Abstract

  The Navier-Stokes evolution of several configurations with evolving, interacting vortices go through a phase where locally orthogonal vortices, dominated one sign of helicity, shed oppositely signed vortex sheets. During this phase the fourth-root of the viscosity ν^{0.25 controls the temporal evolution of all the Ωm(t) moments of the vorticity,  as well as the growth of the computational domain with l∼ν}-0.25 required to accommodate the spread of the vortex sheets. In particular, there is convergence of √νZ(t) at a fixed time tx, with Z =∫dVω^{2=V_llΩ1l2 , the volume-integrated  enstrophy. This convergence can be rewritten as √νZ(t) = Vl (ν}0.25Ω_1)^{2. Note that this is not the convergence of the dissipation rate ε=νZ. That happens later at tε≈2tx. Convergence of √νZ(t) has been found for both perturbed and unperturbed trefoil vortex knots Kerr (2018a,b); Kerr (2023) and interacting coiled vortex rings Kerr (2018c). And now for interacting orthogonal vortices and even for one phase of Taylor-Green vortex evolution. Furthermore for several of  these there is convergence of          V_l (ν}0.25Ω∞(t) )^{2 at their respective tm with t∞ < tm < · · · < t1 = tx and  ν}0.25Ω∞(t) at their respective tm with t∞ < tm < · · · < t1 = tx. Detailed analysis of the orthogonal cases shows that the origin of the ν^{0.25 scaling comes from how the negative helicity vortex sheets are spawned in pairs in the zone of maximum compression between the two orthogonal vortices with a ν}0.5 scaling of the local gradient of the vorticity, which results in the vortex sheets spreading as (√ν)^{0.5 = ν}0.25. Finite energy dissipation  ∆E=∫_0^tε εdt as ν→0 is found for the perturbed trefoils as the vortex sheets generated about the helical knots roll-up.  Kerr, R.M. 2018a Trefoil knot timescales for reconnection and helicity. Fluid Dynamics Res. 50, 011422. Kerr, R.M. 2018bEnstrophy and circulation scaling for Navier-Stokes reconnection. J. Fluid Mech. 839, R2. Kerr, R.M. 2018c Topology of interacting coiled vortex rings. J. Fluid Mech. 854, R2.Kerr, R.M. 2023 Sensitivity of trefoil vortex knot reconnection to the initial vorticity profile. Phys. Rev Fluids 8, .