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Rotational stabilisation of the Rayleigh–Taylor instability at the inner surface of an imploding liquid shell

Published online by Cambridge University Press:  25 June 2019

Justin Huneault Open the ORCID record for Justin Huneault [Opens in a new window] ,
David Plant  and
Andrew J. Higgins
Justin Huneault
Affiliation:
Department of Mechanical Engineering, McGill University, 817 Sherbrooke St. West, Montreal, QC H3A 0C3, Canada
David Plant
Affiliation:
General Fusion Inc., 108-3680 Bonneville Place, Burnaby, BC V3N 4T5, Canada
Andrew J. Higgins*
Affiliation:
Department of Mechanical Engineering, McGill University, 817 Sherbrooke St. West, Montreal, QC H3A 0C3, Canada
*
†Email address for correspondence: andrew.higgins@mcgill.ca
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Abstract

A number of applications utilise the energy focussing potential of imploding shells to dynamically compress matter or magnetic fields, including magnetised target fusion schemes in which a plasma is compressed by the collapse of a liquid metal surface. This paper examines the effect of fluid rotation on the Rayleigh–Taylor (RT) driven growth of perturbations at the inner surface of an imploding cylindrical liquid shell which compresses a gas-filled cavity. The shell was formed by rotating water such that it was in solid body rotation prior to the piston-driven implosion, which was propelled by a modest external gas pressure. The fast rise in pressure in the gas-filled cavity at the point of maximum convergence results in an RT unstable configuration where the cavity surface accelerates in the direction of the density gradient at the gas–liquid interface. The experimental arrangement allowed for visualisation of the cavity surface during the implosion using high-speed videography, while offering the possibility to provide geometrically similar implosions over a wide range of initial angular velocities such that the effect of rotation on the interface stability could be quantified. A model developed for the growth of perturbations on the inner surface of a rotating shell indicated that the RT instability may be suppressed by rotating the liquid shell at a sufficient angular velocity so that the net surface acceleration remains opposite to the interface density gradient throughout the implosion. Rotational stabilisation of high-mode-number perturbation growth was examined by collapsing nominally smooth cavities and demonstrating the suppression of small spray-like perturbations that otherwise appear on RT unstable cavity surfaces. Experiments observing the evolution of low-mode-number perturbations, prescribed using a mode-6 obstacle plate, showed that the RT-driven growth was suppressed by rotation, while geometric growth remained present along with significant nonlinear distortion of the perturbations near final convergence.

JFM classification

Drops and Bubbles: Bubble dynamics Instability: Nonlinear instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 873 , 25 August 2019 , pp. 531 - 567
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Copyright
© 2019 Cambridge University Press 

