Buoyancy instability of homologous implosions
Abstract
With this study, I consider the hydrodynamic stability of imploding ideal gases as an idealized model for inertial confinement fusion capsules, sonoluminescent bubbles and the gravitational collapse of astrophysical gases. For oblate modes (short-wavelength incompressive modes elongated in the direction of the mean flow), a second-order ordinary differential equation is derived that can be used to assess the stability of any time-dependent flow with planar, cylindrical or spherical symmetry. Upon further restricting the analysis to homologous flows, it is shown that a monatomic gas is governed by the Schwarzschild criterion for buoyant stability. Under buoyantly unstable conditions, both entropy and vorticity fluctuations experience power-law growth in time, with a growth rate that depends upon mean flow gradients and, in the absence of dissipative effects, is independent of mode number. If the flow accelerates throughout the implosion, oblate modes amplify by a factor (2C)|N0|ti, where C is the convergence ratio of the implosion, N0 is the initial buoyancy frequency and ti is the implosion time scale. If, instead, the implosion consists of a coasting phase followed by stagnation, oblate modes amplify by a factor exp(π|N0|ts), where N0 is the buoyancy frequency at stagnation and ts is the stagnation time scale. Evenmore »
- Authors:
-
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
- Publication Date:
- Research Org.:
- Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
- Sponsoring Org.:
- USDOE
- OSTI Identifier:
- 1251085
- Report Number(s):
- LLNL-JRNL-671017
Journal ID: ISSN 0022-1120; applab
- Grant/Contract Number:
- AC52-07NA27344
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of Fluid Mechanics
- Additional Journal Information:
- Journal Volume: 774; Journal Issue: 06; Journal ID: ISSN 0022-1120
- Publisher:
- Cambridge University Press
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; compressible flows; instability; sonoluminescence
Citation Formats
Johnson, B. M. Buoyancy instability of homologous implosions. United States: N. p., 2015.
Web. doi:10.1017/jfm.2015.309.
Johnson, B. M. Buoyancy instability of homologous implosions. United States. https://doi.org/10.1017/jfm.2015.309
Johnson, B. M. Mon .
"Buoyancy instability of homologous implosions". United States. https://doi.org/10.1017/jfm.2015.309. https://www.osti.gov/servlets/purl/1251085.
@article{osti_1251085,
title = {Buoyancy instability of homologous implosions},
author = {Johnson, B. M.},
abstractNote = {With this study, I consider the hydrodynamic stability of imploding ideal gases as an idealized model for inertial confinement fusion capsules, sonoluminescent bubbles and the gravitational collapse of astrophysical gases. For oblate modes (short-wavelength incompressive modes elongated in the direction of the mean flow), a second-order ordinary differential equation is derived that can be used to assess the stability of any time-dependent flow with planar, cylindrical or spherical symmetry. Upon further restricting the analysis to homologous flows, it is shown that a monatomic gas is governed by the Schwarzschild criterion for buoyant stability. Under buoyantly unstable conditions, both entropy and vorticity fluctuations experience power-law growth in time, with a growth rate that depends upon mean flow gradients and, in the absence of dissipative effects, is independent of mode number. If the flow accelerates throughout the implosion, oblate modes amplify by a factor (2C)|N0|ti, where C is the convergence ratio of the implosion, N0 is the initial buoyancy frequency and ti is the implosion time scale. If, instead, the implosion consists of a coasting phase followed by stagnation, oblate modes amplify by a factor exp(π|N0|ts), where N0 is the buoyancy frequency at stagnation and ts is the stagnation time scale. Even under stable conditions, vorticity fluctuations grow due to the conservation of angular momentum as the gas is compressed. For non-monatomic gases, this additional growth due to compression results in weak oscillatory growth under conditions that would otherwise be buoyantly stable; this over-stability is consistent with the conservation of wave action in the fluid frame. The above analytical results are verified by evolving the complete set of linear equations as an initial value problem, and it is demonstrated that oblate modes are the fastest-growing modes and that high mode numbers are required to reach this limit (Legendre mode ℓ ≳ 100 for spherical flows). Finally, comparisons are made with a Lagrangian hydrodynamics code, and it is found that a numerical resolution of ~30 zones per wavelength is required to capture these solutions accurately. This translates to an angular resolution of ~(12/ℓ)°, or ≲ 0.1° to resolve the fastest-growing modes.},
doi = {10.1017/jfm.2015.309},
journal = {Journal of Fluid Mechanics},
number = 06,
volume = 774,
place = {United States},
year = {Mon Jun 15 00:00:00 EDT 2015},
month = {Mon Jun 15 00:00:00 EDT 2015}
}
Web of Science
Works referenced in this record:
Explosion Mechanisms of Core-Collapse Supernovae
journal, November 2012
- Janka, Hans-Thomas
- Annual Review of Nuclear and Particle Science, Vol. 62, Issue 1
On shear-layer instability, breakdown and transition
journal, January 1963
- Greenspan, H. P.; Benney, D. J.
