Structural features of the $k_{x}^{-1}$ region of turbulent pipe flow at Re$_{\tau }=$3008

Structural features of a turbulent pipe flow were explored by using the direct numerical simulation data at Re$_{\tau }=$3008 (Ahn et al. 2015). Based on the pre-multiplied streamwise energy spectra of the streamwise velocity fluctuations, three spectral regions were classified: the inner site, the outer site and $k_{x}^{-1}$ region. The inner site was created by the self-sustaining near-wall cycle with $\lambda_{x}^{+}\approx $1000, where $\lambda_{x}$ is the streamwise wavelength. The outer site was made due to very-large-scale motions with $\lambda_{x}$/R$\approx $10, which were generated by the streamwise pseudo-alignment of the adjacent large-scale motions. Between the inner and outer sites, the $k_{x}^{-1}$ region appeared at $y^{+}=$90-300, where $\lambda_{x}\ge $20$y$ and $\lambda_{x}$/R$\le $5. By using the conditional averaging, self-similar structures of the streamwise velocity fluctuations structures in the $k_{x}^{-1}$ region were retained, which were considered as the attached eddies proposed by Townsend (1976). In addition, the vortical structures in the $k_{x}^{-1}$ region were examined by two-point correlation of the velocity components and the vortices in order to find the dominant behavior of the structures.