2016/10/03 by Robert M. Kerr, Kerr, Robert M.
Engineering · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Fluid Dynamics and Turbulent Flows #Fluid Dynamics and Vibration Analysis #Lattice Boltzmann Simulation Studies
paper · pdf · doi:10.48550/arxiv.1610.00398
openalex publication_date 2016/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Perturbed, helical trefoil vortex knots and a set of anti-parallel vortices are examined numerically to identify the scaling of their helicity and vorticity norms during reconnection. For the volume-integrated enstrophy Z=∫ω2 dV, a new scaling regime is identified for both configurations where as the viscosity ν changes, all √νZ(t) cross at ν-independent times tx, identified as when the first reconnection events end. Self-similar linear collapse of Bν(t)=(√νZ)-1/2 can be found for t\lesssim tx by linearly extrapolating Bν(t) to zero at critical times Tc(ν), then plotting (Tc(ν)-tx)(Bν(t)-Bx) where Bx=Bν(tx). The size ℓ3 of the periodic domains must be increased as ν is decreased to maintain this scaling as implied by known Sobolev space bounds. The anti-parallel calculations show that the linear collapse of Bν(t) begins with a quick, viscosity-independent exchange of the circulation Γ between the original vortices and the new vortices. Up to and after the trefoil knots' first reconnection at time tx, their helicity \cal H is preserved, validating the experimental centreline helicity observation of Scheeler et al (2014a). Because the cubic Navier-Stokes velocity norm L3 barely changes and the Navier-Stokes ‖ω‖_∞ are bounded by the Euler values, these flows are never singular. Despite this, the Navier-Stokes Z can, for a brief period, grow faster than the Euler Z and the following increase in the viscous energy dissipation rate ε=νZ shows ν-independent convergence at t≈ 2tx. Taken together, these results could be a new paradigm whereby smooth solutions without singularities or roughness could generate a ν→0 \it dissipation anomaly (finite dissipation in a finite time) as ℓ→∞, as seen in physical turbulent flows.