On determining friction velocity in a spatially developing turbulent boundary layer under zero pressure gradient

Accurate determination of the friction velocity in wall-bounded turbulent flows is crucial for both fundamental research and engineering applications. In this work, the integral relation for friction velocity proposed by Mehdi et al. is modified based on a power-law assumption for the total shear stress within the turbulent boundary layer. The present approach requires only the mean streamwise velocity and Reynolds shear stress profiles in the logarithmic region and beyond, thereby reducing the reliance on near-wall data. Extensive validation against numerical and experimental data shows that the proposed method can accurately predict the friction velocity over a broad range of Reynolds numbers. We further extend the method by deriving a more general relation for the friction velocity through an n-fold repeated integration of the mean streamwise momentum equation. It is found that the accuracy of the present method can be improved to within ±1% when the integral relation is obtained based on a twentyfold repeated integration instead of a threefold integration. The method applies to both smooth- and rough-wall turbulent boundary layers under zero pressure gradient. It is particularly useful in situations where measurements in the near-wall region with y+100 are either unavailable or subject to significant uncertainty.