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)epsilon(2/3)k(-5/3)F(kl) [where C-K is the Kolmogorov constant and F(kl) is a function] and T(k)proportional to epsilon k(-1)psi({F})(kl) [where psi({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 epsilon is the von Karman scale l = u(3)/epsilon. We have assumed that l, u, and epsilon 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 epsilon(t), l(t), u(t) and the constituent integro-differential equation for the scaling function F(kl(t)) = E(k,t)/C(K)epsilon(2/3)k(-5/3) 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. [S1063-651X(98)00210-4]. [References: 22]
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