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

${\displaystyle {\rho }{\sum }_{k=1}^{N}h_{s,k}Y_{k}V_{k,i}}$ 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 :

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

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

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

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

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

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

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

• The heat flows due to mass fraction gradients and the species molecular distribution due to temperature gradients are neglicted (cf : Soret and Dufour).

• The termal conductivity, ${\displaystyle \lambda }$ is given by :

${\displaystyle \lambda ={\dfrac {\mu C_{p}}{P_{r}}}}$

Where ${\displaystyle P_{r}}$ is the Prandtl number :

${\displaystyle P_{r}={\dfrac {\nu }{D_{th}}}}$

This number compares the momemtum distribution with the heat distribution.

• The Lewis number for the species k, ${\displaystyle Le_{k}}$, is :

${\displaystyle Le_{k}={\dfrac {\lambda }{\rho C_{P}D_{k}}}}$ ${\displaystyle ={\dfrac {D_{th}}{D_{k}}}}$

Where ${\displaystyle D_{k}}$ is the species diffusion.

We will consider the Lewis number egals to 1 in order to simplify the physics of the flame of pre-mixture.

• The Schmidt number, ${\displaystyle S_{ck}}$, is :

${\displaystyle S_{ck}=P_{r}Le_{k}}$