Skip to main content Accessibility help

Growth rate of a shocked mixing layer with known initial perturbations

  • Christopher R. Weber (a1) (a2), Andrew W. Cook (a2) and Riccardo Bonazza (a1)

We derive a growth-rate model for the Richtmyer–Meshkov mixing layer, given arbitrary but known initial conditions. The initial growth rate is determined by the net mass flux through the centre plane of the perturbed interface immediately after shock passage. The net mass flux is determined by the correlation between the post-shock density and streamwise velocity. The post-shock density field is computed from the known initial perturbations and the shock jump conditions. The streamwise velocity is computed via Biot–Savart integration of the vorticity field. The vorticity deposited by the shock is obtained from the baroclinic torque with an impulsive acceleration. Using the initial growth rate and characteristic perturbation wavelength as scaling factors, the model collapses the growth-rate curves and, in most cases, predicts the peak growth rate over a range of Mach numbers ( $1. 1\leq {M}_{i} \leq 1. 9$ ), Atwood numbers ( $- 0. 73\leq A\leq - 0. 35$ and $0. 22\leq A\leq 0. 73$ ), adiabatic indices ( $1. 40/ 1. 67\leq {\gamma }_{1} / {\gamma }_{2} \leq 1. 67/ 1. 09$ ) and narrow-band perturbation spectra. The mixing layer at late times exhibits a power-law growth with an average exponent of $\theta = 0. 24$ .

Corresponding author
Email address for correspondence:
Hide All
Alon, U., Hecht, J., Ofer, D. & Shvarts, D. 1995 Power laws and similarity of Rayleigh–Taylor and Richtmyer–Meshkov mixing fronts at all density ratios. Phys. Rev. Lett. 74 (4), 534537.
Aschenbach, B., Egger, R. & Trömper, J 1995 Discovery of explosion fragments outside the Vela supernova remnant shock-wave boundary. Nature 373, 587590.
Barenblatt, G. I. 1983 Selfsimilar turbulence propagation from an instantaneous plane source. In Nonlinear Dynamics and Turbulence (ed. Barenblatt, G. I., Iooss, G. & Joseph, D. D.). pp. 4860. Pitmann.
Barnes, C. W., Batha, S. H., Dunne, A. M., Magelssen, G. R., Rothman, S., Day, R. D., Elliott, N. E., Haynes, D. A., Holmes, R. L., Scott, J. M., Tubbs, D. L., Youngs, D. L., Boehly, T. R. & Jaanimagi, P. 2002 Observation of mix in a compressible plasma in a convergent cylindrical geometry. Phys. Plasmas 9 (11), 44314434.
Brouillette, M. 2002 The Richtmyer–Meshkov instability. Annu. Rev. Fluid Mech. 34, 445468.
Cabot, W. H & Cook, A. W 2006 Reynolds number effects on Rayleigh–Taylor instability with possible implications for type ia supernovae. Nat. Phys. 2 (8), 562568.
Cook, A. W. 2007 Artificial fluid properties for large-eddy simulation of compressible turbulent mixing. Phys. Fluids 19, 055103.
Cook, A. W. 2009 Enthalpy diffusion in multicomponent flows. Phys. Fluids 21, 055109.
Cook, A. W., Cabot, W. & Miller, P. L. 2004 The mixing transition in Rayleigh–Taylor instability. J. Fluid Mech. 511, 333362.
Cook, A. W. & Cabot, W. H. 2005 Hyperviscosity for shock-turbulence interactions. J. Comput. Phys. 203, 379385.
Cook, A. W. & Dimotakis, P. E. 2001 Transition stages of Rayleigh–Taylor instability between miscible fluids. J. Fluid Mech. 443, 6999.
Cotrell, D. L. & Cook, A. W. 2007 Scaling the incompressible Richtmyer–Meshkov instability. Phys. Fluids 19, 078105.
Dimonte, G. 2000 Spanwise homogeneous buoyancy-drag model for Rayleigh–Taylor mixing and experimental evaluation. Phys. Plasmas 7, 22552269.
