Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-29T00:42:31.008Z Has data issue: false hasContentIssue false
  • English
  • Français

Universal mechanism for saturation of vorticity growth in fully developed fluid turbulence

Published online by Cambridge University Press:  09 July 2013

Chakradhar Thantanapally ,
Dhiraj V. Patil ,
Sauro Succi  and
Santosh Ansumali
Chakradhar Thantanapally
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560064, India
Dhiraj V. Patil
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560064, India
Sauro Succi
Affiliation:
Instituto Applicazioni Calcolo ‘Mauro Picone’, C.N.R, Via dei Taurini, 19, 00185, Rome, Italy Freiburg Institute for Advanced Studies, Albertstrasse, 19, D-79104, Freiburg, Germany
Santosh Ansumali*
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560064, India
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Numerical results from large-scale, long-time, simulations of decaying homogeneous turbulence are reported, which indicate that blow-up of inviscid flows is tamed by the emergence of collective dynamics of coherent structures. The simulations also suggest that this collective dynamics might lead to universal behaviour during the transient evolution of turbulence. In particular, simulations with three different initial conditions show evidence of a ${k}^{- 3} \log k$ spectrum in the transient stage, before the Kolmogorov ${k}^{- 5/ 3} $ asymptotic regime is attained. Such a universal transient might serve as a spectral funnel to the time-asymptotic Kolmogorov spectrum, which is invariably observed in the late stage of all three simulations presented in this work. The present work is entirely based on simulation evidence. However, the statistical analysis of the coherent structures suggests an analogy with population dynamics, which might be conducive to new mathematical models of transient decaying turbulence.

JFM classification

Turbulent Flows: Homogeneous turbulence Turbulent Flows: Turbulence simulation Turbulent Flows: Turbulent transition
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 728 , 10 August 2013 , R4
Copyright
©2013 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

Ansumali, S., Karlin, I. V. & Öttinger, H. C. 2003 Minimal entropic kinetic models for simulating hydrodynamics. Europhys. Lett. 63, 798–804.CrossRefGoogle Scholar
Ansumali, S., Karlin, I. V. & Succi, S. 2004 Kinetic theory of turbulence modelling: smallness parameter, scaling and microscopic derivation of smagorinsky model. Physica A 338 (3), 379–394.Google Scholar
Benzi, R., Succi, S. & Vergassola, M. 1992 The lattice Boltzmann equation: theory and applications. Phys. Rep. 222 (3), 145–197.Google Scholar
Brachet, M. E., Meiron, D. I., Orszag, S. A., Nickel, B. G., Morf, R. H. & Frisch, U. 1983 Small-scale structure of the Taylor–Green vortex. J. Fluid Mech. 130, 411–452.Google Scholar
Bustamante, M. D. & Brachet, M. 2012 Interplay between the Beale–Kato–Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem. Phys. Rev. E 86, 066302.Google Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.CrossRefGoogle Scholar
Chen, H., Kandasamy, S., Orszag, S., Shock, R., Succi, S. & Yakhot, V. 2003 Extended Boltzmann kinetic equation for turbulent flows. Science 301 (5633), 633–636.CrossRefGoogle ScholarPubMed
Cichowlas, C. & Brachet, M. E. 2005 Evolution of complex singularities in Kida–Pelz and Taylor–Green inviscid flows. Fluid Dyn. Res. 36 (4), 239–248.CrossRefGoogle Scholar
Davidson, P. A. 2004 Turbulence: An Introduction for Scientists and Engineers. Oxford University Press.Google Scholar
Dhar, S. K., Sain, A. & Pandit, R. 1997 Inertial- and dissipation-range asymptotics in fluid turbulence. Phys. Rev. Lett. 78 (15), 2964–2967.Google Scholar
Donzis, D. A., Yeung, P. K. & Sreenivasan, K. R. 2008 Dissipation and enstrophy in isotropic turbulence: resolution effects and scaling in direct numerical simulations. Phys. Fluids 20 (4), 045108.Google Scholar
Frisch, U. 1995 Turbulence: The Legacy of AN Kolmogorov. Cambridge University Press.Google Scholar
Frisch, U., Matsumoto, T. & Bec, J. 2003 Singularities of Euler flow? not out of the blue!. J. Stat. Phys. 113 (5), 761–781.CrossRefGoogle Scholar
Germano, M. 1992 Turbulence: the filtering approach. J. Fluid Mech. 238, 325–336.Google Scholar
Grafke, T., Homann, H., Dreher, J. & Grauer, R. 2008 Numerical simulations of possible finite time singularities in the incompressible Euler equations: comparison of numerical methods. Physica D 237 (14), 1932–1936.Google Scholar
Holm, D. D. & Kerr, R. 2002 Transient vortex events in the initial value problem for turbulence. Phys. Rev. Lett. 88 (24), 244501.Google Scholar
Hou, T. Y. & Li, R. 2008 Blowup or no blowup? the interplay between theory and numerics. Physica D 237 (14), 1937–1944.CrossRefGoogle Scholar
Kerr, R. M. 1993 Evidence for a singularity of the three-dimensional, incompressible Euler equations. Phys. Fluids 5 (7), 1725–1746.Google Scholar
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10 (7), 1417–1423.Google Scholar
Lesieur, M. & Metais, O. 1996 New trends in large-eddy simulations of turbulence. Annu. Rev. Fluid Mech. 28, 45–82.Google Scholar
Liepmann, H. W. 1979 The rise and fall of ideas in turbulence. Am. Sci. 67 (2), 221–228.Google Scholar
Lumley, J. L. & Yaglom, A. M. 2001 A century of turbulence. Flow Turbul. Combust. 66 (3), 241–286.Google Scholar
Moin, P. & Mahesh, K. 1998 Direct numerical simulation: a tool in turbulence research. Annu. Rev. Fluid Mech. 30, 539–578.Google Scholar
Moxey, D. & Barkley, D. 2010 Distinct large-scale turbulent-laminar states in transitional pipe flow. Proc. Natl Acad Sci. 107 (18), 8091–8096.Google Scholar
Orlandi, P. & Pirozzoli, S. 2010 Vorticity dynamics in turbulence growth. Theor. Comput. Fluid Dyn. 24 (1), 247–251.Google Scholar
Pelz, R. B. 2001 Symmetry and the hydrodynamic blow-up problem. J. Fluid Mech. 444, 299–320.Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Sreenivasan, K. R. 1999 Fluid turbulence. Rev. Mod. Phys. 71 (2), 383–395.Google Scholar
Sreenivasan, K. R. & Antonia, R. A. 1997 The phenomenology of small-scale turbulence. Annu. Rev. Fluid Mech. 29, 435–472.Google Scholar
Succi, S. 2001 The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford University Press.Google Scholar
Toschi, F. & Bodenschatz, E. 2009 Lagrangian properties of particles in turbulence. Annu. Rev. Fluid Mech. 41, 375–404.Google Scholar
Tsinober, A., Shtilman, L. & Vaisburd, H. 1997 A study of properties of vortex stretching and enstrophy generation in numerical and laboratory turbulence. Fluid. Dyn. Res. 21 (6), 477–494.Google Scholar
Yang, Y. & Pullin, D. I. 2010 On Lagrangian and vortex-surface fields for flows with Taylor–Green and Kida–Pelz initial conditions. J. Fluid Mech. 661, 446–481.Google Scholar

