Skip to main content

Analytical linear theory for the shock and re-shock of isotropic density inhomogeneities

  • C. Huete (a1), J. G. Wouchuk (a1), B. Canaud (a2) and A. L. Velikovich (a3)

We present an analytical model that describes the linear interaction of two successive shocks launched into a non-uniform density field. The re-shock problem is important in different fields, inertial confinement fusion among them, where several shocks are needed to compress the non-uniform target. At first, we present a linear theory model that studies the interaction of two successive shocks with a single-mode density perturbation field ahead of the first shock. The second shock is launched after the sonic waves emitted by the first shock wave have vanished. Therefore, in the case considered in this work, the second shock only interacts with the entropic and vortical perturbations left by the first shock front. The velocity, vorticity and density fields are later obtained in the space behind the second shock. With the results of the single-mode theory, the interaction with a full spectrum of random-isotropic density perturbations is considered by decomposing it into Fourier modes. The model describes in detail how the second shock wave modifies the turbulent field generated by the first shock wave. Averages of the downstream quantities (kinetic energy, vorticity, acoustic flux and density) are easily obtained either for two-dimensional or three-dimensional upstream isotropic spectra. The asymptotic limits of very strong shocks are discussed. The study shown here is an extension of previous works, where the interaction of a planar shock wave with random isotropic vorticity/entropy/acoustic spectra were studied independently. It is also a preliminary step towards the understanding of the re-shock of a fully turbulent flow, where all three of the modes, vortical, entropic and acoustic, might be present.

