A DREAM TO SOLVE ALL PHASES OF MATTER, SOLID,
LIQUID, AND GAS, SIMULTANEOUSLY
T. YABE, Y. OGATA, T. NAKAMURA AND Y. ZHANG
Department of Energy Sciences, Tokyo Institute of Technology
4259 Nagatsuta, Midori-ku, Yokohama 226-8502, Japan
1.
Introduction
Various innovations in basic technology have made the developing speed of
computer hardware faster than that of software technology. Similar innovations are required in order to accelerate the developing speed of software.
A universal solver for computational engineering problems would be such
an innovation. Recently developed high technology requires new tools for
the combined analysis of materials in the dierent phase states, i.e., solid,
liquid, and gas. A universal treatment of all phases by one simple algorithm
is essential, and we are at a turning point to achieve this goal. We need
to treat the topology and phase changes of the structure in processes like
welding and cutting. In processes such as freezing, condensation, melting,
and evaporation, the grid system initially aligned to the solid or liquid surface does not provide correct information of the surface at later time, and
the mesh is sometimes distorted or even broken up.
Toward this goal, we have taken the Eulerian-approach based on the
CIP (Takewaki
, 1985; Yabe
, 1991; Wang
, 1993; Yabe and
Xiao, 1995; Ida and Yabe, 1995; Xiao
, 1996a, b) method developed
by the author, which does not need the adaptive grid system and therefore
eliminates the problems of grid distortion caused by structural break-up
and topology change. Almost all the material surface can be captured by
one grid throughout the computation. Furthermore, the code can treat all
the phases of materials from solid through liquid state and two-phase state
to gas without restrictions on the time step from high-sound speed. In
this paper, a historical review of the CIP method and its strategy is given
alongside with some examples.
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350
2.
T. YABE ET AL.
CIP Method
In order to attempt to solve the problems mentioned above, we must rst
nd a method to solve the interaction of compressible gas with incompressible liquid or solid. For compressible uid, elaborate schemes like TVD or
ENO have proved to be quite eective in capturing shock waves. However,
since these schemes employ a conservative form of uid equations, divergence of velocity which becomes zero in the incompressible limit cannot be
treated independently of the advection part. On the contrary, incompressible schemes like QUICK or higher-order upwind schemes combined with
the improved MAC (Marker and Cell) procedure can treat divergence-free
uid vorticity and turbulence. However, these schemes cannot always treat
a shock wave as a sharp discontinuity.
A scheme to treat both compressible and incompressible uids simultaneously in one program is needed to simulate the interaction of gas
with liquid or solid. Fully implicit solvers can treat this procedure, but
the convergence of iteration in highly distorted states remains a problem.
Recently, a new type of scheme CIP was proposed treat shock waves by
a non-conservative scheme. By a simple extension, the CIP can be used
for both compressible and incompressible uids simultaneously. This code,
called the CCUP (CIP Combined and Unied Procedure, Yabe and Wang,
1991; Xiao
, 1996b) can treat incompressible uid with full hydrodynamic equations. In combination with the surface capturing scheme (called
\digitizer" hereafter) presented in a previous paper (Yabe and Xiao, 1993;
Xiao
, 1996b), this scheme provides a useful tool to describe various
physical processes which had never been examined before.
In this section, we review the CIP method and illustrate its underlying
principles. The key issues of the CIP method are the representation of
the advection term and the separate treatment of other terms. By this
separation, the code can be extensible to compressible and incompressible
uids. Let us rst start with a one-dimensional linear advection equation.
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@f
@t
+ u @f
= 0;
(1)
@x
The solution of Eq.(1) gives a simple translational motion of wave with
a velocity u. The initial prole [solid line of Fig.1(a)] moves like a dashed
line in a continuous representation. At this time, the solution at grid points
is denoted by closed circles and is the same as the exact solution. However,
if we eliminate the dashed line as in Fig.1(b), it is dicult to visualize
the original prole, and one may incline to retrieve the original prole like
that shown by a solid line in (c). Thus, numerical diusion arises when we
construct the prole by linear interpolation even with the exact solution as
shown in Fig.1(c). This process is the rst-order upwind scheme. On the
351
other hand, if we use quadratic polynomial for interpolation, it suers from
overshooting. This other process is the Lax-Wendro scheme or the Leith
scheme.
What made this solution worse ? For this reason, we have neglected the
behavior of the solution inside the grid cell and tried to achieve a smooth
solution. To this end, it is necessary to consider how to incorporate the
real solution into the prole within a grid cell. We propose to approximate
the prole as shown below. We dierentiate Eq.(1) with spatial variable x;
then, we get
@g
@g
@u
+
u
=
0
g ;
(2)
@t
@x
@x
where g stands for the spatial derivative of f , @f =@x. In the simplest case,
where the velocity u is constant, Eq.(2) coincides with Eq.(1) and represents
the propagation of spatial derivative with a velocity u. By this equation,
we can trace the time evolution of f and g on the basis of Eq.(1). If g
propagates as shown by the arrows in Fig.1(d), the prole after one step
is limited to a specic prole. The solution is likely to become very close
to the initial prole by this limitation. If two values of f and g are given
at two grid points, the prole
between
these points can be described by
3
2
cubic polynomial F (x) = ax + bx + cx + d. Thus, the nprole
at n +1 step
can
be
obtained
by
shifting
the
prole
by
u1t like f +1 = F (x 0 u1t),
g n+1 = dF (x 0 u1t)=dx.
