0
SPECIAL SECTION PAPERS

Linear Analysis of Converging Richtmyer–Meshkov Instability in the Presence of an Azimuthal Magnetic Field

[+] Author and Article Information
Abeer Bakhsh

Applied Mathematics and
Computational Sciences,
Fluid and Plasma Simulation Laboratory,
King Abdullah University of
Science and Technology,
Thuwal 23955-6900, Saudi Arabia
e-mail: abeer.bakhsh@kaust.edu.sa

Ravi Samtaney

Professor
Mechanical Engineering,
Physical Science and Engineering Division,
Fluid and Plasma Simulation Laboratory,
King Abdullah University of
Science and Technology,
Thuwal 23955-6900, Saudi Arabia
e-mail: ravi.samtaney@kaust.edu.sa

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received November 29, 2016; final manuscript received October 7, 2017; published online December 20, 2017. Assoc. Editor: Ben Thornber.

J. Fluids Eng 140(5), 050901 (Dec 20, 2017) (10 pages) Paper No: FE-16-1784; doi: 10.1115/1.4038487 History: Received November 29, 2016; Revised October 07, 2017

We investigate the linear stability of both positive and negative Atwood ratio interfaces accelerated either by a fast magnetosonic or hydrodynamic shock in cylindrical geometry. For the magnetohydrodynamic (MHD) case, we examine the role of an initial seed azimuthal magnetic field on the growth rate of the perturbation. In the absence of a magnetic field, the Richtmyer–Meshkov growth is followed by an exponentially increasing growth associated with the Rayleigh–Taylor instability (RTI). In the MHD case, the growth rate of the instability reduces in proportion to the strength of the applied magnetic field. The suppression mechanism is associated with the interference of two waves running parallel and antiparallel to the interface that transport vorticity and cause the growth rate to oscillate in time with nearly a zero mean value.

