2006/05/31 by John Gibbon, J. D. Gibbon, Grigorios A. Pavliotis +1
Chemical Engineering · Engineering · Physics and Astronomy · #Fluid Dynamics and Turbulent Flows #Lattice Boltzmann Simulation Studies #Rheology and Fluid Dynamics Studies #nlin.CD #physics.flu-dyn
paper · pdf · doi:10.1063/1.2356912
21 pages, 1 figure, accepted for publication from J. Math. Phys. for the special issue on mathematical fluid mechanics
arxiv created 2006/09/13 · openalex publication_date 2007/06/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The tradition in Navier–Stokes analysis of finding estimates in terms of the Grashof number Gr, whose character depends on the ratio of the forcing to the viscosity ν, means that it is difficult to make comparisons with other results expressed in terms of Reynolds number Re, whose character depends on the fluid response to the forcing. The first task of this paper is to apply the approach of Doering and Foias [C. R. Doering and C. Foias, J. Fluid Mech. 467, 289 (2002)] to the two-dimensional Navier–Stokes equations on a periodic domain [0,L]2 by estimating quantities of physical relevance, particularly long-time averages ⟨∙⟩, in terms of the Reynolds number Re=Uℓ∕ν, where U2=L−2⟨∥u∥22⟩ and ℓ is the forcing scale. In particular, the Constantin–Foias–Temam upper bound [P. Constantin, C. Foias, and R. Temam, Physica D 30, 284 (1988)] on the attractor dimension converts to aℓ2Re(1+lnRe)1∕3, while the estimate for the inverse Kraichnan length is (aℓ2Re)1∕2, where aℓ is the aspect ratio of the forcing. Other inverse length scales, based on time averages, and associated with higher derivatives, are estimated in a similar manner. The second task is to address the issue of intermittency: it is shown how the time axis is broken up into very short intervals on which various quantities have lower bounds, larger than long time averages, which are themselves interspersed by longer, more quiescent, intervals of time.