1 The two-phase flow models
2 =========================
4 We present the homogeneised two phase flow models implemented in CoreFlows.
6 This models are obtained by averaging the balance equations for each separated phase or for the mixture, using space, time or ensemble averaged quantities (\ref ishii and \ref Drew ).
8 \ref The drift model is used in the thermal hydraulics softwares \ref flica4 and \ref flocal, whilst the two-fluid models are used in \ref cathare , \ref neptuneCFD, \ref CobraTF , \ref relap5 .
14 The drift model is a system of four nonlinear equations taking the following conservative form
16 \left\{\begin{array}{lll}
17 \partial_t(\alpha_g\rho_g+\alpha_l\rho_l)&+\nabla\cdot(\alpha_g\rho_g{}^t\vec{u}_g+\alpha_l\rho_l{}^t\vec{u}_l)&=0\\
18 \partial_t(\alpha_g\rho_g)&+\nabla\cdot(\alpha_g\rho_g{}^t\vec{u}_g)&=\Gamma_g(h_m,\Phi)\\
19 \partial_t(\alpha_g\rho_g\vec{u}_g+\alpha_l\rho_l\vec{u}_l)&+\nabla\cdot(\alpha_g\rho_g\vec{u}_g\otimes\vec{u}_g+\alpha_l\rho_l\vec{u}_l\otimes\vec{u}_l+p {I}_d)&=\rho_m\vec{g}-K_g\alpha_g\rho_g||\vec{u}_g||\vec{u}_g-K_l\alpha_l\rho_l||\vec{u}_l||\vec{u}_l\\
20 \partial_t(\alpha_g\rho_g E_g+\alpha_l\rho_l E_l)&+\nabla\cdot(\alpha_g\rho_g H_g{}^t\vec{u}_g+\alpha_l\rho_l H_l{}^t\vec{u}_l)&=\Phi+\rho\vec{g}\cdot\vec{u}-K_g\alpha_g\rho_g||\vec{u}_g||^3-K_l\alpha_l\rho_l||\vec{u}_l||^3
23 where the total energy and total enthalpy are defined by
25 E_k=e_k+\frac{1}{2}|\vec{u}_k|^2,\quad H_k=h_k+\frac{1}{2}|\vec{u}_k|^2,\qquad k=v,l,
27 where $e_k$ is the internal energy, and $h_k=e_k+\frac{p}{\rho_k}$ the enthalpy associated to phase $k$ and
30 \rho_m&=&\alpha_g\rho_g+\alpha_l\rho_l\\
31 \vec{u}_m&=&\frac{\alpha_g\rho_g\vec{u}_g+\alpha_l\rho_l\vec{u}_l}{\alpha_g\rho_g+\alpha_l\rho_l}\\
32 h_m&=&\frac{\alpha_g\rho_g h_g+\alpha_l\rho_l h_l}{\alpha_g\rho_g+\alpha_l\rho_l}.
35 We need a drift correlation for the relative velocity:
37 \vec{u}_r=\vec{u}_g-\vec{u}_l=\vec{f}_r(c_g,\vec{u}_m,\rho_m).
39 The phase change is modeled using the formula
41 \Gamma_g=\left\{\begin{array}{cc}
42 \frac{\Phi}{\mathcal{L}}&\textrm{ if } h_l^{sat}\leq h< h_g^{sat} \textrm{ and } 0<\alpha_g<1\\[1.5ex]
43 0& \textrm{ otherwise }
46 The parameters $\lambda_k, \nu_k,\vec g, K_k $ and $\Phi$ can be set by the user.
48 [More details about the drift model are available here](TwoPhase/DriftModelPage.ipynb)
51 The isothermal two-fluid model
52 -----------------------------------------------
54 The model consists in the phasic mass and momentum balance equations.
56 The main unknowns are $\alpha$, $P$, $\vec{u}_g$, $\vec{u}_l$. The model uses stiffened gas laws $p_g(\rho_g)$ and $p_l(\rho_l)$ for a contant temperature $T_0$ provided by the user.
58 The subscript $k$ stands for $l$ for the liquid phase and $g$ for the gas phase. The common
59 averaged pressure of the two phases is denoted by $p$.
