Self-similar spiral flow structure in low Reynolds number isotropic and decaying turbulence

Abstract

It is rigorously proved for axisymmetric incompressible flows with bounded axial vorticity at infinity that if a spiral-helical streamline has a Kolmogorov capacity (box-counting dimension) ${\mathit{D}}_{\mathit{K}}$\ensuremath{\gtrsim}1, then the velocity field must have a singularity at the axis of symmetry. Furthermore, certain types of singularity with ${\mathit{D}}_{\mathit{K}}$=1 can be excluded. The Burgers and the Lundgren vortices are examples of strained vortices with different types of near-singular structure, and in both cases sections of streamlines have a well-defined ${\mathit{D}}_{\mathit{K}}$\ensuremath{\gtrsim}1. However, the strain severely limits the region in space where ${\mathit{D}}_{\mathit{K}}$ is larger than 1. An algorithm is developed which detects streamlines with persistently strong curvature and calculates both the ${\mathit{D}}_{\mathit{K}}$ of the streamlines and the lower bound scale ${\mathrm{\ensuremath{\delta}}}_{\mathrm{min}}$ of the range of self-similar scaling defined by ${\mathit{D}}_{\mathit{K}}$. Error bounds on ${\mathit{D}}_{\mathit{K}}$ are also computed. The use of this algorithm partly relies on the fact that two to three turns of a spiral are enough to determine a spiral's ${\mathit{D}}_{\mathit{K}}$. 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 ${\mathrm{Re}}_{\ensuremath{\lambda}}$ around 20. The measured values of ${\mathit{D}}_{\mathit{K}}$ vary from ${\mathit{D}}_{\mathit{K}}$=1 to ${\mathit{D}}_{\mathit{K}}$\ensuremath{\approxeq}1.60, and in general the self-similar range of length scales over which ${\mathit{D}}_{\mathit{K}}$ is well defined extends over one decade and ends at one of two well-defined inner scales, one just above and the other just below the Kolmogorov microscale \ensuremath{\eta}. We identify two different types of accumulation of length scales with ${\mathit{D}}_{\mathit{K}}$\ensuremath{\gtrsim}1 on streamlines around the vortex tubes in the simulated turbulence: an accumulation of the streamline towards a central axis of the vortex tube in a spiral-helical fashion, and a helical and axial accumulation of the streamline towards a limit circle at the periphery of the vortex tube. In the latter case, the limit circle lies in a region along the axis of the vortex tube where there is a rapid drop in enstrophy. The existence of spiral-helical streamlines with well-defined ${\mathit{D}}_{\mathit{K}}$\ensuremath{\gtrsim}1 suggests the possibility of a near-singular flow structure in some vortex tubes. Finally, we present some evidence based on the spatial correlation of enstrophy with viscous force indicating that the spatial vorticity profile across vortex tubes is not a well-resolved Gaussian at the resolution of the present simulations. \textcopyright{} 1996 The American Physical Society.