- Journal of Fluid Mechanics, Vol. 15, Issue 1
Adiabatic perturbations in homologous conventional polytropic core collapses of a spherical star
journal, March 2010
- Cao, Yi; Lou, Yu-Qing
- Monthly Notices of the Royal Astronomical Society, Vol. 403, Issue 1
Self-consistent analysis of the hot spot dynamics for inertial confinement fusion capsules
journal, November 2005
- Sanz, J.; Garnier, J.; Cherfils, C.
- Physics of Plasmas, Vol. 12, Issue 11
Self-similar implosions and explosions of radiatively cooling gaseous masses
journal, February 1998
- Basko, Mikhail; Murakami, Masakatsu
- Physics of Plasmas, Vol. 5, Issue 2
On the stability of heterogeneous shear flows
journal, June 1961
- Miles, John W.
- Journal of Fluid Mechanics, Vol. 10, Issue 04
On the Interaction Between Turbulence and a Planar Rarefaction
journal, March 2014
- Johnson, Bryan M.
- The Astrophysical Journal, Vol. 784, Issue 2
Linear Theory of Thin, Radially Stratified Disks
journal, June 2005
- Johnson, Bryan M.; Gammie, Charles F.
- The Astrophysical Journal, Vol. 626, Issue 2
Growth of Perturbations in Gravitational Collapse and Accretion
journal, May 2000
- Lai, Dong; Goldreich, Peter
- The Astrophysical Journal, Vol. 535, Issue 1
Detailed implosion modeling of deuterium-tritium layered experiments on the National Ignition Facility
journal, May 2013
- Clark, D. S.; Hinkel, D. E.; Eder, D. C.
- Physics of Plasmas, Vol. 20, Issue 5
Homologously collapsing stellar cores
journal, June 1980
- Goldreich, P.; Weber, S. V.
- The Astrophysical Journal, Vol. 238
The extension of the Miles-Howard theorem to compressible fluids
journal, October 1970
- Chimonas, G.
- Journal of Fluid Mechanics, Vol. 43, Issue 4
Measurements of collective fuel velocities in deuterium-tritium exploding pusher and cryogenically layered deuterium-tritium implosions on the NIF
journal, April 2013
- Gatu Johnson, M.; Casey, D. T.; Frenje, J. A.
- Physics of Plasmas, Vol. 20, Issue 4
Integrated diagnostic analysis of inertial confinement fusion capsule performance
journal, May 2013
- Cerjan, Charles; Springer, Paul T.; Sepke, Scott M.
- Physics of Plasmas, Vol. 20, Issue 5
Theory of homogeneous isentropic compression and its application to laser fusion
journal, January 1974
- Kidder, R. E.
- Nuclear Fusion, Vol. 14, Issue 1
Drive Asymmetry and the Origin of Turbulence in an ICF Implosion
journal, August 2012
- Thomas, V. A.; Kares, R. J.
- Physical Review Letters, Vol. 109, Issue 7
The effect of turbulent kinetic energy on inferred ion temperature from neutron spectra
journal, July 2014
- Murphy, T. J.
- Physics of Plasmas, Vol. 21, Issue 7
Homologous Contraction of a Sonoluminescing Bubble
journal, June 1996
- Chu, M. -C.
- Physical Review Letters, Vol. 76, Issue 24
Inside a Collapsing Bubble: Sonoluminescence and the Conditions During Cavitation
journal, May 2008
- Suslick, Kenneth S.; Flannigan, David J.
- Annual Review of Physical Chemistry, Vol. 59, Issue 1
Growth rates of the ablative Rayleigh–Taylor instability in inertial confinement fusion
journal, May 1998
- Betti, R.; Goncharov, V. N.; McCrory, R. L.
- Physics of Plasmas, Vol. 5, Issue 5