FIGURES IN THIS ARTICLE
<>
Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

Richtmyer, R. D. , 1960, “ Taylor Instability in Shock Acceleration of Compressible Fluids,” Commun. Pure Appl. Math., 13(2), pp. 297–319. [CrossRef]
Meshkov, E. E. , 1969, “ Instability of the Interface of Two Gases Accelerated by a Shock Wave,” Pure Appl. Math., 4(5), pp. 101–104.
Arnett, D. , 2000, “ The Role of Mixing in Astrophysics,” Astrophys. J., Suppl. Ser., 127(2), pp. 213–217. [CrossRef]
Yang, J. , Kubota, T. , and Zukoski, E. E. , 1993, “ Applications of Shock-Induced Mixing to Supersonic Combustion,” AIAA J., 31(5), pp. 854–862. [CrossRef]
Brouillette, M. , 2002, “ The Richtmyer–Meshkov Instability,” Annu. Rev. Fluid Mech., 34, pp. 445–468. [CrossRef]
Lindl, J. , 1995, “ Development of the Indirect-Drive Approach to Inertial Confinement Fusion and the Target Physics Basis for Ignition and Gain,” Phys. Plasmas, 2(11), pp. 3933–4024. [CrossRef]
Holmes, R. L. , Dimonte, G. , Fryxell, B. , Gittings, M. L. , Grove, J. W. , Schneider, M. , Sharp, D. H. , Velikovich, A. L. , Weaver, R. P. , and Zhang, Q. , 1999, “ Richtmyer–Meshkov Instability Growth: Experiment, Simulation and Theory,” J. Fluid Mech., 389, pp. 55–79. [CrossRef]
Lindl, J. D. , Mccrory, R. L. , and Campbell, E. M. , 1992, “ Progress Toward Ignition and Burn Propagation in Inertial Confinement Fusion,” Phys. Today, 45(9), pp. 32–40. [CrossRef]
Lindl, J. , Landen, O. , Edwards, J. , Moses, E. , and NIC Team, 2014, “ Review of the National Ignition Campaign 2009–2012,” Phys. Plasmas, 21(2), p. 020501. [CrossRef]
Samtaney, R. , 2003, “ Suppression of the Richtmyer–Meshkov Instability in the Presence of a Magnetic Field,” Phys. Fluids, 15(8), pp. L53–L56. [CrossRef]
Wheatley, V. , Pullin, D. I. , and Samtaney, R. , 2005, “ Stability of an Impulsively Accelerated Density Interface in Magnetohydrodynamics,” Phys. Rev. Lett., 95, p. 125002. [CrossRef] [PubMed]
Wheatley, V. , Samtaney, R. , and Pullin, D. I. , 2009, “ The Richtmyer–Meshkov Instability in Magnetohydrodynamics,” Phys. Fluids, 21(8), p. 082102. [CrossRef]
Qiu, Z. , Wu, Z. , Cao, J. , and Li, D. , 2008, “ Effects of Transverse Magnetic Field and Viscosity on the Richtmyer–Meshkov Instability,” Phys. Plasmas, 15(4), p. 42305. [CrossRef]
Cao, J. , Wu, Z. , Ren, H. , and Li, D. , 2008, “ Effects of Shear Flow and Transverse Magnetic Field on Richtmyer–Meshkov Instability,” Phys. Plasmas, 15(4), p. 042102. [CrossRef]
Levy, Y. , Jaouen, S. , and Canaud, B. , 2012, “ Numerical Investigation of Magnetic Richtmyer–Meshkov Instability,” Laser Part. Beams, 30(3), pp. 415–419. [CrossRef]
Wheatley, V. , Samtaney, R. , Pullin, D. I. , and Gehre, R. M. , 2014, “ The Transverse Field Richtmyer–Meshkov Instability in Magnetohydrodynamics,” Phys. Fluids, 26(1), p. 016102. [CrossRef]
Zhang, Q. , and Graham, M. J. , 1998, “ A Numerical Study of Richtmyer–Meshkov Instability Driven by Cylindrical Shocks,” Phys. Fluids, 10(4), pp. 974–992. [CrossRef]
Lombardini, M. , and Pullin, D. I. , 2009, “ Small-Amplitude Perturbations in the Three-Dimensional Cylindrical Richtmyer–Meshkov Instability,” Phys. Fluids, 21(11), p. 114103. [CrossRef]
Mikaelian, K. O. , 1990, “ Rayleigh–Taylor and Richtmyer–Meshkov Instabilities and Mixing in Stratified Spherical Shells,” Phys. Rev. A, 42(6), pp. 3400–3420. [CrossRef] [PubMed]
Mikaelian, K. O. , 2005, “ Rayleigh–Taylor and Richtmyer–Meshkov Instabilities and Mixing in Stratified Cylindrical Shells,” Phys. Fluids, 17(9), p. 094105. [CrossRef]
Pullin, D. I. , Mostert, W. , Wheatley, V. , and Samtaney, R. , 2014, “ Converging Cylindrical Shocks in Ideal Magnetohydrodynamics,” Phys. Fluids, 26(9), p. 097103. [CrossRef]
Whitham, G. B. , 1958, “ On the Propagation of Shock Waves Through Regions of Non-Uniform Area or Flow,” J. Fluid Mech., 4(4), pp. 337–360. [CrossRef]
Chisnell, R. F. , 1998, “ An Analytic Description of Converging Shock Waves,” J. Fluid Mech., 354, pp. 357–375. [CrossRef]
Mostert, W. , Pullin, D. , Samtaney, R. , and Wheatley, V. , 2016, “ Converging Cylindrical Magnetohydrodynamic Shock Collapse Onto a Power-Law-Varying Line Current,” J. Fluid Mech., 793, pp. 414–443. [CrossRef]
Mostert, W. , Wheatley, V. , Samtaney, R. , and Pullin, D. I. , 2014, “ Effects of Seed Magnetic Fields on Magnetohydrodynamic Implosion Structure and Dynamics,” Phys. Fluids, 26(12), p. 126102. [CrossRef]
Mostert, W. , Wheatley, V. , Samtaney, R. , and Pullin, D. , 2015, “ Effects of Magnetic Fields on Magnetohydrodynamic Cylindrical and Spherical Richtmyer–Meshkov Instability,” Phys. Fluids, 27(10), p. 104102. [CrossRef]
Bakhsh, A. , Gao, S. , Samtaney, R. , and Wheatley, V. , 2016, “ Linear Simulations of the Cylindrical Richtmyer–Meshkov Instability in Magnetohydrodynamics,” Phys. Fluids, 28(3), p. 034106. [CrossRef]
Samtaney, R. , 2009, “ A Method to Simulate Linear Stability of Impulsively Accelerated Density Interfaces in Ideal-MHD and Gas Dynamics,” J. Comput. Phys., 228(18), pp. 6773–6783. [CrossRef]
Yang, Y. , Zhang, Q. , and Sharp, D. H. , 1994, “ Small Amplitude Theory of Richtmyer–Meshkov Instability,” Phys. Fluids, 6(5), pp. 1856–1873. [CrossRef]
Meyer, K. A. , and Blewett, P. J. , 1972, “ Numerical Investigation of the Stability of a Shock-Accelerated Interface Between Two Fluids,” Phys. Fluids, 15(5), pp. 753–759. [CrossRef]
Chandrasekhar, S. , 1981, Hydrodynamic and Hydromagnetic Stability ( Dover Classics of Science and Mathematics), Dover Publications, New York, p. 1961.

