On the higher order correction of the turbulent power-law velocity profile: comparison with the log law

For several decades, the classical log law of the wall has been widely regarded as the definitive formulation to describe the turbulent velocity profile near walls. However, in recent years, an alternative power law formulation of the turbulent velocity profile has emerged, challenging the central position of the log law. In a recent study by Anbarlooei et al. (Anbarlooei, H. R., Ramos, F. & Cruz, D. O. A. 2022), they demonstrated that an extended version of the power law expression can accurately replicate the turbulent velocity profile, even within the region where the log law is valid.

In this current study, we present an explicit expression for the turbulent mean velocity profile, which holds true in both the inner and outer regions, including the region of validity for the log law. Crucially, this new equation is derived entirely from first principles (attached eddies hypothesis) and does not rely on any turbulence model or asymptotic methods.