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STIR a
cup of tea, and watch how the tea leaves swirl and dip. Imagine
what it might take to predict that movement, given only the initial
forces and conditions of cup, tea, spoon, and leaves. Now add milk.
The more-or-less orderly motion of tea and leaves suddenly becomes
incredibly more complex, as do the forces that drive the flow and
eddies of the liquids. The pathway to understanding and prediction
becomes less clear as well. Welcome to the world of fluid dynamics—the
study of fluids in motion.
 Lawrence
Livermore physicists Paul Miller and Andrew Cook delved into the
details of fluids on the move to simulate an experiment conducted
at the University of Arizona (UA) and predict interactions of two
dissimilar liquids. With the help of powerful visualization tools
created by Livermore computer scientist Peter Lindstrom, they revealed
the inner workings of a perplexing characteristic that, under certain
situations, is key to the mixing of dissimilar fluids. Termed centrifugal
baroclinic instability, the phenomenon embodies the interaction
of two fluids with varying pressures and densities as they spin
around each other. This fluid dynamic dance occurs in a broad range
of circumstances, from deep ocean eddies to convection currents
in the cores of dying stars.
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Experimental apparatus from
the University of Arizona drop-tank experiment.
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Doing the Bounce
Miller and Cook’s work had its genesis with
a UA experiment to explore what happens at the interface between
two liquids of different densities when that interface is accelerated.
In the experiment, conducted
by UA professor Jeffrey Jacobs and former UA graduate student Charles
Niederhaus (now with the National Aeronautics and Space Administration),
a small rectangular transparent tank was mounted on vertical rails
and suspended above a spring on a platform. The tank contained a
heavier liquid (salt water) and a lighter liquid (an alcohol–water
mix). Initially, it was moved from side to side to set up standing
waves on the liquid interface. Then the tank was released, falling
and bouncing off the spring before coasting up and down to a final
stop. Since the tank was essentially in free-fall before and after
the bounce, the only force the liquids experienced was the sharp
acceleration—50 times that of gravity—of the 30-millisecond
bounce. A video camera documented what happened at the liquid interface
from initial standing wave to the final jolt.
The part of the experiment
that interested Miller and Cook was the 1 second after the bounce
during which the tank is again in free-fall. At bounce time, the
acceleration pushed the peak of the standing wave down, while the
trough moved upward. These opposing actions resulted from a twisting
force (a torque) acting on the liquid interface. In this case, the
twisting is called baroclinic torque because it involves differing
pressures (baro) and inclined (clinic) density interfaces. In stable
configurations—when light fluid is on top of heavy fluid, for
instance—baroclinic torque drives phenomena such as ocean waves.
In unstable configurations—when heavy fluids are on top of
light or when fluids in a stable configuration are accelerated—baroclinic
torque drives Rayleigh–Taylor and Richtmyer–Meshkov instabilities.
These instabilities typically lead to mushroom-shaped structures
forming in fluids. In the UA experiment, the fluids continued to
move after the bounce, forming these mushroom shapes, with the interface
rolling up at the sides of the mushroom.
What happened at the core
of this roll-up drew Miller and Cook’s attention. Rather than
a smooth, continual spiral inward, the roll-up began to disintegrate
because of a small secondary instability. (The primary instability
was the Richtmyer–Meshkov instability that created the large-scale
roll-up.) “This secondary instability happens long after the
bounce,” explains Miller, “so it was not caused by the
acceleration of the bounce itself. The source of these perturbations
deep inside the vortex and how they evolved were not well understood.”
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Flying
through the Data
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Once
Livermore physicists Paul Miller and Andrew Cook ran
their simulation, they were faced with the need to interpret
their results, so they turned to computer scientist
Peter Lindstrom for help in visualizing their data.
Lindstrom explains that he specializes in creating tools
to visualize giant data sets. One of these is a software
tool called Visualization Streams for Ultimate Scalability
(ViSUS). Lindstrom worked closely with Miller and Cook
to create movies that looked at how quantities such
as density and pressure varied over time and space and
how they correlated with the vorticity—that is,
how much local rotation was generated in the fluid,
in what areas, and in what direction. Some of the visualizations
incorporate as many as five variables: two spatial dimensions,
vorticity, vorticity production, and time.
