Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-14T05:49:21.508Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral modelling of high Reynolds number unstably stratified homogeneous turbulence

Published online by Cambridge University Press:  15 January 2015

A. Burlot ,
B.-J. Gréa ,
F. S. Godeferd ,
C. Cambon  and
J. Griffond
A. Burlot
Affiliation:
CEA, DAM, DIF, F-91297 Arpajon, France LMFA, Université de Lyon, École centrale de Lyon, CNRS, INSA, UCBL, F-69134 Écully, France
B.-J. Gréa*
Affiliation:
CEA, DAM, DIF, F-91297 Arpajon, France
F. S. Godeferd
Affiliation:
LMFA, Université de Lyon, École centrale de Lyon, CNRS, INSA, UCBL, F-69134 Écully, France
C. Cambon
Affiliation:
LMFA, Université de Lyon, École centrale de Lyon, CNRS, INSA, UCBL, F-69134 Écully, France
J. Griffond
Affiliation:
CEA, DAM, DIF, F-91297 Arpajon, France
*
†Email address for correspondence: benoit-joseph.grea@cea.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We study unconfined homogeneous turbulence with a destabilizing background density gradient in the Boussinesq approximation. Starting from initial isotropic turbulence, the buoyancy force induces a transient phase toward a self-similar regime accompanied by a rapid growth of kinetic energy and Reynolds number, along with the development of anisotropic structures in the flow in the direction of gravity. We model this with a two-point statistical approach using an axisymmetric eddy-damped quasi-normal Markovian (EDQNM) closure that includes buoyancy production. The model is able to match direct numerical simulations (DNS) in a parametric study showing the effect of initial Froude number and mixing intensity on the development of the flow. We further improve the model by including the stratification timescale in the characteristic relaxation time for triple correlations in the closure. It permits the computation of the long-term evolution of unstably stratified turbulence at high Reynolds number. This agrees with recent theoretical predictions concerning the self-similar dynamics and brings new insight into the spectral energy distribution and anisotropy of the flow.

