Modeling Multiphase Flows
04/23/2009
by Dan Smith & Ed Fontes The design of devices such as inkjet
nozzles and piezoelectric fuel injectors and the control of these
devices, depending on their operating conditions, demand an indepth
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
inkjet printer.
(click for bigger picture)
The shape of the ink droplet is represented by a contour of
the level set function in Comsol Multiphysics software. The color
scale shows the modulus of the velocity vector and the streamlines
show the path of the air flow. Its accurate implementation of the
surface tension force makes the levelset method well suited to the
design and optimization of inkjet printers.
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 levelset method and the phasefield method.
The levelset method represents the interface between the two fluids
by the contour of a smooth function.
Comsol’s implementation of the levelset 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 volumeoffluid (VOF)
method. VOF is a numerical technique for tracking and locating the
interface between two phases using finite volumes.
A relatively new technique, the phasefield 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 levelset method provides better mass conservation,
the phasefield 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.
(click for bigger picture)
In the model of droplet breakup in a Tjunction, red areas
indicate dispersed droplets, while the slice and arrow plots show
the velocity field. Such models are used in emulsification processes
such as in the production of food, cosmetics, and pharmaceutical
products.
The phasefield 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 phasefield gradient. This
capability makes it easy to construct accurate multiphysics
couplings, such as adding surface tension forces to the NavierStokes
equations or modeling fluidstructure interaction by using a moving
frame of reference.
(click for bigger picture)
The plot shows the velocity field and the shape of the
liquid surface in a tank rocked with a sinusoidal motion.
The phasefield 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
phasefield model to other physics modes. For example, when the
phasefield 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
phasefield 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.
(click for bigger picture)
In the model of a twophase fluidstructure interaction, blue
areas represent water while air makes up the rest of the domain.
Orange areas in the rubber obstacle highlight regions where the von
Mises stress is high. The arrows show the direction and magnitude of
the velocity field.
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.
Article edited by Leslie Gordon,
Sr Editor, Machine Design
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Article reprinted by permission of Penton Media,
publisher of Machine Design 
