2024/01/07 by Robert M. Kerr, Kerr, Robert M.
Engineering · #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.2401.03578
openalex publication_date 2024/01/07 · openalex created_date 2024/01/13 · openalex updated_date 2026/07/28
For a perturbed trefoil vortex knot evolving under the Navier-Stokes equations, a sequence of ν-independent times tm are identified corresponding to a set of scaled, volume-integrated vorticity moments ν1/4\it OV1 with this hierarchy t_∞≤…≤ tm… t1=tx≈40 and \it OVm=(∫Vℓ|ω|2mdV)1/2m. For Z(t)=\it O2V1(t) the volume-integrated enstrophy, convergence of √νZ(t) at tx=t1 marks the end of the reconnection scaling phase. Physically, reconnection follows from the formation of a double vortex sheet, then a knot, which splits into spirals. Z then accelerates, leading to approximate finite-time ν-independent convergence of the energy dissipation rate ε(t)=νZ(t) at tε∼ 2tx and sustained over a finite span ΔTε\searrow 0.5 tε, giving Reynolds number independent finite-time, dissipation, ΔEε=∫ΔTεεdt, and thus satisfying one definition for a \it dissipation anomaly. Evidence for transient Kolmogorov-like enstrophy spectra is found over ΔTε. A critical factor in achieving these temporal convergence laws is how the domain V_ℓ=(2ℓπ)3 is increased as ℓ∼ν-1/4, for ℓ=2 to 6, then to ℓ=12, as ν decreases. (2ℓπ)3 domain compatibility with established (2π)3 mathematics in appendix allows small ν Navier-Stokes solutions. Two spans of ν are considered. Over the first factor of 25 decrease in ν, all of the ν1/4\it OVm(t) converge to their respective tm. For the next factor of 5 decrease in ν, ℓ is increased to ℓ=12, there is only convergence of ν1/4ΩV∞(t) to t_∞ and later √νZ(t) convergence at t1=tx and ε(t) over t∼ tε.