Dimonte, G. & Schneider, M. 1997 Turbulent Richtmyer–Meshkov instability experiments with strong radiatively driven shocks. Phys. Plasmas 4 (12), 43474357.
Dimonte, G. & Schneider, M. 2000 Density ratio dependence of Rayleigh–Taylor mixing for sustained and impulsive acceleration histories. Phys. Fluids 12, 304321.
Fraley, G. 1986 Rayleigh–Taylor stability for a normal shock wave-density discontinuity interaction. Phys. Fluids 29 (2), 376386.
Goncharov, V. N. 2002 Analytical model of nonlinear, single-mode, classical Rayleigh–Taylor instability at arbitrary Atwood numbers. Phys. Rev. Lett. 88 (13), 134502.
Gowardhan, A. A., Ristorcelli, J. R. & Grinstein, F. F. 2011 The bipolar behaviour of the Richtmyer–Meshkov instability. Phys. Fluids 23 (7), 071701.
Griffond, J. 2006 Linear interaction analysis for Richtmyer–Meshkov instability at low Atwood numbers. Phys. Fluids 18 (5), 054106.
Hahn, M., Drikakis, D., Youngs, D. L. & Williams, R. J. R. 2011 Richtmyer–Meshkov turbulent mixing arising from an inclined material interface with realistic surface perturbations and reshocked flow. Phys. Fluids 23 (4), 046101.
Hecht, J., Alon, U. & Shvarts, D. 1994 Potential flow models of Rayleigh–Taylor and Richtmyer–Meshkov bubble fronts. Phys. Fluids 6, 40194030.
Hillebrandt, W. & Niemeyer, J. C. 2000 Type ia supernova explosion models. Annu. Rev. Astron. Astrophys. 38 (1), 191230.
Jacobs, J. W. & Krivets, V. V. 2005 Experiments on the late-time development of single-mode Richtmyer–Meshkov instability. Phys. Fluids 17 (3), 034105.
Jacobs, J. W. & Sheeley, J. M. 1996 Experimental study of incompressible Richtmyer–Meshkov instability. Phys. Fluids 8 (2), 405415.
Jun, B. I., Jones, T. W. & Norman, M. L 1996 Interaction of Rayleigh–Taylor fingers and circumstellar cloudlets in young supernova remnants. Astrophys. J. 468, L59L63.
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics, 2nd edn. Elsevier.
Layzer, D. 1955 On the instability of superposed fluids in a gravitational field. Astrophys. J. 122, 112.
Leinov, E., Malamud, G., Elbaz, Y., Levin, L. A, Ben-Dor, G., Shvarts, D. & Sadot, O. 2009 Experimental and numerical investigation of the Richtmyer–Meshkov instability under re-shock conditions. J. Fluid Mech. 626, 449475.
Lele, S. K. 1992 Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 1642.
Likhachev, O. A. & Jacobs, J. W. 2005 A vortex model for Richtmyer–Meshkov instability accounting for finite Atwood number. Phys. Fluids 17, 031704.
Linden, P. F., Redondo, J. M. & Youngs, D. L. 1994 Molecular mixing in Rayleigh–Taylor instability. J. Fluid Mech. 265, 97124.
Lindl, J. D., Amendt, P., Berger, R. L., Glendinning, S. G., Glenzer, S. H., Haan, S. W., Kauffman, R. L., Landen, O. L. & Suter, L. J. 2004 The physics basis for ignition using indirect-drive targets on the National Ignition Facility. Phys. Plasmas 11 (2), 339491.
Lombardini, M., Pullin, D. I. & Meiron, D. I. 2012 Transition to turbulence in shock-driven mixing: a Mach number study. J. Fluid Mech. 690, 203266.
Meshkov, E. E. 1969 Instability of the interface of two gases accelerated by a shock wave. Izv. Akad. Nauk. SSSR Mekh. Zhidk. Gaza. 4, 151157.
Meyer, K. A. & Blewett, P. J. 1972 Numerical investigation of the stability of a shock-accelerated interface between two fluids. Phys. Fluids 15 (5), 753759.
Mikaelian, K. O. 1998 Analytic approach to nonlinear Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Phys. Rev. Lett. 80 (3), 508511.
