Calculation of spectra of turbulence in the energy-containing and inertial ranges

L. Ts. Adzhemyan, M. Hnatich, D. Horváth, and M. Stehlik
Phys. Rev. E 58, 4511 – Published 1 October 1998
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Abstract

A statistical theory for the power law stage of freely decaying homogeneous and isotropic developed turbulence is proposed. Attention is focused on the velocity field statistics in the energy-containing and inertial scales. The kinetic energy spectrum E(k,t) and energy transfer spectrum T(k,t) are calculated as functions of wave number k and decay time t. The scaling properties of the spectra of the stationary model of the randomly stirred fluid have been chosen as the starting point for the approximate derivation of time-dependent spectra E(k,t) and T(k,t). The stationary model analyzed by means of the renormalization group and short-distance expansion methods has provided the spectra E(k)=CKɛ2/3k5/3F(kl) [where CK is the Kolmogorov constant and F(kl) is a function] and T(k)ɛk1ψ{F}(kl) [where ψ{F}(kl) is functionally dependent on F]. The characteristic length scale of these spectra defined from the mean square root velocity u and mean energy dissipation ɛ is the von Kármán scale l=u3/ɛ. We have assumed that l, u, and ɛ as well as E(k) and T(k) are no longer constants but unknown functions of t. Scaling forms constructed in this way are consistent with the basic assumption of George’s closure [W. M. George, Phys. Fluids A 4, 1492 (1992)]. Power decay laws for ɛ(t), l(t), u(t) and the constituent integro-differential equation for the scaling function F(kl(t))=E(k,t)/CKɛ2/3k5/3 have been obtained using the equation of the spectral energy budget. The equation for F(kl(t)) has been investigated numerically for the three-dimensional system with Saffman’s invariant [P. G. Saffman, J. Fluid Mech. 27, 581 (1967); Phys. Fluids 10, 1349 (1967)]. The calculated longitudinal energy spectrum has been compared with the available experimental data.

  • Received 14 August 1997

DOI:https://doi.org/10.1103/PhysRevE.58.4511

©1998 American Physical Society

Authors & Affiliations

L. Ts. Adzhemyan1, M. Hnatich2, D. Horváth3, and M. Stehlik2

  • 1Department of Theoretical Physics, St. Petersburg University, Uljanovskaja 1, 198904 St. Petersburg, Petrodvorez, Russia
  • 2Institute of Experimental Physics of Slovak Academy of Sciences, Watsonova 47, 04353 Košice, Slovakia
  • 3Department of Theoretical Physics, Faculty of Science, P.J. Šafárik University, Moyzesova 16, 04154 Košice, Slovakia

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Vol. 58, Iss. 4 — October 1998

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