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The influence of initial conditions on turbulent mixing due to Richtmyer–Meshkov instability†

Published online by Cambridge University Press:  12 May 2010

B. THORNBER ,
D. DRIKAKIS ,
D. L. YOUNGS  and
R. J. R. WILLIAMS
B. THORNBER*
Affiliation:
Fluid Mechanics and Computational Science Group, Aerospace Sciences Department, Cranfield University, Cranfield MK43 0AL, UK
D. DRIKAKIS
Affiliation:
Fluid Mechanics and Computational Science Group, Aerospace Sciences Department, Cranfield University, Cranfield MK43 0AL, UK
D. L. YOUNGS
Affiliation:
AWE, Aldermaston, Reading RG7 4PR, UK
R. J. R. WILLIAMS
Affiliation:
AWE, Aldermaston, Reading RG7 4PR, UK
*
‡Email address for correspondence: b.j.r.thornber@cranfield.ac.uk
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Abstract

This paper investigates the influence of different three-dimensional multi-mode initial conditions on the rate of growth of a mixing layer initiated via a Richtmyer–Meshkov instability through a series of well-controlled numerical experiments. Results are presented for large-eddy simulation of narrowband and broadband perturbations at grid resolutions up to 3 × 109 points using two completely different numerical methods, and comparisons are made with theory and experiment. It is shown that the mixing-layer growth is strongly dependent on initial conditions, the narrowband case giving a power-law exponent θ ≈ 0.26 at low Atwood and θ ≈ 0.3 at high Atwood numbers. The broadband case uses a perturbation power spectrum of the form P(k) ∝ k−2 with a proposed theoretical growth rate of θ = 2/3. The numerical results confirm this; however, they highlight the necessity of a very fine grid to capture an appropriately broad range of initial scales. In addition, an analysis of the kinetic energy decay rates, fluctuating kinetic energy spectra, plane-averaged volume fraction profiles and mixing parameters is presented for each case.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 654 , 10 July 2010 , pp. 99 - 139
Copyright
Copyright © Cambridge University Press 2010

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Footnotes

†

Contains material ©British Crown Copyright 2009/MoD.

