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SUMMARY:How helicity helps create finite dissipation without singularities
 . - Robert Kerr (University of Warwick)
DTSTART:20240620T150000Z
DTEND:20240620T160000Z
UID:TALK218080@talks.cam.ac.uk
DESCRIPTION:&nbsp\;\nThe Navier-Stokes evolution of several configurations
  with evolving\, interacting vortices go through a phase where locally ort
 hogonal vortices\, dominated one sign of helicity\, shed oppositely signed
  vortex sheets. During this phase the fourth-root of the viscosity &nu\;^0
 .25 controls the temporal evolution of all the &Omega\;m(t) moments of the
  vorticity\,&nbsp\; as well as the growth of the computational domain with
  l&sim\;&nu\;^-0.25 required to accommodate the spread of the vortex sheet
 s. In particular\, there is convergence of &radic\;&nu\;Z(t) at a fixed ti
 me tx\, with Z =&int\;dV&omega\;^2=V_ll&Omega\;_1l2 \, the volume-integrat
 ed&nbsp\; enstrophy. This convergence can be rewritten as &radic\;&nu\;Z(t
 ) = Vl (&nu\;^0.25&Omega\;_1)^2. Note that this is not the convergence of 
 the dissipation rate &epsilon\;=&nu\;Z. That happens later at t&epsilon\;&
 asymp\;2tx. Convergence of &radic\;&nu\;Z(t) has been found for both pertu
 rbed and unperturbed trefoil vortex knots Kerr (2018a\,b)\; Kerr (2023) an
 d interacting coiled vortex rings Kerr (2018c). And now for interacting or
 thogonal vortices and even for one phase of Taylor-Green vortex evolution.
  Furthermore for several of&nbsp\; these there is convergence of &nbsp\; &
 nbsp\; &nbsp\; &nbsp\; &nbsp\;V_l (&nu\;^0.25&Omega\;&infin\;(t) )^2 at th
 eir respective tm with t&infin\; < tm < &middot\; &middot\; &middot\; < t1
  = tx and&nbsp\; &nu\;^0.25&Omega\;&infin\;(t) at their respective tm with
  t&infin\; < tm < &middot\; &middot\; &middot\; < t1 = tx. Detailed analys
 is of the orthogonal cases shows that the origin of the &nu\;^0.25 scaling
  comes from how the negative helicity vortex sheets are spawned in pairs i
 n the zone of maximum compression between the two orthogonal vortices with
  a &nu\;^0.5 scaling of the local gradient of the vorticity\, which result
 s in the vortex sheets spreading as (&radic\;&nu\;)^0.5 = &nu\;^0.25. Fini
 te energy dissipation&nbsp\; ∆E=&int\;_0^t_&epsilon\; &epsilon\;dt as &n
 u\;&rarr\;0 is found for the perturbed trefoils as the vortex sheets gener
 ated about the helical knots roll-up.&nbsp\;\nKerr\, R.M. 2018a Trefoil kn
 ot timescales for reconnection and helicity. Fluid Dynamics Res. 50\, 0114
 22. Kerr\, R.M. 2018bEnstrophy and circulation scaling for Navier-Stokes r
 econnection. J. Fluid Mech. 839\, R2. Kerr\, R.M. 2018c Topology of intera
 cting coiled vortex rings. J. Fluid Mech. 854\, R2.Kerr\, R.M. 2023 Sensit
 ivity of trefoil vortex knot reconnection to the initial vorticity profile
 . Phys. Rev Fluids 8\, .
LOCATION:Seminar Room 2\, Newton Institute
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