Intermediate scaling for streamwise mean velocity and variance in a turbulent boundary layer

In this work, we use an intermediate length scale ($y_{m}^{+})$ proportional to the square root of friction Reynolds number and an associated velocity scale ($u_{m})$ to scale the streamwise mean velocity and variance in the intermediate region of a turbulent boundary layer (TBL; Diwan and Morrison TSFP-11, 2019). Towards this we make use of the high-Reynolds-number TBL data available in the literature. The mean velocity is plotted in defect formulation, i.e., $U-U(y_{m})/u_{m}$. The intermediate-scaled mean velocity and variance exhibit Reynolds-number invariance over a certain region around $y/y_{m}=$1. This implies existence of a Reynolds-number independent log law in terms of the intermediate variables for the mean velocity and variance. The classical ``Karman constant'' and ``Townsend-Perry constant'' are expressed in terms of the log-law constants obtained from the intermediate scaling and their dependence on Reynolds number examined. The present work suggests a three-layer asymptotic structure for the TBL, with each layer governed by distinct length and velocity scales, which results in two overlap layers (Afzal, Ing.-Arch. 1982). We discuss the relevance of the power/log law behavior of the mean velocity in the overlap layers, based on the ratio of the velocity scales of the adjacent layers. .