Logarithmic scaling in the longitudinal velocity variance explained by a spectral budget

A logarithmic scaling for the streamwise turbulent intensity sigma(2)(upsilon) /u(*)(2) = B-1 - A(1) ln(z/delta) was reported across several high Reynolds number laboratory experiments as predicted from Townsend's attached eddy hypothesis, where u(*) is the friction velocity and z is the height normalized by the boundary layer thickness delta. Aphenomenological explanation for the origin of this log-law in the intermediate region is provided here based on a solution to a spectral budget where the production and energy transfer terms are modeled. The solution to this spectral budget predicts A1 = (18/55) C-0/k(2/3) 3 and B-1 = (2.5)A(1), where C-0 and k are the Kolmogorov and von Karman constants. These predictions hold when very large scale motions do not disturb the k(-1) scaling existing across all wavenumbers 1/delta < k < 1/z in the streamwise turbulent velocity spectrum E-u(k). Deviations from a k-1 scaling along with their effects on A1 and B1 are discussed using published data and field experiments. (C) 2013 AIP Publishing LLC.