Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-05-21T08:13:26.432Z Has data issue: false hasContentIssue false
  • English
  • Français

Effects of the secondary baroclinic vorticity on the energy cascade in the Richtmyer–Meshkov instability

Published online by Cambridge University Press:  31 August 2021

Naifu Peng Open the ORCID record for Naifu Peng [Opens in a new window] ,
Yue Yang Open the ORCID record for Yue Yang [Opens in a new window]  and
Zuoli Xiao Open the ORCID record for Zuoli Xiao [Opens in a new window]
Naifu Peng
Affiliation:
State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, PR China Center for Applied Physics and Technology and HEDPS, Peking University, Beijing 100871, PR China
Yue Yang*
Affiliation:
State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, PR China Center for Applied Physics and Technology and HEDPS, Peking University, Beijing 100871, PR China Beijing Innovation Center for Engineering Science and Advanced Technology, Peking University, Beijing 100871, PR China
Zuoli Xiao
Affiliation:
State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, PR China Center for Applied Physics and Technology and HEDPS, Peking University, Beijing 100871, PR China Beijing Innovation Center for Engineering Science and Advanced Technology, Peking University, Beijing 100871, PR China
*
†Email address for correspondence: yyg@pku.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the effect of the secondary baroclinic vorticity (SBV) on the energy cascade in the mixing induced by the multi-mode Richtmyer–Meshkov instability (RMI). With the aid of vorticity-based simplified models and the vortex-surface field, we find that the effect of the SBV peaks at a critical time when the vortex reconnection widely occurs in the mixing zone. Before the critical time, spikes and bubbles evolve almost independently, and we demonstrate that the variation of the kinetic energy spectrum induced by the SBV has the $-1$ scaling law at intermediate wavenumbers using the model of vortex rings. This SBV effect causes the slope of the total energy spectrum at intermediate wavenumbers to evolve towards $-3/2$ at the critical time. Subsequently, the SBV effect diminishes and the energy spectrum decays to the $-5/3$ law. Inspired by the vortex dynamics, we develop a model for estimating the mixing width and validate the model using numerical simulations of the multi-mode RMI with various modes of initial perturbations. This model captures the nonlinear growth of the mixing width before the self-similar growth stage.

JFM classification

Vortex Flows: Vortex dynamics Compressible Flows: Shock waves Mixing: Turbulent mixing
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 925 , 25 October 2021 , A39
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Aglitskiy, Y., et al. 2010 Basic hydrodynamics of Richtmyer–Meshkov-type growth and oscillations in the inertial confinement fusion-relevant conditions. Phil. Trans. R. Soc. Lond. A 368, 1739–1768.Google ScholarPubMed
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, 534.CrossRefGoogle ScholarPubMed
Arnett, D. 2000 The role of mixing in astrophysics. Astrophys. J. Suppl. 127, 213.CrossRefGoogle Scholar
Arnett, W.D., Bahcall, J.N., Kirshner, R.P. & Woosley, S.E. 1989 Supernova 1987A. Annu. Rev. Astron. Astrophys. 27, 629–700.CrossRefGoogle Scholar
Brouillette, M. 2002 The Richtmyer–Meshkov instability. Annu. Rev. Fluid Mech. 34, 445–468.CrossRefGoogle Scholar
Chapman, S. & Cowling, T.G. 1990 The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity. Cambridge University Press.Google Scholar
Cook, A.W. 2007 Artificial fluid properties for large-eddy simulation of compressible turbulent mixing. Phys. Fluids 19, 055103.CrossRefGoogle Scholar
Cook, A.W. 2009 Enthalpy diffusion in multicomponent flows. Phys. Fluids 21, 055109.CrossRefGoogle Scholar