References

REFERENCES

Almgren, A. S., Bell, J. B., Rendleman, C. A. & Zingale, M. 2006 Low Mach number modelling of type Ia supernovae. Part I. Hydrodynamics. Astrophys. J. 637, 922–936.Google Scholar
Amendt, P., Colvin, J. D., Tipton, R. E., Hinkel, D. E., Edwards, M. J., Landen, O. L., Ramshaw, J. D., Suter, L. J., Varnum, W. S. & Watt, R. G. 2002 Indirect-drive noncryogenic double-shell ignition targets for the National Ignition Facility: design and analysis. Phys. Plasmas 9 (5), 2221–2233.Google Scholar
Barenblatt, G. I., Looss, G. & Joseph, D. D. 1983 Nonlinear Dynamics and Turbulence. Pitman Publishing.Google Scholar
Barnes, C. W., Batha, S. H., Dunne, A. M., Magelssen, G. R., Rothman, S., Day, R. D., Elliott, N. E., Haynes, D. A., Holmes, R. L., Scott, J. M., Tubbs, D. L., Youngs, D. L., Boehly, T. R. & Jaanimagi, P. 2002 Observation of mix in a compressible plasma in a convergent cylindrical geometry. Phys. Plasmas 9 (11), 4431–4438.Google Scholar
Bogey, C., de Cacqueray, N. & Bailly, C. 2009 A shock-capturing methodology based on adaptative spatial filtering for high-order nonlinear computations. J. Comput. Phys. 228, 1447–1465.Google Scholar
Chapman, P. R. & Jacobs, J. W. 2006 Experiments on the three-dimensional incompressible Richtmyer–Meshkov instability. Phys. Fluids 18, 074101.Google Scholar
Clark, T. T. & Zhou, Y. 2006 Growth rate exponents of Richtmyer–Meshkov mixing layers. J. Appl. Mech. 73, 461–268.Google Scholar
Cohen, R. H., Dannevik, W. P., Dimits, A. M., Eliason, D. E., Mirin, A. A., Zhou, Y., Porter, D. H. & Woodward, P. R. 2002 Three-dimensional simulation of a Richtmyer–Meshkov instability with a two-scale initial perturbation. Phys. Fluids 14 (10), 3692–3709.Google 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.Google Scholar
Cook, A. W., Cabot, W. & Miller, P. L. 2004 The mixing transition in Rayleigh–Taylor instability. J. Fluid Mech. 511, 333–362.Google Scholar
Cook, A. W. & Zhou, Y. 2002 Energy transfer in Rayleigh–Taylor instability. Phys. Rev. E 66, 026312.Google Scholar
Dahlburg, J. P., Fyfe, D. E., Gardner, J. H., Haan, S. W., Bodner, S. E. & Doolen, G. D. 1995 Three-dimensional multimode simulations of the ablative Rayleigh–Taylor instability. Phys. Plasmas 2 (6), 2453–2472.Google Scholar
Debar, R. 1974 A method in 2-D Eulerian hydrodynamics. Tech. Rep. UCID-196831. Lawrence Livermore National Laboratory.Google Scholar
Dimonte, G., Frerking, C. E. & Schneider, M. 1995 Richtmyer–Meshkov instability in the turbulent regime. Phys. Rev. Lett. 74, 4855–4858.Google Scholar
Dimonte, G. & Schneider, M. 1997 Turbulent Richtmyer–Meshkov instability experiments with strong radiatively driven shocks. Phys. Plasmas 4 (12), 4347–4357.Google Scholar
Dimonte, G. & Schneider, M. 2000 Density ratio dependence of Rayleigh–Taylor mixing for sustained and impulsive acceleration histories. Phys. Fluids 12, 304–321.Google Scholar
Dimonte, G., Youngs, D. L., Dimits, A., Wunsch, S., Garasi, C., Andrews, M. J., Calder, A. C., MacNeice, P., Ricker, P., Weber, S., Marinak, M., Robinson, A., Ramaprabhu, P., Fryxell, B., Olson, K., Rosner, R., Biello, J., Dursi, L., Timmes, F., Tufo, H., Young, Y.-N. & Zingale, M. 2004 A comparative study of the turbulent Rayleigh–Taylor instability using high resolution three-dimensional numerical simulations: the alpha-group collaboration. Phys. Fluids 16 (5), 1668–1693.Google Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Drikakis, D. 2003 Advances in turbulent flow computations using high-resolution methods. Prog. Aerosp. Sci. 39, 405–424.Google Scholar
Drikakis, D., Hahn, M., Mosedale, A. & Thornber, B. 2009 Large eddy simulation using high-resolution and high-order methods. Phil. Trans. R. Soc. A 367 (1899), 2985–2997.Google Scholar
Drikakis, D. & Rider, W. 2004 High-Resolution Methods for Incompressible and Low-Speed Flows. Springer.Google Scholar
Drikakis, D. & Tsangaris, S. 1993 On the solution of the compressible Navier–Stokes equations using improved flux vector splitting methods. Appl. Math. Model. 17 (6), 282.Google Scholar
Fraley, G. 1986 Rayleigh–Taylor stability for a normal shock wave–density discontinuity interaction. Phys. Fluids 29, 376–386.Google Scholar
Gauthier, S. & Bonnet, M. 1990 A k - ϵ model for turbulent mixing in shock-tube flows induced by Rayleigh–Taylor instability. Phys. Fluids A 2 (9), 1685–1694.Google Scholar
Godunov, S. K. 1959 A finite-difference method for the computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb. 47, 271–295.Google Scholar
Goncharov, V. N. 2002 Analytical model of nonlinear, single-mode, classical Rayleigh–Taylor instability at arbitrary Atwood numbers. Phys. Rev. Lett. 88 (13), 134502.Google Scholar
Grinstein, F. F., Margolin, L. G. & Rider, W. J. (Ed.) 2007 Implicit Large Eddy Simulation: Computing Turbulent Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Hill, D. J., Pantano, C. & Pullin, D. I. 2006 Large-eddy simulation and multi-scale modelling of a Richtmyer–Meshkov instability with reshock. J. Fluid. Mech 557, 29–61.Google Scholar
