Difference between revisions of "Contrib/CompressibleMixingPhaseChangeFoam"

Latest revision as of 19:45, 21 March 2015

Solver for two fluids with phase change (for example - water <---> steam), pressure and temperature density dependence Valid versions:

1 Model Equations Derivation

VALID OPENFOAM VERSION - 2.1.0

Equation of state

Low-compressible fluid: $\rho = \rho_0 + \frac{\partial \rho}{\partial T} \Delta T + \frac{\partial \rho}{\partial p} \Delta p$

Ideal gas: $\rho=\frac{p}{(R/M)T}$

By combining this equations, we can get general relation:

$\rho = \hat \rho + \frac{\partial \rho}{\partial p} \Delta p$

where $\hat \rho$ computed with respect to previous formulations

mixture density $\rho$ calculated as

$\rho = \alpha_l \rho_l + \alpha_v \rho_v$

Here, and afterthere indices:

1,l,f - for liquid (heavy media with low compressibility)

2,g,s - for gas (light media (like steam) with big compressibility)

without index - mixture variable (or all variables local to some phase)

Liquid volume transport

Let us consider transport of liquid (heavy phase) volume fraction $\alpha_l$ :

$\frac{\partial \alpha_l \rho_l}{\partial t} + \nabla \cdot \left ( \alpha_l \rho_l \mathbf{U} \right ) = \dot m_l$

By converting to volume fluxes we get:

$\frac {\partial \alpha_l}{\partial t} + \nabla \cdot \left ( \alpha_l \textbf{U} \right ) = -\frac {\alpha_l}{\rho_l} \frac {d \rho_l}{dt} + \frac {\dot m_l}{\rho_l}$

Using equation of state, we can reformulate substantial derivative for density in terms of pressure for any phase:

$\frac {d \rho}{dt}= \frac {d \hat \rho} {dt} + \psi \frac {dp} {dt} + p \frac {d \psi}{dt}$

General rule for converting from mass to volume fluxes in transport equation

$\frac{\partial \alpha_i \rho_i \zeta}{\partial t} + \nabla \cdot \left ( \alpha_i \rho_i \textbf{U} \zeta \right ) = \rho_i \alpha_i \left ( \frac {\partial \zeta}{\partial t} + \nabla \cdot \left ( \textbf{U} \zeta \right ) \right ) + \zeta \alpha_l \frac {d \rho_i}{dt} + \zeta \rho_i \frac {d \alpha_i}{dt}$

Momentum equation (velocity prediction)

$\frac {\partial \rho \textbf{U}}{\partial t} + \nabla \cdot \rho \textbf{U} \textbf{U} = \nabla \cdot R^{Eff} - \nabla p + \rho \textbf{g}$

by substituting piezometric pressure $\hat p = p - \rho \textbf{g} \textbf{x}$ we get:

$\frac {\partial \rho \textbf{U}}{\partial t} + \nabla \cdot \rho \textbf{U} \textbf{U} = \nabla \cdot R^{Eff} - \nabla \hat p - \textbf{g} \cdot \nabla \rho$

Pressure equation obtained by summation of equation for volume phase fraction of liquid and gas phases:

$\nabla \cdot \left ( \textbf{U} \right ) = - \left ( \frac {\alpha_1 \psi_1}{\rho_1} + \frac {\alpha_2 \psi_2}{\rho_2} \right ) \frac {d p}{d t} - \left ( \frac {\alpha_1}{\rho_1} \frac {d \psi_1}{d t} + \frac {\alpha_2}{\rho_2} \frac {d \psi_2}{d t} \right ) p - \left ( \frac {\alpha_1}{\rho_1} \frac {d \hat \rho_1}{d t} + \frac {\alpha_2}{\rho_2} \frac {d \hat \rho_2}{d t} \right )+ \dot m_1 \left ( \frac{1}{\rho_1} - \frac{1}{\rho_2} \right )$

