Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-14T07:03:50.843Z Has data issue: false hasContentIssue false
  • English
  • Français

The mixing region in freely decaying variable-density turbulence

Published online by Cambridge University Press:  05 May 2015

Pooya Movahed  and
Eric Johnsen
Pooya Movahed*
Affiliation:
Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA
Eric Johnsen
Affiliation:
Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA
*
†Email address for correspondence: pooyam@umich.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

A novel set-up is proposed to numerically study turbulent multimaterial mixing, starting from an unperturbed material interface between a light and a heavy fluid. We conduct direct numerical simulation (DNS) to better understand the role of density gradient alone on the turbulence, specifically with regard to the mixing region dynamics and anisotropy across scales. Freely decaying isotropic turbulent fields of different densities but identical kinematic viscosities are juxtaposed. The rationale for this strategy is that conventional turbulence scalings are based on kinetic energy per unit mass and kinematic viscosity. Thus, by matching the initial kinematics (root-mean-square velocity) and the dissipation (kinematic viscosity), the turbulence (kinetic energy per unit mass) decays at the same rate in both fluids. With this set-up, the effect of the density gradient alone on the turbulence can be considered, independently from other contributions (e.g. mismatch in kinetic energy per unit mass, acceleration field, etc.). We examine the mixing region dynamics at large and small scales for different density ratios and Reynolds numbers. After an initial transient, we observe a self-similar growth of the mixing region, which we explain via theoretical arguments verified by the DNS results. Inside the mixing region, the momentum of the heavier eddies causes the mean interface location to shift toward the light fluid. A higher density ratio leads to a wider, less molecularly mixed mixing region. Although anisotropy is evident at the large scales, the dissipation scales remain essentially isotropic, even at the highest density ratio under consideration (12:1). The intermittency of the velocity field exhibits isotropy, while the mass fraction field is more intermittent in the direction of the density gradient.

JFM classification

Turbulent Flows: Intermittency Turbulent Flows: Isotropic turbulence Mixing: Turbulent mixing
Type
Papers
Information
Journal of Fluid Mechanics , Volume 772 , 10 June 2015 , pp. 386 - 426
Check for updates
Copyright
© 2015 Cambridge University Press 

