Distance-from-the-wall scaling of turbulent motions in wall-bounded flows

Abstract

An assessment of self-similarity in the inertial sublayer is presented by considering the wall-normal velocity, in addition to the streamwise velocity component. The novelty of the current work lies in the inclusion of the second velocity component, made possible by carefully conducted subminiature ×-probe experiments to minimise the errors in measuring the wall-normal velocity. We show that not all turbulent stress quantities approach the self-similar asymptotic state at an equal rate as the Reynolds number is increased, with the Reynolds shear stress approaching faster than the streamwise normal stress. These trends are explained by the contributions from attached eddies. Furthermore, the Reynolds shear stress cospectra, through its scaling with the distance from the wall, are used to assess the wall-normal limits where self-similarity applies within the wall-bounded flow. The results are found to be consistent with the recent prediction from the work of Wei et al. [“Properties of the mean momentum balance in turbulent boundary layer, pipe and channel flows,” J. Fluid Mech. 522, 303–327 (2005)], Klewicki [“Reynolds number dependence, scaling, and dynamics of turbulent boundary layers,” J. Fluids Eng. 132, 094001 (2010)], and others that the self-similar region starts and ends at $z^{+}\sim \mathcal{O}(\sqrt{{\delta }^{+}})$ and $O({\delta }^{+})$ , respectively. Below the self-similar region, empirical evidence suggests that eddies responsible for turbulent stresses begin to exhibit distance-from-the-wall scaling at a fixed $z^{+}$ location; however, they are distorted by viscous forces, which remain a leading order contribution in the mean momentum balance in the region $z^{+}\lesssim \mathcal{O}(\sqrt{{\delta }^{+}})$ , and thus result in a departure from self-similarity.