Mouton@LaBergerie

Casalingo for one year

Latex dans WP

mai 22, 2008 à 9:45 · Classé sous LaTex and taggé: ,

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}}

Neglecting molecular diffusion leads to

\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}}

or

\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}}

Voilà. Désolé. Je pense pas qu’il y ait d’erreurs, j’ai copié/collé un vieux draft. En général dans les bouquins on trouve des phrases du style “après quelques manipulations, on trouve que” ou alors “the well-know variance budget equation”. Faut au moins le faire une fois.

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