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Journal of Turbulence

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Breakdown of Kolmogorov's first similarity hypothesis in grid turbulence

L. Djenidia, S.F. Tardub, R.A. Antoniaa & L. Danailac a Discipline of Mechanical Engineering, School of Engineering,

University of Newcastle, Newcastle, Australia b Laboratoires des Ecoulements Geophysiques et Industriels, UMR 5519, University of Grenoble, Grenoble, France c CORIA – UMR 6614, University of Rouen, Rouen, France

Published online: 12 Jun 2014.

To cite this article: L. Djenidi, S.F. Tardu, R.A. Antonia & L. Danaila (2014) Breakdown of

Kolmogorov's first similarity hypothesis in grid turbulence, Journal of Turbulence, 15:9, 596-610,

DOI: 10.1080/14685248.2014.913848

To link to this article: http://dx.doi.org/10.1080/14685248.2014.913848

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Journal of Turbulence, 2014

Vol. 15, No. 9, 596–610, http://dx.doi.org/10.1080/14685248.2014.913848

Breakdown of Kolmogorov’s first similarity hypothesis in grid turbulence

L. Djenidia∗, S.F. Tardub, R.A. Antoniaa and L. Danailac aDiscipline of Mechanical Engineering, School of Engineering, University of Newcastle, Newcastle,

Australia; bLaboratoires des Ecoulements Geophysiques et Industriels, UMR 5519, University of

Grenoble, Grenoble, France; cCORIA – UMR 6614, University of Rouen, Rouen, France (Received 7 January 2014; accepted 6 April 2014)

Kolmogorov’s first similarity hypothesis (or KSH1) stipulates that two-point statistics have a universal form which depends on two parameters, the kinematic viscosity ν and the mean energy dissipation rate 〈〉. KSH1 is underpinned by two assumptions: the

Reynolds number is very large and local isotropy holds. To disentangle the intricacies of these two requirements, we assess the validity of KSH1 in a flow where local isotropy is a priori tenable, i.e. decaying grid turbulence. The main question we address is how large should the Reynolds number be for KSH1 to be valid over a range of scales wider than, say, five Kolmogorov scales. To this end, direct numerical simulations based on the lattice Boltzmann method are carried out in low Reynolds number grid turbulence. The results show that when the Taylor microscale Reynolds number Rλ drops below about 20, the Kolmogorov normalised spectra deviate from those at higher Rλ; the deviation increases with decreasing Rλ. It is shown that at Rλ 20, the contribution of the energy transfer in the scale-by-scale energy budget becomes smaller than the contributions from the viscous and (large-scale) non-homogeneous terms at all scales, but never vanishes, at least for the range of Reynolds investigated here. A phenomenological argument based on the ratio N between the energy-containing timescale and the dissipative range timescale leads to the condition Rλ ≥ N √ 15 for KSH1 to hold. The numerical data indicate thatN= 5, yieldingRλ 20, thus confirming our numerical finding. The present results show that KSH1, unlike the second Kolmogorov similarity hypothesis (KSH2,) does not require the existence of an inertial range. While it may seem remarkable that

KSH1 is validated at much lower Reynolds numbers than required for KSH2 in grid turbulence (Rλ ≥ 1000,), KSH1 applies to small scales which include both dissipative scales and inertial range (if it exists). One can expect that, as the Reynolds number increases, the dissipative scales should satisfy KSH1 first; then, as the Reynolds number attains very high values, the inertial range is established in conformity with KSH2.

Keywords: direct numerical simulation; homogeneous turbulence; isotropic turbulence 1. Introduction

The first similarity hypothesis of Kolmogorov (or KSH1, [1]) stipulates that when the

Reynolds number Re is large enough, and r L (r and L are the spatial separation between two points and the integral length scale, respectively), the statistics of δu(r) = u(x + r)− u(x) (the longitudinal velocity structure function of u, the longitudinal velocity fluctuation) have a universal form which depends on 〈〉, r and ν (〈〉 = ν2 〈( ∂ui∂xj + ∂uj ∂xi )2〉 is ∗Corresponding author. Email: lyazid.djenidi@newcastle.edu.au

C© 2014 Taylor & Francis

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Journal of Turbulence 597 the turbulent kinetic energy dissipation rate and ν the fluid kinematic viscosity (summation convention applies for double indices, angular brackets signify averaging). Two assumptions underpin KSH1: the Reynolds number is very large and the small scales are isotropic. One consequence of KSH1 is that second-order statistics can be expressed in a dimensionless form as 〈 (δu)2 〉 = u2Kf (r/η), (1) where vK and η are the Kolmogorov velocity (uK = (ν〈〉)1/4) and length (η = (ν3/〈〉)1/4) scales, respectively, and f a universal function; for convenience, 〈(δu)2〉(r) is replaced by 〈(δu)2〉 in the remainder of the text. In spectral form, expression (1) becomes