Thantanapally et al. supplementary movie

This movie refers to the KP initial condition at Re = 5000, with a spatial resolution of 10243, using a PS code. It shows three plots of maximum vorticity versus time, energy spectra and Taylor microscale Reynolds number versus time, respectively. The k-3 log k scaling (middle plot) in energy spectrum, in relation with the attainment of the first maxima in maximum vorticity at t ~ 2.1, is well apparent in the movie. The spectra quickly aligns with the k-3 log k line when the first maxima in maximum vorticity is reached, and then moves towards the k^-5/3 scaling, when the magnitude of maximum vorticity decreases and the flow enters a constant decay stage. The movie runs from t = 0 - 10.

Download Thantanapally et al. supplementary movie(Video)
Video 1.8 MB

Thantanapally et al. supplementary movie

Like the previous one, this movie shows the simulation result of TG initial condition at Re = 5000 and N = 10243, from t = 0-12, using a PS code. Unlike the KP flow, the first maximum in maximum vorticity is attained at t~4, where the k-3log k scaling is observed. From the movie, it is also evident that the global maxima in maximum vorticity is reached long after the first local maximum. The flow then goes towards the steady decay, where k-5/3 scaling in energy spectra is observed.

Download Thantanapally et al. supplementary movie(Video)
Video 1.5 MB

Thantanapally et al. supplementary movie

This movie refers to the simulation of a group of six Hills vortices, one from centre of each side, colliding at the centre of the periodic box. This initial condition is different from the previous ones, as it shows no symmetry. Notwithstanding this lack of symmetry, the movie confirms the existence of k-3 log k scaling, in accordance with the first maxima of maximum vorticity. It should also be noted that PS methods with periodic boundary conditions cannot represent exactly the jump in vorticity across the boundary of the vortex. This implies a small violation of incompressibility which shows however no appreciable effect on the spectral dynamics of the flow.

Download Thantanapally et al. supplementary movie(Video)
Video 1.4 MB

Thantanapally et al. supplementary movie

Indication from tracking of maximum vorticity that vorticity saturation is driven by collision-like events among high vorticity blob (0.9 of maximum vorticity at a given time, shown in movie in yellow color) and low vorticity blobs (0.5 of maximum vorticity at a given time, shown in movie in green color).

Download Thantanapally et al. supplementary movie(Video)
Video 3.5 MB