McKee, S.

English

Mathematics in Industry Information Service Eprints Archive

Implementation of masslheat transfer boundary
conditions on a moving boundary
SMcKee
University of Strathclyde. Glasgow
Industrial applications of fluid flows with free surfaces are ubiquitous: applications
include casting, container filling, extrusion and fluid jetting devices. The accurate
determination of these free surfaces is important especially if the flow itself is
determined by the position and curvature of the free surface, as it would be if surface
tension were significant. It is also essential that any numerical algorithm can cope with
merging, folding or separation of free surfaces.
Over the years a number of computational techniques have been developed for solving
free surface flows (see, e.g. Shyy et al. (1996)). These may be broadly divided into two
categories: interface tracking methods and front-capturing methods. Front or shock-
capturing methods are usually associated with compressible fluids; these methods are
now extremely sophisticated, explicitly enforcing monotonicity through a nonlinear
step while simultaneously maintaining high order. The reader is referred, for instance,
to the books by Leveque (1992) and Hirsch (1998).
The Unilever Study Group problem was however, concerned with incompressible
flows and we shall focus on front tracking methods. The purpose of this report is two-
fold: firstly it provides an up to date survey of the literature of the many approaches
that have been proposed; but, knowing that foodstuff is generally a low(ish) viscosity
non-Newtonian shear thinning fluid, it also provides a simple, but we believe a
perfectly adequate approach to relatively low Reynolds number multiple free surface
flows.
Tracking methods may be subdivided into front-tracking and volume-tracking. If high
accuracy is required it is generally accepted that front-tracking is needed, when the
interface itself is described by additional computational elements. Although the basic
idea goes back to Richtmyer and Morton (1967), its primary implementation has been
through the work of Glimm and his co-workers (see, eg Glimm et al. (1988)). They
represent the moving front by a connected set of points, which form a moving internal
boundary. To calculate the evolution inside the fluid in the vicinity of the interface, an
irregular grid is constructed and a special finite difference stencil is used on these
irregular grids. Authors who have used this approach to different flow regimes include
Chern et al. (1986), Daripa et al. (1988), Moretti (1987) and Peskin (1977) (also Fauci
and Peskin (1988), and Fogelson and Peskin (1988)). In Peskin's work the connected
set of particles carry forces which are adjusted to achieve a specific velocity at the
interface. Within this category one might include the so-called boundary integral or
boundary element methods and the vortex-in-cell (VIC) method. Boundary integral
methods can be effective when inertia forces are negligible (see, for instance, Baker
and Moore (1989) or Tsai and Miksis (1994) who solve successfully the axisymmetric
problem of gas bubbles rising in a liquid). The vortex-in-cell method, normally used for
homogeneous flows, has been extended to cope with weakly stratified flows (Meng
and Thomson (1978» and arbitrary stratification (Tryggvason (1988». More recently
Unverdi and Tryggvason (1992) described a front-tracking method for incompressible,
viscous, multi-fluid flows in which the interface is explicitly tracked but maintains a
distinct thickness dependent upon the mesh size. The main advantage of this approach
is that interfaces can interact in a rather natural way, since gradients simply add or
cancel as the grid distribution is constructed from the information carried by the
tracked front.
Another approach which has found favour is the level set approach. This would appear
to have been first introduced by Osher and Sethian (1988). The level set function is
typically a smooth function which eliminates the sorts of problems, like oscillations,
that conventional difference schemes often have. It also gets rid of having to add or
subtract points to a moving grid and it automatically takes care of merging and
breaking up of an interface. More recently, Sussman et al. (1994) has combined the
level set method with projection methods (see, eg Bell and Marcus (1992» to avoid
explicitly tracking the interface. A level set approach has also been applied to three-
dimensional two-phase flows by Beux and Banerjee (1996).
Volume-tracking methods can be further subdivided into Marker and Cell (MAC)
methods and Volume-of-fluid (VOF) methods. Indeed the original MAC method was
one of the first such tracking methods dating back to Harlow and Welsh (1965). Both
these classes of methods are still popular and, although they suffer from not being able
to accurately provide a surface interface, arguably this is less important today - a 100
x 100 x 100 grid is possible on a good workstation and will certainly be easily feasible
on even a modest one in the next few years. With the MAC method virtual marker
particles are pushed forward according to the Eulerian fluid calculation (with
appropriate bilinear interpolation for the velocity components) and it is these that
define the fluid region and hence the interface. Ths Simplified Marker and Cell
(SMAC) was introduced by Amsden and Harlow (1970). Over the intervening years
research into this method has continued, see for example Viecelli (1971), Rirt and
Shannon (1971) who also use the immersed boundary technique to handle its
interaction with the underlying grid.
Possibly the first Volume-in-fluid type code was the Simple Line Interface Calculation
(SLIC) of Noh and Woodward (1976). This was employed by Chorin (1980) to model
flame propagation, and later by Ghoniem et al. (1982) and Sethian (1984) to model
turbulent combustion. However, one usually associates Rirt and Nichols (1981) with
the VOF method, whereby a volume fraction is convected forward with the fluid. This
then led to many variants and descendants, namely, SOLA-VOF (Nichols et al.
(1980», NASA-VOF 2D (Torrey et al. (1985», NASA-VOF3D (Torrey et al.
(1987», RIPPLE (Kothe and Mjolsness (1992), Koth et al. (1991» and Flow 3D (Rirt
(1988». These have been widely used in industrial applications. Most recently, an
interesting idea of a second order VOF tracking method, employing an approximate
projection operator, has been put forward by Puckett et al. (1997».
Recently, Tome and McKee (1996) (see also Tome et al. (1997», motivated by
industrial filling processes, returned to the SMAC methodology and developed the
GENSMAC code. GENSMAC simulates incompressible time dependent fluid flows in
Cartesian coordinates within arbitrary, user specified two-dimensional domains. In
addition, it can handle free-slip and no-slip boundary conditions, there can be a number
of inflows and outflows, and a number of arbitrary shaped obstacles can be contained
within the general flow domain. GENSMAC has been modified to cope with
axisymmetric flow (Tome et a1.); and the techniques of solid modelling have been
applied to permit, through a graphic interface, enhanced flow visualization (Castelo et
a1.). A full three dimensional code in an arbitrary domain is under development.

