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

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) , 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.