Hostname: page-component-7c8c6479df-94d59 Total loading time: 0 Render date: 2024-03-29T05:19:27.552Z Has data issue: false hasContentIssue false

Transition to turbulence in shock-driven mixing: a Mach number study

Published online by Cambridge University Press:  21 November 2011

M. Lombardini*
Affiliation:
Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, USA
D. I. Pullin
Affiliation:
Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, USA
D. I. Meiron
Affiliation:
Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, USA
*
Email address for correspondence: manuel@caltech.edu

Abstract

Large-eddy simulations of single-shock-driven mixing suggest that, for sufficiently high incident Mach numbers, a two-gas mixing layer ultimately evolves to a late-time, fully developed turbulent flow, with Kolmogorov-like inertial subrange following a power law. After estimating the kinetic energy injected into the diffuse density layer during the initial shock–interface interaction, we propose a semi-empirical characterization of fully developed turbulence in such flows, based on scale separation, as a function of the initial parameter space, as , which corresponds to late-time Taylor-scale Reynolds numbers . In this expression, represents the post-shock perturbation amplitude, the change in interface velocity induced by the shock refraction, the characteristic kinematic viscosity of the mixture, the inner diffuse thickness of the initial density profile, the post-shock Atwood ratio, and for the gas combination and post-shock perturbation amplitude considered. The initially perturbed interface separating air and SF6 (pre-shock Atwood ratio ) was impacted in a heavy–light configuration by a shock wave of Mach number , 1.25, 1.56, 3.0 or 5.0, for which is fixed at about 25 % of the dominant wavelength of an initial, Gaussian perturbation spectrum. Only partial isotropization of the flow (in the sense of turbulent kinetic energy and dissipation) is observed during the late-time evolution of the mixing zone. For all Mach numbers considered, the late-time flow resembles homogeneous decaying turbulence of Batchelor type, with a turbulent kinetic energy decay exponent and large-scale () energy spectrum , and a molecular mixing fraction parameter, . An appropriate time scale characterizing the Taylor-scale Reynolds number decay, as well as the evolution of mixing parameters such as and the effective Atwood ratio , seem to indicate the existence of low- and high-Mach-number regimes.