We shall briefly describe the MAC and VOF Eulerian codes. Briefly the technique
involves the following steps:
a- 1
1. Use the Euler method to compute ii from ;: + (ii· V)ii = - Vp + Re V2ii + f.
2. Solve V2'l/J = - V . ii where u = ii +V"p ( NB: V· u = 0).
3. Compute new particle positions using : = u. (MAC approach).
To illustrate how to construct a simple (but in our view for our purposes, adequate) free
surface, we shall consider the sessile drop.
t<; 5
~ F f- 'r"
~ r F F F ,\S
sf ~ F f- F f F '~
~ F F F F F f ~
,S ,5 F F F F Y
st l= F F F $
Surface cells have Fi(O ~ Fi ~ 1), i = 1,2, ... , n.
For i = 1(1)n do
Determine the orientation
N, S => line liE, W
E, W => linellN, S
The application of the heat/mass transfer boundary condition may be achieved as follows:
As an example consider diagram 2 and the boundary condition
(
1 aT 1 aT)K --+-- =h(T-T)V2 ax V2 ay 9
We may consider (2) applied at node 0 as a consistent approximation to (1) - straight-
forward Taylor series expansion verifies this.
With the orientation illustrated in diagram 3, choose a straight line with gradient - ~
and passing through any particle. Iterate on all particles in S to obtain the best one-sided
linear fit.
A similar approach of using a straight line with gradient - ~ may be used for the VOF
method.
The same ideas may be extended to surface tension where we require a C2 curve (at least
locally).
For the MAC method consider diagram 4.
-,----_ .•. ,
:\;
• :1/1
$t
Consider the union of all particles (points) in 51, 52 and 53'
Determine the best (say L2) one-sided approximation
N
ie. min L[Yi - (ax~ + bXi + cW
a,b,c .
1=1
Yi - (ax~ + bXi + c) 2: 0
Evaluate the curvature at the mid-point of the arc passing through S2.
Now consider S2, S3, and S4 and continue.
Note: £1 approximation gives rise to a linear programming problem.
Consider
N m
m~nL IYj - Laix~1
j=1 i=O
m
subject to Yj - Laix~ ~ O,j = 1,2, ... ,N.
i=O
Clearly the objective function may be rewritten as
N m N
LYj - Nao - Lai Lx~.
j=1 i=O j=1
m N
m:x[Nao + Lai(Lx~)]
i=1 j=1
-----rt- -,i ... \... \
" ..•. '~
( "
s,
Determine the points 1,2, ... ,6. Form a quadratic through points 1 and 6 and specify it
exactly by making it pass through all points in turn, checking at each time to see if it is
one-sided.
Evaluate the curvature @ S2 (mid-point of arc), parametrically or otherwise, and apply
the boundary condition on the free surface
8Ui 8uj
-pOij + J1( -8 + -8 ) = Pa + 2,11:
Xj Xi
To evaluate the curvature at S3 consider S2, S3 and S4.
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Implementation of mass/heat transfer boundary conditions on a moving boundary

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