Shock Waves

, Volume 23, Issue 6, pp 603–617 | Cite as

Viscous effects on the non-classical Rayleigh–Taylor instability of spherical material interfaces

  • M. R. Mankbadi
  • S. BalachandarEmail author
Original Article


The viscous and conductivity effects on the instability of a rapidly expanding material interface produced by a spherical shock tube are investigated through the employment of a high-order WENO scheme. The instability is influenced by various mechanisms, which include (a) classical Rayleigh–Taylor (RT) effects, (b) Bell–Plesset or geometry/curvature effects, (c) the effects of impulsively accelerating the interface, (d) compressibility effects, (e) finite thickness effects, and (f) viscous effects. Henceforth, the present instability studied is more appropriately referred to as non-classical RT instability to distinguish it from classical RT instability. The linear regime is examined and the development of the viscous three-dimensional perturbations is obtained by solving a one-dimensional system of partial differential equations. Numerical simulations are performed to illustrate the viscous effects on the growth of the disturbances for various conditions. The inviscid analysis does not show the existence of a maximum amplification rate. The present viscous analysis, however, shows that the growth rate increases with increasing the wave number, but there exists a peak wavenumber beyond which the growth rate decreases with increasing the wave number due to viscous effects.


Rayleigh–Taylor instability  Spherical shock tube  Viscous effects 



This work is supported by a Fellowship from the Department of Energy’s Sandia National Laboratories and AFOSR under grant number FA9550-10-1-0309. Special thanks to Dr. A. Brown and Dr. T. Aselage of Sandia National Labs for their guidance and advice.


  1. 1.
    Rayleigh, L.: Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc. 14, 170–177 (1883)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Taylor, G.I.: The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. Proc. R. Soc. A. 201, 192–196 (1950)CrossRefzbMATHGoogle Scholar
  3. 3.
    Prosperetti, A.: Viscous effects on perturbed spherical flows. Q. Appl. Math. 34, 339–351 (1977)zbMATHGoogle Scholar
  4. 4.
    Hao, Y., Prosperetti, A.: The effect of viscosity on the spherical stability of oscillating gas bubbles. Phys. Fluids 11, 1309–1317 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Lin, H., Storey, B.D., Szeri, A.J.: Rayleigh–Taylor instability of violently collapsing bubbles. Phys. Fluids 14, 2925–2928 (2002)Google Scholar
  6. 6.
    Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Dover Publications Inc., NY (1981)Google Scholar
  7. 7.
    Bell, G.I.: Taylor instability on cylinders and spheres in the small amplitude approximation. LA-1321, LANL (1951).Google Scholar
  8. 8.
    Plesset, M.S.: On the stability of fluid flows with spherical symmetry. J. Appl. Phys. 25, 96–98 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grun, J., Stamper, J., Manka, C., Resnick, J., Burris, R., Crawford, J., Ripin, B.H.: Instability of Taylor-Sedov blast waves propagating through a uniform gas. Phys. Rev. Lett. 67, 3200–3200 (1991)CrossRefGoogle Scholar
  10. 10.
    Mankbadi, M.R., Balachandar, S.: Compressible inviscid instability of rapidly expanding spherical material interfaces. Phys. Fluids 24, 034106 (2012)CrossRefGoogle Scholar
  11. 11.
    Frost, D.L., Zarei, Z., Zhang, F.: Instability of combustion products interface from detonation of heterogeneous explosives. 20th International Colloquium on the Dynamics of Explosions and Reactive Systems. Montreal, Canada (2005) Google Scholar
  12. 12.
    Roe, P.L., Pike, J.: Efficient construction and utilization of approximate Riemann solutions. In: Computational Methods in Applied Sciences and Engineering. North-Holland, Amsterdam (1984)Google Scholar
  13. 13.
    Taylor, G.I.: The formation of a blast wave by a very intense explosion I. Proc. R. Soc. A. 201, 159–174 (1950)CrossRefzbMATHGoogle Scholar
  14. 14.
    Sedov, L.I.: Similarity and Dimensional Methods in Mechanics, Chapter 4. Academic Press, NY (1959)Google Scholar
  15. 15.
    Brode, H.L.: Numerical solutions of spherical blast waves. J. Appl. Phys. 26, 766–775 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Friedman, M.P.: A simplified analysis of spherical and cylindrical blast waves. J. Fluid Mech. 11, 1–15 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ling, Y., Haselbacher, A., Balachandar, S.: Importance of unsteady contributions to force and heating for particles in compressible flows: Part 1: Modeling and analysis for shock-particle interaction. Int. J. Multiphase Flows 37, 1026–1044 (2011)CrossRefGoogle Scholar
  18. 18.
    Ling, Y., Haselbacher, A., Balachandar, S.: Importance of unsteady contributions to force and heating for particles in compressible flows: Part 2: Application to particle dispersal by blast waves. Int. J. Multiphase Flows 37, 1013–1025 (2011)CrossRefGoogle Scholar
  19. 19.
    Jian, G.S., Shu, C.W.: Efficient implementation of weighted ENO scheme. J. Comput. Phys. 126, 202–228 (1996)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Haselbacher, A.: A WENO reconstruction algorithm for unstructured grids based on explicit stencil construction. AIAA paper 2005–0879 (2005)Google Scholar
  21. 21.
    Richtmyer, D.H.: Taylor Instability in shock acceleration of compressible fluids. Commun. Pure Appl. Math. 13, 297–319 (1960)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Meshkov, E.E.: Instability of the interface of two gases accelerated by a shock wave. Soviet Fluid Dyn. 4, 101–104 (1969)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Cherfils, C., Lafitte, O.: Analytic solutions of the Rayleigh equation for linear density profiles. Phys. Rev. E. 62, 2967–2970 (2000)CrossRefGoogle Scholar
  24. 24.
    Lindl, J.D.: Inertial Confinement Fusion. Springer, New York (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace EngineeringUniversity of FloridaGainesvilleUSA

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