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

A phenomenological description is presented to explain the intermediate and low-frequency/large-scale contributions to the wall-shear-stress ( ${\tau }_w$ ) and wall-pressure ( $\,{p}_w$ ) 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 ${\tau }_w$ , ${p}_w$ and velocity-fluctuation time series in the log region are considered, across a friction-Reynolds-number ( $Re_{\tau }$ ) range of $ {O}(10^3) \lesssim Re_{\tau } \lesssim {O}(10^6)$ . 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 ${\tau }_w$ - and $p_w$ -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 ${\tau }_w$ -spectrum, which grow with $Re_{\tau }$ 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 $p_w$ -spectrum. Both these contributions are found to increase with increasing $Re_{\tau }$ owing to the broadening and energization of the wall-scaled (attached) eddy hierarchy. This potentially explains the rapid $Re_{\tau }$ -growth of the $p_w$ -spectra relative to ${\tau }_w$ , given the dependence of the latter only on the inactive contributions.