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

Abarzhi, S. I. 2010 On fundamentals of Rayleigh–Taylor turbulent mixing. Europhys. Lett. 91, 35001.Google Scholar
Abarzhi, S. I., Gorobets, A. & Sreenivasan, K. R. 2005 Rayleigh–Taylor turbulent mixing of immiscible, miscible and stratified fluids. Phys. Fluids 17, 081705.CrossRefGoogle Scholar
Anand, M. S. & Pope, S. B. 1983 Diffusion behind a line source in grid turbulence. Turbulent Shear Flows 4, 46–61.Google Scholar
Antonia, R. A., Lee, S. K., Djenidi, L., Lavoie, P. & Danaila, L. 2013 Invariants for slightly heated decaying grid turbulence. J. Fluid Mech. 727, 379–406.Google Scholar
Batchelor, G. K. & Proudman, I. 1956 The large-scale structure of homogeneous turbulence. Phil. Trans. R. Soc. Lond. A 248, 369–405.Google Scholar
Bhagatwala, A. & Lele, S. K. 2011 Interaction of a Taylor blast wave with isotropic turbulence. Phys. Fluids 23, 035103.Google Scholar
Bhagatwala, A. & Lele, S. K. 2012 Interaction of a converging spherical shock wave with isotropic turbulence. Phys. Fluids 24, 085102.CrossRefGoogle Scholar
Blaisdell, G. A., Spyropoulos, E. T. & Qin, J. H. 1996 The effect of the formulation of nonlinear terms on aliasing errors in spectral methods. Appl. Numer. Maths 21, 207–219.Google Scholar
Brouillette, M. 2002 The Richtmyer–Meshkov instability. Annu. Rev. Fluid Mech. 34, 445–468.Google Scholar
Cabot, W. H. & Cook, A. W. 2006 Reynolds number effects on Rayleigh–Taylor instability with possible implications for type-Ia supernovae. Nat. Phys. 2, 562–568.Google Scholar
Cabot, W. & Zhou, Y. 2013 Statistical measurements of scaling and anisotropy of turbulent flows induced by Rayleigh–Taylor instability. Phys. Fluids 25, 015107.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Comte-Bellot, G. & Corrsin, S. 1966 Use of a contraction to improve isotropy of grid-generated turbulence. J. Fluid Mech. 25 (4), 657–682.Google Scholar
Cook, A. W. & Dimotakis, P. E. 2001 Transition stages of Rayleigh–Taylor instability between miscible fluids. J. Fluid Mech. 443, 69–99.Google Scholar
Corrsin, S. 1951 On the spectrum os isotropic temperature fluctuations in an isotropic turbulence. J. Appl. Phys. 22 (4), 469–473.Google Scholar
Danckwerts, P. V. 1952 The definition and measurement of some characteristics of mixtures. Appl. Sci. Res. 3 (4), 279–296.Google Scholar
Danckwerts, P. V. 1958 The effect of incomplete mixing on homogeneous reactions. Chem. Engng Sci. 8 (1), 93–102.Google Scholar
Dimonte, G., Youngs, D. L., Dimits, A., Weber, S., Marinak, M., Wunsch, S., Garasi, C., Robinson, A., Andrews, M. J., Ramaprabhu, P., Calder, A. C., Fryxell, B., Biello, J., Dursi, L., MacNeice, P., Olson, K., Ricker, P., Rosner, R., 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.CrossRefGoogle Scholar
Dimotakis, P. E. 2000 The mixing transition in turbulent flows. J. Fluid Mech. 409, 69–98.Google Scholar
Dimotakis, P. E. 2005 Turbulent mixing. Annu. Rev. Fluid Mech. 37, 329–356.Google Scholar
Drake, R. P. 2006 High-Energy Density Physics, 1st edn. Springer.Google Scholar
Ducros, F., Laporte, F., Souleres, T., Guinot, V., Moinat, P. & Caruelle, B. 2000 High-order fluxes for conservative skew-symmetric-like schemes in structured meshes: application to compressible flows. J. Comput. Phys. 161, 114–139.CrossRefGoogle Scholar
Eyink, G. L. & Aluie, H. 2009 Localness of energy cascade in hydrodynamic turbulence. I. Smooth coarse graining. Phys. Fluids 21 (11), 115107.Google Scholar
George, W. K. 1992 The decay of homogeneous isotropic turbulence. Phys. Fluids A 4 (7), 1492–1509.Google Scholar
Gottlieb, S. & Shu, C. W. 1998 Total variation diminishing Runge–Kutta schemes. Maths Comput. 67, 73–85.Google Scholar
Hill, D. J., Pantano, C. & Pullin, D. I. 2006 Large-eddy simulation and multiscale modelling of a Richtmyer–Meshkov instability with reshock. J. Fluid Mech. 557, 29–61.Google Scholar
Huq, P. & Britter, R. E. 1995a Mixing due to grid-generated turbulence of a two-layer scalar profile. J. Fluid Mech. 285, 17–40.Google Scholar
Huq, P. & Britter, R. E. 1995b Turbulence evolution and mixing in a two-layer stably stratified fluid. J. Fluid Mech. 285, 41–67.Google Scholar