Mikaelian, K. O. 2011 Extended model for Richtmyer–Meshkov mix. Physica D 240 (11), 935942.
Motl, B., Oakley, J., Ranjan, D., Weber, C., Anderson, M. & Bonazza, R. 2009 Experimental validation of a Richtmyer–Meshkov scaling law over large density ratio and shock strength ranges. Phys. Fluids 21 (12), 126102.
Mueschke, N. J., Andrews, M. J. & Schilling, O. 2006 Experimental characterization of initial conditions and spatio-temporal evolution of a small-Atwood-number Rayleigh–Taylor mixing layer. J. Fluid Mech. 567, 2763.
Mueschke, N. J., Schilling, O., Youngs, D. L. & Andrews, M. J. 2009 Measurements of molecular mixing in a high-Schmidt-number Rayleigh–Taylor mixing layer. J. Fluid Mech. 632, 1748.
Poinsot, T. J. & Lele, S. K. 1992 Boundary conditions for direct simulations of compressible viscous flows. J. Comput. Phys. 101, 104129.
Prasad, J. K., Rasheed, A., Kumar, S. & Sturtevant, B. 2000 The late-time development of the Richtmyer–Meshkov instability. Phys. Fluids 12 (8), 21082115.
Rayleigh, Lord 1883 Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. London Math. Soc. 14, 170177.
Richtmyer, R. D. 1960 Taylor instability in shock acceleration of compressible fluids. Commun. Pure Appl. Maths 23, 297319.
Rikanati, A., Alon, U. & Shvarts, D. 1998 Vortex model for the nonlinear evolution of the multimode Richtmyer–Meshkov instability at low Atwood numbers. Phys. Rev. E 58 (6), 74107418.
Taylor, G. I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their plane. Proc. R. Soc. Lond. Ser. A 201, 192.
Thornber, B., Drikakis, D., Youngs, D. L. & Williams, J. R. 2010 The influence of initial conditions on turbulent mixing due to Richtmyer–Meshkov instability. J. Fluid Mech. 654, 99139.
Vandenboomgaerde, M., Mügler, C. & Gauthier, S. 1998 Impulsive model for the Richtmyer–Meshkov instability. Phys. Rev. E 58 (2), 18741882.
Velikovich, A. L. 1996 Analytic theory of Richtmyer–Meshkov instability for the case of reflected rarefaction wave. Phys. Fluids 8 (6), 16661679.
Velikovich, A. L. & Dimonte, G. 1996 Nonlinear perturbation theory of the incompressible Richtmyer–Meshkov instability. Phys. Rev. Lett. 76 (17), 31123115.
Vetter, M. & Sturtevant, B. 1995 Experiments on the Richtmyer–Meshkov instability of an air/ F6 interface. Shock Waves 4, 247252.
Wouchuk, J. G. 2001a Growth rate of the linear Richtmyer–Meshkov instability when a shock is reflected. Phys. Rev. E 63 (5), 056303.
Wouchuk, J. G. 2001b Growth rate of the Richtmyer–Meshkov instability when a rarefaction is reflected. Phys. Plasmas 8 (6), 28902907.
Wouchuk, J. G. & Nishihara, K. 1996 Linear perturbation growth at a shocked interface. Phys. Plasmas 3 (10), 37613776.
Wouchuk, J. G. & Nishihara, K. 1997 Asymptotic growth in the linear Richtmyer–Meshkov instability. Phys. Plasmas 4 (4), 10281038.
Wright, J. K. 1961 Shock Tubes. Methuen.
Yang, Y., Zhang, Q. & Sharp, D. H. 1994 Small amplitude theory of Richtmyer–Meshkov instability. Phys. Fluids 6 (5), 18561873.
Zhang, Q. & Sohn, S. I. 1996 An analytical nonlinear theory of Richtmyer–Meshkov instability. Phys. Lett. A 212 (3), 149155.
Youngs, D. L. 1994 Numerical simulation of mixing by Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Laser Part. Beams 12, 725750.
Zhang, Q. & Sohn, S. I. 1997 Nonlinear theory of unstable fluid mixing driven by shock wave. Phys. Fluids 9 (4), 11061124.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed