Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-05-11T19:17:21.078Z Has data issue: false hasContentIssue false
  • English
  • Français

A computational parameter study for the three-dimensional shock–bubble interaction

Published online by Cambridge University Press:  14 December 2007

JOHN H. J. NIEDERHAUS ,
J. A. GREENOUGH ,
J. G. OAKLEY ,
D. RANJAN ,
M. H. ANDERSON  and
R. BONAZZA
JOHN H. J. NIEDERHAUS
Affiliation:
Department of Engineering Physics, University of Wisconsin-Madison, Madison, WI 53706, USA
J. A. GREENOUGH
Affiliation:
Lawrence Livermore National Laboratory, Livermore, CA 94550, USA
J. G. OAKLEY
Affiliation:
Department of Engineering Physics, University of Wisconsin-Madison, Madison, WI 53706, USA
D. RANJAN
Affiliation:
Department of Engineering Physics, University of Wisconsin-Madison, Madison, WI 53706, USA
M. H. ANDERSON
Affiliation:
Department of Engineering Physics, University of Wisconsin-Madison, Madison, WI 53706, USA
R. BONAZZA
Affiliation:
Department of Engineering Physics, University of Wisconsin-Madison, Madison, WI 53706, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The morphology and time-dependent integral properties of the multifluid compressible flow resulting from the shock–bubble interaction in a gas environment are investigated using a series of three-dimensional multifluid-Eulerian simulations. The bubble consists of a spherical gas volume of radius 2.54 cm (128 grid points), which is accelerated by a planar shock wave. Fourteen scenarios are considered: four gas pairings, including Atwood numbers −0.8 < A < 0.7, and shock strengths 1.1 < M ≤ 5.0. The data are queried at closely spaced time intervals to obtain the time-dependent volumetric compression, mean bubble fluid velocity, circulation and extent of mixing in the shocked-bubble flow. Scaling arguments based on various properties computed from one-dimensional gasdynamics are found to collapse the trends in these quantities successfully for fixed A. However, complex changes in the shock-wave refraction pattern introduce effects that do not scale across differing gas pairings, and for some scenarios with A > 0.2, three-dimensional (non-axisymmetric) effects become particularly significant in the total enstrophy at late times. A new model for the total velocity circulation is proposed, also based on properties derived from one-dimensional gasdynamics, which compares favourably with circulation data obtained from calculations, relative to existing models. The action of nonlinear-acoustic effects and primary and secondary vorticity production is depicted in sequenced visualizations of the density and vorticity fields, which indicate the significance of both secondary vorticity generation and turbulent effects, particularly for M > 2 and A > 0.2. Movies are available with the online version of the paper.

JFM classification

Compressible Flows: Shock waves Compressible Flows: Gas dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 594 , 10 January 2008 , pp. 85 - 124
Copyright
Copyright © Cambridge University Press 2008