Courant, R. & Friedrichs, K.O. 1999 Supersonic Flow and Shock Waves. Springer.Google Scholar
Dimotakis, P.E. 2000 The mixing transition in turbulent flows. J. Fluid Mech. 409, 69–98.CrossRefGoogle Scholar
Fraley, G. 1986 Rayleigh–Taylor stability for a normal shock wave-density discontinuity interaction. Phys. Fluids 29, 376–386.CrossRefGoogle Scholar
Gottlieb, S. & Shu, C.-W. 1998 Total variation diminishing Runge–Kutta schemes. Maths Comput. 67, 73–85.CrossRefGoogle Scholar
Graves, R.E. & Argrow, B.M. 1999 Bulk viscosity: past to present. J. Thermophys. Heat Transfer 13, 337–342.CrossRefGoogle Scholar
Groom, M. & Thornber, B. 2019 Direct numerical simulation of the multimode narrowband Richtmyer–Meshkov instability. Comput. Fluids 194, 104309.CrossRefGoogle Scholar
Groom, M. & Thornber, B. 2021 Reynolds number dependence of turbulence induced by the Richtmyer–Meshkov instability using direct numerical simulations. J. Fluid Mech. 908, A31.CrossRefGoogle Scholar
Hao, J., Xiong, S. & Yang, Y. 2019 Tracking vortex surfaces frozen in the virtual velocity in non-ideal flows. J. Fluid Mech. 863, 513–544.CrossRefGoogle Scholar
Hill, D.J., Pantano, C. & Pullin, D.I. 2006 Large-eddy simulation and multiscale modelling of a Richtmyer–Meshkov instability with reshock. J. Fluid Mech. 557, 29–61.CrossRefGoogle Scholar
Jacobs, J.W., Krivets, V.V., Tsiklashvili, V. & Likhachev, O.A. 2013 Experiments on the Richtmyer–Meshkov instability with an imposed, random initial perturbation. Shock Waves 23, 407–413.CrossRefGoogle Scholar
Kawai, S. & Lele, S.K. 2008 Localized artificial diffusivity scheme for discontinuity capturing on curvilinear meshes. J. Comput. Phys. 227, 9498–9526.CrossRefGoogle Scholar
Khokhlov, A.M., Oran, E.S. & Thomas, G.O. 1999 Numerical simulation of deflagration-to-detonation transition: the role of shock-flame interactions in turbulent flames. Combust. Flame 117, 323–339.CrossRefGoogle Scholar
Kokkinakis, I.W., Drikakis, D. & Youngs, D.L. 2020 Vortex morphology in Richtmyer–Meshkov-induced turbulent mixing. Physica D 407, 132459.CrossRefGoogle Scholar
Kramer, R.M.J., Pullin, D.I., Meiron, D.I. & Pantano, C. 2010 Shock-resolved Navier–Stokes simulation of the Richtmyer–Meshkov instability start-up at a light–heavy interface. J. Fluid Mech. 642, 421–443.CrossRefGoogle Scholar
Li, H., He, Z., Zhang, Y. & Tian, B. 2019 On the role of rarefaction/compression waves in Richtmyer–Meshkov instability with reshock. Phys. Fluids 31, 054102.Google Scholar
Lindl, J.D., McCrory, R.L. & Campbell, E.M. 1992 Progress toward ignition and burn propagation in inertial confinement fusion. Phys. Today 45, 32–40.CrossRefGoogle Scholar
Liu, H. & Xiao, Z. 2016 Scale-to-scale energy transfer in mixing flow induced by the Richtmyer–Meshkov instability. Phys. Rev. E 93, 053112.CrossRefGoogle ScholarPubMed
Lombardini, M. & Pullin, D.I. 2009 Small-amplitude perturbations in the three-dimensional cylindrical Richtmyer–Meshkov instability. Phys. Fluids 21, 114103.CrossRefGoogle Scholar
Lombardini, M., Pullin, D.I. & Meiron, D.I. 2014 Turbulent mixing driven by spherical implosions. Part 1. Flow description and mixing-layer growth. J. Fluid Mech. 748, 85–112.CrossRefGoogle Scholar
Luo, X., Li, M., Ding, J., Zhai, Z. & Si, T. 2019 Nonlinear behaviour of convergent Richtmyer–Meshkov instability. J. Fluid Mech. 877, 130–141.CrossRefGoogle Scholar
Meshkov, E.E. 1969 Instability of the interface of two gases accelerated by a shock wave. Fluid Dyn. 4, 101–104.CrossRefGoogle Scholar
Mikaelian, K.O. 1989 Turbulent mixing generated by Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Physica D 36, 343–357.CrossRefGoogle Scholar
Mikaelian, K.O. 2011 Extended model for Richtmyer–Meshkov mix. Physica D 240, 935–942.CrossRefGoogle Scholar
Mikaelian, K.O. 2015 Testing an analytic model for Richtmyer–Meshkov turbulent mixing widths. Shock Waves 25, 35–45.CrossRefGoogle Scholar
