Integral relation in zero-pressure-gradient boundary layer flows

In their 2016 paper, Wei and Klewicki [Phys. Rev. Fluids 1, 082401 (2016)] developed an integral relation, UeVe/u2 tau = H12, which connects key parameters in zero-pressure-gradient (ZPG) boundary layer flows: mean velocity components Ue and Ve at the boundary layer edge, and friction velocity u tau to shape factor H12. While this relation holds exactly for ZPG laminar boundary layers featuring self-similar streamwise velocity profiles, it is an approximation for ZPG turbulent boundary layers (TBLs), with its accuracy improving as the Reynolds number increases. In this paper, we present a correction to the original integral relation, providing an exact integral relation that is applicable to ZPG boundary layer flows at arbitrary Reynolds numbers. The correction comprises two terms: one addressing deviations from self-similarity in mean streamwise velocity, and the other considering the impact of Reynolds normal stresses. Experimental and numerical data are shown to support the relative insignificance of the newly identified correction terms, except for Reynolds numbers in the transitional regime.