CALCULATION OF SPECTRA OF TURBULENCE IN THE ENERGY-CONTAINING AND INERTIAL RANGES

A statistical theory for the power law stage of freely decaying homoge neous and isotropic developed turbulence is proposed. Attention is foc used on the velocity field statistics in the energy-containing and ine rtial scales. The kinetic energy spectrum E(k, t) and energy transfer spectrum T(k, t) are calculated as functions of wave number k and deca y time t. The scaling properties of the spectra of the stationary mode l 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 g roup 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 characte ristic length scale of these spectra defined from the mean square root velocity u and mean energy dissipation epsilon is the von Karman scal e 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. Scal ing forms constructed in this way are consistent with the basic assump tion 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 inte gro-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 sp ectral 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].