Wall-attached structures of velocity fluctuations in a turbulent boundary layer

Wall turbulence is a ubiquitous phenomenon in nature and engineering application, yet predicting such turbulence is difficult due to its complexity. High-Reynolds-number turbulence, which includes most practical flows, is particularly complicated because of its wide range of scales. Although the attached-eddy hypothesis postulated by Townsend can be used to predict turbulence intensities and serves as a unified theory for the asymptotic behaviors of turbulence, the presence of attached structures has not been confirmed.Here, we demonstrate the logarithmic region of turbulence intensity by identifying wall-attached structures of velocity fluctuations ($u_i$) through direct numerical simulation of a moderate-Reynolds-number boundary layer ($Re_\tau \approx 1000$). The wall-attached structures are self-similar with respect to their heights ($l_y$), and in particular the population density of the streamwise component ($u$) scales inversely with $l_y$, which is reminiscent of the hierarchy of attached eddies. The turbulent intensities contained within the wall-parallel components ($u$ and $w$) exhibit the logarithmic behavior. The tall attached structures ($l_y^+ > 100$) of $u$ are composed of multiple uniform momentum zones (UMZs) with a long streamwise extent, whereas those of the cross-stream components ($v$ and $w$) are relatively short with a comparable width, suggesting the presence of tall vortical structures associated with multiple UMZs. The magnitudes of the near-wall peak observed in the streamwise turbulent intensity increase with increasing $l_y$, reflecting nested hierarchies of the attached $u$ structures. These findings suggest that the identified structures are prime candidates for Townsend's attached-eddy hypothesis and serve as cornerstones for understanding the multiscale phenomena of high-Reynolds-number boundary layers.