Empirical equations for the structure of isotropic turbulence

Abstract

An empirical equation for the double velocity-correlation function f(r, t) in the Kármán-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/√2 v (λ is the longitudinal microscale and v the kinematic viscosity), in the form of f(φ, RλA), where (φ=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 Rλ10 to 104. Further, the triple correlation function k(r, t) is computed numerically based on the K-Hequation with the knownvariable U′2f, and consequently can be expressed in the form of k(φ, Rλ, Iλ), where Iλis 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 measurementsand somerelevant measurements reported for 10≲λ≲104. The energy spectrum function E(k) and the transfer function T(k) calculated from U′2f(r) and U′3k(r), respectively, are also considered quantitatively to be compared with some measurementsand Kolmogorov’s-5/3 powerspectrum on E(k). In addition, the application of these empirical equations to grid turbulence is described. © 1983, The Society of Chemical Engineers, Japan. All rights reserved.