Work in progress. These notes describe OpenFOAM 9. Sections marked “Coming soon” are incomplete.
CompressibleInterFoam is a default multiphase solver in OpenFOAM. It is capable of solving two compressible, non-isothermal immiscible fluids, with/without cavitation. The phase interface is captured using volume of fluid (VoF) approach. The following analysis is based on OpenFOAM 9.
Conservation equations
The phase-wise mass-conservation equations are
\begin{equation} \label{eq:mass_alpha1}
\frac{\partial\left(\rho_1 \alpha_1\right)}{\partial t}+\nabla \cdot\left(\rho_1 \boldsymbol{U} \alpha_1\right)=\dot{m}
\end{equation}
\begin{equation} \label{eq:mass_alpha2}
\frac{\partial\left(\rho_2 \alpha_2\right)}{\partial t}+\nabla \cdot\left(\rho_2 \boldsymbol{U} \alpha_2\right)=-\dot{m}
\end{equation}
where \(\alpha_{1}+\alpha_{2} = 1\); \(\dot{m}\) is the source term from phase change. If there is no phase change, \(\dot{m}\) is zero.
Consider only the volume fraction in the l.h.s., and move other parts to the r.h.s. to be the source term. Eq. \(\eqref{eq:mass_alpha1}\) and \(\eqref{eq:mass_alpha2}\) were re-written as
\begin{equation} \label{eq:mass_alpha1_comp}
\frac{\partial \alpha_1}{\partial t}+\nabla \cdot\left(\boldsymbol{U} \alpha_1\right)=-\frac{\alpha_1}{\rho_1} \frac{D \rho_1}{D t} + \frac{\dot{m}}{\rho_1}
\end{equation}
\begin{equation} \label{eq:mass_alpha2_comp}
\frac{\partial \alpha_2}{\partial t}+\nabla \cdot\left(\boldsymbol{U} \alpha_2\right)=-\frac{\alpha_2}{\rho_2} \frac{D \rho_2}{D t} - \frac{\dot{m}}{\rho_2}
\end{equation}
Adding these two equations gives
\begin{equation} \label{eq:divU}
\nabla \cdot\boldsymbol{U}=-\left(\frac{\alpha_1}{\rho_1} \frac{D \rho_1}{D t}+\frac{\alpha_2}{\rho_2} \frac{D \rho_2}{D t}\right)+\dot{m}\left(\frac{1}{\rho_1}-\frac{1}{\rho_2}\right)
\end{equation}
The 1st and 2nd term in the r.h.s. are the compressibility and phase change effects.
Substitute this expression into Eq. \(\eqref{eq:mass_alpha1_comp}\) by adding and subtracting \(\alpha_{1}\nabla \cdot\boldsymbol{U}\):
\begin{equation} \label{eq:mass_alpha1_final}
\frac{\partial \alpha_1}{\partial t}+\nabla \cdot\left(\boldsymbol{U} \alpha_1\right)=\alpha_1 \alpha_2 \left(\frac{1}{\rho_1} \frac{D \rho_1}{D t}-\frac{1}{\rho_2} \frac{D \rho_2}{D t}\right)
+\alpha_1 \nabla \cdot \boldsymbol{U}+\dot{m}\left(\frac{\alpha_2}{\rho_1}+\frac{\alpha_1}{\rho_2}\right)
\end{equation}
The compressibility, \(\alpha_1 \nabla \cdot \boldsymbol{U}\) and phase change are source terms for the alpha equation. They are separated into \(S_{p}\) and \(S_{u}\) in the MULES algorithm. \(S_{p}\) is implicit source term, \(S_{u}\) is explicit source term.
MULES method
Source-term decomposition
In Eq. \(\eqref{eq:mass_alpha1_final}\), it was defined
\begin{equation}
\mathrm{dgdt}=-\alpha_1 \alpha_2 \left(\frac{1}{\rho_1} \frac{D \rho_1}{D t}-\frac{1}{\rho_2} \frac{D \rho_2}{D t}\right)
\end{equation}
dgdt was calculated in the pEqn, i.e.
