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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) |