Jc. Vassilicos et Jg. Brasseur, SELF-SIMILAR SPIRAL FLOW STRUCTURE IN LOW-REYNOLDS-NUMBER ISOTROPIC AND DECAYING TURBULENCE, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 54(1), 1996, pp. 467-485
It is rigorously proved for axisymmetric incompressible flows with bou
nded axial vorticity at infinity that if a spiral-helical streamline h
as a Kolmogorov capacity (box-counting dimension) D-K>1 then the veloc
ity field must have a singularity at the axis of symmetry. Furthermore
, certain types of singularity with D-K=1 can be excluded. The Burgers
and the Lundgren vortices are examples of strained vortices with diff
erent types of near-singular structure, and in both cases sections of
streamlines have a well-defined D-K>1. However, the strain severely li
mits the region in space where D-K is larger than 1. An algorithm is d
eveloped which detects streamlines with persistently strong curvature
and calculates both the D-K of the streamlines and the lower bound sca
le delta(min) of the range of self-similar scaling defined by D-K. Err
or bounds on D-K are also computed. The use of this algorithm partly r
elies on the fact that two to three turns of a spiral are enough to de
termine a spiral's D-K. We detect well-defined self-similar scaling in
the geometry of streamlines around vortex tubes in decaying isotropic
direct numerical simulation turbulence with exceptionally fine small-
scale resolution and Re-lambda around 20. The measured values of D-K v
ary from D-K=1 to D-K approximate to 1.60, and in general the self-sim
ilar range of length scales over which D-K is well defined extends ove
r one decade and ends at one of two well-defined inner scales, one jus
t above and the other just below the Kolmogorov microscale eta. We ide
ntify two different types of accumulation of length scales with D-K>1
on streamlines around the vortex tubes in the simulated turbulence: an
accumulation of the streamline towards a central axis of the vortex t
ube in a spiral-helical fashion, and a helical and axial accumulation
of the streamline towards a limit circle at the periphery of the vorte
x tube. In the latter case, the limit circle lies in a region along th
e axis of the vortex tube where there is a rapid drop in enstrophy. Th
e existence of spiral-helical streamlines with well-defined D-K>1 sugg
ests the possibility of a near-singular flow structure ih some vortex
tubes. Finally, we present some evidence based on the spatial correlat
ion of enstrophy with viscous force indicating that the spatial vortic
ity profile across vortex tubes is not a well-resolved Gaussian at the
resolution of the present simulations.