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

TL;DR

This paper presents a phenomenological model explaining how active and inactive motions contribute to wall pressure and shear stress spectra in turbulent boundary layers, supported by new data and energy decomposition methods across a wide Reynolds number range.

Contribution

It introduces a novel application of energy decomposition to distinguish active and inactive motions' contributions to wall spectra in turbulent boundary layers.

Findings

Inactive motions drive non-local energy transport to the wall.

Large-scale signatures in shear stress spectra grow with Reynolds number.

Both active and inactive motions increase with Reynolds number in wall pressure spectra.

Abstract

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, which are well known to increase with Reynolds number. The explanation is based on the concept of active and inactive motions (Townsend, J. Fluid Mech., vol. 11, 1961) 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 friction-Reynolds-number ($Re_{\tau}$) range of $\mathcal{O}$($10^3$) $\lesssim$ $Re_{\tau}$ $\lesssim$ $\mathcal{O}$($10^6$). A recently proposed energy-decomposition methodology (Deshpande et al., J. Fluid Mech., vol. 914, 2021) 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, 1967) 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.