Movahed, P. & Johnsen, E. 2013 A solution-adaptive method for efficient compressible multifluid simulations, with application to the Richtmyer–Meshkov instability. J. Comput. Phys. 239, 166–186.CrossRefGoogle Scholar
Moyal, J.E. 1952 The spectra of turbulence in a compressible fluid; eddy turbulence and random noise. Math. Proc. Camb. Phil. Soc. 48, 329–344.CrossRefGoogle Scholar
Niederhaus, C.E. & Jacobs, J.W. 2003 Experimental study of the Richtmyer–Meshkov instability of incompressible fluids. J. Fluid Mech. 485, 243–277.CrossRefGoogle Scholar
Oggian, T., Drikakis, D., Youngs, D.L. & Williams, R.J.R. 2015 Computing multi-mode shock-induced compressible turbulent mixing at late times. J. Fluid Mech. 779, 411–431.CrossRefGoogle Scholar
Olson, B.J. & Lele, S.K. 2013 A mechanism for unsteady separation in over-expanded nozzle flow. Phys. Fluids 25, 239–249.CrossRefGoogle Scholar
Peng, G., Zabusky, N.J. & Zhang, S. 2003 Vortex-accelerated secondary baroclinic vorticity deposition and late-intermediate time dynamics of a two-dimensional Richtmyer–Meshkov interface. Phys. Fluids 15, 3730–3744.CrossRefGoogle Scholar
Peng, N. & Yang, Y. 2018 Effects of the Mach number on the evolution of vortex-surface fields in compressible Taylor–Green flows. Phys. Rev. Fluids 3, 013401.CrossRefGoogle Scholar
Peng, N., Yang, Y., Wu, J. & Xiao, Z. 2021 Mechanism and modelling of the secondary baroclinic vorticity in the Richtmyer–Meshkov instability. J. Fluid Mech. 911, A56.CrossRefGoogle Scholar
Ramshaw, J.D. 1990 Self-consistent effective binary diffusion in multicomponent gas mixtures. J. Non-Equilib. Thermodyn. 15, 295–300.CrossRefGoogle Scholar
Rasmus, A.M., et al. 2018 Shock-driven discrete vortex evolution on a high-Atwood number oblique interface. Phys. Plasma 25, 032119.CrossRefGoogle Scholar
Read, K.I. 1984 Experimental investigation of turbulent mixing by Rayleigh–Taylor instability. Physica D 12, 45–58.CrossRefGoogle Scholar
Reid, R.C., Pransuitz, J.M. & Poling, B.E. 1987 The Properties of Gases and Liquids. McGraw-Hill.Google Scholar
Reilly, D., McFarland, J., Mohaghar, M. & Ranjan, D. 2015 The effects of initial conditions and circulation deposition on the inclined-interface reshocked Richtmyer–Meshkov instability. Exp. Fluids 56, 168.CrossRefGoogle Scholar
Richtmyer, R.D. 1960 Taylor instability in shock acceleration of compressible fluids. Commun. Pure Appl. Maths 13, 297–319.CrossRefGoogle Scholar
Roberts, P.D., Rose, S.J., Thompson, P.C. & Wright, R.J. 1980 The stability of multiple-shell ICF targets. J. Phys. D 13, 1957.CrossRefGoogle Scholar
Samtaney, R., Ray, J. & Zabusky, N.J. 1998 Baroclinic circulation generation on shock accelerated slow/fast gas interfaces. Phys. Fluids 10, 1217–1230.CrossRefGoogle Scholar
Samtaney, R. & Zabusky, N.J. 1993 On shock polar analysis and analytical expressions for vorticity deposition in shock-accelerated density-stratified interfaces. Phys. Fluids 5, 1285–1287.CrossRefGoogle Scholar
Samtaney, R. & Zabusky, N.J. 1994 Circulation deposition on shock-accelerated planar and curved density-stratified interfaces: models and scaling laws. J. Fluid Mech. 269, 45–78.CrossRefGoogle Scholar
Shankar, S.K., Kawai, S. & Lele, S.K. 2011 Two-dimensional viscous flow simulation of a shock accelerated heavy gas cylinder. Phys. Fluids 23, 131.CrossRefGoogle Scholar
Soulard, O., Guillois, F., Griffond, J., Sabelnikov, V. & Simoëns, S. 2018 Permanence of large eddies in Richtmyer–Meshkov turbulence with a small Atwood number. Phys. Rev. Fluids 3, 104603.CrossRefGoogle Scholar
Soulard, O., Guillois, F., Griffond, J., Sabelnikov, V. & Simoëns, S. 2020 A two-scale Langevin PDF model for Richtmyer–Meshkov turbulence with a small Atwood number. Physica D 403, 132276.CrossRefGoogle Scholar
Taccetti, J.M., Batha, S.H., Fincke, J.R., Delamater, N.D., Lanier, N.E., Magelssen, G.R., Hueckstaedt, R.M., Rothman, S.D., Horsfield, C.J. & Parker, K.W. 2005 Richtmyer–Meshkov instability reshock experiments using laser-driven double-cylinder implosions. Astrophys. Space Sci. 298, 327–331.CrossRefGoogle Scholar
