An invariant representation of mean inertia: theoretical basis for a log law in turbulent boundary layers

Abstract

A refined scaling analysis of the two-dimensional mean momentum balance (MMB) for the zero-pressure-gradient turbulent boundary layer (TBL) is presented and experimentally investigated up to high friction Reynolds numbers, $\unicode[STIX]{x1D6FF}^{+}$. For canonical boundary layers, the mean inertia, which is a function of the wall-normal distance, appears instead of the constant mean pressure gradient force in the MMB for pipes and channels. The constancy of the pressure gradient has led to theoretical treatments for pipes/channels, that are more precise than for the TBL. Elements of these analyses include the logarithmic behaviour of the mean velocity, specification of the Reynolds shear stress peak location, the square-root Reynolds number scaling for the log layer onset and a well-defined layer structure based on the balance of terms in the MMB. The present analyses evidence that similarly well-founded results also hold for turbulent boundary layers. This follows from transforming the mean inertia term in the MMB into a form that resembles that in pipes/channels, and is constant across the outer inertial region of the TBL. The physical reasoning is that the mean inertia is primarily a large-scale outer layer contribution, the ‘shape’ of which becomes invariant of $\unicode[STIX]{x1D6FF}^{+}$ with increasing $\unicode[STIX]{x1D6FF}^{+}$, and with a ‘magnitude’ that is inversely proportional to $\unicode[STIX]{x1D6FF}^{+}$. The present analyses are enabled and corroborated using recent high resolution, large Reynolds number hot-wire measurements of all the terms in the TBL MMB.