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arXiv:astro-ph/0102239v1 14 Feb 2001
(This manuscript will appear in the proceedings of the 20th Texas Symposium on Rela-
tivistic Astrophysics.)
Simulations of Astrophysical Fluid
Instabilities
A. C. Calder∗,†, B. Fryxell∗,‡, R. Rosner∗,†,‡, L. J. Dursi∗,†,
K. Olson∗,‡,§, P. M. Ricker∗,†, F. X. Timmes∗,†, M. Zingale∗,†,
P. MacNeice§, and H. M. Tufo∗
∗Center for Astrophysical Thermonuclear Flashes1, University of Chicago Chicago, IL 60637
†Department of Astronomy and Astrophysics, University of Chicago Chicago, IL 60637
‡Enrico Fermi Institute, The University of Chicago, Chicago, IL 60637
§NASA Goddard Space Flight Center, Greenbelt, MD 20771
Abstract. We present direct numerical simulations of mixing at Rayleigh-Taylor un-
stable interfaces performed with the FLASH code, developed at the ASCI/Alliances
Center for Astrophysical Thermonuclear Flashes at the University of Chicago. We
present initial results of single-mode studies in two and three dimensions. Our re-
sults indicate that three-dimensional instabilities grow significantly faster than two-
dimensional instabilities and that grid resolution can have a significant effect on insta-
bility growth rates. We also find that unphysical diffusive mixing occurs at the fluid
interface, particularly in poorly resolved simulations.
INTRODUCTION
Many of the problems of interest in relativistic astrophysics involve fluid instabil-
ities. The shock of a core-collapse supernova propagating through the outer layers
of the collapsing star, for example, is subject to Rayleigh-Taylor instabilities occur-
ring at the boundaries of the layers. A fluid interface is said to be Rayleigh-Taylor
unstable if either the system is accelerated in a direction perpendicular to the in-
terface such that the acceleration opposes the density gradient or if the pressure
gradient opposes the density gradient [1,2]. Growth of these instabilities can lead
to mixing of the layers. The early observation of 56Co, an element formed in the
core, in SN 1987A strongly suggested that mixing did indeed play a fundamental
role in the dynamics. Following this observation, supernova modelers embraced
multi-dimensional models with the goal of understanding the role of fluid instabil-
ities in the core collapse supernova process [3]. Despite years of modeling these
1) This work is supported by the U.S. Department of Energy under Grant No. B341495 to the
Center for Astrophysical Thermonuclear Flashes at the University of Chicago.
events, many fundamental questions remain concerning fluid instabilities and mix-
ing. In this manuscript, we present early results of our research into resolving fluid
instabilities with FLASH, our simulation code for astrophysical reactive flows.
The FLASH code [4] is an adaptive mesh, parallel simulation code for study-
ing multi-dimensional compressible reactive flows in astrophysical environments.
It uses a customized version of the PARAMESH library [5] to manage a block-
structured adaptive grid, placing resolution elements only where needed in order
to track flow features. FLASH solves the compressible Euler equations by an ex-
plicit, directionally split version of the piecewise-parabolic method [6] and allows
for general equations of state using the method of Colella & Glaz [7]. FLASH solves
a separate advection equation for the partial density of each chemical or nuclear
species as required for reactive flows. The code does not explicitly track interfaces
between fluids, so a small amount of numerical mixing can be expected during
the course of a calculation. FLASH is implemented in Fortran 90 and uses the
Message-Passing Interface library to achieve portability. Further details concern-
ing the algorithms used in the code, the structure of the code, verification tests,
and performance may be found in Fryxell et al. [4] and Calder et al. [8].
RESULTS
From our single-mode Rayleigh-Taylor studies, we find significantly faster in-
stability growth rates in three-dimensional simulations than in two-dimensional
simulations. In addition, we find that obtaining a converged growth rate requires
at least 25 grid points per wavelength of the perturbation, that grid noise seeds
small scale structure, and that the amount of small scale structure increases with
resolution due to the lack of a physical dissipation mechanism (such as a viscosity).
Another result is that poorly-resolved simulations exhibit a significant unphysical
diffusive mixing. Figure 1 shows the growth of bubble and spike amplitudes for two
well-resolved simulations beginning from equivalent initial conditions. The three-
dimensional result (left panel) shows faster growth than the two-dimensional result
(right panel). Results of our single-mode studies will appear in Calder et al. [9].
Our single-mode studies serve as a prelude to multi-mode studies, which are works
in progress; our single-mode results strongly suggest that using sufficient resolution
is essential in order to obtain physically-sensible results for these calculations. In
the multi-mode case, bubble and spike mergers are thought to lead to an instability
growth according to a t2scaling law, which for the case of a dense fluid over a lighter
fluid in a gravitational field may be written as [10]
hb,s =αb,sgAt2(1)
where hb,s is the height of a bubble or spike, gis the acceleration due to gravity,
A= (ρ2−ρ1)/(ρ2+ρ1) is the Atwood number where ρ1,2is the density of the
lighter (heavier) fluid, and tis the time. αis a proportionality ‘constant’ that may
be thought of as a measure of the efficiency of potential energy release. Experiments
0.0 1.0 2.0 3.0 4.0
Time
−0.8
−0.4
0.0
0.4
0.8
Amplitude
0.0 1.0 2.0 3.0 4.0
Time
−0.8
−0.4
0.0
0.4
0.8
Amplitude
FIGURE 1. Bubble and spike amplitudes for two-dimensional (right) and three-dimensional
(left) simulations of single-mode instabilities. The resolutions are 128 X 768 (2-d) and 128 X 128
X 768 (3-d). The amplitudes are measured by tracking the advection of each fluid. The initial
conditions consisted of a dense fluid (ρ= 2) over a lighter fluid (ρ= 1) and g= 1. The initial
perturbation consisted of a sinusoidal vertical velocity perturbation of 2.5% of the local sound
speed with the horizontal components chosen so the initial velocity field was divergence-free.
and simulations indicate that αlies in the range of 0.03 to 0.06, and it is thought to
depend on Atwood number, evolution time, initial conditions, and dimensionality.
See Young et al. [11] and references therein for a discussion of experimental results.
Results of our multi-mode studies will appear in publications of the Alpha Group,
a consortium formed by Guy Dimonte in 1998 to determine if the t2scaling law
holds for the growth of the Rayleigh-Taylor instability mixing layer, and if so, to
determine the value of α[12].
REFERENCES
1. Taylor, G., Proc. Roy. Soc.,A 201, 192 (1950)
2. Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability, New York: Dover,
1961, ch. X, pp. 428-480.
3. Arnett, D., Fryxell, B., and M¨uller, E., Ap. J.,341, L63 (1989)
4. Fryxell, B. A., et al.,Ap. J. S. 131, 273 (2000)
5. MacNeice, P., et al.,Comp. Phys. Comm.,126, 330 (2000)
6. Colella, P. and Woodward, P., J. Comp. Phys. 54, 174 (1984)
7. Colella, P. and Glaz, H. M., J. Comp. Phys. 59, 264 (1985)
8. Calder, A. C., et al., in Proc. Supercomputing 2000, IEEE Computer Soc., 2000
9. Calder, A. C. et al., in prep. (2001)
10. Youngs, D. L., Lasers and Particle Beams,12, no. 4, 725 (1994)
11. Young, Y.-N., et al.,J. Fluid Mech., in press (2001)
12. Dimonte, G. et al., in prep. (2001)