Holmes, R. L., Dimonte, G., Fryxell, B., Gittings, M. L., Grove, J. W., Schneider, M. S., Sharp, D. H., Velikovich, A. L., Weaver, R. P. & Zhang, Q. 1999 Richtmyer–Meshkov instability growth: experiment, simulation and theory. J. Fluid Mech. 389, 55–79.Google Scholar
Huang, M.-J. & Leonard, A. 1994 Power-law decay of homogeneous turbulence at low Reynolds numbers. Phys. Fluids 6 (11), 3765–3775.Google Scholar
Inogamov, N. A. 1999 The role of Rayleigh–Taylor and Richtmyer–Meshkov instabilities in astrophysics: an introduction. Astrophys. Space Phys. 10, 1–335.Google Scholar
Inogamov, N. 2006 Richtmyer–Meshkov turbulence. In Proceedings of the International Workshop on the Physics of Compressible Turbulent Mixing 10. Available online at: http://www.iwpctm.org/.Google Scholar
Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid. Mech. 285, 69–94.Google Scholar
Kim, K. H. & Kim, C. 2005 Accurate, efficient and monotonic numerical methods for multi-dimensional compressible flows. Part II. Multi-dimensional limiting process. J. Comput. Phys. 208, 570–615.Google Scholar
Kolmogorov, A. N. 1941 The local structure of turbulence in an incompressible fluid at very high Reynolds numbers. Dokl. Akad. Nauk. SSSR 30, 299.Google Scholar
van Leer, B. 1977 Towards the ultimate conservative difference scheme. Part IV. A new approach to numerical convection. J. Comput. Phys. 23, 276–299.Google Scholar
Lesieur, M. & Metais, O. 1996 New trends in large-eddy simulations of turbulence. Annu. Rev. Fluid Mech. 28, 45–82.CrossRefGoogle Scholar
Llor, A. 2006 Invariants of free turbulent decay. arXiv:physics/0612220v1.Google Scholar
Meshkov, E. E. 1969 Instability of the interface of two gases accelerated by a shock wave. Fluid Dyn. 43 (5), 101–104.Google Scholar
Mikaelian, K. O. 1989 Turbulent mixing generated by Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Physica D 36, 343–357.Google Scholar
Mosedale, A. & Drikakis, D. 2007 Assessment of very high-order of accuracy in LES models. J. Fluids Engng 129, 1497–1503.Google Scholar
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), 2883–2889.Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Poujade, O. & Peybernes, M. 2010 Growth rate of Rayleigh–Taylor turbulent mixing layers from first principles. Phys. Rev. E 81 (1), 016316.CrossRefGoogle Scholar
Prasad, J. K., Rasheed, A., Kumar, S. & Sturtevant, B. 2000 The late-time development of the Richtmyer–Meshkov instability. Phys. Fluids 12 (8), 2108–2115.Google Scholar
Ramaprabu, P. K. 2003 On the dynamics of Rayleigh–Taylor mixing. PhD thesis, Texas A&M University.Google Scholar
Ramshaw, J. D. 1998 Simple model for linear and nonlinear mixing at unstable fluid interfaces with variable acceleration. Phys. Rev. E 58 (5), 5834–5840.Google Scholar
Richtmyer, R. D. 1960 Taylor instability in shock acceleration of compressible fluids. Comm. Pure Appl. Math. 13, 297–319.Google Scholar
Spiteri, R. J. & Ruuth, S. J. 2002 A class of optimal high-order strong-stability preserving time discretization methods. SIAM J. Numer. Anal. 40 (2), 469–491.Google Scholar
Thornber, B. & Drikakis, D. 2007 Large-eddy simulation of multi-component compressible turbulent flows using high resolution methods. Comput. Fluids doi:10.1016/j.compfluid.2007.04.009.Google Scholar
Thornber, B. & Drikakis, D. 2008 Implicit large eddy simulation of a deep cavity using high resolution methods. AIAA J. 46 (10), 2634–2645.Google Scholar
Thornber, B., Mosedale, A. & Drikakis, D. 2007 On the implicit large eddy simulation of homogeneous decaying turbulence. J. Comput. Phys. 226, 1902–1929.Google Scholar
Thornber, B., Mosedale, A., Drikakis, D., Youngs, D. & Williams, R. 2008 An improved reconstruction method for compressible flows with low Mach number features. J. Comput. Phys. 227, 4873–4894.Google Scholar
Toro, E. F. 1997 Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer.Google Scholar
Vincent, A. & Meneguzzi, M. 1991 The spatial structure and statistical properties of homogeneous turbulence. J. Fluid Mech. 225, 1–20.Google Scholar
Youngs, D. L. 1982 Time-dependent multimaterial flow with large fluid distortion. In Numerical Methods for Fluid Dynamics, pp. 273–285. Academic Press.Google Scholar
Youngs, D. L. 1991 Three-dimensional numerical simulation of turbulent mixing by Rayleigh–Taylor instability. Phys. Fluids A 3 (5), 1312–1320.Google Scholar
Youngs, D. L. 1994 Numerical simulation of mixing by Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Laser Part. Beams 12, 725–750.Google Scholar
Youngs, D. L. 2003 Application of MILES to Rayleigh–Taylor and Richtmyer–Meshkov mixing. Paper 2003-4102. AIAA.Google Scholar
Youngs, D. L. 2004 Effect of initial conditions on self-similar turbulent mixing. In Proceedings of the International Workshop on the Physics of Compressible Turbulent Mixing 9. Available online at: http://www.iwpctm.org/.Google Scholar
Zhang, Q. & Sohn, S.-I. 1997 Nonlinear theory of unstable fluid mixing driven by shock wave. Phys. Fluids 9 (4), 1106–1124.Google Scholar
Zhou, Y. 2001 A scaling analysis of turbulent flows driven by Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Phys. Fluids 13 (2), 538–543.Google Scholar