Type
Papers
Copyright
Copyright © Cambridge University Press 2011

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

1. Balakumar, B. J., Orlicz, G. C., Tomkins, C. D. & Prestridge, K. P. 2008 Simultaneous particle-image velocimetry–planar laser-induced fluorescence measurements of Richtmyer–Meshkov instability growth in a gas curtain with and without reshock. Phys. Fluids 20 (12), 124103.CrossRefGoogle Scholar
2. Batchelor, G. K. & Proudman, I. 1956 The large-scale structure of homogeneous turbulence. Phil. Trans. R. Soc. Lond. A 248, 369405.Google Scholar
3. Blaisdell, G. A. 1991 Numerical simulation of compressible homogeneous turbulence. PhD thesis, Stanford University.Google Scholar
4. Chapman, D. R. 1979 Computational aerodynamics development and outlook. AIAA J. 17, 12931313.CrossRefGoogle Scholar
5. Chung, D. & Pullin, D. I. 2010 Direct numerical simulation and large-eddy simulation of stationary buoyancy-driven turbulence. J. Fluid Mech. 643 279308.CrossRefGoogle Scholar
6. Collins, B. D. & Jacobs, J. W. 2002 PLIF flow visualization and measurements of the Richtmyer–Meshkov instability of an interface. J. Fluid Mech. 464, 113136.CrossRefGoogle Scholar
7. Cook, A. W., Cabot, W. & Miller, P. L. 2004 The mixing transition in Rayleigh–Taylor instability. J. Fluid Mech. 511, 333362.CrossRefGoogle Scholar
8. Dimonte, G. & Schneider, M. 2000 Density ratio dependence of Rayleigh–Taylor mixing for sustained and impulsive acceleration histories. Phys. Fluids 12 (2), 304321.CrossRefGoogle Scholar
9. Dimotakis, P. E. 2000 The mixing transition in turbulent flows. J. Fluid Mech. 409, 6997.CrossRefGoogle Scholar
10. Drikakis, D., Fureby, C., Grinstein, F. F. & Youngs, D. 2007 Simulation of transition and turbulence decay in the Taylor–Green vortex. J. Turbul. 8 112.CrossRefGoogle Scholar
11. Ghosal, S. 1996 An analysis of numerical errors in large-eddy simulations of turbulence. J. Comput. Phys. 125 (1), 187206.CrossRefGoogle Scholar
12. Glezer, A. 1988 The formation of vortex rings. Phys. Fluids 31 (12), 35323542.CrossRefGoogle Scholar
13. Gottlieb, S., Shu, C. -W. & Tadmor, E. 2001 Strong stability-preserving high-order time discretization methods. SIAM Rev. 43 (1), 89112.CrossRefGoogle Scholar
14. Grinstein, F. F., Margolin, L. G. & Rider, W. G. 2007 Implicit Large-Eddy Simulation: Computing Turbulent Flow Dynamics. Cambridge University Press.CrossRefGoogle Scholar
15. Hill, D. J., Pantano, C. & Pullin, D. I. 2006 Large-eddy simulation and multi-scale modelling of Richtmyer–Meshkov instability with reshock. J. Fluid Mech. 557, 2961.CrossRefGoogle Scholar
16. Hill, D. J. & Pullin, D. I. 2004 Hybrid tuned centre-difference-WENO method for large eddy simulations in the presence of strong shocks. J. Comput. Phys. 194 (2), 435450.CrossRefGoogle Scholar
17. Honein, A. E. & Moin, P. 2004 Higher entropy conservation and numerical stability of compressible turbulence simulations. J. Comput. Phys. 201 (2), 531545.CrossRefGoogle Scholar
18. 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
19. Johnsen, E., Larsson, J., Bhagatwala, A. V., Cabot, W. H., Moin, P., Olson, B. J., Rawat, P. S., Shankar, S. K., Sjögreen, B., Yee, H. C., Zhong, X. & Lele, S. K. 2010 Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves. J. Comput. Phys. 229, 12131237.CrossRefGoogle Scholar
20. Konrad, J. H. 1976 An experimental investigation of mixing in two-dimensional turbulent shear flows with applications to diffusion-limited chemical reactions. PhD thesis, California Institute of Technology.Google Scholar
21. Kosovic, B., Pullin, D. I. & Samtaney, R. 2002 Subgrid-scale modelling for large-eddy simulations of compressible turbulence. Phys. Fluids 14 (4), 15111522.CrossRefGoogle Scholar
22. Krogstad, P. Å. & Davidson, P. A. 2010 Is grid turbulence Saffman turbulence? J. Fluid Mech. 642, 373394.CrossRefGoogle Scholar
23. Lavoie, P., Djenidi, L. & Antonia, R. A. 2007 Effects of initial conditions in decaying turbulence generated by passive grids. J. Fluid Mech. 585, 395420.CrossRefGoogle Scholar
24. Leinov, E., Malamud, G., Elbaz, Y., Levin, L. A., Ben-Dor, G., Shvarts, D. & Sadot, O. 2009 Experimental and numerical investigation of the Richtmyer–Meshkov instability under re-shock conditions. J. Fluid Mech. 626, 449475.CrossRefGoogle Scholar
25. Lombardini, M. 2008 Richtmyer–Meshkov instability in converging geometries. PhD thesis, California Institute of Technology.Google Scholar
26. Lombardini, M., Hill, D. J., Pullin, D. I. & Meiron, D. I. 2011 Atwood ratio dependence of Richtmyer–Meshkov flows under reshock conditions using large-eddy simulations. J. Fluid Mech. 670, 439480.CrossRefGoogle Scholar
