2006/03/31 by John Gibbon, John D. Gibbon, Darryl D. Holm · 2 citations
Engineering · Mathematics · Physics and Astronomy · #Fluid Dynamics and Turbulent Flows #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations #nlin.CD
paper · pdf · doi:10.1016/j.physd.2006.06.012
16 pages, no figures, final version accepted for Physica D
arxiv created 2006/06/29 · openalex publication_date 2006/08/01 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Foias, Holm & Titi \citeFHT2 have settled the problem of existence and uniqueness for the 3D \lans equations on periodic box [0,L]3. There still remains the problem, first introduced by Doering and Foias \citeDF for the Navier-Stokes equations, of obtaining estimates in terms of the Reynolds number \Rey, whose character depends on the fluid response, as opposed to the Grashof number, whose character depends on the forcing. \Rey is defined as \Rey = Uℓ/ν where U is a bounded spatio-temporally averaged Navier-Stokes velocity field and ℓ the characteristic scale of the forcing. It is found that the inverse Kolmogorov length is estimated by ℓλk-1 ≤ c (ℓ/α)1/4\Rey5/8. Moreover, the estimate of Foias, Holm & Titi for the fractal dimension of the global attractor, in terms of \Rey, comes out to be dF(A) ≤ c \fracVαVℓ1/2(L2λ1)9/8 \Rey9/4 where Vα = (L/(ℓα)1/2)3 and Vℓ = (L/ℓ)3. It is also shown that there exists a series of time-averaged inverse squared length scales whose members, <κn,02>, %, are related to the 2nth-moments of the energy spectrum when α→ 0. are estimated as (n≥ 1) ℓ2<κn,02> ≤ cn,αVα(n-1)/(n) \Rey^11/4 - (7)/(4n)(ln\Rey)(1)/(n) + c1\Rey(ln\Rey) . The upper bound on the first member of the hierarchy <κ1,02> coincides with the inverse squared Taylor micro-scale to within log-corrections.