Reconciling the Zagarola/smits scaling with the george/castillo theory for the zero pressure gradient turbulent boundary layer

Recently two new approaches to the scaling of the velocity deficit for turbulent boundary layers have been set forth, both of which depart from using the friction velocity as an outer velocity scale. George and Castillo (1997) argue on theoretical grounds that the proper asymptotically invariant velocity deficit is given by (U - U_∞) = U_∞f_o(y/δ,δ⁺) where = u*δ/ν. As δ⁺ → ∞, f_o(y/δ,δ⁺) → f_o∞(y/δ). Only in this limit, which is well beyond existing experimental data, will profiles truly collapse. Similarly, the ratio, δ*/δ is shown to be dependent on δ⁺, but is constant asymptotically. Zagarolaand Smits (1998), in apparent contradiction, show striking experimental collapse of the velocity deficit normalized with a new scale defined as U_s = U_∞δ_* /δ, where δ* is the displacement thickness. In fact, using this scaling, near perfect collapse is also achieved for a range of δ⁺ from 10³ to 10⁶ (and beyond) for velocity profiles calculated from the composite solution suggested by George and Castillo. This paper shows how to reconcile the two points of view. If it is assumed that f_o is separable, fo(y/δ,δ⁺) - G(δ⁺)f_o∞(y/δ), then the Zagarola/ Smits result follows directly from the definition of the displacement thickness, since for δ⁺>>1 G(δ⁺) = δ*/δ. Hence both proposals are in essential agreement on two points: (1) the need for including the Reynolds number dependence in the velocity deficit and (2) what this Reynolds number dependence is.