An alternative outer velocity scale for adverse-pressure-gradient turbulent boundary layer

It is widely recognized that the Zagarola-Smits velocity scale U_ZS serves as a suitable outer velocity scale for adverse-pressure-gradient (APG) turbulent boundary layers (TBLs). In this paper, a relationship between U_ZS, the edge velocity U_e of the boundary layer, and the skin friction coefficient C_f is established. An alternative outer velocity scale, denoted as U_f, is proposed for APG TBLs based on U_e and C_f. This proposition is validated based on several high-quality databases from the literature, along with the presently obtained experimental APG database. As U_ZS, the U_f-normalized mean-velocity-defect distributions collapse well together at small to moderate values of the Clauser pressure-gradient parameter. As such, a new and simply empirical method is proposed to determine C_f, or equivalently the friction velocity, for APG TBLs.