61 In our model, pressure equilibrium between the two phases is postulated, and the resulting system to solve is:
65 \frac{\partial m_g}{\partial t}& +& \nabla \cdot \vec{q}_g &= 0,\\[1.5ex]
66 \frac{\partial m_l}{\partial t} &+ &\nabla \cdot \vec{q}_l &= 0,\\[1.5ex]
67 \frac{\partial \vec{q}_g}{\partial t}& +& \nabla \cdot (\vec{q}_g\otimes\frac{\vec{q}_g}{m_g})+ \alpha_g \vec\nabla p&\\[1.5ex]
68 &+&\Delta p \nabla \alpha_g -\nu_g\Delta \vec{u}_g &= m_g\vec{g}-K_gm_g||\vec{u}_g||\vec{u}_g\\[1.5ex]
69 \frac{\partial \vec{q}_l}{\partial t}& +& \nabla \cdot (\vec{q}_l\otimes\frac{\vec{q}_l}{m_l})+ \alpha_l \vec\nabla p&\\[1.5ex]
70 &+&\Delta p \nabla \alpha_l -\nu_l\Delta \vec{u}_l &= m_l\vec{g}-K_lm_l||\vec{u}_l||\vec{u}_l,\\
76 - $\nu_k$ is the viscosity of phase $k$,
77 - $\Delta p$ denotes the pressure default $p-p_k$ between the bulk average pressure and the interfacial average pressure.
81 \left\{\begin{array}{clc}
82 \alpha_g +\alpha_l &=& 1 \\[1.5ex]
83 m_k &=& \alpha_k \rho_k \\[1.5ex]
84 \vec{q}_k &=& \alpha_k \rho_k \vec{u}_k \\[1.5ex]
88 The parameters $\lambda_k, \nu_k,\vec g, K_k $ and $\Phi$ can be set by the user.
90 [More details about the isothermal two-fluid model are available here](IsothermalPage.ipynb)
93 The five equation two-fluid model
94 -----------------------------------------------
97 The model consists in the phasic mass and momentum balance equations and one mixture total energy balance equation.
99 The main unknowns are $\alpha$,$P$,$\vec{u}_g$,$\vec{u}_l$ and $T=T_g=T_l$.
101 The model uses stiffened gas laws $p_g(\rho_g,T)$ and $p_l(\rho_l,T)$.
106 \frac{\partial m_g}{\partial t}& +& \nabla \cdot \vec{q}_g &= \Gamma_g(h_g,\Phi),\\[1.5ex]
107 \frac{\partial m_l}{\partial t} &+ &\nabla \cdot \vec{q}_l &= \Gamma_l(h_l,\Phi),\\[1.5ex]
108 \frac{\partial \vec{q}_g}{\partial t}& +& \nabla \cdot (\vec{q}_g\otimes\frac{\vec{q}_g}{m_g})+ \alpha_g \nabla p&\\[1.5ex]
109 &+&\Delta p \nabla \alpha_g -\nu_g(\Delta \frac{\vec{q}_g}{m_g}) &= m_g\vec{g}-K_gm_g||\vec{u}_g||\vec{u}_g\\[1.5ex]
110 \frac{\partial \vec{q}_l}{\partial t}& +& \nabla \cdot (\vec{q}_l\otimes\frac{\vec{q}_l}{m_l})+ \alpha_l \nabla p&\\[1.5ex]
111 &+&\Delta p \nabla \alpha_l -\nu_l(\Delta \frac{\vec{q}_l}{m_l}) &= m_l\vec{g}-K_lm_l||\vec{u}_l||\vec{u}_l,\\[1.5ex]
112 \partial_t\rho_mE_m&+&\nabla\cdot(\alpha_g\rho_g H_g{}^t\vec{u}_g+\alpha_l\rho_l H_l{}^t\vec{u}_l)&=\Phi+\rho\vec{g}\cdot\vec{u}-K_gm_g||\vec{u}_g||^3-K_lm_l||\vec{u}_l||^3
119 \rho_m&=&\alpha_g\rho_g+\alpha_l\rho_l\\
120 E_m&=&\frac{\alpha_g\rho_g E_g+\alpha_l\rho_l E_l}{\alpha_g\rho_g+\alpha_l\rho_l}.
124 The phase change is modeled using the formula
127 \Gamma_g=\left\{\begin{array}{cc}
128 \frac{\Phi}{\mathcal{L}}&\textrm{ if } h_l^{sat}\leq h< h_g^{sat} \textrm{ and } 0<\alpha_g<1\\[1.5ex]
129 0& \textrm{ otherwise }
133 The parameters $\lambda_k, \nu_k,\vec g, K_k $ and $\Phi$ can be set by the user.
135 [More details about the five equation two-fluid model are available here](TwoPhase/FiveEqPage.ipynb)