  Two-dimensional (2-D) spectra of the streamwise velocity component, measured
at friction Reynolds numbers ranging from 2400 to 26000, are used to refine a
model for the logarithmic region of turbulent boundary layers. Here, we focus
on the attached eddy model (AEM). The conventional AEM assumes the boundary
layer to be populated with hierarchies of self-similar wall-attached
($Type\,A$) eddies alone. While $Type\,A$ eddies represent the dominant
energetic large-scale motions at high Reynolds numbers, the scales that are not
represented by such eddies are observed to carry a significant proportion of
the total kinetic energy. Therefore, in the present study, we propose an
extended AEM that incorporates two additional representative eddies. These
eddies, named $Type\,C_A$ and $Type\,SS$, represent the self-similar but
wall-incoherent low-Reynolds number features, and the non-self-similar
wall-coherent superstructures, respectively. The extended AEM is shown to
better predict a greater range of energetic length scales and capture the low-
and high-Reynolds number scaling trends in the 2-D spectra of all three
velocity components. A discussion on spectral self-similarity and the
associated $k^{-1}$ scaling law is also presented.

Fluid Dynamics