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

Bell, J. B., Colella, P. & Trangenstein, J. A. 1989 Higher order Godunov methods for general systems of hyperbolic conservation laws. J. Comput. Phys. 82, 362–397.CrossRefGoogle Scholar
Bell, J. B., Berger, M. J., Saltzman, J. S. & Welcome, M. 1994 Three dimensional adaptive mesh refinement for hyperbolic conservation laws. SIAM J. Sci. Comput. 15, 127–138.CrossRefGoogle Scholar
Berger, M. & Colella, P. 1989 Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys. 82, 64–84.CrossRefGoogle Scholar
Berger, M. & Oliger, J. 1984 Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53, 484–512.CrossRefGoogle Scholar
Colella, P. 1985 A direct Eulerian MUSCL scheme for gas dynamics. SIAM J. Sci. Stat. Comput. 6, 104–117.CrossRefGoogle Scholar
Colella, P. & Glaz, H. M. 1985 Efficient solution algorithms for the Riemann problem for real gases. J. Comput. Phys. 59, 264–289.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
Crutchfield, W. Y. & Welcome, M. L. 1993 Object-oriented implementations of adaptive mesh refinement algorithms. Sci. Prog. 2, 145–156.Google Scholar
Davy, B. A. & Blackstock, D. T. 1971 Measurements of the refraction and diffraction of a short N wave by a gas-filled soap bubble. J. Acoust. Soc. Am. 49, 732–737.CrossRefGoogle Scholar
Delale, C. F., Nas, S. & Tryggvason, G. 2005 Direct numerical simulations of shock propagation in bubbly liquids. Phys. Fluids 17, 121705.CrossRefGoogle Scholar
Fureby, C. & Grinstein, F. F. 2002 Large eddy simulation of high-Reynolds-number free and wall-bounded flows. J. Comput. Phys. 181, 68–97.CrossRefGoogle Scholar
Giordano, J. & Burtschell, Y. 2006 Richtmyer–Meshkov instability induced by shock–bubble interaction: numerical and analytical studies with experimental validation. Phys. Fluids 18, 036102.CrossRefGoogle Scholar
Gordon, S. & McBride, B. J. 1976 Computer program for computation of complex chemical equilibrium compositions, rocket performance, incident and reflected shocks, and Chapman–Jouguet detonations. Spec. Publ. SP-273. Lewis Research Center, NASA.Google Scholar
Greenough, J. A., Beckner, V., Pember, R. B., Crutchfield, W. Y., Bell, J. B. & Colella, P. 1995 An adaptive multifluid interface-capturing method for compressible flow in complex geometries. AIAA Paper 95-1718.CrossRefGoogle Scholar
Greenough, J. A., deSupinski, B. Supinski, B., Yates, R. K., Rendleman, C. A., Skinner, D., Beckner, V. E., Lijewski, M. & Bell, J. B. 2005 Performance of a block structured hierarchical adaptive mesh refinement code on the 64K node IBM BlueGene/L computer. LBNL Rep. LBNL-57500. Lawrence Berkeley National Laboratory, Berkeley, CA.Google Scholar
Haas, J.-F. & Sturtevant, B. 1987 Interaction of weak shock waves with cylindrical and spherical inhomogeneities. J. Fluid Mech. 181, 41–76.CrossRefGoogle Scholar
Hansen, J. F., Robey, H. F., Klein, R. I. & Miles, A. R. 2006 Mass-stripping analysis of an interstellar cloud by a supernova shock. Astrophys. Space Sci. Online version.CrossRefGoogle Scholar
Henderson, L. F. 1966 The refraction of a plane shock wave at a gas interface. J. Fluid Mech. 26, 607–637.CrossRefGoogle Scholar
Henderson, L. F. 1989 On the refraction of shock waves. J. Fluid Mech. 198, 365–386.CrossRefGoogle Scholar
Henderson, L. F., Colella, P. & Puckett, E. G. 1991 On the refraction of shock waves at a slow–fast gas interface. J. Fluid Mech. 224, 1–27.CrossRefGoogle Scholar
Jacobs, J. W. 1993 The dynamics of shock accelerated light and heavy gas cylinders. Phys. Fluids A 5, 1993.CrossRefGoogle Scholar
Jamaluddin, A. R., Ball, G. J. & Leighton, T. J. 2005 Free-Lagrange simulations of shock/bubble interaction in shock wave lithotripsy. In Shock Waves: Proc. 24th Intl Symp. on Shock Waves (ed. Z. L. Jiang).Google Scholar
Klein, R. I., McKee, C. F. & Colella, P. 1994 On the hydrodynamic interaction of shock waves with interstellar clouds. I. Nonradiative shocks in small clouds. Astrophys. J. 420, 213–236.CrossRefGoogle Scholar
Klein, R. I., Budil, K. S., Perry, T. S. & Bach, D. R. 2003 The interaction of supernova remnants with interstellar clouds: experiments on the NOVA laser. Astrophys. J. 583, 245–259.CrossRefGoogle Scholar
Latini, M., Schilling, O. & Don, W. S. 2006 Effects of WENO flux reconstruction order and spatial resolution on reshocked two-dimensional Richtmyer–Meshkov instability. J. Comput. Phys. 221, 805–836.CrossRefGoogle Scholar
Layes, G. 2005 Etude expérimentale de l'interaction d'une onde de choc avec une bulle de gaz. PhD thesis, Université de Provence (Aix-Marseille I).Google Scholar
Layes, G., Jourdan, G. & Houas, L. 2003 Distortion of a spherical gaseous interface accelerated by a plane shock wave. Phys. Rev. Lett. 91 (17).CrossRefGoogle ScholarPubMed
Layes, G., Jourdan, G. & Houas, L. 2005 Experimental investigation of the shock wave interaction with a spherical gas inhomogeneity. Phys. Fluids 17, 028103.CrossRefGoogle Scholar
