A Correction Term for the Asymptotic Scaling of Drag in Flat-Plate Turbulent Boundary Layers

Dixit et al. proposed an asymptotic drag scaling for zero-pressure-gradient flat-plate turbulent boundary layers based on the approximation $M\sim U_{\tau}^2\delta$, where $M$ is the kinematic momentum rate through the boundary layer, $U_{\tau}$ is the friction velocity, and $\delta$ is the boundary-layer thickness. In the present paper, an explicit Reynolds-number-dependent correction to this approximation is derived from the logarithmic mean-velocity profile. Integration of the log law across the layer yields $M\sim U_{\tau}^2\delta\,f(Re_{\tau})$, where $Re_{\tau}=\delta U_{\tau}/\nu$ is the friction Reynolds number and $f(Re_{\tau})$ is given analytically. Application of the correction to the dataset compiled by Dixit et al. shows that the corrected scaling gives an exponent consistent with the asymptotic value $-1/2$ within bootstrap confidence intervals, whereas the uncorrected formulation does not. The correction should be viewed as a leading-order amendment, since the derivation uses the logarithmic law outside its strict range of validity.