Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-28T06:55:12.320Z Has data issue: false hasContentIssue false
  • English
  • Français

A stochastic view of isotropic turbulence decay

Published online by Cambridge University Press:  26 January 2011

MARCELLO MELDI ,
PIERRE SAGAUT  and
DIDIER LUCOR
MARCELLO MELDI*
Affiliation:
DIMeC Dipartimento di Ingegneria Meccanica e Civile, Università degli studi di Modena e Reggio Emilia, 41100 Modena, Italy
PIERRE SAGAUT
Affiliation:
Institut Jean Le Rond d'Alembert, UMR 7190, 4 Place Jussieu, Case 162, Université Pierre et Marie Curie, Paris 6, F-75252 ParisCEDEX 5, France
DIDIER LUCOR
Affiliation:
Institut Jean Le Rond d'Alembert, UMR 7190, 4 Place Jussieu, Case 162, Université Pierre et Marie Curie, Paris 6, F-75252 ParisCEDEX 5, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A stochastic eddy-damped quasi-normal Markovian (EDQNM) approach is used to investigate self-similar decaying isotropic turbulence at a high Reynolds number (400 ≤ Reλ ≤ 104). The realistic energy spectrum functional form recently proposed by Meyers & Menevau (Phys. Fluids, vol. 20, 2008, p. 065109) is generalized by considering some of the model constants as random parameters, since they escape measure in most experimental set-ups. The induced uncertainty on the solution is investigated, building response surfaces for decay power-law exponents of usual physical quantities. Large-scale uncertainties are considered, the emphasis being put on Saffman and Batchelor turbulences. The sensitivity of the solution to initial spectrum uncertainties is quantified through probability density functions of the decay exponents. It is observed that the initial spectrum shape at very large scales governs the long-time evolution, even at a high Reynolds number, a parameter which is not explicitly taken into account in many theoretical works. Therefore, a universal asymptotic behaviour in which kinetic energy decays as t−1 is not detected. However, this decay law is observed at finite Reynolds numbers with low probability for some initial conditions.

JFM classification

Turbulent Flows: Isotropic turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 668 , 10 February 2011 , pp. 351 - 362
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

