A consistent diffuse-interface model for two-phase flow problems with rapid evaporation

Abstract

We present accurate and mathematically consistent formulations of a diffuse-interface model for two-phase flow problems involving rapid evaporation. The model addresses challenges including discontinuities in the density field by several orders of magnitude, leading to high velocity and pressure jumps across the liquid-vapor interface, along with dynamically changing interface topologies. To this end, we integrate an incompressible Navier-Stokes solver combined with a conservative level-set formulation and a regularized, i.e., diffuse, representation of discontinuities into a matrix-free adaptive finite element framework. The achievements are three-fold: First, we propose mathematically consistent definitions for the level-set transport velocity in the diffuse interface region by extrapolating the velocity from the liquid or gas phase. They exhibit superior prediction accuracy for the evaporated mass and the resulting interface dynamics compared to a local velocity evaluation, especially for strongly curved interfaces. Second, we show that accurate prediction of the evaporation-induced pressure jump requires a consistent, namely a reciprocal, density interpolation across the interface, which satisfies local mass conservation. Third, the combination of diffuse interface models for evaporation with standard Stokes-type constitutive relations for viscous flows leads to significant pressure artifacts in the diffuse interface region. To mitigate these, we propose to introduce a correction term for such constitutive model types. Through selected analytical and numerical examples, the aforementioned properties are validated. The presented model promises new insights in simulation-based prediction of melt-vapor interactions in thermal multiphase flows such as in laser-based powder bed fusion of metals.

Figures (23)

• Figure 1: Physical domain of interest for the two-phase flow with phase change problem. The domain is decomposed into a liquid and a gaseous phase, represented by ${\Omega}_{{ \mathord{\hbox{$\ell$}} }}$ and ${\Omega}_{\text{g}}$, respectively, separated by an interface $\Gamma$. The spatial discretization of the domain is performed by a finite element mesh $\mathcal{T}_\Omega$. Based on the level-set function $\phi$, the two phases are implicitly distinguished.
• Figure 2: Distribution of (left) the effective dynamic viscosity (Eq. (\ref{['eq:muEff']})) and (right) the effective density using a (standard) arithmetic phase-fraction weighted average versus the employed reciprocal interpolation function (Eq. (\ref{['eq:rhoEff']})). The values for the liquid phase are chosen to represent Ti-6Al-4V, i.e. ${\rho}_{{ \mathord{\hbox{$\ell$}} }}=4133kg/m^3$ and ${\mu}_{{ \mathord{\hbox{$\ell$}} }}=3.5e-3Pa\cdot s$. The ratios between the phases are artificially varied for the sake of demonstration.
• Figure 3: Schematic illustration of different predictions of the melt pool geometry, indicated by the dashed lines, obtained for different level-set transport velocities.
• Figure 4: Comparison of sharp and diffuse models based on the analytical solution for the fluid velocity normal to the interface of a flat (top) and an axisymmetric curved (bottom) interface subject to evaporation. The parameters are chosen as $\dot{m}_\text{v}=0.01kg/m^2s, {\rho}_{{ \mathord{\hbox{$\ell$}} }}=1000kg/m^3$, ${\rho}_{\text{g}}=1kg/m^3$. For the curved interface, the diffuse model results in a slightly lower peak velocity compared to the sharp model through the inherent diffusion of the velocity over the curved interface zone.
• Figure 5: Evaluation of diffuse models based on the analytical solution for the level-set transport velocity of a flat (top) and an axisymmetric curved (bottom) interface subject to evaporation (cf. Fig. \ref{['fig:transport_velocity_profiles_velocity']}): The considered variants 1-3 yield identical results corresponding to the sharp reference solution for the flat interface. For the curved interface, only variant 2, i.e., considering an extension velocity from the liquid end of the interface zone, yields a good approximation of the exact transport velocity for finite values of the interface thickness.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3