Energy equation

Energy equation for mixture temperature obtained from sum of energy equations for each phase. Consider internal energy equation for phase-1:

$\frac{\partial \alpha_1 \rho_1 e_1} {\partial t} + \nabla \cdot \left ( \alpha_1 \rho_1 e_1 \textbf{U} \right ) + \nabla \cdot \textbf{q}_1 = -\alpha_1 p (\nabla \cdot \textbf{U} - \dot m_1 / \rho_1) + \dot m_1 e_1$

By combining equations of phases, we get internal energy balance for mixture in temperatures:

$\frac{\partial T}{\partial t} + \nabla \cdot \left ( T \textbf{U} \right ) - T \nabla \cdot \textbf{U} -\frac{1}{\alpha_1 \rho_1 C_{v,1} + \alpha_2 \rho_2 C_{v,2}} (\nabla \cdot \kappa_1 \nabla T + \nabla \cdot \kappa_2 \nabla T) =$

$-\left ( \nabla \cdot \textbf{U} - \left(\frac{1}{\rho_1}-\frac{1}{\rho_2}\right) \dot m_1 \right ) \frac{p}{\alpha_1 \rho_1 C_{v,1} + \alpha_2 \rho_2 C_{v,2}} - \left ( \frac{\partial L}{\partial t} + \nabla \cdot (\textbf{U} L) - L \nabla \cdot \textbf{U} \right ) \frac{\alpha_2 \rho_2}{\alpha_1 \rho_1 C_{v,1} + \alpha_2 \rho_2 C_{v,2}}$

Where $L$ - latent heat of evaporation

Linking liquid volume transport to pressure equation is done by introducing $+\alpha_1 \nabla \cdot \textbf{U}$ and $-\alpha_1 \nabla \cdot \textbf{U}$ at r.h.s of volume fraction balance equation. Then, $-\nabla \cdot \textbf{U}$ replaced by value from pressure equation

$\frac {\partial \alpha_l}{\partial t} + \nabla \cdot \left ( \alpha_l \textbf{U} \right ) = \alpha_1 \nabla \cdot \textbf{U} + \dot m_1 \left (\frac{1}{\rho_1} - \alpha_1 \left (\frac{1}{\rho_1} - \frac{1}{\rho_2} \right ) \right ) +$

$\alpha_1 \alpha_2 \left ( -\frac{1}{\rho_1} \frac{d \hat \rho_1}{d t} + \frac{1}{\rho_2} \frac{d \hat \rho_2}{d t}\right ) +\alpha_1 \alpha_2 p \left ( -\frac{1}{\rho_1} \frac{d \psi_1}{d t} + \frac{1}{\rho_2} \frac{d \psi_2}{d t}\right ) +\alpha_1 \alpha_2 \frac{d p}{d t}\left ( -\frac{\psi_1}{\rho_1} + \frac{\psi_2}{\rho_2} \right )$

2 Implemented phase change models

Phase change model Schnerr-Sauer
Phase change model Zwart
Phase change model Singhal
Phase change model Merkle
Phase change model Kunz

3 Model Source Code For OpenFOAM 2.1.0

Source code for solver located here

4 Tutorial cases

Case #1 - cavitation on the hemispherical body

Problem description, initial and boundary conditions are shown on the figure

Figure 1.1

Volume fraction of liquid (water) is shown on the next figure Figure 1.2

Case #2 - hot water flashing in the nozzle

Problem description, initial and boundary conditions are shown on the figure 2.1

Figure 2.1

Volume and mass fractions of vapour are shown on the figures 2.2 and 2.3 Pressure distribution is shown on figure 2.4 Velocity magnitude and Mach number for compressible case are shown on figures 2.5 and 2.6 Velocity magnitude for incompressible case is shown on the case 2.7 Temperature distribution is located of figure 2.8

Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6

Figure 2.7

Figure 2.8

@@ Line 1: / Line 1: @@
 Solver for two fluids with phase change (for example - water <---> steam), pressure and temperature density dependence
+{{VersionInfo}}{{Version2.1.0}}
 == Model Equations Derivation ==
+VALID OPENFOAM VERSION - 2.1.0
 * '''Equation of state'''
 Low-compressible fluid: <math>
@@ Line 7: / Line 8: @@
 </math>
-Ideal gas: <math> \rho=\frac{p}{(C_p/C_v)(R/M)T}
+Ideal gas: <math> \rho=\frac{p}{(R/M)T}
 </math>
@@ Line 107: / Line 108: @@
 * '''Energy equation'''
-Energy equation for mixture temperature obtained from sum of energy equations for each phase. Consider energy equation for phase-1:
+Energy equation for mixture temperature obtained from sum of energy equations for each phase. Consider internal energy equation for phase-1:
 <math>
 \frac{\partial \alpha_1 \rho_1 e_1} {\partial t} +
-\nabla \cdot \left ( \alpha_1 \rho_1 e_1 \textbf{U} \right ) - \nabla \cdot \textbf{q}_1
+\nabla \cdot \left ( \alpha_1 \rho_1 e_1 \textbf{U} \right ) + \nabla \cdot \textbf{q}_1
 =
+-\alpha_1 p (\nabla \cdot \textbf{U} - \dot m_1 / \rho_1)
++
 \dot m_1 e_1
 </math>
-by converting to enthalpies we get:
-<math>
+By combining equations of phases, we get internal energy balance for mixture in temperatures:
-\frac{\partial \alpha_1 \rho_1 h_1} {\partial t} +
-\nabla \cdot \left ( \alpha_1 \rho_1 h_1 \textbf{U} \right )-
-\left (
-\frac {\partial \alpha_1 p}{\partial t} + \nabla \cdot \left ( \alpha_1 p \textbf{U} \right )
-\right )
-- \nabla \cdot \textbf{q}_1
-=
-\dot m_1 h_1 - \dot m_1 \frac{p}{\rho_1}
-</math>
-By substituting temperature instead of enthalpy, after conversion to volume fluxes we get equation for temperature (divided by
-<math>\rho_1 C_{p,1} </math> )
-<math>
-\alpha_1 \left (\frac{\partial T}{\partial t} + \nabla \cdot \left ( T \textbf{U} \right ) \right )
-+ T \frac{\alpha_1}{\rho_1} \frac{d \rho_1}{d t} + T \frac{d \alpha_1}{d t} - \frac{1}{\rho_1 C_{p,1}} \nabla \cdot \textbf{q}_1
-=
-T \frac{\dot m_1}{\rho_1} - \frac{1}{\rho_1 C_{p,1}} \frac{p}{\rho_1} \dot m_1 +
-\frac{1}{\rho_1 C_{p,1}}
-\left (
-\frac {\partial \alpha_1 p}{\partial t} + \nabla \cdot \left ( \alpha_1 p \textbf{U} \right )
-\right )
-</math>
-By combining equations of phases, we get energy balance for mixture:
 <math>
 \frac{\partial T}{\partial t} + \nabla \cdot \left ( T \textbf{U} \right ) - T \nabla \cdot \textbf{U}
--\frac{1}{\rho_1 C_{p,1}} \nabla \cdot \kappa_1 \nabla T -\frac{1}{\rho_2 C_{p,2}} \nabla \cdot \kappa_2 \nabla T
+-\frac{1}{\alpha_1 \rho_1 C_{v,1} + \alpha_2 \rho_2 C_{v,2}} (\nabla \cdot \kappa_1 \nabla T + \nabla \cdot \kappa_2 \nabla T)
 =
 </math>
 <math>
-\left ( \frac{\alpha_1}{\rho_1 C_{p,1}} + \frac{\alpha_2}{\rho_2 C_{p,2}} \right ) \frac{d p}{d t}
+-\left ( \nabla \cdot \textbf{U} - \left(\frac{1}{\rho_1}-\frac{1}{\rho_2}\right) \dot m_1 \right )