JFM classification

Turbulent Flows: Stratified turbulence Turbulent Flows: Turbulence modelling Mixing: Turbulent mixing
Type
Papers
Information
Journal of Fluid Mechanics , Volume 765 , 25 February 2015 , pp. 17 - 44
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
André, J.-C. & Lesieur, M. 1977 Influence of helicity on the evolution of isotropic turbulence at high Reynolds number. J. Fluid Mech. 81, 187–207.Google Scholar
Batchelor, G. K. 1949 The role of big eddies in homogeneous turbulence. Proc. R. Soc. Lond. A 195 (1043), 513–532.Google Scholar
Batchelor, G. K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.Google Scholar
Batchelor, G. K., Canuto, V. M. & Chasnov, J. R. 1992 Buoyancy-driven variable-density turbulence. J. Fluid Mech. 235, 349–378.Google Scholar
Cambon, C. & Jacquin, L. 1989 Spectral approach to non-isotropic turbulence subjected to rotation. J. Fluid Mech. 202, 295–317.CrossRefGoogle Scholar
Cambon, C., Mansour, N. N. & Godeferd, F. S. 1997 Energy transfer in rotating turbulence. J. Fluid Mech. 337, 303–332.Google Scholar
Canuto, V. M., Dubovikov, M. S. & Dienstfrey, A. 1997 A dynamical model for turbulence. IV. Buoyancy-driven flows. Phys. Fluids 9, 2118–2131.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability, Dover.Google Scholar
Chen, S., Doolen, G., Herring, J. R., Kraichnan, R. H., Orszag, S. A. & She, Z. S. 1993 Far-dissipation range of turbulence. Phys. Rev. Lett. 70, 3051–3054.Google Scholar
Chertkov, M. 2003 Phenomenology of Rayleigh–Taylor turbulence. Phys. Rev. Lett. 91, 115001.Google Scholar
Chung, D. & Pullin, D. I. 2010 Direct numerical simulation and large-eddy simulation of stationary buoyancy-driven turbulence. J. Fluid Mech. 643, 279–308.Google Scholar
Cook, A. & Cabot, W. 2006 Reynolds number effects on Rayleigh–Taylor instability with possible implications for type-ia supernovae. Nat. Phys. 2, 562–568.Google Scholar
Craya, A.1958 Contribution à l’analyse de la turbulence associée à des vitesses moyennes. Thèse d’État. http://tel.archives-ouvertes.fr/tel-00684659.Google Scholar
Cui, A. & Street, R. L. 2004 Large-eddy simulation of coastal upwelling flow. Environ. Fluid Mech. 4 (2), 197–223.CrossRefGoogle Scholar
Delache, A., Cambon, C. & Godeferd, F. S. 2014 Scale by scale anisotropy in freely decaying rotating turbulence. Phys. Fluids 26 (2), 025104.Google Scholar
Dimonte, G., Youngs, D. L., Dimits, A., Weber, S., Mariak, M., Wunch, 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, H., 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, 1668.Google Scholar
Favier, B., Godeferd, F. S., Cambon, C., Delache, A. & Bos, W. J. T. 2011 Quasi-static magnetohydrodynamic turbulence at high Reynolds number. J. Fluid Mech. 681, 434–461.CrossRefGoogle Scholar
Godeferd, F. S. & Cambon, C. 1994 Detailed investigation of energy transfers in homogeneous stratified turbulence. Phys. Fluids 6 (6), 2084–2100.Google Scholar
Godeferd, F. S. & Staquet, C. 2003 Statistical modelling and direct numerical simulations of decaying stably stratified turbulence. Part 2. Large-scale and small-scale anisotropy. J. Fluid Mech. 486, 115–159.Google Scholar
Gréa, B.-J. 2013 The rapid acceleration model and growth rate of a tubulent mixing zone induced by Rayleigh–Taylor instability. Phys. Fluids 25, 015118.Google Scholar
Gréa, B.-J., Griffond, J. & Burlot, A. 2014 The effects of variable viscosity on the decay of homogeneous isotropic turbulence. Phys. Fluids 26 (3), 035104.CrossRefGoogle Scholar
Griffond, J., Gréa, B.-J. & Soulard, O. 2014 Unstably stratified homogeneous turbulence as a tool for turbulent mixing modeling. Trans. ASME: J. Fluids Engng 136, 091201.Google Scholar
Herring, J. R. 1974 Approach of axisymmetric turbulence to isotropy. Phys. Fluids 17, 859.Google Scholar
Kaneda, Y., Ishihara, T., Yokokawa, M., Itakura, K. & Uno, A. 2003 Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box. Phys. Fluids 15 (2), L21–L24.Google Scholar
Leith, C. E. 1971 Atmospheric predictability and two-dimensional turbulence. J. Atmos. Sci. 28, 145–161.Google Scholar
Lesieur, M. 2008 Turbulence in Fluids, Springer.Google Scholar
Lesieur, M. & Ossia, S. 2000 3d isotropic turbulence at very high Reynolds numbers: EDQNM study. J. Turbul. 1, N7.Google Scholar
Liechtenstein, L., Godeferd, F. S. & Cambon, C. 2005 Nonlinear formation of structures in rotating stratified turbulence. J. Turbul. 6, 1–18.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.Google Scholar
Livescu, D. & Ristorcelli, J. R. 2007 Buoyancy-driven variable-density turbulence. J. Fluid Mech. 591, 43–71.Google Scholar
Lumley, J. L. 1964 The spectrum of nearly inertial turbulence in a stably stratified fluid. J. Atmos. Sci. 21, 99–102.Google Scholar
Molchanov, O. A. 2004 On the origin of low- and middler-latitude ionospheric turbulence. Phys. Chem. Earth A/B/C 29 (4–9), 559–567.Google Scholar
Orszag, S. A. 1969 Analytical theories of turbulence. J. Fluid Mech. 41, 363–386.CrossRefGoogle Scholar
Orszag, S. A.1973 Lectures on the statistical theory of turbulence, In Les Houches Summer School 1973, pp. 273–374.Google Scholar
Orszag, S. A. & Patterson, G. S. 1972 Numerical simulation of three-dimensional homogeneous isotropic turbulence. Phys. Rev. Lett. 28, 76–79.Google Scholar
Pope, S. B. 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
Poujade, O. & Peybernes, M. 2010 Growth rate of Rayleigh–Taylor turbulent mixing layers with the foliation approach. Phys. Rev. E 81, 016316.Google Scholar
Pouquet, A., Lesieur, M., André, J. C. & Basdevant, C. 1975 Evolution of high Reynolds number two-dimensional turbulence. J. Fluid Mech. 72, 305–319.Google Scholar
Rayleigh, Lord 1882 Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc. s1–14 (1), 170–177.Google Scholar
Rogallo, R. S.1981 Numerical experiments in homogeneous turbulence. NASA Tech. Rep. 81315.Google Scholar
Sagaut, P. & Cambon, C. 2008 Homogeneous Turbulence Dynamics. Cambridge University Press.Google Scholar
Sharp, D. H. 1984 An overview of Rayleigh–Taylor instability. Physica D 12 (1–3), 3–18.Google Scholar
Soulard, O. 2012 Implications of the Monin–Yaglom relation for Rayleigh–Taylor turbulence. Phys. Rev. Lett. 109, 254501.CrossRefGoogle ScholarPubMed
Soulard, O. & Griffond, J. 2012 Inertial-range anisotropy in Rayleigh–Taylor turbulence. Phys. Fluids 24 (2), 025101.CrossRefGoogle Scholar
Soulard, O., Griffond, J. & Gréa, B.-J. 2014 Large-scale analysis of self-similar unstably stratified homogeneous turbulence. Phys. Fluids 26, 015110.Google Scholar
Taylor, G. I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. Lond. A 201 (1065), 192–196.Google Scholar
Thoroddsen, S. T., Van Atta, C. W. & Yampolsky, J. S. 1998 Experiments on homogeneous turbulence in an unstably stratified fluid. Phys. Fluids 10 (12), 3155–3167.Google Scholar
Vladimirova, N. & Chertkov, M. 2009 Self-similarity and universality in Rayleigh–Taylor, Boussinesq turbulence. Phys. Fluids 21, 015102.Google Scholar
Youngs, D. L. 1984 Numerical simulation of turbulent mixing by Rayleigh–Taylor instability. Physica D 12 (1–3), 32–44.Google Scholar
Youngs, D. L. 2013 The density ratio dependence of self-similar Rayleigh–Taylor mixing. Phil. Trans. R. Soc. Lond. A 371, 20120173.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.CrossRefGoogle Scholar
Zhou, Y. 2010 Renormalization group theory for fluid and plasma turbulence. Phys. Rep. 488, 1–49.Google Scholar
Zhou, Y., Matthaeus, W. H. & Dmitruk, P. 2004 Colloquium: magnetohydrodynamic turbulence and time scales in astrophysical and space plasmas. Rev. Mod. Phys. 76, 1015–1035.Google Scholar
Zhou, Y., Robey, H. F. & Buckingham, A. C. 2003 Onset of turbulence in accelerated high-Reynolds-number flow. Phys. Rev. E 67, 056305.Google Scholar