Geoscience Reference
In-Depth Information
MAC numerical discretization and solution procedure, but uses a continuous function,
the fluid volume fraction
f
, instead of discrete marker particles to identify the domain
of fluid (Fig. 7.2). The values of
f
are set to 1 and 0 for the cells with water and air,
respectively. The interfacial cells are then identified as those with fractional values
of
f
. The volume fraction
f
is advected with the local flow velocity and governed by
the following kinematic equation:
∂
f
u
x
∂
f
u
y
∂
f
u
z
∂
f
+
x
+
y
+
=
0
(7.7)
∂
t
∂
∂
∂
z
Figure 7.2
Computational grid in VOF method.
The VOF method has the advantages of the MACmethod, but uses less memory and
CPU time. It can also handle very complicated interfacial phenomena, such as droplet
and wave breaking. Furthermore, because this method uses a continuous function, it
does not suffer from the lack of divisibility that discrete markers exhibit.
However, the VOF method has a disadvantage in that it cannot precisely treat the
water surface at which the distribution of volume fraction
f
has sharp, step-function-
like features. A straightforward numerical approximation cannot be used to solve
Eq. (7.7) because numerical diffusion and dispersion errors may destroy this sharp dis-
tribution. Special care needs to be taken to recover the surface shape. Grid refinements
are sometimes needed along the water surface.
7.1.3.2 SIMPLE algorithm
Numerical discretization
Since the location of water surface is part of the solution and can in general also
change with time, a moving, curvilinear grid that adjusts to the changing free surface
is used. Eqs. (7.1), (7.2), (2.55), and (2.56) can be written as Eq. (4.152) in the moving,
curvilinear coordinate system, with
φ
standing for 1,
u
i
,
k
, and
ε
depending on the
equation considered and
ξ
m
(
m
=
1, 2, 3
)
being the curvilinear coordinates.
