Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-18T23:17:10.865Z Has data issue: false hasContentIssue false
  • English
  • Français

Shock-resolved Navier–Stokes simulation of the Richtmyer–Meshkov instability start-up at a light–heavy interface

Published online by Cambridge University Press:  09 December 2009

R. M. J. KRAMER ,
D. I. PULLIN ,
D. I. MEIRON  and
C. PANTANO
R. M. J. KRAMER*
Affiliation:
Graduate Aeronautical Laboratories, California Institute of Technology, MC 205-45, Pasadena, CA 91125, USA
D. I. PULLIN
Affiliation:
Graduate Aeronautical Laboratories, California Institute of Technology, MC 205-45, Pasadena, CA 91125, USA
D. I. MEIRON
Affiliation:
Applied and Computational Mathematics, California Institute of Technology, MC 205-45, Pasadena, CA 91125, USA
C. PANTANO
Affiliation:
Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
*
†Email address for correspondence: rmjkramer@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

The single-mode Richtmyer–Meshkov instability is investigated using a first-order perturbation of the two-dimensional Navier–Stokes equations about a one-dimensional unsteady shock-resolved base flow. A feature-tracking local refinement scheme is used to fully resolve the viscous internal structure of the shock. This method captures perturbations on the shocks and their influence on the interface growth throughout the simulation, to accurately examine the start-up and early linear growth phases of the instability. Results are compared to analytic models of the instability, showing some agreement with predicted asymptotic growth rates towards the inviscid limit, but significant discrepancies are noted in the transient growth phase. Viscous effects are found to be inadequately predicted by existing models.

JFM classification

Mathematical Foundations: Computational methods Compressible Flows: Shock waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 642 , 10 January 2010 , pp. 421 - 443
Copyright
Copyright © Cambridge University Press 2009

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

Brouillette, M. & Sturtevant, B. 1994 Experiments on the Richtmyer–Meshkov instability: single-scale perturbations on a continuous interface. J. Fluid Mech. 263, 271–292.CrossRefGoogle Scholar
Butcher, J. C. 2003 Numerical Methods for Ordinary Differential Equations. Wiley.CrossRefGoogle Scholar
Carlès, P. & Popinet, S. 2001 Viscous nonlinear theory of Richtmyer–Meshkov instability. Phys. Fluids 13 (7), 1833–1836.CrossRefGoogle Scholar
Carpenter, M. H., Gottlieb, D. & Abarbanel, S. 1994 Time-stable boundary conditions for finite difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes. J. Comput. Phys. 111 (2), 220–236.CrossRefGoogle Scholar
Collins, B. D. & Jacobs, J. W. 2002 PLIF flow visualization and measurements of the Richtmyer–Meshkov instability of an air/SF6 interface. J. Fluid Mech. 464, 113–136.CrossRefGoogle Scholar
Duff, R. E., Harlow, F. H. & Hirt, C. W. 1962 Effects of diffusion on interface instability between gases. Phys. Fluids 5 (4), 417–425.CrossRefGoogle Scholar
Engquist, B. & Sjögreen, B. 1998 The convergence rate of finite difference schemes in the presence of shocks. SIAM J. Numer. Anal. 35 (6), 2464–2485.CrossRefGoogle Scholar
Erpenbeck, J. J. 1962 Stability of step shocks. Phys. Fluids 5 (10), 1181–1187.CrossRefGoogle Scholar
Fornberg, B. 1988 Generation of finite-difference formulas on arbitrarily spaced grids. Math. Comput. 51, 699–706.CrossRefGoogle Scholar
Herrmann, M., Moin, P. & Abarzhi, S. I. 2008 Nonlinear evolution of the Richtmyer–Meshkov instability. J. Fluid Mech. 612, 311–338.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. 2005 Experiments on the late-time development of single-mode Richtmyer–Meshkov instability. Phys. Fluids 17, 034105.CrossRefGoogle Scholar
Jones, M. A. & Jacobs, J. W. 1997 A membraneless experiment for the study of Richtmyer–Meshkov instability of a shock-accelerated gas interface. Phys. Fluids 9 (10), 3078–3085.CrossRefGoogle Scholar
Kramer, R. M. J. 2009 Stable high-order finite-difference interface schemes with application to the Richtmyer–Meshkov instability. PhD thesis, California Institute of Technology, Pasadena.Google Scholar
Kramer, R. M. J., Pantano, C. & Pullin, D. I. 2007 A class of energy stable, high-order finite-difference interface schemes suitable for adaptive mesh refinement of hyperbolic problems. J. Comput. Phys. 226, 1458–1484.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Pergamon.Google Scholar
Lombardini, M. 2008 Richtmyer–Meshkov instability in converging geometries. PhD thesis, California Institute of Technology, Pasadena.Google Scholar
Meshkov, E. E. 1969 Instability of the interface of two gases accelerated by a shock wave. Sov. Fluid Dyn. 4 (5), 151–157.Google Scholar
Mikaelian, K. O. 1991 Density gradient stabilization of the Richtmyer–Meshkov instability. Phys. Fluids A 3 (11), 2638–2643.CrossRefGoogle Scholar
Mikaelian, K. O. 1993 Effect of viscosity on Rayliegh–Taylor and Richtmyer–Meshkov instabilities. Phys. Rev. E 47, 375–383.CrossRefGoogle ScholarPubMed
Richtmyer, R. D. 1960 Taylor instability in shock acceleration of compressible fluids. Commun. Pure Appl. Math. 13, 297–319.CrossRefGoogle Scholar
Saffman, P. G. & Meiron, D. I. 1989 Kinetic energy generated by the incompressible Richtmyer–Meshkov instability in a continuously stratified fluid. Phys. Fluids A 1 (11), 1767–1771.CrossRefGoogle Scholar
Svärd, M., Carpenter, M. H. & Nordström, J. 2007 A stable high-order finite difference scheme for the compressible Navier–Stokes equations, far-field boundary conditions. J. Comput. Phys. 225, 1020–1038.CrossRefGoogle Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar
Wouchuk, J. G. 2001 Growth rate of the linear Richtmyer–Meshkov instability when a shock is reflected. Phys. Rev. E 63 (5), 056303.CrossRefGoogle Scholar
Wouchuk, J. G. & Nishihara, K. 1997 Asymptotic growth in the linear Richtmyer–Meshkov instability. Phys. Plasmas 4 (4), 1028–1038.CrossRefGoogle Scholar
Yang, Y., Zhang, Q. & Sharp, D. H. 1994 Small amplitude theory of Richtmyer–Meshkov instability. Phys. Fluids 6 (5), 1856–1873.CrossRefGoogle Scholar
Yee, H. C. & Sjögreen, B. 2007 Simulation of Richtmyer–Meshkov instability by sixth-order filter methods. Shock Waves 17, 185–193.CrossRefGoogle Scholar
Zhang, Q. & Sohn, S. 1997 Nonlinear theory of unstable fluid mixing driven by shock wave. Phys. Fluids 9 (4), 1106–1124.CrossRefGoogle Scholar