REFERENCES

Antonia, R. A., Smalley, R. J., Zhou, T., Anselmet, F. & Danaila, L. 2003 Similarity of energy structure functions in decaying homogeneous isotropic turbulence. J. Fluid Mech. 487, 245–269.CrossRefGoogle Scholar
Batchelor, G. K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.Google Scholar
Batchelor, G. K. & Proudman, I. 1965 The large-scale structure of homogeneous turbulence. Phil. Trans. R. Soc. Lond. A 248, 369–405.Google Scholar
Batchelor, G. K. & Townsend, A. A. 1948 Decay of turbulence in the final period. Proc. R. Soc. Lond. A 194, 527–543.Google Scholar
Burattini, P., Lavoie, P., Agrawal, A., Djenidi, L. & Antonia, R. A. 2006 Power law of decaying homogeneous isotropic turbulence at low Reynolds number. Phys. Rev. E 73, 066304.CrossRefGoogle ScholarPubMed
Clark, T. T. & Zemach, C. 1998 Symmetries and the approach to statistical equilibrium in isotropic turbulence. Phys. Fluids 10 (11), 2846–2858.CrossRefGoogle Scholar
Comte-Bellot, G. & Corrsin, S. 1966 The use of a contraction to improve the isotropy of grid-generated turbulence. J. Fluid Mech. 25, 657–682.CrossRefGoogle Scholar
Davidson, P. A. 2004 Turbulence: An Introduction for Scientists and Engineers. Oxford University Press.Google Scholar
Frenkel, A. L. 1984 Decay of homogeneous turbulence in three-range models. Phys. Lett. A 102 (7), 298–302.CrossRefGoogle Scholar
Frenkel, A. L. & Levich, E. 1983 Statistical helicity invariant and decay of inertial turbulence. Phys. Lett. A 98 (1–2), 25–27.CrossRefGoogle Scholar
Hinze, J. O. 1975 Turbulence. McGraw-Hill Series in Mechanical Engineering.Google Scholar
George, W. K. 1992 The decay of homogeneous isotropic turbulence. Phys. Fluids A 4 (7), 1492–1509.CrossRefGoogle Scholar
Ghanem, R. G. & Spanos, P. 1991 Stochastic Finite Elements: A Spectral Approach. Springer.CrossRefGoogle Scholar
Ishida, T., Davidson, P. A. & Kaneda, Y. 2006 On the decay of isotropic turbulence. J. Fluid Mech. 564, 455–475.CrossRefGoogle Scholar
Ko, J., Lucor, D. & Sagaut, P. 2008 Sensitivity of two-dimensional spatially developing mixing layers with respect to uncertain inflow conditions. Phys. Fluids 20 (7), 077102–077120.CrossRefGoogle Scholar
Kolmogorov, A. N. 1941 On the degeneration of isotropic turbulence in an incompressible viscous fluid. Dokl. Akad. Nauk. SSSR 31 (6), 538–541.Google Scholar
Krogstad, P.Å. & Davidson, P. A. 2010 Is grid turbulence Saffman turbulence? J. Fluid Mech. 642, 373–394.CrossRefGoogle Scholar
Lavoie, P., Djenedi, L. & Antonia, R. A. 2007 Effects of initial conditions in decaying turbulence generated by passive grids. J. Fluid Mech. 585, 395–420.CrossRefGoogle Scholar
Lucor, D., Meyers, J. & Sagaut, P. 2007 Sensitivity analysis of large-eddy simulations to subgrid-scale-model parametric uncertainty using polynomial chaos. J. Fluid Mech. 585, 255–279.CrossRefGoogle Scholar
Meyers, J. & Menevau, C. 2008 A functional form for the energy spectrum parametrizing bottleneck and intermittency effects. Phys. Fluids 20 (6), 065109.CrossRefGoogle Scholar
Mohamed, M. S. & LaRue, J. C. 1990 The decay power law in grid-generated turbulence. J. Fluid Mech. 219, 195–214.CrossRefGoogle Scholar
Monin, A. S. & Yaglom, A. M. 1975 Statistical Fluid Mechanics: Mechanics of Turbulence, vol. 2. MIT Press.Google Scholar
Oberlack, M. 2002 On the decay exponent of isotropic turbulence. Proc. Appl. Math. Mech. 1 (1), 294–297.3.0.CO;2-W>CrossRefGoogle Scholar
Orszag, S. A. 1970 Analytical theories of turbulence. J. Fluid Mech. 41, 363–386.CrossRefGoogle Scholar
Saffman, P. J. 1967 a The large-scale structure of homogeneous turbulence. J. Fluid Mech. 27, 581–593.CrossRefGoogle Scholar
Saffman, P. J. 1967 b A note on decay of homogeneous turbulence. Phys. Fluids 10, 1349–1352.CrossRefGoogle Scholar
Sagaut, P. & Cambon, C. 2008 Homogenous Turbulence Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Skrbrek, L. & Stalp, S. R. 2000 On the decay of homogeneous isotropic turbulence. Phys. Fluids 12 (8), 1997–2019.CrossRefGoogle Scholar
Sobol, I. 1993 Sensitivity estimates for nonlinear mathematical models. Math. Modeling Comput. Exp. 1, 407–414.Google Scholar
Speziale, C. G. & Bernard, P. S. 1992 The energy decay in self-preserving isotropic turbulence revisited. J. Fluid Mech. 241, 645–667.CrossRefGoogle Scholar
Taylor, G. I. 1935 Statistical theory of turbulence. Proc. R. Soc. Lond. A 151, 421–444.CrossRefGoogle Scholar
Uberoi, M. S. 1963 Energy transfer in isotropic turbulence. Phys. Fluids 6 (8), 1048–1056.CrossRefGoogle Scholar
Wang, H. & George, W. K. 2002 The integral scale in homogeneous isotropic turbulence. J. Fluid Mech. 459, 429–443.CrossRefGoogle Scholar
Warhaft, Z. & Lumley, J. L. 1978 An experimental study of the decay of temperature fluctuations in grid-generated turbulence. J. Fluid Mech. 88, 659–684.CrossRefGoogle Scholar