dgdt =
(
alpha1*(p_rghEqnComp2 & p_rgh)
- alpha2*(p_rghEqnComp1 & p_rgh)
);
In solving Eq. \(\eqref{eq:mass_alpha1_final}\), dgdt was separated according to its sign in each cell
if (dgdt[celli] > 0.0)
{
SpRef[celli] -= dgdt[celli]/max(1.0 - alpha1[celli], 1e-4);
SuRef[celli] += dgdt[celli]/max(1.0 - alpha1[celli], 1e-4);
}
else if (dgdt[celli] < 0.0)
{
SpRef[celli] += dgdt[celli]/max(alpha1[celli], 1e-4);
}
The code means, if dgdt > 0
\[S_p = \text{phaseChangeS}[1] -\frac{\mathrm{dgdt}}{\alpha_{2}}, S_u = \text{phaseChangeS}[0] + \frac{\mathrm{dgdt}}{\alpha_{2}}\]
if dgdt < 0
\[S_p = \text{phaseChangeS}[1] + \frac{dgdt}{\alpha_{1}}, S_u = \text{phaseChangeS}[0]\]
\(S_p\), \(S_u\) together with other terms were sending to
// In src/twoPhaseModels/twoPhaseMixture/VoF/alphaEqn.H
if (divU.valid())
{
MULES::explicitSolve
(
geometricOneField(), // 1
alpha1,
phiCN, // phi for Euler ddt method, i.e. U
alphaPhi10, // alpha*U
Sp(),
(Su() + divU()*min(alpha1(), scalar(1)))(),
oneField(),
zeroField()
);
}
Explicit update and limiting
In MULES::explicitSolve, the reciprocal time step is calculated, the limiter is applied, and another explicitSolve overload is called.
// In src/finiteVolume/fvMatrices/solvers/MULES/MULESTemplates.C
const scalar rDeltaT = 1.0/mesh.time().deltaTValue();
limit(rDeltaT, rho, psi, phi, phiPsi, Sp, Su, psiMax, psiMin, false);
explicitSolve(rDeltaT, rho, psi, phiPsi, Sp, Su);
// rho - geometricOneField()
// psi - alpha1
// phiPsi - alphaPhi10
// Sp - SpRef
// Su - SuRef + alpha1*divU()
In the latter explicitSolve call, \(\alpha_1\) for next time step was obtained via
// In src/finiteVolume/fvMatrices/solvers/MULES/MULESTemplates.C
psiIf =
(
rho.oldTime().field()*psi0*rDeltaT
+ Su.field()
- psiIf
)/(rho.field()*rDeltaT - Sp.field());
This means:
\[\alpha_1(t+\Delta t)=\frac{\alpha_1(t)/\Delta t + \mathrm{Su.field} - \frac{1}{V_P} \sum_f U\alpha_1 S_{f}}{1/\Delta t-\mathrm{Sp.field} }\]
The corresponding equation was
\[\frac{\partial \alpha_1}{\partial t}+\nabla \cdot(\boldsymbol{U} \alpha_1)=\alpha_1 S_{p} + S_{u} + \alpha_{1}\nabla\cdot\boldsymbol{U}\]
Implementation note: Su.field() = \(S_u+\alpha_{1}\nabla\cdot\boldsymbol{U}\), \(\alpha_1 S_{p} + S_{u}\) is implemented in alphaSuSp.H in the solver.
Pressure-velocity coupling
The velocity field is firstly estimated from the momentum equation, then the velocity field is inserted to Eq. \(\eqref{eq:divU}\).
\begin{equation}
\nabla \cdot (\mathbf{H}_{\boldsymbol{U}} - \frac{1}{a_p} \nabla p_{\mathrm{rgh}}) = -\left(\frac{\alpha_1}{\rho_1} \frac{D \rho_1}{D t} + \frac{\alpha_2}{\rho_2} \frac{D \rho_2}{D t}\right) + \dot{m}\left(\frac{1}{\rho_1} - \frac{1}{\rho_2}\right)
\end{equation}
The source code for the pressure–Poisson equation is
fvScalarMatrix p_rghEqnIncomp
(
fvc::div(phiHbyA) - fvm::laplacian(rAUf, p_rgh)
== Sp_rgh
);
solve
(
p_rghEqnComp1() + p_rghEqnComp2() + p_rghEqnIncomp
);
where p_rghEqnIncomp means the incompressible part while p_rghEqnComp1 and p_rghEqnComp2 are compressible parts corresponding to the two phases.
fvc::div(phiHbyA): \(\nabla \cdot \mathbf{H}_{\boldsymbol{U}}\)fvm::laplacian(rAUf, p_rgh): \(\nabla \cdot ( \frac{1}{a_p} \nabla p_{\mathrm{rgh}})\)Sp_rgh: source term from phase change, i.e. \(\dot{m}\left(\frac{1}{\rho_1}-\frac{1}{\rho_2}\right)\)p_rghEqnComp1(): \(\frac{\alpha_1}{\rho_1} \frac{D \rho_1}{D t}\)p_rghEqnComp2(): \(\frac{\alpha_2}{\rho_2} \frac{D \rho_2}{D t}\)
Compressibility treatment
Coming soon. This section is under development.