“We also
worked up a tool that allows researchers to interact
with a 3D simulation,” Lindstrom explains. “Basically,
we put them in the driver’s seat, giving them full
control over the visualization parameters so they can
explore and interact with their data in ways that are
useful to them. This is potentially so much more powerful
than having someone such as me create a single image
or canned movie where all the parameters
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are fixed. It is not likely that a single setting of
many parameters is sufficient or that I know exactly
what to emphasize in the visualization. Also, for large
data sets—and in particular three-dimensional data
where things might be occluded or hidden deep within
the data—the scientist needs to be able to move
around the data set to obtain the most meaningful picture
of the data. With ViSUS, the scientist can zoom in on
small features, look at more global trends in the data,
and explore it from many different vantage points, while
at the same time turning the control knobs for the visualization
itself. This control is possible only with interactive
visualization.”
Such tools allow
researchers to look at the simulation while it is progressing,
so they can stop it—to tinker with the mesh, for
instance—and correct it as needed. No longer do
they need to wait two weeks for a visual result to make
corrections. Tools such as ViSUS are beginning to show
up on physicists’ desktops and will, in the long
run, only make it easier for scientists to stay on top
of complex simulations created on the Advanced Simulation
and Computing Program’s supercomputing systems.
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Turning
and Turning in the Widening Gyre
To gain insights into the
nature of these secondary instabilities, Miller and Cook used MIRANDA,
a direct numerical simulation code created by Cook. MIRANDA’s
hybrid spectral and compact-finite-difference algorithms resolve
all scales of motion in a flow, down to the viscous and diffusive
scales. “These were direct numerical simulations, meaning we
tried to work from first principles—or as close as we could
get—without making assumptions or using models for some of
the smaller dynamics of the system,” says Miller.
The computational mesh was
a two-dimensional slab one cell thick (1,025 by 1 by 5,000 grid
points). Each computational cell was 41 micrometers across, or less
than half the width of a human hair. “Since the experiment
was essentially two dimensional,” says Miller, “we were
able to increase the resolution by running a two-dimensional simulation.
Particularly in the timeframe we were interested in, three-dimensional
physics—such as three-dimensional tilt or stretch in the vortices—doesn’t
play an important role.”
The simulation ran on 64
of the 1,088 processors that make up ASCI Frost, the unclassified
portion of the Advanced Simulation and Computing (ASCI) Program’s
White supercomputer system. The simulation re-creates 2.5 seconds
from the experiment, starting with the motion of the initial standing
wave and continuing for about 1 second after the bounce. Re-creating
the details of the wave allowed Miller and Cook to replicate the
low-level velocity from the wave that was present when the bounce
occurred.
The results of that calculation
were then used to simulate the instabilities that developed during
and after the bounce, particularly the secondary instability in
the core of the vortex. By using visualization software developed
by Peter Lindstrom of the Center for Applied Scientific Computing
(see the box on p. 23), Miller and Cook discovered the cause of
this secondary instability—the interaction of the low-pressure
field in the center of the vortex (similar to the low-pressure “eye”
in the center of a cyclone or the “well” that appears
in the cup of vigorously stirred tea) with the varying densities
in the fluid whirling around in the vortex.
According to Miller, this
instability evolves as follows. The interface of the two liquids
begins to roll up because of the vorticity deposited by the bounce.
The simulation (see the box on p. 24) shows that at the start of
this process, the two liquids remain mostly unmixed, curling around
the center of the roll-up and forming a spiral pattern. As the liquid
interface spirals inward, centrifugal force (the pseudoforce that
appears to push matter outward from the center of rotation) comes
into play, producing a low-pressure well at the center of the evolving
“jelly roll.” The pressure increases up the sides of this
well, while the density alternates between light and heavy. Prior
to the secondary instability, all of the fluid spins counterclockwise.
The interaction of varying pressure and density generates new vortices—some
spinning clockwise, others counterclockwise—on the sides of
the pressure well. These tiny harbingers of disorder increase in
number, spread, and grow, eventually leading to the breakdown of
the orderly spiral of fluids and an increase in fluid mix.
The visualizations created
by Lindstrom allowed Miller and Cook to more easily see correlations
and relationships in their numerical results, which included data
on pressure, density, vorticity, and vorticity production (baroclinic
torque) at different points in time during the experiment.
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One image from a set taken
during a University of Arizona experiment exploring the interface
between two liquids—one of lighter density (black) and
one of heavier density (white)—when the interface was accelerated.