Jayesh, Tong, C. N. & Warhaft, Z. 1994 On temperature spectra in grid turbulence. Phys. Fluids 6 (1), 306–312.Google Scholar
Johnsen, E., Larsson, J., Bhagatwala, A. V., Cabot, W. H., Moin, P., Olson, B. J., Rawat, P. S., Shankar, S. K., Sjogreen, 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, 1213–1237.Google Scholar
Joseph, D. D. 1990 Fluid dynamics of two miscible liquids with diffusion and gradient stresses. Eur. J. Mech. (B/Fluids) 9 (6), 565–596.Google Scholar
Kifonidis, K., Plewa, T., Scheck, L., Janka, H. T. & Muller, E. 2006 Non-spherical core collapse supernovae – II. The late-time evolution of globally anisotropic neutrino-driven explosions and their implications for SN 1987 A. Astron. Astrophys. 453, 661–678.Google Scholar
Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 301.Google Scholar
Krogstad, P. A. & Davidson, P. A. 2010 Is grid turbulence Saffman turbulence? J. Fluid Mech. 642, 373–394.Google Scholar
Lamriben, C., Cortet, P. & Moisy, F. 2011 Direct measurements of anisotropic energy transfers in a rotating turbulence experiment. Phys. Rev. Lett. 107 (2), 024503.Google Scholar
Larsson, J. & Lele, S. K. 2009 Direct numerical simulation of canonical shock/turbulence interaction. Phys. Fluids 21, 126101.Google Scholar
Lawrie, A. G. W. & Dalziel, S. B. 2011 Turbulent diffusion in tall tubes. I. Models for Rayleigh–Taylor instability. Phys. Fluids 23, 085109.CrossRefGoogle Scholar
Layzer, D. 1955 On the instability of superposed fluids in a gravitational field. Astrophys. J. 122, 1–12.Google Scholar
Lee, S., Lele, S. K. & Moin, P. 1991 Eddy shocklets in decaying compressible turbulence. Phys. Fluids 3, 657–664.Google Scholar
Linden, P. F. 1980 Mixing across a density interface produced by grid turbulence. J. Fluid Mech. 100, 691–703.Google 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.Google Scholar
Livescu, D., Jaberi, F. A. & Madnia, C. K. 2000 Passive-scalar wake behind a line source in grid turbulence. J. Fluid Mech. 416, 117–149.Google Scholar
Livescu, D. & Ristorcelli, J. R. 2007 Buoyancy-driven variable-density turbulence. J. Fluid Mech. 591, 43–71.Google Scholar
Livescu, D. & Ristorcelli, J. R. 2008 Variable-density mixing in buoyancy-driven turbulence. J. Fluid Mech. 605, 145–180.Google Scholar
Livescu, D., Ristorcelli, J. R., Petersen, M. R. & Gore, R. A. 2010 New phenomena in variable-density Rayleigh–Taylor turbulence. Phys. Scr. T 2010 (142), 014015.Google Scholar
Monin, A. S. & Yaglom, A. M. 1975 Statistical Fluid Mechanics. MIT Press.Google Scholar
Movahed, P.2014 High-fidelity numerical simulations of compressible turbulence and mixing generated by hydrodynamic instabilities. PhD thesis, University of Michigan.Google Scholar
Movahed, P. & Johnsen, E.2011 Numerical simulations of the Richtmyer–Meshkov instability with reshock. AIAA Paper 2011-3680.Google Scholar
Movahed, P. & Johnsen, E. 2013a A solution-adaptive method for efficient compressible multifluid simulations, with application to the Richtmyer–Meshkov instability. J. Comput. Phys. 239, 166–186.Google Scholar
Movahed, P. & Johnsen, E.2013b Turbulence diffusion effects at material interfaces, with application to the Rayleigh–Taylor instability. AIAA Paper 2013-3121.Google Scholar
Mueschke, N. J. & Schilling, O. 2009 Investigation of Rayleigh–Taylor turbulence and mixing using direct numerical simulation with experimentally measured initial conditions. I. Comparison to experimental data. Phys. Fluids 21, 014106.Google Scholar
Mydlarski, L. & Warhaft, Z. 1998 Passive scalar statistics in high-Peclet-number grid turbulence. J. Fluid Mech. 358, 135–175.Google Scholar
Naso, A. & Pumir, A. 2005 Scale dependence of the coarse-grained velocity derivative tensor structure in turbulence. Phys. Rev. E 72 (5), 056318.CrossRefGoogle ScholarPubMed
Naso, A., Pumir, A. & Chertkov, M. 2006 Scale dependence of the coarse-grained velocity derivative tensor: influence of large-scale shear on small-scale turbulence. J. Turbul. 7, 1–11.Google Scholar
Obukhov, A. M. 1949 Structure of the temperature field in turbulent flows. Izv. Akad. Nauk SSSR Geogr. Geofiz. 13, 1–58.Google Scholar
Pirozzoli, S. 2011 Numerical methods for high-speed flows. Annu. Rev. Fluid Mech. 43, 163–194.Google Scholar
