x_j}}+2\overline{u_j}c^{\prime}\frac{\partial {c^{\prime}}}{\partial {x_j}}+2{u^{\prime}_j}c^{\prime}\frac{\partial {c^{\prime}}}{\partial {x_j}}=+2c^{\prime}\frac{\partial {\overline{{u^{\prime}_j}c^{\prime}}}}{\partial {x_j}}+2c^{\prime}\nu_c\frac{\partial^2 c^{\prime}}{\partial {x^2_j}}$

We use to product rule of calculus to convert term like $2c^{\prime}\frac{\partial {c^{\prime}}}{\partial t}$ into $\frac{\partial {c^{\prime2}}}{\partial t}$

$\label{eq5}\frac{\partial {c^{\prime2}}}{\partial t}+2{u^{\prime}_j}c^{\prime}\frac{\partial {\overline{c}}}{\partial {x_j}}+\overline{u_j}\frac{\partial {c^{\prime2}}}{\partial {x_j}}+{u^{\prime}_j}\frac{\partial {c^{\prime2}}}{\partial {x_j}}=+2c^{\prime}\frac{\partial {\overline{{u^{\prime}_j}c^{\prime}}}}{\partial {x_j}}+2c^{\prime}\nu_c\frac{\partial^2 c^{\prime}}{\partial {x^2_j}}$

Then we average and apply Reynolds averaging rules

$\label{eq6}\frac{\partial {\overline{c^{\prime2}}}}{\partial t}+2\overline{{u^{\prime}_j}c^{\prime}}\frac{\partial {\overline{c}}}{\partial {x_j}}+\overline{u_j}\frac{\partial{\overline{c^{\prime2}}}}{\partial {x_j}}+\overline{{u^{\prime}_j}\frac{\partial {c^{\prime2}}}{\partial {x_j}}}=+2\overline{c^{\prime}\nu_c\frac{\partial^2 c^{\prime}}{\partial {x^2_j}}}$

adding $\overline{c^{\prime2}\frac{\partial {u_j^{\prime2}}}{\partial {x_j}}}=0$ leads to

$\label{eq7}\frac{\partial {\overline{c^{\prime2}}}}{\partial t}+\overline{u_j}\frac{\partial {\overline{c^{\prime2}}}}{\partial {x_j}}=-2\overline{{u^{\prime}_j}c^{\prime}}\frac{\partial {\overline{c}}}{\partial {x_j}}-\frac{\partial {\overline{u^{\prime}_j c^{\prime2}}}}{\partial {x_j}}+2\nu_c\overline{c^{\prime}\frac{\partial^2 c^{\prime}}{\partial {x^2_j}}}$

The dissipation can be written as the sum of molecular diffusion and molecular dissipation i.e.

$2\nu_c\overline{c^{\prime}\frac{\partial^2 c^{\prime}}{\partial {x^2_j}}}=\nu_c\frac{\partial^2 \overline{c^{\prime2}}}{\partial {x^2_j}}-2\nu_c\overline{\left(\frac{\partial c^{\prime}}{\partial {x_j}}\right)^{2}}$

$\frac{\partial {\overline{c^{\prime2}}}}{\partial t}+\overline{u_j}\frac{\partial{\overline{c^{\prime2}}}}{\partial{x_j}}=-2\overline{{u^{\prime}_j}c^{\prime}}\frac{\partial {\overline{c}}}{\partial {x_j}}-\frac{\partial {\overline{u_j c^{\prime2}}}}{\partial {x_j}}-2\nu_c\overline{\left(\frac{\partial c^{\prime}}{\partial {x_j}}\right)^{2}}$
$\frac{\partial {\overline{c^{\prime2}}}}{\partial t}+\overline{u_j}\frac{\partial{\overline{c^{\prime2}}}}{\partial {x_j}}=-2\overline{{u^{\prime}_j}c^{\prime}}\frac{\partial {\overline{c}}}{\partial {x_j}}-\frac{\partial {\overline{u_j c^{\prime2}}}}{\partial {x_j}}-2\epsilon_{\overline{c\prime^{2}}}$
where $\epsilon_{\overline{c\prime^{2}}}=\nu_c\overline{\left(\frac{\partial c^{\prime}}{\partial {x_j}}\right)^{2}}$