Summary of the incompressible flow equations

Conservation of momentum:

  $\displaystyle \rho_P ^{(m-1)} V_P \frac{3(v_i)_P ^{(m)} - 4(v_i)_P^{n-1}+(v_i)_P^{n-2}}{2 \Delta t}$    
  $\displaystyle + \sum_{f}^{} \dot{m} _f ^{(m-1)} \left [ (\overrightarrow{v_i})_...
...verline{v_i})_f ^{(m-1)} - (\overrightarrow{ v_i})_f^{UD(m-1)} \right) \right ]$    
  $\displaystyle = \sum_{f}^{} \mu^T_f A_f \left[ \frac{(v_i)_F ^{(m)} - (v_i)_P ^...
...} + (\nabla v_i)_f ^{(m-1)} \cdot (\boldsymbol{n}_f - \boldsymbol{j}_f) \right]$    
  $\displaystyle - \frac{2}{3} A_f \overrightarrow{\rho }_f ^{(m-1)} k_f ^{(m-1)} \delta_{ij} (n_j)_f$    
  $\displaystyle - \sum_{f}^{} \overline{p}_f ^{(m-1)} (n_i)_f A_f + \rho_P ^{(m-1)} g_i V_P$ (774)

Conservation of mass:

  $\displaystyle \sum_{f\setminus BC}^{} \overrightarrow{\rho }_f ^{(m-1)} A_f \ov...
...bol{r}_P) }{(\boldsymbol{r}_F - \boldsymbol{r}_P)\cdot \boldsymbol{n}} \right )$    
  $\displaystyle = \sum_{f\setminus BC}^{} \overrightarrow{\rho }_f ^{(m-1)} A_f \boldsymbol{v}_f^{\Box} \cdot \boldsymbol{n}_f +\sum_{BC}^{} \dot{m} _f ^{(m)},$ (775)

Conservation of energy:

  $\displaystyle \rho_P ^{(m-1)} (c_v)_P ^{(m-1)} V_P \frac{3 T_P ^{(m)} - 4 T_P ^{n-1}+T_P^{n-2}}{2 \Delta t}$    
  $\displaystyle + \sum_{f}^{} \dot{m}_f ^{(m)} (c_v)_f ^{(m-1)} \left \{ \overrig...
...t [ \overline{T}_f ^{(m-1)} - \overrightarrow{T}_f^{UD(m-1)} \right ] \right \}$    
  $\displaystyle + \sum_{f}^{} \lambda_f^T A_f \left [ \frac{T_F ^{(m)} - T_P ^{(m...
...}} + (\nabla T)_f ^{(m-1)} \cdot (\boldsymbol{n}_f - \boldsymbol{j}_f) \right ]$    
  $\displaystyle +V_P \mu ^{(m-1)} \left( d_{kl}t_{kl}/\mu \right )_P ^{(m)} + V_P \Phi \left ( \overrightarrow{\rho }_P^{(m-1)} h_P , T_P \right ) ,$ (776)

Equation for the turbulent kinetic energy:

  $\displaystyle \rho_P ^{(m-1)} V_P \frac{3k_P ^{(m)} - 4k_P ^{n-1}+k_P^{n-2}}{2 \Delta t}$    
  $\displaystyle + \sum_{f}^{} \dot{m}_f ^{(m)} \left \{ \overrightarrow{k}_f^{UD(...
...t [ \overline{k}_f ^{(m-1)} - \overrightarrow{k}_f^{UD(m-1)} \right ] \right \}$    
  $\displaystyle + \sum_{f}^{} (\mu+\sigma_k \mu_t)_f^{(m-1)} A_f \left [ \frac{k_...
...}} + (\nabla k)_f ^{(m-1)} \cdot (\boldsymbol{n}_f - \boldsymbol{j}_f) \right ]$    
  $\displaystyle +V_P \mu_t ^{(m-1)} \left( {d_{kl}t_{kl}^l}/{\mu} \right )_P ^{(m)} - V_P \beta^* {\rho }_P^{(m)} \omega_P ^{(m-1)} k_P ^{(m)} ,$ (777)

Equation for the turbulence frequency $ \omega=k/\nu_t$:

  $\displaystyle \rho_P ^{(m-1)} V_P \frac{3\omega _P ^{(m)} - 4\omega _P ^{n-1}+\omega_P^{n-1}}{2 \Delta t}$    
  $\displaystyle + \sum_{f}^{} \dot{m}_f ^{(m)} \left \{ \overrightarrow{\omega }_...
...e{\omega }_f ^{(m-1)} - \overrightarrow{\omega }_f^{UD(m-1)} \right ] \right \}$    
  $\displaystyle + \sum_{f}^{} (\mu+\sigma_\omega \mu_t)_f^{(m-1)} A_f \left [ \fr...
...\nabla \omega )_f ^{(m-1)} \cdot (\boldsymbol{n}_f - \boldsymbol{j}_f) \right ]$    
  $\displaystyle +V_P \rho _P ^{(m)} \gamma_P ^{(m-1)} \left( \frac{d_{kl}t_{kl}^l...
...P \beta_P ^{(m-1)} {\rho }_P^{(m)} \omega_P ^{(m-1)} \omega _P ^{(m)} \nonumber$    
  $\displaystyle + V_P \Phi \left ( 2 \rho_P ^{(m)} (1-F_1 ^{(m-1)} )_P {\sigma_\o...
...(m)} \cdot \nabla \omega_P ^{(m-1)} }{\omega_P ^{(m-1)} } , \omega_P \right ) ,$ (778)