Active and inactive contributions to the wall pressure and wall-shear stress in turbulent boundary layers

Abstract

A phenomenological description is presented to explain the intermediate and low-frequency/large-scale contributions to the wall-shear-stress (τw) and wall-pressure (pw) spectra of canonical turbulent boundary layers, both of which are well known to increase with Reynolds number, albeit in a distinct manner. The explanation is based on the concept of active and inactive motions (Townsend, J. Fluid Mech., vol. 11, issue 1, 1961, pp. 97-120) associated with the attached-eddy hypothesis. Unique data sets of simultaneously acquired τw, pw and velocity-fluctuation time series in the log region are considered, across a friction-Reynolds-number (Reτ) range of O(103) ≲ Reτ ≲ O(106). A recently proposed energy-decomposition methodology (Deshpande et al., J. Fluid Mech., vol. 914, 2021, A5) is implemented to reveal the active and inactive contributions to the τw- and pw-spectra. Empirical evidence is provided in support of Bradshaw's (J. Fluid Mech., vol. 30, issue 2, 1967, pp. 241-258) hypothesis that the inactive motions are responsible for the non-local wall-ward transport of the large-scale inertia-dominated energy, which is produced in the log region by active motions. This explains the large-scale signatures in the τw-spectrum, which grow with Reτ despite the statistically weak signature of large-scale turbulence production, in the near-wall region. For wall pressure, active and inactive motions respectively contribute to the intermediate and large scales of the pw-spectrum. Both these contributions are found to increase with increasing Reτ owing to the broadening and energization of the wall-scaled (attached) eddy hierarchy. This potentially explains the rapid Reτ-growth of the pw-spectra relative to τw, given the dependence of the latter only on the inactive contributions.