  Energy transfers from larger to smaller scales in turbulence. This energy
cascade is a process of the creation of smaller-scale coherent vortices by
larger ones. In our recent study (Yoneda, Goto and Tsuruhashi 2021), we
reformulated the energy cascade in terms of this stretching process and derived
the $-5/3$ law of the energy spectrum under physically reasonable assumptions.
In the present study, we provide a quantitative verification of these
assumptions by using direct numerical simulations. We decompose developed
turbulence in a periodic cube into scales by using the band-pass filter and
identify the axes of coherent tubular vortices by the low-pressure method. Even
when the turbulent kinetic energy and its dissipation rate temporally fluctuate
about their temporal means, the total length of the vortices at each scale
varies little with time. This result is consistent with our assumption of the
temporal stationarity on the vorticity decomposition. The present numerical
analysis also shows that the hierarchy of vortex axes is self-similar in a wide
range of scales, i.e. in the inertial range and a lower part of the dissipation
range and that the volume fraction occupied by the tubular vortices at each
scale is independent of the scale.