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References

Alikhanov, S. G., Belan, V. G., Ivanchenko, A. I., Karasjuk, V. N. & Kichigin, G. N. 1968 The production of pulsed megagauss fields by compression of the metallic cylinder in Z-pinch configuration. J. Sci. Instrum. 1 (5), 543.Google Scholar
Avital, E. J., Suponitsky, V., Khalzov, I. H., Zimmermann, J. & Plant, D. 2019 On the hydrodynamic stability of an imploding rotating circular cylindrical liquid liner. Phys. Fluids (submitted).Google Scholar
Baldwin, K. A., Scase, M. M. & Hill, R. J. A. 2015 The inhibition of the Rayleigh–Taylor instability by rotation. Sci. Rep. 5, 11706.Google Scholar
Barcilon, A., Book, D. L. & Cooper, A. L. 1974 Hydrodynamic stability of a rotating liner. Phys. Fluids 17 (9), 1707–1718.Google Scholar
Bell, G. I.1951 Taylor instability on cylinders and spheres in the small amplitude approximation. Tech. Rep. LA-1321, Los Alamos Scientific Laboratory.Google Scholar
Betti, R., Goncharov, V. N., McCrory, R. L. & Verdon, C. P. 1998 Growth rates of the ablative Rayleigh–Taylor instability in inertial confinement fusion. Phys. Plasmas 5 (5), 1446–1454.Google Scholar
Book, D. L. & Bernstein, I. B. 1979 Soluble model for the analysis of stability in an imploding compressible liner. Phys. Fluids 22 (1), 79–88.Google Scholar
Book, D. L. & Turchi, P. J. 1979 Dynamics of rotationally stabilized implosions of compressible cylindrical liquid shells. Phys. Fluids 22 (1), 68–78.Google Scholar
Book, D. L. & Winsor, N. K. 1974 Rotational stabilization of a metallic liner. Phys. Fluids 17 (3), 662–663.Google Scholar
Buyko, A. M., Garanin, S. F., Mokhov, V. N. & Yakubov, V. B. 1997 Possibility of low-dense magnetized DT plasma ignition threshold achievement in a MAGO system. Laser Part. Beams 15 (1), 127–132.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Cnare, E. C. 1966 Magnetic flux compression by magnetically imploded metallic foils. J. Appl. Phys. 37 (10), 3812–3816.Google Scholar
Epstein, R. 2004 On the Bell–Plesset effects: the effects of uniform compression and geometrical convergence on the classical Rayleigh–Taylor instability. Phys. Plasmas 11 (11), 5114–5124.Google Scholar
Fowler, C. M., Garn, W. B. & Caird, R. S. 1960 Production of very high magnetic fields by implosion. J. Appl. Phys. 31 (3), 588–594.Google Scholar
Gol’berg, S. M. & Velikovich, A. L. 1993 Suppression of Rayleigh–Taylor instability by the snowplow mechanism. Phys. Fluids B 5 (4), 1164–1172.Google Scholar
Goncharov, V. N. 2002 Analytical model of nonlinear, single-mode, classical Rayleigh–Taylor instability at arbitrary Atwood numbers. Phys. Rev. Lett. 88, 134502.Google Scholar
Haan, S. W. 1989 Onset of nonlinear saturation for Rayleigh–Taylor growth in the presence of a full spectrum of modes. Phys. Rev. A 39, 5812–5825.Google Scholar
Harris, E. G. 1962 Rayleigh–Taylor instabilities of a collapsing cylindrical shell in a magnetic field. Phys. Fluids 5 (9), 1057–1062.Google Scholar
Hsing, W. W., Barnes, C. W., Beck, J. B., Hoffman, N. M., Galmiche, D., Richard, A., Edwards, J., Graham, P., Rothman, S. & Thomas, B. 1997 Rayleigh–Taylor instability evolution in ablatively driven cylindrical implosions. Phys. Plasmas 4 (5), 1832–1840.Google Scholar
Huang, Y. M. & Hassam, A. B. 2001 Velocity shear stabilization of centrifugally confined plasma. Phys. Rev. Lett. 87, 235002.Google Scholar
Kirkpatrick, R. C., Lindemuth, I. R. & Ward, I. R. 1995 Magnetized target fusion: an overview. Fusion Technol. 27 (3), 201–214.Google Scholar
Kull, H. J. 1991 Theory of the Rayleigh–Taylor instability. Phys. Rep. 206 (5), 197–325.Google Scholar
Laberge, M. 2008 An acoustically driven magnetized target fusion reactor. J. Fusion Energy 27 (1), 65–68.Google Scholar
Mikaelian, K. O. 1990 Rayleigh–Taylor and Richtmyer–Meshkov instabilities and mixing in stratified spherical shells. Phys. Rev. A 42, 3400–3420.Google Scholar
Mikaelian, K. O. 2005 Rayleigh–Taylor and Richtmyer–Meshkov instabilities and mixing in stratified cylindrical shells. Phys. Fluids 17 (9), 094105.Google Scholar
Mikaelian, K. O. 2010 Analytic approach to nonlinear hydrodynamic instabilities driven by time-dependent accelerations. Phys. Rev. E 81, 016325.Google Scholar
Mjolsness, R. C. & Ruppel, H. M. 1986 Stability of an accelerated shear layer. Phys. Fluids 29 (7), 2202–2209.Google Scholar
Plesset, M. S. 1954 On the stability of fluid flows with spherical symmetry. J. Appl. Phys. 25 (1), 96–98.Google Scholar
Rayleigh, L. 1883 Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc. 14, 170–177.Google Scholar
Reid, R. R., Romero-Talamás, C. A., Young, W. C., Ellis, R. F. & Hassam, A. B. 2014 100 eV electron temperatures in the Maryland centrifugal experiment observed using electron Bernstein emission. Phys. Plasmas 21 (6), 063305.Google Scholar
Robson, A. E. 1982 The linus concept. In Unconventional Approaches to Fusion (ed. Brunelli, B. & Leotta, G. G.), pp. 257–279. Springer.Google Scholar
Ruden, E. L. 2002 Rayleigh–Taylor instability with a sheared flow boundary layer. IEEE Trans. Plasma Sci. 30 (2), 611–615.Google Scholar
Ryutov, D. D., Derzon, M. S. & Matzen, M. K. 2000 The physics of fast Z pinches. Rev. Mod. Phys. 72, 167–223.Google Scholar
Scase, M. M., Baldwin, K. A. & Hill, R. J. A. 2017 Rotating Rayleigh–Taylor instability. Phys. Rev. Fluids 2, 024801.Google Scholar
Scase, M. M. & Hill, R. J. A. 2018 Centrifugally forced Rayleigh–Taylor instability. J. Fluid Mech. 852, 543–577.Google Scholar
Shumlak, U., Golingo, R. P., Nelson, B. A. & Den Hartog, D. J. 2001 Evidence of stabilization in the Z -pinch. Phys. Rev. Lett. 87, 205005.Google Scholar
Shumlak, U. & Hartman, C. W. 1995 Sheared flow stabilization of the m = 1 kink mode in Z pinches. Phys. Rev. Lett. 75, 3285–3288.Google Scholar
Shumlak, U. & Roderick, N. F. 1998 Mitigation of the Rayleigh–Taylor instability by sheared axial flows. Phys. Plasmas 5 (6), 2384–2389.Google Scholar
Somon, J. P. 1969 The dynamical instabilities of cylindrical shells. J. Fluid Mech. 38 (4), 769–791.Google Scholar
Suponitsky, V., Froese, A. & Barsky, S. 2014 Richtmyer–Meshkov instability of a liquid–gas interface driven by a cylindrical imploding pressure wave. Comput. Fluids 89, 1–19.Google Scholar
Sweeney, M. A. & Perry, F. C. 1981 Investigation of shell stability in imploding cylindrical targets. J. Appl. Phys. 52 (7), 4487–4502.Google Scholar
Tao, J. J., He, X. T., Ye, W. H. & Busse, F. H. 2013 Nonlinear Rayleigh–Taylor instability of rotating inviscid fluids. Phys. Rev. E 87, 013001.Google Scholar
Taylor, G. I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. Lond. A 201 (1065), 192–196.Google Scholar
Teodorescu, C., Ellis, R. F., Case, A., Cothran, C., Hassam, A., Lunsford, R. & Messer, S. 2005 Experimental verification of the dielectric constant of a magnetized rotating plasma. Phys. Plasmas 12 (6), 062106.Google Scholar
Turchi, P. J. 2008 Imploding liner compression of plasma: concepts and issues. IEEE Trans. Plasma Sci. 36 (1), 52–61.Google Scholar
Turchi, P. J., Cooper, A. L., Ford, R. & Jenkins, D. J. 1976 Rotational stabilization of an imploding liquid cylinder. Phys. Rev. Lett. 36, 1546–1549.Google Scholar
Turchi, P. J., Cooper, A. L., Ford, R. D., Jenkins, D. J. & Burton, R. L. 1980 Review of the NRL liner implosion program. In Megagauss Physics and Technology (ed. Turchi, P. J.), pp. 375–386. Springer.Google Scholar
Velikovich, A. L., Cochran, F. L. & Davis, J. 1996 Suppression of Rayleigh–Taylor instability in Z -pinch loads with tailored density profiles. Phys. Rev. Lett. 77, 853–856.Google Scholar
Velikovich, A. L. & Schmit, P. F. 2015 Bell–Plesset effects in Rayleigh–Taylor instability of finite-thickness spherical and cylindrical shells. Phys. Plasmas 22 (12), 122711.Google Scholar
Wang, L. F., Wu, J. F., Guo, H. Y., Ye, W. H., Liu, J., Zhang, W. Y. & He, X. T. 2015 Weakly nonlinear Bell–Plesset effects for a uniformly converging cylinder. Phys. Plasmas 22 (8), 082702.Google Scholar
Weir, S. T., Chandler, E. A. & Goodwin, B. T. 1998 Rayleigh–Taylor instability experiments examining feedthrough growth in an incompressible, convergent geometry. Phys. Rev. Lett. 80, 3763–3766.Google Scholar

