338
BRACKBILL, KOTHE, AND ZEMACH
direction to A, fi(x,,,), passes through x. Then xSO is the
B. Properties of F,,(x)
surface point closest to x. The integral in (21) is, approxi-
mately,
In summary, the volume force, F,,(x), has the following
lr
properties:
I
72
J
ri(x,)Y(x-x,)dA
1.
A
The volume force in the transition region, where the
color varies smoothly from c1 to c2, is designed to simulate
1
= 2 i(%o)
s
h 2
A
Y(x-x,)dA+Q z
(( >>
3 (22)
the surface pressure on the interface between the fluids.
Thus, the line integral of F,,(x) across the transition region,
e.g., from P, to P, in Fig. 1, is approximately equal to the
where R is the radius of curvature of the surface at xSO. One
conventional surface pressure (x, is the interface point on
can bound the integral in (22) by
the line P, P,):
1
h2
I
Wx -x,) dA <9(x -x,,,).
A
As h + 0, 9(x - x,,,) is zero everywhere but x = xSO, and
d?(x)
the corresponding limit of the integral of V?(x) across the
CI
Mx) fi(x) ccl
interface is given by
= mc(x,) fi(x,) for h>O.
(29)
lim
h-0
s
6(x,,) -W(x) dx= [cl.
(24)
2. In the limit that the width of the transition region in
a direction normal to the interface goes to zero (h + 0), the
Thus, the limit h -+ 0 of V?(x) can be written
volume force becomes
lim VC(x)=A[c] S[ri.(x-x,)] =Vc(x).
(25)
lim F,,(x) = F,,(x) X%x,). (x -x,)1,
(30)
h-0
h-0
This delta function can be used to rewrite F,,(x,) as a
which yields the conventional surface pressure at x, given
volume integral for h = 0:
by (9).
=
s
F,,(x) b[fi(x,) . (x - xS)] d3x
V
=
s
m(x) ii(x) cS[iz(x,). (x - x,)] d3x. (26)
V
The delta function converts the integral of F,,(x) over a
volume V containing the interface A to an integral over A of
F,,(x,) evaluated at that surface. The integral relation in
(26), an identity for discontinuous interfaces (h = 0), can be
used to approximate interfaces having a finite thickness h
when (25) is substituted for the delta function. Upon
substituting (25) into (26), we find that
I
F,,(x,) dA = lim
s
Wx) d3X.
4x) ccl
(27)
A
h-0 y
By comparing (27) with (13), we identify the volume force,
F,,(x), for finite h as
V?(x)
F,,(x)= 04x1 ccl . (28)
III. NUMERICAL IMPLEMENTATION
A. Choice of the Color Function
For cases of incompressible flow, there is a natural
alternative to the definition of Z(a) by (16) in terms of an
interpolation function. Instead we set
S(x) = P(X)
(31)
at grid points, with p(x) derived from the evolution
equation (15). The volume force is still given by (28).
The transition region thickness is then of the order of the
grid spacing and, at points outside the transition region,
F(x) has the values p, , p2 in fluids 1, 2, respectively. The
interface between the fluids is given by the surface
P(xs)=th +Pz)‘<P).
One can multiply the integrand on the right side of (27)
by a function g(x) = Z(x)/(c) without changing the value
of the integral in the limit h + 0, since at the interface
x =x, and g(xS) = 1. If, for incompressible flow, we use
E(x) = p(x), then g(x) is given by
g(x) = P(X)/(P),
(32)