Corresponding author
Email address for correspondence:
Hide All
1. Barre S. 1996 Experimental study of a normal shock/homogeneous turbulence interaction. AIAA J. 34 (5), 968.
2. Boehly T. R., Goncharov V. N., Seka W., Barrios M. A., Celliers P. M., Hicks D. G., Collins G. W., Hu S. X., Marozas J. A. & Meyerhofer D. D. 2011 Velocity and timing of multiple spherically converging shock waves in liquid deuterium. Phys. Rev. Lett. 106 (19), 195005.
3. Clarisse J.-M., Jaouen S. & Raviart P.-A. 2004 A Godunov-type method in Lagrangian coordinates for computing linearly-perturbed planar-symmetric flows of gas dynamics. J. Comput. Phys. 198 (1), 80105.
4. Collins T. J. B., Poludnenko A., Cunningham A. & Frank A. 2005 Shock propagation in deuterium–tritium-saturated foam. Phys. Plasmas 12 (6), 062705.
5. Davies B. 2002 Integral Transforms and Their Applications. Springer.
6. Desselberger M., Jones M. W., Edwards J., Dunne M. & Willi O. 1995 Use of X-ray preheated foam layers to reduce beam structure imprint in laser-driven targets. Phys. Rev. Lett. 74 (15), 29612964.
7. Dimonte G. & Tipton R. 2006 K-L turbulence model for the self-similar growth of the Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Phys. Fluids 18 (8), 085101.
8. Drake R. P. 2006 High-Energy-Density Physics: Fundamentals, Inertial Fusion, and Experimental Astrophysics. Springer.
9. Elbaz D., Dias F., Canaud B. & Ballereau P. 2010 Modified shock velocity in heterogeneous wetted foams in the strong shock limit. Phys. Plasmas 17 (1), 012702.
10. Goncharov V. N., Gotchev O. V., Vianello E., Boehly T. R., Knauer J. P., McKenty P. W., Radha P. B., Regan S. P., Sangster T. C., Skupsky S., Smalyuk V. A., Betti R., McCrory R. L., Meyerhofer D. D. & Cherfils-Clérouin C. 2006 Early stage of implosion in inertial confinement fusion: Shock timing and perturbation evolution. Phys. Plasmas 13 (1), 012702.
11. Hazak G., Velikovich A. L., Gardner J. H. & Dahlburg J. P. 1998 Shock propagation in a low-density foam filled with fluid. Phys. Plasmas 5 (12), 43574365.
12. Huang W., Qin H., Luo S. & Wang Z. 2010 Research status of key techniques for shock-induced combustion ramjet (shcramjet) engine. Sci. China Technol. Sci. 53, 220226.
13. Huete C., Wouchuk J. G. & Velikovich A. L. 2012 Analytical linear theory for the interaction of a planar shock wave with a two- or three-dimensional random isotropic acoustic wave field. Phys. Rev. E 85, 026312.
14. Huete Ruiz de Lira C. 2010 Turbulence generation by a shock wave interacting with a random density inhomogeneity field. Phys. Scr. 2010 (T142), 014022.
15. Huete Ruiz de Lira C., Velikovich A. L. & Wouchuk J. G. 2011 Analytical linear theory for the interaction of a planar shock wave with a two- or three-dimensional random isotropic density field. Phys. Rev. E 83 (5), 056320.
16. Johnson B. M. & Schilling O. 2011 Reynolds-averaged Navier–Stokes model predictions of linear instability. II. Shock-driven flows. J. Turbul. N37.
17. Kevlahan N. & Pudritz R. E. 2009 Shock-generated vorticity in the interstellar medium and the origin of the stellar initial mass function. Astrophys. J. 702 (1), 39.
18. Kotelnikov A. D. & Montgomery D. C. 1998 Shock induced turbulence in composite materials at moderate reynolds numbers. Phys. Fluids 10 (8), 20372054.
19. Landau L. D & Lifshitz E. M. 1987 Fluid Mechanics. Butterworth-Heinemann.
20. Larsson J. & Lele S. K. 2009 Direct numerical simulation of canonical shock/turbulence interaction. Phys. Fluids 21, 126101.
21. Lee S., Lele S. K. & Moin P. 1993 Direct numerical simulation of isotropic turbulence interacting with a weak shock wave. J. Fluid Mech. 251, 533562.
22. Lee S., Lele S. K. & Moin P. 1997 Interaction of isotropic turbulence with shock waves: effect of shock strength. J. Fluid Mech. 340, 225247.
23. Mahesh K., Lee S., Lele S. K. & Moin P. 1995 The interaction of an isotropic field of acoustic waves with a shock wave. J. Fluid Mech. 300, 383407.
24. Mahesh K., Lele S. K. & Moin P. 1997 The influence of entropy fluctuations on the interaction of turbulence with a shock wave. J. Fluid Mech. 334, 353379.
25. Mason R. J., Kopp R. A., Vu H. X., Wilson D. C., Goldman S. R., Watt R. G., Dunne M. & Willi O. 1998 Computational study of laser imprint mitigation in foam-buffered inertial confinement fusion targets. Phys. Plasmas 5 (1), 211221.
26. Moody J. D., MacGowan B. J., Glenzer S. H., Kirkwood R. K., Kruer W. L., Montgomery D. S., Schmitt A. J., Williams E. A. & Stone G. F. 2000 Experimental investigation of short scalelength density fluctuations in laser-produced plasmas. Phys. Plasmas 7 (5), 21142125.
27. Moore F. K. 1954 Unsteady oblique interaction of a shock wave with a plane disturbance. NACA Report 1165.
28. Morice J. & Jaouen S. 2003 Perturbations linéaires d’ècoulements monodimensionnels á geèometriques plane, cylindrique et sphèrique. CEA-Report. CEA-R-6040 2003 R, 6040.
29. Philippe F., Canaud B., Fortin X., Garaude F. & Jourdren H. 2004 Effects of microstructure on shock propagation in foams. Laser Part. Beams 22 (02), 171174.
30. Poludnenko A. Y., Frank A. & Blackman E. G. 2002 Hydrodynamic interaction of strong shocks with inhomogeneous media. I. Adiabatic case. Astrophys. J. 576 (2), 832.
31. Remington B. A., Drake R. P., Takabe H. & Arnett D. 2000 A review of astrophysics experiments on intense lasers. Phys. Plasmas 7 (5), 16411652.
32. Ribner H. S. 1969 Acoustic energy flux from shock–turbulence interaction. J. Fluid Mech. 35 (02), 299310.
33. Ribner H. S. 1987 Spectra of noise and amplified turbulence emanating from shock-turbulence interaction. AIAA J. 25 (3), 436.
34. Ribner H. S. 1998 Comment on experimental study of a normal shock/homogeneous turbulence interaction. AIAA J. 36 (3), 494.
35. Rotman D. 1991 Shock wave effects on a turbulent flow. Phys. Fluids A: Fluid Dyn. 3 (7), 17921806.
36. Sacks R. A. & Darling D. H. 1987 Direct drive cryogenic ICF capsules employing D-T wetted foam. Nucl. Fusion 27 (3), 447.
37. 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, 99139.
38. 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 (5), 16621671.
39. Velikovich A. L., Huete C. & Wouchuk J. G. 2012 Effect of shock-generated turbulence on the Hugoniot jump conditions. Phys. Rev. E 85, 016301.
40. Velikovich A. L., Wouchuk J. G., de Lira C. H. R., Metzler N., Zalesak S. & Schmitt A. J. 2007 Shock front distortion and Richtmyer–Meshkov-type growth caused by a small preshock non-uniformity. Phys. Plasmas 14 (7), 072706.
41. Wouchuk J. G. 2001 Growth rate of the linear Richtmyer–Meshkov instability when a shock is reflected. Phys. Rev. E 63 (5), 056303.
42. Wouchuk J. G., Huete Ruiz de Lira C. & Velikovich A. L. 2009 Analytical linear theory for the interaction of a planar shock wave with an isotropic turbulent vorticity field. Phys. Rev. E 79 (6), 066315.
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? *


Type Description Title
Supplementary Materials

Huete et al. supplementary material

 PDF (582 KB)
582 KB


Full text views

Total number of HTML views: 1
Total number of PDF views: 15 *
Loading metrics...

Abstract views

Total abstract views: 125 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 25th November 2017. This data will be updated every 24 hours.