Figures

Grahic Jump Location
Fig. 2

Initial conditions of the base flow variables: (a) IS position ρ∘, (b) modified magnetic field B∘θ1, (c) pressure p∘, and (d) radial velocity u∘r

Grahic Jump Location
Fig. 1

Schematic diagram of the physical setup: two fluids of densities ρ1 and ρ2 are separated by a perturbed interface located at r = R0. A fast magnetosonic MHD shock is initially located at r = Rs, with Mach number M.

Grahic Jump Location
Fig. 8

Spacetime diagram of density field of the base state. IS denotes the incident shock, TF denotes the TF MHD shock, RR denotes the rarefaction fan, and CD is the contact discontinuity.

Grahic Jump Location
Fig. 6

Spacetime diagram of density field of the base state. IS denotes the incident shock, TF denotes the TF MHD shock, RF denotes the reflected fast MHD shock, and CD is the contact discontinuity.

Grahic Jump Location
Fig. 7

Density profiles of base state at t = 0.0 (initial conditions) and t = 0.2 for β = 4. IS denotes the incident shock, TF denotes the TF MHD shock, RR denotes the reflected rarefaction fan, and CD is the contact discontinuity.

Grahic Jump Location
Fig. 3

Time history of unscaled growth rate of the shocked interface for β = 16 wavenumber m = 128 using different mesh sizes: 800, 1600, 3200, 6400, and 12,800 zones in the radial domain

Grahic Jump Location
Fig. 4

Error of unscaled growth rate of the shocked m = 128 wavenumber interface for β = 16 at time t = 0.3

Grahic Jump Location
Fig. 5

Density profiles of base state at t = 0.0 (initial conditions) and t = 0.5 for β = 16. IS denotes the incident shock, TF denotes the TF MHD shock, RF denotes the reflected fast MHD shocks, and CD is the contact discontinuity.

Grahic Jump Location
Fig. 12

Time history of the growth rates for different m and β = 4: (a) effect of different m and β = 4 for A > 0 and (b) effect of different m and β = 4 for A < 0

Grahic Jump Location
Fig. 10

Time history of growth rates for different β and fixed azimuthal wavenumber m = 256: (a) effect of different β and m = 256 for A > 0 and (b) effect of different β and m = 256 for A < 0

Grahic Jump Location
Fig. 11

2D vorticity ωz in the interface for β = 16, m = 256 at different scaled times chosen to correspond to a full oscillation period of the growth rate. The arrows in the growth rate plot (a) indicate the times chosen h˙/h˙∞ for β = 16 and m = 256, (b) t = 18, (c) t = 23, (d) t = 27, (e) t = 33, and (f) t = 40.

Grahic Jump Location
Fig. 13

Time history of growth rates for high β values, and fixed azimuthal wavenumber m = 256: (a) effect of different β on RTI for A > 0, m = 256 and (b) effect of different β on RTI for A < 0, m = 256

Grahic Jump Location
Fig. 9

Scaled h˙ for HD and m = 256 for A < 0 and A > 0

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In