Model of Navier-Stokes/CAC with Liquid-Gas phase change

Incompressible two-phase flows with phase change

The mathematical model is composed of incompressible Navier-Stokes equations coupled with the phase-field equation and temperature equation. The basic model is the previous one NSAC_Comp except that the mass balance is modified with a source term and the temperature equation replaces the composition one. The modifications are highlighted in the orange boxes below.

Mass balance with source term

The mass balance writes:

(75)\[\boldsymbol{\nabla}\cdot\boldsymbol{u}=\dot{m}'''\left(\frac{1}{\rho_{g}}-\frac{1}{\rho_{l}}\right)\]

where \(\boldsymbol{u}\equiv\boldsymbol{u}(\boldsymbol{x},t)=(u_x,u_y,u_z)^T\) is the fluid velocity, and \(\dot{m}'''\) is the volumic production of one phase at the interface. It is defined in Eq. (78).

Impulsion balance equation

The impulsion balance equation writes:

(76)\[\varrho(\phi,c)\left[\frac{\partial\boldsymbol{u}}{\partial t}+\boldsymbol{\nabla}\cdot(\boldsymbol{u}\boldsymbol{u})\right]=-\boldsymbol{\nabla}p_{h}+\boldsymbol{\nabla}\cdot\left[\eta(\phi)\left(\boldsymbol{\nabla}\boldsymbol{u}+\boldsymbol{\nabla}\boldsymbol{u}^{T}\right)\right]+\boldsymbol{F}_{tot}\]

where \(p_{h}\equiv p_{h}(\boldsymbol{x},t)\) is the hydrodynamic pressure, \(\varrho\) is the total densiy and \(\eta\) is the dynamic viscosity. Those quantities are obtained from the interpolation of bulk properties with the phase-field \(\phi\). Their expressions will be given in the subsection “Closure”.

Phase-field equation

The interface is followed by an additional PDE on the phase-field:

(77)\[\frac{\partial\phi}{\partial t}+\boldsymbol{\nabla}\cdot(\boldsymbol{u}\phi)=\boldsymbol{\nabla}\cdot\left[M_{\phi}\left(\boldsymbol{\nabla}\phi-\frac{4}{W}\phi(1-\phi)\boldsymbol{n}\right)\right]+\frac{\dot{m}'''}{\rho_{g}}\]

where \(\phi\equiv\phi(\boldsymbol{x},t)\) is the phase-field, \(M_{\phi}\) is the mobility and \(W\) is the interface width. \(\dot{m}'''\) is a production term at interface defined by

(78)\[\frac{\dot{m}'''}{\rho_{g}}=-\frac{4\alpha}{\mathscr{A}W^{2}}(\theta_{I}-\theta)\phi(1-\phi)\]

where the coefficient \(\mathscr{A}\) is equal to \(\mathscr{A}=10/48\approx0.21\).

Finally, the temperature equation is also considered in that model:

Temperature equation

(79)\[\frac{\partial T}{\partial t}+\boldsymbol{\nabla}\cdot(\boldsymbol{u}T)=\alpha\boldsymbol{\nabla}^{2}T-\frac{\mathcal{L}}{\mathcal{C}_{p}}\left[\frac{\partial\phi}{\partial t}+\boldsymbol{\nabla}\cdot(\boldsymbol{u}\phi)\right].\]

where \(T\equiv T(\boldsymbol{x},t)\) is the temperature, \(\mathcal{C}_{p}\) is the specific heat, \(\alpha\) is the thermal diffusivity, \(\mathcal{L}\) is the latent heat..

Force and source terms

Several forces are defined in NSAC_Comp kernel.

Total force

The total force \(\boldsymbol{F}_{tot}\) in Eq. (76) is defined by

(80)\[\boldsymbol{F}_{tot}=\boldsymbol{F}_{c}+\boldsymbol{F}_{g}\]

where \(\boldsymbol{F}_{c}\) is the capillary force, \(\boldsymbol{F}_{g}\) is the gravity force and \(\boldsymbol{F}_{M}\) is the Marangoni force. They are detailed below.

Capillary force

The capillary force \(\boldsymbol{F}_{c}\) is defined by

(81)\[\boldsymbol{F}_{c}=\mu_{\phi}\boldsymbol{\nabla}\phi\]

where the chemical potential \(\mu_{\phi}\) is defined by

(82)\[\mu_{\phi}=\frac{3}{2}\sigma\left[\frac{16}{W}\phi(1-\phi)(1-2\phi)-W\boldsymbol{\nabla}^{2}\phi\right]\]

\(\sigma\) is the surface tension and \(W\) is the interface width.

\(\hspace{5mm}\)

Gravity force

The gravity force \(\boldsymbol{F}_{g}\) is defined by

(83)\[\boldsymbol{F}_{g}=\varrho(\phi,c)\boldsymbol{g}\]

where \(\varrho(\phi,c)\) is an interpolation of bulk densities by Eq. (66)

Section author: Alain Cartalade