Thornber, B., Drikakis, D., Youngs, D.L. & Williams, R.J.R. 2010 The influence of initial conditions on turbulent mixing due to Richtmyer–Meshkov instability. J. Fluid Mech. 654, 99–139.CrossRefGoogle Scholar
Tritschler, V.K., Hickel, S., Hu, X.Y. & Adams, N.A. 2013 On the Kolmogorov inertial subrange developing from Richtmyer–Meshkov instability. Phys. Fluids 25, 071701.CrossRefGoogle Scholar
Tritschler, V.K., Olson, B.J., Lele, S.K., Hickel, S., Hu, X.Y. & Adams, N.A. 2014 a On the Richtmyer–Meshkov instability evolving from a deterministic multimode planar interface. J. Fluid Mech. 755, 429–462.CrossRefGoogle Scholar
Tritschler, V.K., Zubel, M., Hickel, S. & Adams, N.A. 2014 b Evolution of length scales and statistics of Richtmyer–Meshkov instability from direct numerical simulations. Phys. Rev. E 90, 063001.CrossRefGoogle ScholarPubMed
Van Riper, K.A. 1982 Stellar core collapse. II-Inner core bounce and shock propagation. Astrophys. J. 257, 793–820.CrossRefGoogle Scholar
Velikovich, A.L., Dahlburg, J.P., Schmitt, A.J., Gardner, J.H., Phillips, L., Cochran, F.L., Chong, Y.K., Dimonte, G. & Metzler, N. 2000 Richtmyer–Meshkov-like instabilities and early-time perturbation growth in laser targets and Z-pinch loads. Phys. Plasmas 7, 1662–1671.CrossRefGoogle Scholar
Wang, T., Bai, J.S., Li, P., Wang, B., Du, L. & Tao, G. 2016 Large-eddy simulations of the multi-mode Richtmyer–Meshkov instability and turbulent mixing under reshock. High Energ. Dens. Phys. 19, 65–75.CrossRefGoogle Scholar
Weber, C.R., Cook, A.W. & Bonazza, R. 2013 Growth rate of a shocked mixing layer with known initial perturbations. J. Fluid Mech. 725, 372–401.CrossRefGoogle Scholar
Xiong, S. & Yang, Y. 2019 Construction of knotted vortex tubes with the writhe-dependent helicity. Phys. Fluids 31, 047101.CrossRefGoogle Scholar
Yang, J., Kubota, T. & Zukoski, E.E. 1993 Applications of shock-induced mixing to supersonic combustion. AIAA J. 31, 854–862.CrossRefGoogle Scholar
Yang, Y. & Pullin, D.I. 2010 On Lagrangian and vortex-surface fields for flows with Taylor–Green and Kida–Pelz initial conditions. J. Fluid Mech. 661, 446–481.CrossRefGoogle Scholar
Yang, Y. & Pullin, D.I. 2011 Evolution of vortex-surface fields in viscous Taylor–Green and Kida–Pelz flows. J. Fluid Mech. 685, 146–164.CrossRefGoogle Scholar
Yang, Y., Zhang, Q. & Sharp, D.H. 1994 Small amplitude theory of Richtmyer–Meshkov instability. Phys. Fluids 6, 1856–1873.CrossRefGoogle Scholar
Yeung, P.K. & Pope, S.B. 1989 Lagrangian statistics from direct numerical simulations of isotropic turbulence. J. Fluid Mech. 207, 531–586.CrossRefGoogle Scholar
Zabusky, N.J. 1999 Vortex paradigm for accelerated inhomogeneous flows: visiometrics for the Rayleigh–Taylor and Richtmyer–Meshkov environments. Annu. Rev. Fluid Mech. 31, 495–536.CrossRefGoogle Scholar
Zabusky, N.J., Kotelnikov, A.D., Gulak, Y. & Peng, G. 2003 Amplitude growth rate of a Richtmyer–Meshkov unstable two-dimensional interface to intermediate times. J. Fluid Mech. 475, 147–162.CrossRefGoogle Scholar
Zhao, Y., Yang, Y. & Chen, S. 2016 Vortex reconnection in the late transition in channel flow. J. Fluid Mech. 802, R4.CrossRefGoogle Scholar
Zhou, H., You, J., Xiong, S., Yang, Y., Thévenin, D. & Chen, S. 2019 Interactions between the premixed flame front and the three-dimensional Taylor–Green vortex. Proc. Combust. Inst. 37, 2461–2468.CrossRefGoogle Scholar
Zhou, Y. 2001 A scaling analysis of turbulent flows driven by Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Phys. Fluids 13, 538–543.CrossRefGoogle Scholar
Zhou, Y. 2017 a Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. I. Phys. Rep. 720, 1–136.Google Scholar
Zhou, Y. 2017 b Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. II. Phys. Rep. 723, 1–160.Google Scholar