Difference between revisions of "User talk:Ferrandm"

Revision as of 16:55, 6 June 2012

Hypotheses

The hypotheses to simplify the Navier-Stokes equations are :

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

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

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

This release can be defined by this equations : ${\displaystyle {\dot {\omega _{T}}}=-\sum _{k=1}^{N}\Delta h_{f,k}^{0}{\dot {\omega _{k}}}=\nu _{F}M_{F}{\dot {\omega }}Q_{m}}$

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

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

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

Thanks to these hypotheses, the total energy balance becomes :

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