Layes, G. & LeMétayer, O. 2007 Quantitative numerical and experimental studies of the shock accelerated heterogeneous bubbles motion. Phys. Fluids 19, 042105.CrossRefGoogle Scholar
Lee, D.-K., Peng, G. & Zabusky, N. J. 2006 Circulation rate of change: a vortex approach for understanding accelerated inhomogeneous flows through intermediate times. Phys. Fluids 18, 097102.CrossRefGoogle Scholar
vanLeer, B. Leer, B. 1979 Towards the ultimate conservative difference scheme. J. Comput. Phys. 32, 101–136.Google Scholar
Liepmann, H. W. & Roshko, A. 1957 Elements of Gasdynamics. John Wiley.CrossRefGoogle Scholar
Lindl, J. 1995 Development of the indirect-drive approach to inertial confinement fusion and the target physics basis for ignition and gain. Phys. Plasmas 2, 3933–4024.CrossRefGoogle Scholar
Marquina, A. & Mulet, P. 2003 A flux-split algorithm applied to conservative models for multicomponent compressible flows. J. Comput. Phys. 185, 120–138.CrossRefGoogle Scholar
Meshkov, Ye. Ye. 1970 Instability of a shock wave accelerated interface between two gases. NASA TT F-13074.Google Scholar
Miller, G. H. & Puckett, E. G. 1996 A higher-order Godunov method for multiple condensed phases. J. Comput. Phys. 128, 134–164.CrossRefGoogle Scholar
Niederhaus, J. H. J. 2007 A computational parameter study for three-dimensional shock–bubble interactions. PhD thesis, University of Wisconsin-Madison.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
Picone, J. M. & Boris, J. P. 1988 Vorticity generation by shock propagation through bubbles in a gas. J. Fluid Mech. 189, 23–51.CrossRefGoogle Scholar
Picone, J. M., Oran, E. S., Boris, J. P. & Young, T. R. 1985 Theory of vortiity generation by shock wave and flame interactions. In Dynamics of Shock Waves, Explosions, and Detonations (ed. Bowen, J. R., Manson, N., Oppenheim, A. K. & Soloukhin, R. I.), Progress in Astronautics and Aeronautics, vol. 94, chap. 4, pp. 429–448. AIAA.CrossRefGoogle Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Quirk, J. J. & Karni, S. 1996 On the dynamics of a shock–bubble interaction. J. Fluid Mech. 318, 129–163.CrossRefGoogle Scholar
Ranjan, D., Anderson, M., Oakley, J. & Bonazza, R. 2005 Experimental investigation of a strongly shocked gas bubble. Phys. Rev. Lett. 94, 184507.CrossRefGoogle ScholarPubMed
Ranjan, D., Niederhaus, J., Motl, B., Anderson, M., Oakley, J. & Bonazza, R. 2007 Experimental investigation of primary and secondary features in high-Mach-number shock–bubble interaction. Phys. Rev. Lett. 98, 024502.CrossRefGoogle ScholarPubMed
Ray, J., Samtaney, R. & Zabusky, N. J. 2000 Shock interactions with heavy gaseous elliptic cylinders: two leeward-side shock competition modes and a heuristic model for interfacial circulation deposition at early times. Phys. Fluids 12, 707–716.CrossRefGoogle Scholar
Rendleman, C. A., Beckner, V. E., Lijewski, M., Crutchfield, W. Y. & Bell, J. B. 1998 Parallelization of structured, hierarchical adaptive mesh refinement algorithms. Comput. Visualiz. Sci. 3, 147–157.CrossRefGoogle Scholar
Richtmyer, R. D. 1960 Taylor instability in shock accleration of compressible fluids. Commun. Pure Appl. Maths 13, 297–319.CrossRefGoogle Scholar
Rudinger, G. & Somers, L. M. 1960 Behaviour of small regions of different gases carried in accelerated gas flows. J. Fluid Mech. 7, 161–176.CrossRefGoogle Scholar
Samtaney, R. & Pullin, D. I. 1996 On initial-value and self-similar solutions of the compressible Euler equations. Phys. Fluids 8, 2650–2655.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. 1994 Circulation deposition on shock-accelerated planar and curved density-stratified interfaces: models and scaling laws. J. Fluid Mech. 269, 45–78.CrossRefGoogle Scholar
Strang, G. 1968 On the construction and comparison of different schemes. SIAM J. Numer. Anal. 5, 506–517.CrossRefGoogle Scholar
Widnall, S. E., Bliss, D. B. & Tsai, C. Y. 1974 The instability of short waves on a vortex ring. J. Fluid Mech. 66, 35–47.CrossRefGoogle Scholar
Winkler, K.-H., Chalmers, J. W., Hodson, S. W., Woodward, P. R. & Zabusky, N. J. 1987 A numerical laboratory. Phys. Today 40 (10), 28–37.CrossRefGoogle Scholar
Yang, J., Kubota, T. & Zukoski, E. E. 1994 A model for characterization of a vortex pair formed by shock passage over a light-gas inhomogeneity. J. Fluid Mech. 258, 217–244.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. & Zeng, S. M. 1998 Shock cavity implosion morphologies and vortical projectile generation in axisymmetric shock-spherical fast/slow bubble interactions. J. Fluid Mech. 362, 327–346.CrossRefGoogle Scholar
Zhang, S., Peng, G. & Zabusky, N. J. 2005 Vortex dynamics and baroclinically forced inhomogeneous turbulence for shock-planar heavy curtain interactions. J. Turbulence 6, 1–29.CrossRefGoogle Scholar