This image was taken 749 milliseconds after acceleration. Livermore
physicists wanted to uncover the mechanism that destroyed the
orderly roll-up in the sides of the mushroom shape (boxed in
white). |
About
the Simulations
The high-fidelity computer simulations developed by
Livermore’s Andrew Cook and Paul Miller were
carried out
on a 1,025- by 5,000-node mesh, at a Schmidt number
of 100, and a circulation Reynolds number of 3,800.
A suite of two-dimensional animations of calculated
quantities and a fly-over of a three-dimensional animated
rendering of the vorticity field allowed researchers
to visualize the fluid flow as it developed. All of
the still shots below were taken at the same time
(0.75 second after the bounce). The award-winning
movie “Visualizations of the Dynamics of a Vortical
Flow” is available online at the VIEWS Visualization
Project: Image and Movie Gallery www.llnl.gov/icc/sdd/img/images.shtml.
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Fluid density visualization.
Red shows the higher density liquid and blue the lower.
Green shows where the two liquids have mixed. The interface
between the two liquids is rolled up around a large
vortex in the middle. In the core of the vortex, where
a low-pressure region exists, the secondary instability
has led to increased mixing of the two fluids.
View
simulation.
Pressure (shown by height
and contour lines) visualization with a superimposed
color map of the vorticity production (or baroclinic
torque). Areas of fluid rotation are springing up on
the sides of the pressure well where the change in pressure
is steepest. Clockwise and counterclockwise vortices
are generated in alternating thin sheets. Eventually,
these small areas of oppositely rotating fluid will
break down the orderly structure of the rolled-up fluids,
resulting in increased mixing.
View
simulation.
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Vorticity (local fluid rotation)
visualization. Red and its variants (yellow and orange)
indicate areas where liquid is rotating in a counterclockwise
direction; blue indicates areas where it is rotating
in a clockwise direction. Green represents areas where
no rotation is taking place. The pockets and strips
of blue in the midst of the red core indicate areas
that have been affected by the secondary instability
and are rotating in the opposite direction from the
surrounding fluid.
View
simulation.
Vorticity production visualized
onto a heightmap of the vorticity field, with the visualization
mesh partially exposed. Peaks are areas of most counterclockwise
rotation; valleys are areas of highest clockwise rotation.
Flat surrounding areas are locations of little or no
rotation. The colors show vorticity being produced by
the secondary instability. Red indicates counterclockwise
rotation; blue indicates clockwise rotation. Blue patches
on high peaks are particularly telling. Since the rotation
being produced (clockwise) does not correlate with the
rotation that is ongoing (counterclockwise), the secondary
instability is responsible.
View
simulation.
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From
Fusion Pellets to Planet Rotation
Cook
and Miller validated their simulations using data from the UA experiment
and presented the results of their research at the 55th Annual Meeting
of the American Physical Society, Division of Fluid Dynamics, held
in Dallas, Texas, in November 2002. A video created with Lindstrom
describing their work and highlighting the visualizations was honored
in the meeting’s “Gallery of Fluid Motion.”
Understanding such fluid
instabilities—how and why they form and evolve and being able
to predict them—is important to understanding how fluids, including
both liquids and gases, behave. Such instabilities occur on scales
from the microscopic to astronomical and can have a dramatic effect.
Richtmyer– Meshkov instabilities, for instance, may affect
the performance of laser fusion pellets and nuclear weapons and
can occur in the explosions of supernovas. “After all,”
Miller concludes, “the same physical laws that apply to supernovas
govern a cup of tea.”
—Ann Parker
Key Words:
centrifugal baroclinic torque, fluid dynamics, Richtmyer–Meshkov
instability, secondary instability.
For further information contact Paul Miller (925) 423-6455 (miller3@llnl.gov).
To view a video of the University of Arizona experiment, see:
info-center.ccit.arizona.edu/~fluidlab/incomp.html
To view the “Visualizations of the Dynamics of a Vortical Flow,”
the award-winning video on the work described in this article, see:
www.llnl.gov/icc/sdd/img/images/aps02.mov
To view examples from the American Physical Society’s “Gallery
of Fluid Motion,” see:
ojps.aip.org/phf/gallery/index1.jsp
[The work discussed in this article is scheduled to be posted to the
APS site during 2003.]
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