Huneault et al. supplementary movie 1

Off-axis view of the shell inner surface for unperturbed cavity experiments at initial angular velocities of 79, 105, and 131 rad/s. The shell inner surface only comes into view at a radius of approximately 40 mm to focus on the final stages of collapse near turnaround.

Download Huneault et al. supplementary movie 1(Video)
Video 7.8 MB

Huneault et al. supplementary movie 2

Normal view of the unperturbed cavity experiments at initial angular velocities of 79, 105, and 131 rad/s. The dark edge which delineates the cavity surface is notably broadened in the 79 and 105 rad/s experiments due to perturbation growth.

Download Huneault et al. supplementary movie 2(Video)
Video 4 MB

Huneault et al. supplementary movie 3

Normal view of mode-6 perturbed implosion experiments showing the entire implosion at initial angular velocities of 79, 105, and 131 rad/s. The obstruction plate fins can be seen on the periphery.

Download Huneault et al. supplementary movie 3(Video)
Video 6.8 MB

Huneault et al. supplementary movie 4

Normal view of mode-6 perturbed implosion experiments at initial angular velocities of 79, 105, and 131 rad/s, zoomed-in to focus on perturbation growth near turnaround.

Download Huneault et al. supplementary movie 4(Video)
Video 7 MB

Huneault et al. supplementary movie 5

Off-axis view of the shell inner surface for mode-6 perturbed implosion experiments at initial angular velocities of 79, 105, and 131 rad/s. The shell inner surface only comes into view at a radius of approximately 40 mm to focus on the final stages of collapse near turnaround.

Download Huneault et al. supplementary movie 5(Video)
Video 6.8 MB

Huneault et al. supplementary movie 6

Normal view of a mode-24 perturbed implosion experiment at an initial angular velocity of 131 rad/s.

Download Huneault et al. supplementary movie 6(Video)
Video 4.5 MB