A DREAM METHOD TO SOLVE ALL PHASES SIMULTANEOUSLY
3.
Compressible and Incompressible Fluid
Since we are treating hydrodynamic equations in a non-conservative form,
it is easy to extend them to include both incompressible and compressible uids. In Fig.2, we plot the iso-temperature contour of aluminum automatically generated from the semi-analytical formula. In the gas phase
where density is suciently low, the pressure is in proportion to the density.
Therefore, we may solve the density rst. Then, after the temperature is
obtained, we use EOS
(equation of state) in Fig.2. However, near the solid
density of 2.7 g cm03, the pressure rises very sharply. If we use the same
procedure there, the pressure can easily change by 3-4 orders of magnitude
even with a small error of density of about a few tens of percent. Therefore,
the strategy to solve the density rst is not applicable in this area. This
reinforces the idea of the diculty in nding a universal treatment for solid,
liquid, and gas matter. An interesting technique was devised by physicists
in incompressible uids to try to solve this problem. We will translate the
strategy they used from a dierent viewpoint and reconsider the technique.
If the pressure is very sensitive to the density, pressure has to be solved rst.
This means that we should rotate Fig.2 by 90 degrees. If we have a way
352
T. YABE ET AL.
Figure 1
Figure 3
Figure 2
Figure 1.
The principle of the CIP method. (a) the solid line is the initial prole, and the
dashed line is an exact solution after advection, whose solution (note: "whose solution"
is an incomplete phrase) (b) at discretized points (note: "at discretized points is also a
fragment.) (c) When (b) is linearly interpolated, numerical diusion appears. (d) In the
CIP, the spatial derivative also propagates and the prole inside a grid cell is retrieved.
Figure 2. Equation of state of aluminum. Each line represents isotherms.
Figure 3. Density contour of the rim for the gas density of (A and D) 0.001, (B and E)
0.005 and (C and F) 0.01 g cm03 . The left three gures (A, B, C) show the contours at
80 ms, and the right three gures (D, E, F) show those at 250 ms.
to solve pressure rst, then we obtain a very accurate density at the solid
density. Since the pressure is proportional to the density in the gas phase,
this strategy does not jeopardize the solution there either. Our method to
realize this strategy starts with the thermodynamic relation:
1p =
@p @
T
1 +
@p @T
1T;
(3)
where 1p = pn+1 0 p3 and * represents a prole after advection. The same
expression is used for and T . Therefore, if 1 and 1T are predicted, 1p
can be obtained since (@p=@)T and (@p=@T ) are already given from EOS.
The advantage of the CIP is the separate treatment of the non-advection
353
term. Therefore, we can limit our discussion here to the non-advection term
only. We should note that this advantage is crucial to obtain the nal result.
Then, we nally get
A DREAM METHOD TO SOLVE ALL PHASES SIMULTANEOUSLY
!
n+1
r rp3 =
p
n+1
0 p3
+ r1u
2
1t
PT H
1t2 3Cs2 + Cv T
3
;
(4)
where Cv is the specic heat for constant volume and PT H = T (@p=@T ),
(Cs)2 = (@p=@)T . It is very important to note that in Eq.(4), is inside the
derivative on the left-hand side. At the interface between materials having
large density dierences, the continuity of acceleration rp= is very important because the denominator can change by several orders of magnitude
in one grid. Equation (4) guarantees the continuity of rp= at the discontinuity. By this procedure, we can treat all the material at once by simply
changing its equation of state. Again, this property is a consequence of the
separate treatment of advection and non-advection terms; otherwise, the
continuity of rp= is not guaranteed and large densities can not be traced.
4.
Application
Since the pioneering work by Worthington (1908), the coronet or the socalled milk-crown has been investigated mostly by experiments that revealed many interesting features. However, simulation of the coronet has
long been a dream in the eld of computational physics because it can not
only demonstrate the power of the scheme but also investigate the physics
of coronet formation by choosing a situation which can be hardly realized
by experiments. Although the present-day technique is already close to this
goal, nobody has reported the three-dimensional coronet formation before
except for several two-dimensional works pioneered by Harlow and Shannon (1967), in which instability and therefore break-up of rim leading to
the coronet were not simulated. This is because the coronet is not merely a
consequence of a free surface problem; ambient gas must be treated as well.
This requires a special treatment at the complex boundary of the coronet
where density changes by 1000 times. In solving pressure p, it is quite a
dicult task to guarantee the continuity of force (rp)= across the complex
interface, which has a very large density ratio. If this is not done, however,
quite a large acceleration
can occur because of the change of denominator
03
from 1 to 0.001 g cm across the interface.