Pope, S. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Poujade, O. 2006 Rayleigh–Taylor turbulence is nothing like Kolmogorov turbulence in the self-similar regime. Phys. Rev. Lett. 97, 185002.Google Scholar
Ramaprabhu, P., Karkhanis, V. & Lawrie, A. G. W. 2013 The Rayleigh–Taylor Instability driven by an accel–decel–accel profile. Phys. Fluids 25 (11), 115104.Google Scholar
Reid, R. C., Prausnitz, J. M. & Poling, B. E. 1987 The Properties of Gases and Liquids, 4th edn. McGraw-Hill.Google Scholar
Ristorcelli, J. R. & Blaisdell, G. A. 1997 Consistent initial conditions for the DNS of compressible turbulence. Phys. Fluids 9, 4–6.Google Scholar
Ristorcelli, J. R. & Clark, T. T. 2004 Rayleigh–Taylor turbulence: self-similar analysis and direct numerical simulations. J. Fluid Mech. 507, 213–253.Google Scholar
Rogers, M. M. & Moser, R. D. 1992 The three-dimensional evolution of a plane mixing layer: the Kelvin–Helmholtz rollup. J. Fluid Mech. 243, 183–226.Google Scholar
Saffman, P. J. 1967 The large-scale structure of homogeneous turbulence. J. Fluid Mech. 27, 581–593.Google Scholar
Samtaney, R., Pullin, D. I. & Kosovic, B. 2001 Direct numerical simulation of decaying compressible turbulence and shocklet statistics. Phys. Fluids 13 (5), 1415–1430.Google Scholar
Sawford, B. 2001 Turbulent relative dispersion. Annu. Rev. Fluid Mech. 33, 289–317.Google Scholar
Sharp, D. H. 1984 An overview of Rayleigh–Taylor instability. Physica D 12, 3–18.Google Scholar
Soulard, O. & Griffond, J. 2012 Inertial-range anisotropy in Rayleigh–Taylor turbulence. Phys. Fluids 24, 025101.Google Scholar
Sreenivasan, K. R. & Antonia, R. A. 1997 The phenomenology of small-scale turbulence. Annu. Rev. Fluid Mech. 29, 435–472.Google Scholar
Stapountzis, H., Sawford, B. L., Hunt, J. C. R. & Britter, R. E. 1986 Structure of the temperature-field downwind of a line source in grid turbulence. J. Fluid Mech. 165, 401–424.Google Scholar
Tao, B., Katz, J. & Meneveau, C. 2002 Statistical geometry of subgrid-scale stresses determined from holographic particle image velocimetry measurements. J. Fluid Mech. 457, 35–78.Google Scholar
Thomas, V. A. & Kares, R. J. 2012 Drive asymmetry and the origin of turbulence in an ICF implosion. Phys. Rev. Lett. 109, 075004.Google Scholar
Thompson, K. W. 1987 Time-dependent boundary-conditions for hyperbolic systems. J. Comput. Phys. 68, 1–24.Google Scholar
Thornber, B., Drikakis, D., Youngs, D. L. & Williams, R. J. R. 2011 Growth of a Richtmyer–Meshkov turbulent layer after reshock. Phys. Fluids 23, 095107.Google Scholar
Tong, C. N. & Warhaft, Z. 1994 On passive scalar derivative statistics in grid turbulence. Phys. Fluids 6 (6), 2165–2176.Google Scholar
Tordella, D. & Iovieno, M. 2006 Numerical experiments on the intermediate asymptotics of shear-free turbulent transport and diffusion. J. Fluid Mech. 549, 429–441.Google Scholar
Tordella, D. & Iovieno, M. 2011 Small-scale anisotropy in turbulent shearless mixing. Phys. Rev. Lett. 107, 194501.CrossRefGoogle ScholarPubMed
Tordella, D. & Iovieno, M. 2012 Decaying turbulence: What happens when the correlation length varies spatially in two adjacent zones. Physica D 241, 178–185.Google Scholar
Tordella, D., Iovieno, M. & Bailey, P. R. 2008 Sufficient condition for Gaussian departure in turbulence. Phys. Rev. E 77, 016309.Google Scholar
Tritschler, V. K., Olson, B. J., Lele, S. K., Hickel, S., Hu, X. Y. & Adams, N. A. 2014 On the Richtmyer–Meshkov instability evolving from a deterministic multimode planar interface. J. Fluid Mech. 755, 429–462.Google Scholar
Turner, J. S. 1968 The influence of molecular diffusivity on turbulent entrainment across a density interface. J. Fluid Mech. 33, 639–656.Google Scholar
Warhaft, Z. 1984 The interference of thermal fields from line sources in grid turbulence. J. Fluid Mech. 144, 363–387.Google Scholar
Warhaft, Z. 2000 Passive scalars in turbulent flows. Annu. Rev. Fluid Mech. 32, 203–240.Google Scholar
Watanabe, T. & Gotoh, T. 2006 Intermittency in passive scalar turbulence under the uniform mean scalar gradient. Phys. Fluids 18, 058105.Google Scholar
Watanabe, T. & Gotoh, T. 2007 Inertial-range intermittency and accuracy of direct numerical simulation for turbulence and passive scalar turbulence. J. Fluid Mech. 590, 117–146.Google Scholar