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

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 ${E(k)=C}_{K}{\ensuremath{\varepsilon}}^{2/3}{k}^{\ensuremath{-}5/3}\mathcal{F}(\mathrm{kl})$ [where ${C}_{K}$ is the Kolmogorov constant and $\mathcal{F}(\mathrm{kl})$ is a function] and $T(k)\ensuremath{\propto}\ensuremath{\varepsilon}{k}^{\ensuremath{-}1}{\ensuremath{\psi}}^{{\mathcal{F}}}(\mathrm{kl})$ [where ${\ensuremath{\psi}}^{{\mathcal{F}}}(\mathrm{kl})$ is functionally dependent on F]. The characteristic length scale of these spectra defined from the mean square root velocity $u$ and mean energy dissipation \ensuremath{\varepsilon} is the von K\'arm\'an scale ${l=u}^{3}/\ensuremath{\varepsilon}.$ We have assumed that $l,$ $u,$ and \ensuremath{\varepsilon} 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 $\ensuremath{\varepsilon}(t),$ $l(t),$ $u(t)$ and the constituent integro-differential equation for the scaling function $F(\mathrm{kl}(t)){=E(k,t)/C}_{K}{\ensuremath{\varepsilon}}^{2/3}{k}^{\ensuremath{-}5/3}$ have been obtained using the equation of the spectral energy budget. The equation for $F(\mathrm{kl}(t))$ 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.