The CIP has the advantage of being capable of solving all the phases of
matter simultaneously from solid through liquid to gas because it can treat
very large density changes while keeping the continuity of (rp)= at the
interface without any special boundary condition. These advantages of the
354
CIP method have been demonstrated in the simulation of the melting and
evaporation dynamics of metal irradiated by laser light (Yabe and Xiao,
1995; Yabe
, 1996). This capability is essential for application to the
instability during the dynamics of coronet formation.
We used 1002100(horizontal)234 (vertical) xed, equally spaced Cartesian grids with 0.1 cm grid spacing. A thin water lm of 0.4 cm thick was
placed on a solid plate, and a01water drop of 0.8 cm radius impacted from
the top at a speed of 50 cm s . We solved gas as well as water. The density ratio at the interface was almost 1000, as already mentioned. At t =
52 and 88 ms, the irregular ring, which we call \ornament" of the coronet,
appeared at rst, a laminar belt developing below the ornament later on.
If the break-up of the rim had been caused by the eect of nite-sized
rectangular grids, its eect would have already appeared at an early stage
(t 44 ms). However, break-up abruptly occurred just at the time when
deceleration started, and thus some instability might have taken place.
Furthermore, as will be shown later, the wavelength of the irregularity along
the rim changed depending on the density of ambient gas. This proves that
the breakup observed here is not an artifact of nite grid size although it
might have provided a reason for the instability.
From the simulation results,
we
can estimate the deceleration of the
3
02
motion to be a = 2:3 2 10 cm s . From typical wavelength 0.67 cm (k
= 10 cm01) of the irregularity1=2along the01rim during deceleration, we get
the growth rate of RT = (ak) = 151 s for the R-T (Rayleigh-Taylor)
instability. This growth rate seems to be sucient to account for the evolution of the irregularity because of RT1t = 3.93 during the deceleration
time 1t = 26 ms.
Since computer capability is limited, the validity of the theory cannot
be checked by largely changing the parameters. The easiest and most interesting way to alter the situation is to change the gas density gas . Although
denser gas seems to be unrealistic, it might be helpful to check the physics
of situations that can only be realized by computer simulation. Since gas
density is 1000 times less than that of water, it may not contribute to the
evolution of the coronet. Surprisingly, however, it sensitively changed its
evolution, as shown in this paper.
Figure 3 shows the density contour of water sliced horizontally in the
middle of the rim. Interestingly, unstable wavelength became longer03when
gas density was increased, and nally at the gas density of 0.01 g cm , the
mode number was reduced to 4. Figures 3A-C show the density contour at
t = 80 ms, and Figures 3D-F show that at t = 250 ms. As shown in the
gure, the dierent nature of the breakup is already seen at this early stage
of t= 80 ms when the deceleration has just stopped.
We have conrmed that the time evolution was similar for any density
T. YABE ET AL.
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355
of gas, and the deceleration time was slightly longer for denser gas. If this
deceleration had caused the breakup, then denser gas and hence long deceleration time would have made the irregularity be slightly larger. This
contradicts the observation in Fig.3. Although the gas density comes into
the growth rate as the Attwood number = [(water 0 gas )=(water + gas )]1=2,
it is 03
already too small to change the number even for denser gas of 0.01
g cm compared with the water density water = 1 g cm03.
If the instability is caused by the K-H (Kelvin-Helmholtz) instability,
the growth rate is KH = kjU j(watergas)1=2=(water + gas ), where U is the
relative velocity between gas and water and can be aected by the ambient
gas density by a factor of 3 for 10 times dierence of density. However, the
growth rate is larger for denser gas, which contradicts the simulation result.
One possible explanation for this result is that the K-H instability drives
the swelling-up of the rim to the vertical direction rather than to break-up,
while the R-T instability owing to radial acceleration causes the break-up
of the thin lm of the rim along the azimuthal direction, thus causing the
\ornament." If KH RT, then the break-up becomes slower than the
swelling-up, and the coronet is thus not formed. Therefore, the criterion for
the break-up should be
kwatergas
a
RT KH =)
(5)
(water 0 gas )(water + gas ) U 2 ;
If this relation is correct, the maximum wave number is inversely proportional to gas density. In fact, the wavelength of the ornament was approximately proportional to gas density. This justies the relation (5) because
a and U were similar regardless of gas density.
A DREAM METHOD TO SOLVE ALL PHASES SIMULTANEOUSLY
5.
Summary
We have proposed a new tool to achieve the simultaneous solution of all the
phases of matter. The success of the code is due to its high ability to trace
sharp interfaces even with xed grids and exibility of extension to various
materials and physics. The code's scope of application is quite large. It has
already been applied, for example, to the break-up of comet ShoemakerLevy 9 (Yabe
, 1995). A list of other applications includes bubbly
ow, combustion and chemical reaction, volcanic explosion, and cavitation
(Yabe and Xiao, 1995). Recently, the CIP method has been extended to
many kinds of equations, such as diusion equations, Poisson equations,
and KdV equations. One of the most interesting applications will be the
6-dimensional phase space solver by the CIP method. Vlasov solution in 6dimension is now possible on a DEC Alpha machine with 256 MB memory.
Some of the above applications are not yet available in open literature, but
they will soon be.
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356
T. YABE ET AL.
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