  Townsend's attached-eddy hypothesis (AEH) provides a theoretical description
of turbulence statistics in the logarithmic region in terms of coherent motions
that are self-similar with the wall-normal distance (y). Here, we show the
self-similar behaviour of turbulence motions contained within wall-attached
structures of streamwise velocity fluctuations using the direct numerical
simulation dataset of turbulent boundary layer, channel, and pipe flows
($Re_\tau \approx 1000$) The physical sizes of the identified structures are
geometrically self-similar in terms of height, and the associated turbulence
intensity follows the logarithmic variation in all three flows. Moreover, the
corresponding two-dimensional energy spectra are aligned along a linear
relationship between the streamwise and spanwise wavelengths ($\lambda_x$ and
$\lambda_z$, respectively) in the large-scale range ($12y < \lambda_x <$
3--4$\delta$), which is reminiscent of self-similarity. Consequently,
one-dimensional spectra obtained by integrating the two-dimensional spectra
over the self-similar range show some evidence for self-similar scaling
$\lambda_x \sim \lambda_z$ and the possible existence of $k_x^{-1}$ and
$k_z^{-1}$ scaling regions in a similar subrange. The present results reveal
that the asymptotic behaviours can be obtained by identifying the self-similar
coherent structures in canonical wall turbulence, albeit in low Reynolds number
flows.
