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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
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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

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