Wall-attached and detached structures in turbulent boundary layers

3-D coherent u clusters are explored from the perspective of the attached-eddy model. We extract clusters of the streamwise velocity fluctuations (u) by using the connectivity of six-orthogonal neighbors in Cartesian coordinates without any assumptions. To this end, DNS datasets for adverse pressure gradient (APG) (β = 1.43) and zero pressure gradient (β = 0) turbulent boundary layers (TBLs) with Re_τ ≈ 800 are used. Here, β is the non-dimensional pressure gradient parameter. The identified structures can be decomposed into attached self-similar, attached non-self-similar, detached self-similar and detached non-self-similar structures with respect to y_min and l_y , which are the minimal distance from the wall and the height of clusters, respectively. The sizes of the attached self-similar structures (100 < l_y⁺ < 0.6δ⁺) scale with l_y, and their population density is inversely proportional to l_y in the region l_y/δ = 0.4−0.58. Here, δ is the boundary layer thickness. They contribute to the logarithmic variation in the streamwise Reynolds stress and to the presence of the k_z⁻¹ region in the pre-multiplied energy spectra of u, where k_z is the spanwise wavenumber. The attached self-similar structures are universal wall motions in the logarithmic region that are not dependent on the pressure gradient. The attached non-self-similar structures (l_y/δ > 0.6) are responsible for the enhanced large scales in the outer region in the presence of the APG. Their streamwise sizes are Ο(3−6δ), and some of them extend over 6δ in the streamwise direction and penetrate deeply into the near-wall region, reminiscent of very-large-scale motions. In addition, wall-detached structures are also decomposed into self-similar and non-self-similar ones. We establish new insights into coherent structures in TBLs and the development of the attached-eddy model.