27. Lombardini, M. & Pullin, D. I. 2009 Small-amplitude perturbations in the three-dimensional cylindrical Richtmyer–Meshkov instability. Phys. Fluids 21 (11), 114103.CrossRefGoogle Scholar
28. Lundgren, T. S. 1982 Strained spiral vortex model for turbulence fine structure. Phys. Fluids 25 (12), 21932203.CrossRefGoogle Scholar
29. Miles, A. R., Blue, B., Edwards, M. J., Greenough, J. A., Hansen, J. F., Robey, H. F., Drake, R. P., Kuranz, C. & Leibrandt, D. R. 2005 Transition to turbulence and effect of initial conditions on three-dimensional compressible mixing in planar blast-wave-driven systems. Phys. Plasmas 12, 056317.CrossRefGoogle Scholar
30. Misra, A. & Pullin, D. I. 1997 A vortex-based model for large-eddy simulation. Phys. Fluids 9 (8), 24432454.CrossRefGoogle Scholar
31. Motl, B., Oakley, J., Ranjan, D., Weber, C., Anderson, M. & Bonazza, R. 2009 Experimental validation of a Richtmyer–Meshkov scaling law over large density ratio and shock strength ranges. Phys. Fluids 21 (12), 126102.CrossRefGoogle Scholar
32. Orlicz, G. C., Balakumar, B. J., Tomkins, C. D. & Prestridge, K. P. 2009 A Mach number study of the Richtmyer–Meshkov instability in a varicose, heavy-gas curtain. Phys. Fluids 21 (6), 064102.CrossRefGoogle Scholar
33. Oron, D., Arazi, L., Kartoon, D., Rikanati, A., Alon, U. & Shvarts, D. 2001 Dimensionality dependence of the Rayleigh–Taylor and Richtmyer–Meshkov instability late-time scaling laws. Phys. Plasmas 8 (6), 28832889.CrossRefGoogle Scholar
34. Prestridge, K., Rightley, P. M., Vorobieff, P., Benjamin, R. F. & Kurnit, N. A. 2000 Simultaneous density-field visualization and PIV of a shock-accelerated gas curtain. Exp. Fluids 29, 339346.CrossRefGoogle Scholar
35. Pullin, D. I. 2000 A vortex-based model for the subgrid flux of a passive scalar. Phys. Fluids 12 (9), 23112319.CrossRefGoogle Scholar
36. Pullin, D. I., Buntine, J. D. & Saffman, P. G. 1994 On the spectrum of a stretched spiral vortex. Phys. Fluids 6, 30103027.CrossRefGoogle Scholar
37. Reid, R. C., Prausnitz, J. M. & Polling, B. E. 1987 The Properties of Gases and Liquids. McGraw-Hill.Google Scholar
38. Richtmyer, R. D. 1960 Taylor instability in shock acceleration of compressible fluids. Commun. Pure Appl. Maths 13, 297319.CrossRefGoogle Scholar
39. Rightley, P. M., Vorobieff, P., Martin, R. & Benjamin, R. F. 1999 Experimental observations of the mixing transition in a shock-accelerated gas curtain. Phys. Fluids 11 (1), 186200.CrossRefGoogle Scholar
40. Saddoughi, S. G. & Veeravalli, S. V. 1994 Local isotropy in turbulent boundary layers at high reynolds number. J. Fluid Mech. 268, 333372.CrossRefGoogle Scholar
41. Saffman, P. G. 1967 The large-scale structure of homogeneous turbulence. J. Fluid Mech. 27, 581593.CrossRefGoogle Scholar
42. 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), 17671771.CrossRefGoogle Scholar
43. Samtaney, R., Pullin, D. I. & Kosovic, B. 2001 Direct numerical simulation of decaying compressible turbulence and shocklet statistics. Phys. Fluids 13 (5), 14151430.CrossRefGoogle Scholar
44. 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, 4578.CrossRefGoogle Scholar
45. Sreenivasan, K. R. 1984 On the scaling of the turbulent energy dissipation rate. Phys. Fluids 27, 10481050.CrossRefGoogle Scholar
46. Tomkins, C., Kumar, S., Orlicz, G. & Prestridge, K. 2008 An experimental investigation of mixing mechanisms in shock-accelerated flow. J. Fluid Mech. 611, 131150.CrossRefGoogle Scholar
47. Vetter, M. & Sturtevant, B. 1995 Experiments on the Richtmyer–Meshkov instability of an interface. Shock Waves 4, 247252.CrossRefGoogle Scholar
48. Vorobieff, P., Rightley, P. M. & Benjamin, R. F. 1998 Power-law spectra of incipient gas-curtain turbulence. Phys. Rev. Lett. 81 (11), 22402243.CrossRefGoogle Scholar
49. Vorobieff, P., Rightley, P. M. & Benjamin, R. F. 1999 Shock-driven gas curtain: fractal dimension evolution in transition to turbulence. Physica D 133, 469476.CrossRefGoogle Scholar
50. Youngs, D. L. 1994 Numerical simulations of mixing by Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Laser Part. Beams 12, 725750.CrossRefGoogle Scholar
51. Zhou, Y. 2001 A scaling analysis of turbulent flows driven by Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Phys. Fluids 13 (2), 538544.CrossRefGoogle Scholar
52. Zhou, Y., Robey, H. F. & Buckingham, A. C. 2003 Onset of turbulence in accelerated high-Reynolds-number flow. Phys. Rev. E 67 (5).CrossRefGoogle ScholarPubMed