-+ \frac{p}{\rho_1 C_{p,1}}\frac{d \alpha_1}{d t} + \frac{p}{\rho_2 C_{p,2}}\frac{d \alpha_2}{d t}
+\frac{p}{\alpha_1 \rho_1 C_{v,1} + \alpha_2 \rho_2 C_{v,2}}
-+ \dot m_1 p \left (-\frac{1}{\rho_1 \rho_1 C_{p,1}} + \frac{1}{\rho_2 \rho_2 C_{p,2}}\right )
+- \left ( \frac{\partial L}{\partial t} + \nabla \cdot (\textbf{U} L) - L \nabla \cdot \textbf{U} \right )
+\frac{\alpha_2 \rho_2}{\alpha_1 \rho_1 C_{v,1} + \alpha_2 \rho_2 C_{v,2}}
 </math>
+Where <math> L </math> - latent heat of evaporation
 * '''Linking liquid volume transport to pressure equation''' is done by introducing <math>+\alpha_1 \nabla \cdot \textbf{U}</math> and <math>-\alpha_1 \nabla \cdot \textbf{U}</math> at r.h.s of volume fraction balance equation. Then, <math>-\nabla \cdot \textbf{U}</math> replaced by value from pressure equation
@@ Line 166: / Line 146: @@
 =
 \alpha_1 \nabla \cdot \textbf{U} +
-\dot m_1 \left (\frac{1}{\rho_1} \alpha_1 - \left (\frac{1}{\rho_1} - \frac{1}{\rho_2} \right ) \right ) +
+\dot m_1 \left (\frac{1}{\rho_1} - \alpha_1 \left (\frac{1}{\rho_1} - \frac{1}{\rho_2} \right ) \right ) +
 </math>
@@ Line 175: / Line 155: @@
 </math>
-* Phase change model
+== Implemented phase change models ==
+* Phase change model Schnerr-Sauer
+* Phase change model Zwart
+* Phase change model Singhal
+* Phase change model Merkle
+* Phase change model Kunz
+== Model Source Code For OpenFOAM 2.1.0 ==
+[http://www.os-cfd.ru/compressibleMixingPhaseChangeFoam/compressibleMixingPhaseChangeFoam-New.tgz Source code for solver located here]
+== Tutorial cases ==
+* '''Case #1 - cavitation on the hemispherical body'''
+Problem description, initial and boundary conditions are shown on the figure
+[[File:cavitator.jpg]]
+'''Figure 1.1'''
+Volume fraction of liquid (water) is shown on the next figure
+[[File:H-alpha1.png]]
+'''Figure 1.2'''
+* '''Case #2 - hot water flashing in the nozzle'''
+Problem description, initial and boundary conditions are shown on the figure 2.1
+[[File:Water-flashing.jpg]]
+'''Figure 2.1'''
+Volume and mass fractions of vapour are shown on the figures 2.2 and 2.3
+Pressure distribution is shown on figure 2.4
+Velocity magnitude and Mach number for compressible case are shown on figures 2.5 and 2.6
+Velocity magnitude for incompressible case is shown on the case 2.7
+Temperature distribution is located of figure 2.8
+[[File:f-alpha2.png]]
+'''Figure 2.2'''
+[[File:f-X2.png]]
+'''Figure 2.3'''
+[[File:f-pprofile.png]]
+'''Figure 2.4'''
+[[File:f-magU.png]]
+'''Figure 2.5'''
+[[File:f-Mach.png]]
+'''Figure 2.6'''
+[[File:f-magU-ico.png]]
+'''Figure 2.7'''
+[[File:f-T.png]]
+'''Figure 2.8'''
-== Model Equations Summary ==
+[[Category:Multiphase flow solvers]]
-[http://www.os-cfd.ru/compressibleMixingPhaseChangeFoam/Solver.tgz Solver sources and tutorials located here]