Authored by: Dan Smith Ed Fontes Comsol Inc. Burlington, Mass.
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The design of devices such as ink-jet nozzles and piezoelectric fuel injectors and the control of these devices, depending on their operating conditions, demand an in-depth understanding of multiphase flow. Of interest are the balances between surface tension and other forces acting on the fluids, such as stresses and inertial forces. These forces determine the shape of the interface between the different phases, for instance, between air and a liquid. The modeling of multiphase flows makes it possible to understand these systems and optimize designs to, for example, cut fuel consumption of an engine or control the droplet size in an ink-jet printer.

Models that thoroughly describe the interface between different phases in a multiphase flow can be solved in multiphysics software such as **Comsol** using either a moving or fixed mesh. When solving such problems with a fixed mesh, the velocity and position of the interface is governed by a transport equation.

The software provides two approaches for modeling multiphase flow with a fixed mesh — the level-set method and the phase-field method. The level-set method represents the interface between the two fluids by the contour of a smooth function.

Comsol’s implementation of the level-set method is noteworthy in that the formulation of the equations yields excellent conservation of mass. The software uses a modified method, in essence a compromise between the level set method and the volume-offluid (VOF) method. VOF is a numerical technique for tracking and locating the interface between two phases using finite volumes.

A relatively new technique, the phase-field method, is based on the Cahn Hilliard equation. In this case, the free energy of the system is minimized within a timescale. This is a more sophisticated approach compared to the level set method, yet it comes with the extra cost of an additional transport equation.

Although the level-set method provides better mass conservation, the phase-field method gives decent results with a minimum of effort. Both methods provide an improved implementation of surface tension forces in comparison to the VOF method, which must reconstruct the interface from a discontinuous function (intuitively, a function that “jumps” with each input) at each time step. This means the curvature of the interface can, in particular, be difficult to compute accurately using VOF.

The phase-field approach is attractive because it keeps track of the position of the interface as well as its local properties obtained through the value of the phase-field gradient. This capability makes it easy to construct accurate multiphysics couplings, such as adding surface tension forces to the Navier-Stokes equations or modeling fluid-structure interaction by using a moving frame of reference.

The phase-field model can also be coupled to various turbulence models. Users can thereby employ the multiphysics software to solve turbulent multiphase flow problems such as the impingement of highspeed liquid jets. The software also makes it easy to couple the phase-field model to other physics modes. For example, when the phase-field model is coupled to electrostatics, users can model phenomena such as electrocoalescence, which causes the elongation of liquid droplets from electric stresses on the surface. When the phase-field model is coupled to heat transfer, users can model complex events such as film boiling.

In some applications it is crucial to conserve mass perfectly and to be able to track the exact position of the boundary. In such cases, the Arbitrary Lagrangian Eulerian (ALE) method is usually preferred. Since there is an exact boundary separating the phases, auxiliary physics can be modeled in either of the phases individually. For example, ALE can handle the modeling of a mixing tank where chemical reactions are exclusively happening in the liquid phase — a problem not possible with the level set and phase field methods.

Comsol Multiphysics software provides several different approaches to solving the equations that describe multiphase flow. These are fully coupled and segregated approaches, giving the user the advantage of trading computational speed for lower memory consumption and vice versa.