Navier-Stokes bounds and early turbulent scaling for compact trefoils in (2ℓπ)3 domains.

Abstract A principle largely unknown outside of the applied analysis fluids community is that ν≠0Navier-Stokes solutions in (2π)3 periodic domains are bounded by their ν ≡ 0 Euler solutions as ν → 0 for ν < νs, with the νs critical viscosities coming from analysis in higher-s Sobolev Hs spaces. This result also bounds lower-s norms. Could modest Reynolds number viscous calculations in larger domains be influenced by this result? Numerically, for (2lπ)3 domains, empirical critical viscosities νc(l) have been identified using compact trefoil vortex knots. √νZ(t), Z(t) enstrophy could be disrupted if ν < νc (l). And then restored if l is increased. That type of convergence has now been extended to higher-order vorticity moments with temporal convergence of ν1/4Ωm, m = 9,∞, at tm < tx, scaling that is consistent with the empirical observation that the domain should increase as l ∼ ν-1/4 as the observed vortex sheets grow with decreasing ν. Once formed, enstrophy growth accelerates as those sheets roll-up, leading to convergent dissipation rates with ε = νZ at t ∼ teps ∼ 2tx. Simultaneously, between tx and teps, a Zv ∼ k1/3 enstrophy spectrum develops, analogous to a k-5/3 energy spectrum. To explain these observations, the Sobolev νs critical viscosity analysis in (2π)3 domains is extended to larger (2lπ)3, l > 1, domains. The method squeezes the ∥u)∥s ̇+1 of Navier- Stokes solutions in a large (2lπ)3 domain into a (2π)3 domain, then in (2π)3 applies the resulting ∥u_λ∥.s+1+1 (λ = l) to approximations of the Hs2pi Sobolev mathematics developed in (2π)3 domains. Finally, the resulting νs2pi are rescaled back into the original (2lπ)3 domain. Resulting in decreasing νs ∼ exp( ∫λs+1∥ul(τ)∥.s+1dτ)λ3 as λ = l increases, with the ∥ul(τ)∥.s+1coming from the original domain and the exponential factor dominating over the algebraic λ3 factor. In the discussion, mathematics that might fill gaps between the high-s Sobolev analysis and the numerical analysis is mentioned.