Phase-change model
Coupling to the volume-fraction and pressure equations
The phase change model was included in the \(\alpha\) equation via
Pair<tmp<volScalarField::Internal>> phaseChangeS
(
phaseChange.Salpha(alpha1)
);
if (phaseChangeS[0].valid())
{
Su = phaseChangeS[0];
Sp = phaseChangeS[1];
}
and pressure equation via
fvScalarMatrix Sp_rgh(phaseChange.Sp_rgh(rho, gh, p_rgh));
Schnerr–Sauer model: volume-fraction source
Foam::Pair<Foam::tmp<Foam::volScalarField::Internal>>
Foam::twoPhaseChangeModels::SchnerrSauer::mDotAlphal() const
{
...
const volScalarField::Internal pCoeff(this->pCoeff(p));
...
return Pair<tmp<volScalarField::Internal>>
(
Cc_*limitedAlpha1*pCoeff*max(p - pSat(), p0_),
Cv_*(1.0 + alphaNuc() - limitedAlpha1)*pCoeff*min(p - pSat(), p0_)
);
}
// Note:
// pCoeff = (3*rho1()*rho2())*sqrt(2/(3*rho1()))
// *rRb(limitedAlpha1)/(rho*sqrt(mag(p - pSat()) + 0.01*pSat()));
Foam::Pair<Foam::tmp<Foam::volScalarField::Internal>>
Foam::twoPhaseChangeModels::cavitationModel::Salpha
(
volScalarField& alpha
) const
{
...
const volScalarField::Internal vDotcAlphal(alphalCoeff*mDotAlphal[0]());
const volScalarField::Internal vDotvAlphal(alphalCoeff*mDotAlphal[1]());
return Pair<tmp<volScalarField::Internal>> // mdot = \alpha_{l}S_{p} + S_{u}
(
1.0*vDotcAlphal, // Su
vDotvAlphal - vDotcAlphal // Sp
);
}
The source term in \(\alpha_{l}\) equation, \(S_{\alpha}\) was separated as
\[\begin{aligned}
S_{\alpha} &= \alpha_{l,coeff}\dot{m} \\
&= \alpha_{l,coeff}(\alpha_{l}\dot{m}_{v} + \alpha_{v}\dot{m}_{c}) \\
&= \alpha_{l,coeff}[\alpha_{l}(\dot{m}_{v}-\dot{m}_{c}) + \dot{m}_{c}]\\
&= \underbrace{\alpha_{l}\alpha_{l,coeff}(\dot{m}_{v}-\dot{m}_{c})}_{\alpha_{l} \text{Sp} <0} + \underbrace{\alpha_{l,coeff}\dot{m}_{c}}_{\text{Su} >0} \\
&= \alpha_{l}\text{Sp} + \text{Su}
\end{aligned}\]
\[\alpha_{l,coeff} = \left(\frac{1}{\rho_l}-\alpha_{l}\left(\frac{1}{\rho_l}-\frac{1}{\rho_v}\right)\right) > 0\]
In OpenFOAM, fvm::Sp handles negative source term. fvm::Sp makes the negative source term implicit so it contributes to the diagonal and increases its diagonal contribution. This can help convergence when the source term is negative on the rhs (sink term). (ref.: OpenFOAM® Beginner training session)
Schnerr–Sauer model: pressure source
The mass transfer rate in pEqn, \(\dot{m}_{p}\) was defined as
\[\dot{m}_{p} = -\frac{\dot{m}}{P-P_{s}} = \underbrace{-\frac{\alpha_{l}\dot{m}_{v}}{P-P_{s}}}_{\dot{m}_{p,v} \leq 0 } - \underbrace{\frac{(1-\alpha_{l})\dot{m}_{c}}{P-P_{s}}}_{\dot{m}_{p,c} \geq 0 } =
\dot{m}_{p,v} - \dot{m}_{p,c}\]
Foam::Pair<Foam::tmp<Foam::volScalarField::Internal>>
Foam::twoPhaseChangeModels::SchnerrSauer::mDotP() const
{
const volScalarField::Internal apCoeff(limitedAlpha1*pCoeff);
return Pair<tmp<volScalarField::Internal>>
(
Cc_*(1.0 - limitedAlpha1)*pos0(p - pSat())*apCoeff,
(-Cv_)*(1.0 + alphaNuc() - limitedAlpha1)*neg(p - pSat())*apCoeff
);
}
Foam::tmp<Foam::fvScalarMatrix>
Foam::twoPhaseChangeModels::cavitationModel::Sp_rgh
(
const volScalarField& rho,
const volScalarField& gh,
volScalarField& p_rgh
) const
{
const volScalarField::Internal pCoeff(1.0/rho1() - 1.0/rho2());
const Pair<tmp<volScalarField::Internal>> mDotP = this->mDotP();
const volScalarField::Internal vDotcP(pCoeff*mDotP[0]);
const volScalarField::Internal vDotvP(pCoeff*mDotP[1]);
return
(vDotvP - vDotcP)*(pSat() - rho()*gh())
- fvm::Sp(vDotvP - vDotcP, p_rgh);
}
(Note: \({1}/{\rho_{l}}-{1}/{\rho_{v}} <0\))
Temperature equation
Coming soon. This section is under development.
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