Niederhaus et al. supplementary movie

Movie 1. Density (bottom) and vorticity magnitude (top) fields for the interaction of a M=1.68 shock wave with a 2.54-cm-radius helium bubble in an air environment. For this gas pairing, the Atwood number is A=-0.757, and visible here is the development of irregular and divergent shock refraction patterns and the formation of a characteristic vortex ring.

Download Niederhaus et al. supplementary movie(Video)
Video 2.3 MB

Niederhaus et al. supplementary movie

Movie 2. Density (bottom) and vorticity magnitude (top) fields for the interaction of a M=3.38 shock wave with a 2.54-cm-radius argon bubble in a nitrogen environment. For this gas pairing, the Atwood number is A=0.163, and visible here are the very weak nonlinear-acoustic effects associated with low Atwood-number magnitude, resulting in low density contrast at late times and a flowfield dominated by the development of a single large vortex ring. Also visible is the emergence of a small upstrem-directed axial jet.

Download Niederhaus et al. supplementary movie(Video)
Video 892.7 KB

Niederhaus et al. supplementary movie

Movie 3. Density (bottom) and vorticity magnitude (top) fields for the interaction of a M=1.68 shock wave with a 2.54-cm-radius krypton bubble in an air environment. For this gas pairing, the Atwood number is A=0.486, and visible here are the stronger nonlinear-acoustic and turbulence-like effects associated with higher Atwood numbers. Also visible is the development of a prominent upstream-directed axial jet and strong secondary vortices around the primary vortex ring.

Download Niederhaus et al. supplementary movie(Video)
Video 6 MB

Niederhaus et al. supplementary movie

Movie 4. Density (bottom) and vorticity magnitude (top) fields for the interaction of a M=5 shock wave with a 2.54-cm-radius R12 bubble in an air environment. For this gas pairing, the Atwood number is A=0.613, and visible here are the development of very strong nonlinear-acoustic and turbulence-like effects associated with high Atwood numbers, including strong secondary shock waves scattered through the mixing region at intermediate times, and the development of a disordered vorticity field and a large plume of well-mixed fluid at late times.

Download Niederhaus et al. supplementary movie(Video)
Video 5.9 MB