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

A logarithmic scaling for the streamwise turbulent intensity [12pt]{minimal}{document}σ ᵤ²/u_*²=B₁-A₁ln (z/δ ){document}σu2/u*2=B1−A1lnz/δ 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 δ. A phenomenological 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)Co/κ2/3 and B1 = (2.5)A1, where Co and κ are the Kolmogorov and von Kármán constants. These predictions hold when very large scale motions do not disturb the k−1 scaling existing across all wavenumbers 1/δ < k < 1/z in the streamwise turbulent velocity spectrum Eu(k). Deviations from a k−1 scaling along with their effects on A1 and B1 are discussed using published data and field experiments.