Empirical equations for the structure of isotropic turbulence.

Abstract

An empirical equation for the double velocity-correlation function f(r, t) in the Karman-Howarth (K-H) equation, which represents the dynamic behavior of homogeneous isotropic turbulence, is proposed by use of two parameters: the rms fluctuating velocity u'' and the turbulent Reynolds number Rλ=u''λ/√2v (λ is the longitudinal microscale and v the kinematic viscosity), in the form of f(ψ, Rλ), where ψ = r/λ. The empirical equation is obtained by rearranging the data measured in a nearly isotropic field behind a grid with reference to other investigations reported previously. The validity of this empirical equation is confirmed by comparison of calculated results with experimental data over a wide range of Rλ, 10 to 104. Further, the triple correlation function k(r, t) is computed numerically based on the K-H equation with the known variable u''2 f, and consequently can be expressed in the form of k(ψ, Rλ Iλ), where Iλ is introduced to describe the decay state, as defined by Iλ = (1/4v)dλ2/dt. The calculated results are consistent with data of k(r) obtained by direct measurements and some relevant measurements reported for 10 ?? Rλ ?? 104. The energy spectrum function E(κ) and the transfer function T(κ) calculated from u''2f(r) and u''3k(r), respectively, are also considered quantitatively to be compared with some measurements and Kolmogorov''s - 5/3 power spectrum on E(κ). In addition, the application of these empirical equations to grid turbulence is described.