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Conservation of momentum:
| \displaystyle \rho_P ^{(m-1)} V_P \frac{(v_i)_P ^{(m)} - (v_i)_P^{n-1}}{\Delta t} | ||
| \displaystyle + \sum_{f}^{} \dot{m} _f ^{(m-1)} \left [ (\overrightarrow{v_i})_... ...htarrow{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 - \sum_{f}^{} \mu^T_f A_f (v_{j,i})_f ^{(m-1)} (n_j)_f | ||
| \displaystyle - \frac{2}{3} A_f \left [ \mu^T_f (v_{k,k})_f ^{(m-1)} + \overrightarrow{\rho }_f ^{(m-1)} k_f ^{(m-1)} \right] \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 | (750) |
Conservation of mass:
| \displaystyle \frac{V_P p'_P}{r T_P ^{(m-1)} \Delta t} + \sum_{f}^{} {\overrigh... ...}'}_f^{UD} \frac{\dot{m}_f^*}{r T_f ^{(m-1)} \overrightarrow{\rho}_f ^{(m-1)} } | ||
| \displaystyle - \sum_{f}^{} \overrightarrow{\rho }_f ^{(m-1)} A_f \overline{D}_... ...P}{l_{PF}} + (\nabla p')_f \cdot (\boldsymbol{n}_f - \boldsymbol{j}_f) \right ] | ||
| \displaystyle = V_P \frac{\rho_P ^{n-1} - \rho_P ^{(m-1)} }{\Delta t} - \sum_{f}^{} \dot{m}_f^* | (751) |
Conservation of energy:
| \displaystyle \rho_P ^{(m-1)} (c_v)_P ^{(m-1)} V_P \frac{T_P ^{(m)} - T_P ^{n-1}}{\Delta t} | ||
| \displaystyle + \sum_{f}^{} \dot{m}_f ^{(m)} (c_v)_f ^{(m-1)} \left \{ \overrig... ...verrightarrow{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 + \frac{2}{3}(\nabla \c... ...3} \frac{(\nabla \cdot \boldsymbol{v}_P^{(m)} )^{2} }{T_P ^{(m-1)} } T_P ^{(m)} | ||
| \displaystyle + V_P \Phi \left ( - {p }_P ^{(m)} (\nabla \cdot \boldsymbol{v})_... ...t ) + V_P \Phi \left ( \overrightarrow{\rho }_P^{(m-1)} h_P , T_P \right ) = 0, | (752) |
where
| \Phi(x,y):= \left ( \frac{1+\text{sign}(x)}{2} \right ) x + \left ( \frac{1-\text{sign}(x)}{2} \right ) \frac{y ^{(m)} }{y ^{(m-1)} } x | (753) |
Equation for the turbulent kinetic energy:
| \displaystyle \rho_P ^{(m-1)} V_P \frac{k_P ^{(m)} - k_P ^{n-1}}{\Delta t} | ||
| \displaystyle + \sum_{f}^{} \dot{m}_f ^{(m)} \left \{ \overrightarrow{k}_f^{UD(... ...verrightarrow{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)_P ^{(m-1)} \left( \frac{d_{kl}t_{kl}^l}{\mu} + \frac... ...} \frac{(\nabla \cdot \boldsymbol{v}_P ^{(m)} )^{2} }{k_P ^{(m-1)} } k_P ^{(m)} | ||
| \displaystyle - V_P \beta^* {\rho }_P^{(m)} \omega_P ^{(m-1)} k_P ^{(m)} + V_P ... ... ^{(m)} {k }_P ^{(m-1)} (\nabla \cdot \boldsymbol{v})_p ^{(m)} , k_P \right ) , | (754) |
Equation for the turbulence frequency \omega=k/\nu_t:
| \displaystyle \rho_P ^{(m-1)} V_P \frac{\omega _P ^{(m)} - \omega _P ^{n-1}}{\Delta t} | ||
| \displaystyle + \sum_{f}^{} \dot{m}_f ^{(m)} \left \{ \overrightarrow{\omega }_... ...w{\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... ...} \frac{(\nabla \cdot \boldsymbol{v}_P ^{(m)} )^{2} }{k_P ^{(m-1)} } k_P ^{(m)} | ||
| \displaystyle + V_P \Phi \left( - \frac{2}{3} \gamma_P ^{(m-1)} \rho_P ^{(m)} \... ...ight) - V_P \beta_P ^{(m-1)} {\rho }_P^{(m)} \omega_P ^{(m-1)} \omega _P ^{(m)} | ||
| \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 ) , | (755) |
For flows with no viscosity, no heat conduction, no body forces and no heat sources these equations reduce to (Euler equations):
Conservation of momentum:
| \displaystyle \rho_P ^{(m-1)} V_P \frac{(v_i)_P ^{(m)} - (v_i)_P^{n-1}}{\Delta t} | ||
| \displaystyle + \sum_{f}^{} \dot{m} _f ^{(m-1)} \left [ (\overrightarrow{v_i})_... ...htarrow{v_i})_f ^{(m-1)} - (\overrightarrow{ v_i})_f^{UD(m-1)} \right) \right ] | ||
| \displaystyle = - \sum_{f}^{} \overline{p}_f ^{(m-1)} (n_i)_f A_f | (756) |
Conservation of mass:
| \displaystyle \frac{V_P p'_P}{r T_P ^{(m-1)} \Delta t} + \sum_{f}^{} {\overrigh... ...}'}_f^{UD} \frac{\dot{m}_f^*}{r T_f ^{(m-1)} \overrightarrow{\rho}_f ^{(m-1)} } | ||
| \displaystyle - \sum_{f}^{} \overrightarrow{\rho }_f ^{(m-1)} A_f \overline{D}_... ...P}{l_{PF}} + (\nabla p')_f \cdot (\boldsymbol{n}_f - \boldsymbol{j}_f) \right ] | ||
| \displaystyle = V_P \frac{\rho_P ^{n-1} - \rho_P ^{(m-1)} }{\Delta t} - \sum_{f}^{} \dot{m}_f^* | (757) |
Conservation of energy:
| \displaystyle \rho_P ^{(m-1)} (c_v)_P ^{(m-1)} V_P \frac{T_P ^{(m)} - T_P ^{n-1}}{\Delta t} | ||
| \displaystyle + \sum_{f}^{} \dot{m}_f ^{(m)} (c_v)_f ^{(m-1)} \left \{ \overrig... ...verrightarrow{T}_f ^{(m-1)} - \overrightarrow{T}_f^{UD(m-1)} \right ] \right \} | ||
| \displaystyle + V_P \Phi \left ( - {p }_P ^{(m)} (\nabla \cdot \boldsymbol{v})_p ^{(m)} \right ) = 0 | (758) |