User talk:Ferrandm

Hypotheses

The hypotheses to simplify the Navier-Stokes equations are :

• The total energy balance according to Poinsot-Veynante is descreibed below, if $E=e_{s}+{\dfrac {1}{2}}u_{i}^{2}$ :

${\rho }{\dfrac {DE}{Dt}}={\dfrac {\partial {{\rho }E}}{\partial {t}}}+{\dfrac {\partial {{\rho }u_{i}E}}{\partial {x_{i}}}}={\dot {\omega _{T}}}+{\dfrac {\partial }{\partial {x_{i}}}}({\lambda }{\dfrac {\partial {T}}{\partial {x_{i}}}})-{\dfrac {\partial }{\partial {x_{i}}}}({\rho }{\sum }_{k=1}^{N}h_{s,k}Y_{k}V_{k,i})+{\dfrac {\partial {\sigma _{ij}}u_{i}}{\partial {x_{j}}}}+{\dot {Q}}+{\rho }{\sum }_{k-1}^{N}Y_{k}f_{k,i}(u_{i}+V_{k,i})$

The term ${\dfrac {\partial }{\partial {x_{i}}}}({\rho }{\sum }_{k=1}^{N}h_{s,k}Y_{k}V_{k,i})$ can be ignored with regard to ${\dot {\omega _{T}}}$, which is the heat release due to combustion.

This release can be defined by this equations : ${\dot {\omega _{T}}}=-\sum _{k=1}^{N}\Delta h_{f,k}^{0}{\dot {\omega _{k}}}=\nu _{F}M_{F}{\dot {\omega }}Q_{m}$, where $Q_{m}$ is the specific heat of reaction, $\Delta h_{f,k}^{0}$ is the standard enthalpy of formation and ${\dot {\omega _{k}}}$ is the reaction rate.

${\rho }{\sum }_{k=1}^{N}h_{s,k}Y_{k}V_{k,i}$ is the power produced by volume forces $f_{k}$ on species k.

${\dot {Q}}$ is the heat source term which is null here since there is no electric spark, laser or radiative flux.

The tensor $\sigma _{ij}$ is composed of the tensor of constraints and the tensor of pressure : $\sigma _{ij}=\tau _{ij}-P\delta _{ij}$

Thanks to these hypotheses, the total energy balance becomes :

${\dfrac {\partial \rho E}{\partial t}}+{\dfrac {\partial (P+\rho E)U_{j}}{\partial x_{j}}}={\dfrac {\partial q_{j}}{\partial x_{j}}}+{\dfrac {\partial \tau _{ij}U_{i}}{\partial x_{j}}}+S_{5}$.

With $S_{5}$ which corresponds to the heat release due to combustion, ${\dot {\omega _{T}}}$.

• The term $S_{i}^{*}$ is not used. Thanks to it we can define other forces like electromagnetical force or weight.

• $S_{k}$ is either a fuel or a combustive source term.

$S_{6}=\nu _{F}M_{F}{\dot {\omega }}$ if it is a fuel source term or $S_{7}=\nu _{O}M_{O}{\dot {\omega }}$ if it is an oxidizer source term.

• ${\dot {\omega }}$ can be determined by the Arrhenius law :

${\dot {\omega }}=B_{1}(\phi )\rho Y_{F}Y_{O}e^{\dfrac {\beta }{\alpha }}e^{\dfrac {-T_{a}}{T}},B_{1}(\phi )=cst$

• The heat flows due to mass fraction gradients and the species molecular distribution due to temperature gradients are neglicted.