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Estimates for the two-dimensional Navier-Stokes equations in terms of the Reynolds number. (English) Zbl 1144.81348

Summary: 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 \(v\), 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 C. R. Doering and J. D. Foias [Physica D 165, No. 3–4, 163–175 (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 \(\left<\cdot\right>\), in terms of the Reynolds number \(Re = U\ell/\nu\), where \(U^2= L^{-2}\left<\|u\|_2^2\right>\) and \(\ell\) is the forcing scale. In particular, the Constantin-Foias-Temam upper bound [Physica D 30, No. 3, 284–296 (1988; Zbl 0658.58030)] on the attractor dimension converts to \(a_{\ell}^{2}Re(1 + \ln Re)^{1/3}\), while the estimate for the inverse Kraichnan length is \((a_{\ell}^2 Re)^{1/2}\), where \(a_\ell\) 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.

MSC:

76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
76F20 Dynamical systems approach to turbulence

Citations:

Zbl 0658.58030
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References:

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