Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-06-18T22:43:17.745Z Has data issue: false hasContentIssue false
  • English
  • Français

Comments on the quasi-normal Markovian approximation for fully-developed turbulence

Published online by Cambridge University Press:  19 April 2006

U. Frisch ,
M. Lesieur  and
D. Schertzer
U. Frisch
Affiliation:
CNRS, Observatoire de Nice, France
Work performed in part at the Division of Applied Sciences, Harvard University.
M. Lesieur
Affiliation:
Institut de Mécanique de Grenoble, BP 53 Centre de Tri, 38041 Grenoble, France
D. Schertzer
Affiliation:
Direction de la Météorologie, EERM-GMD, 73 rue de Sèvres, 92100 Boulogne, France
Get access Rights & Permissions [Opens in a new window]

Abstract

In a recent paper, Tatsumi, Kida & Mizushima (1978) have made a numerical study of the quasi-normal Markovian (QNM) equation for homogeneous isotropic incompressible turbulence at Reynolds numbers R up to 800.

Analytical investigations of the QNM equation support the contention of Tatsumi et al. that, at R = ∞, the decay of an initial energy spectrum of the form ka exp (− k2) leads to an initial energy-conserving regularity phase followed by a self-similar decay phase. During the former we give explicit expressions for the enstrophy and skewness. During the latter we show that for 1 < a < 4 the energy follows, for t → ∞, a t−b law with the usual value b = 2(a + 1)/(a + 3); when a [ges ] 4 deviations from Kolmogorov's (1941) $t^{\frac{10}{7}}$ law originate from non-local ‘beating’ interactions between eddies with sizes of the order of the integral scale.

We also show, analytically, that the QNM equation has a k−2, not a $k^{-\frac{5}{3}}$, inertial range and that its dissipation range is of the form k3e−k/kD, rather than e−σk1.5.

Our results are illustrated by numerical integration of the QNM equation for R up to 106 and by comparison with results from the eddy-damped quasi-normal Markovian equation which is known to produce a k−5/3 spectrum.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 97 , Issue 1 , 11 March 1980 , pp. 181 - 192
Copyright
© 1980 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

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
Bardos, C., Penel, P., Frisch, U. & Sulem, P. L. 1979 Modified dissipativity for a nonlinear evolution equation arising in turbulence. Arch. Rat. Mech. Anal. 71, 237–256.Google Scholar
DeDominicis, C. & Martin, P. C. 1979 Energy spectra of certain randomly-stirred fluids. Phys. Rev. A 19, 419–422.Google Scholar
Forster, D., Nelson, D. & Stephen, M. 1977 Large-distance and long-time properties of a randomly-stirred fluid. Phys. Rev. A 16, 732–749.Google Scholar
Fournier, J.-D. & Frisch, U. 1978 d-Dimensional turbulence. Phys. Rev. A 17, 747–762.Google Scholar
Frisch, U., Lesieur, M. & Brissaud, A. 1974 A Markovian random coupling model for turbulence. J. Fluid Mech. 65, 145–152.Google Scholar
Frisch, U., Sulem, P. L. & Nelkin, M. 1978 A simple dynamical model of intermittent fully-developed turbulence. J. Fluid Mech. 87, 719–736.Google Scholar
Kida, S. 1979 Asymptotic properties of Burgers’ turbulence. J. Fluid Mech. 93, 337–377.Google Scholar
Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. C.R. Acad. Sci. U.R.S.S. 30, 301–305.Google Scholar
Kolmogorov, A. N. 1962 A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 12, 82–85.Google Scholar
Kraichnan, R. H. 1959 The structure of isotropic turbulence at very high Reynolds numbers. J. Fluid Mech. 5, 497–543.Google Scholar
Kraichnan, R. H. 1971a An almost-Markovian Galilean-invariant turbulence model. J. Fluid Mech. 47, 513–524.Google Scholar
Kraichnan, R. H. 1971b Inertial-range transfer in two- and three-dimensional turbulence. J. Fluid Mech. 47, 525–535.Google Scholar
Kraichnan, R. H. 1974 On Kolmogorov's inertial range theories. J. Fluid Mech. 62, 305–330.Google Scholar
Leith, C. E. 1971 Atmospheric predictability and two-dimensional turbulence. J. Atmos. Sci. 28, 145–161.Google Scholar
Lesieur, M. 1973 Contribution à l’étude de quelques problèmes en turbulence pleinement développéo. Thèse d'Etat. Université de Nice.
Lesieur, M. & Schertzer, D. 1978 Amortissement auto-similaire d'une turbulence à grand nombre de Reynolds. J. Méc. 17, 610–646.Google Scholar
Leslie, D. C. 1973 Developments in the theory of turbulence. Clarendon.
Mandelbrot, B. 1976 Intermittent turbulence and fractal dimension. In Turbulence and Navier Stokes Equations (ed. R. Temam), Lecture notes in Maths. 565, pp. 121–145, Springer.
Orszag, S. A. 1966 Theory of turbulence. Princeton University Ph.D. (Available from Xerox University Microfilms, Ann Arbor, Michigan.)
Orszag, S. A. 1970 Analytical theories of turbulence. J. Fluid Mech. 41, 363–386.Google Scholar
Orszag, S. A. 1976 Statistical theory of turbulence. In Fluid Dynamics, Les Houches 1973 Lectures (ed. R. Balian & J. L. Peube), pp. 235–274. Gordon and Breach.
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
Proudman, I. & Reid, W. H. 1954 On the decay of normally distributed and homogeneous turbulent velocity fields. Phil. Trans. Roy. Soc. A 247, 163–189.Google Scholar
Rose, H. A. & Sulem, P. L. 1978 Fully-developed turbulence and statistical mechanics. J. Phys. 39, 441–484.Google Scholar
Saffman, P. G. 1967 Note on decay of homogeneous turbulence. Phys. Fluids 10, 1349.Google Scholar
Sulem, P. L., Fournier, J. D. & Pouquet, A. 1979 Fully developed turbulence and Renormalization Group. In Dynamical Critical Phenomena and Related Topics (ed. C. P. Enz). Lecture notes in Physics, vol. 104, pp. 321–335. Springer.
Sulem, P. L., Lesieur, M. & Frisch, U. 1975 Le ‘Test Field Model’ interpreté comme méthode de fermeture des équations de la turbulence. Ann. Géophys. 31, 487–495.Google Scholar
Tatsumi, T. 1960 Energy spectra in magneto-fluid-dynamic turbulence. Rev. Mod. Phys. 32, 807–812.Google Scholar
Tatsumi, T., Kida, S. & Mizushima, J. 1978 The multiple-scale cumulant expansion for isotiopic turbulence. J. Fluid Mech. 85, 97–142.Google Scholar
Tatsumi, T. & Yakase, S. 1978 The multiple-scale cumulant expansion for two-dimensional isotropic turbulence. Proc. EUROMECH 105, Institut Méc. Grenoble, p. 37.