23 research outputs found
Extensivity of two-dimensional turbulence
This study is concerned with how the attractor dimension of the
two-dimensional Navier--Stokes equations depends on characteristic length
scales, including the system integral length scale, the forcing length scale,
and the dissipation length scale. Upper bounds on the attractor dimension
derived by Constantin--Foias--Temam are analysed. It is shown that the optimal
attractor-dimension estimate grows linearly with the domain area (suggestive of
extensive chaos), for a sufficiently large domain, if the kinematic viscosity
and the amplitude and length scale of the forcing are held fixed. For
sufficiently small domain area, a slightly ``super-extensive'' estimate becomes
optimal. In the extensive regime, the attractor-dimension estimate is given by
the ratio of the domain area to the square of the dissipation length scale
defined, on physical grounds, in terms of the average rate of shear. This
dissipation length scale (which is not necessarily the scale at which the
energy or enstrophy dissipation takes place) can be identified with the
dimension correlation length scale, the square of which is interpreted,
according to the concept of extensive chaos, as the area of a subsystem with
one degree of freedom. Furthermore, these length scales can be identified with
a ``minimum length scale'' of the flow, which is rigorously deduced from the
concept of determining nodes.Comment: No figures, 14 page
Secondary fluid flows driven electromagnetically in a two-dimensional extended duct
This study focuses on two-dimensional fluid flows in a straight duct with free-slip boundary conditions applied on the channel walls y=0 and y=2?N with N>1. In this extended wall-bounded fluid motion problem, secondary fluid flow patterns resulting from steady-state and Hopf bifurcations are examined and shown to be dependent on the choice of longitudinal wave numbers. Some secondary steady-state flows appear at specific wave numbers, whereas at other wave numbers, both secondary steady-state and self-oscillation flows coexist. These results, derived through analytical arguments and truncation series approximation, are confirmed by simple numerical experiments supporting the findings observed from laboratory experiments
Constraints on the spectral distribution of energy and enstrophy dissipation in forced two-dimensional turbulence
We study two-dimensional (2D) turbulence in a doubly periodic domain driven by a monoscale-like forcing and damped by various dissipation mechanisms of the form ΜΌ(âÎ)ÎŒ. By âmonoscale-likeâ we mean that the forcing is applied over a finite range of wavenumbers kminâ€kâ€kmax, and that the ratio of enstrophy injection ηâ„0 to energy injection Δâ„0 is bounded by kmin2Δâ€Î·â€kmax2Δ. Such a forcing is frequently considered in theoretical and numerical studies of 2D turbulence. It is shown that for ÎŒâ„0 the asymptotic behaviour satisfies â„uâ„12â€kmax2â„uâ„2, where â„uâ„2 and â„uâ„12 are the energy and enstrophy, respectively. If the condition of monoscale-like forcing holds only in a time-mean sense, then the inequality holds in the time mean. It is also shown that for NavierâStokes turbulence (ÎŒ=1), the time-mean enstrophy dissipation rate is bounded from above by 2Îœ1kmax2. These results place strong constraints on the spectral distribution of energy and enstrophy and of their dissipation, and thereby on the existence of energy and enstrophy cascades, in such systems. In particular, the classical dual cascade picture is shown to be invalid for forced 2D NavierâStokes turbulence (ÎŒ=1) when it is forced in this manner. Inclusion of Ekman drag (ÎŒ=0) along with molecular viscosity permits a dual cascade, but is incompatible with the log-modified â3 power law for the energy spectrum in the enstrophy-cascading inertial range. In order to achieve the latter, it is necessary to invoke an inverse viscosity (ÎŒ<0). These constraints on permissible power laws apply for any spectrally localized forcing, not just for monoscale-like forcing
Hopf bifurcation of the three-dimensional Navier-Stokes equations
This paper is concerned with the three-dimensional NavierâStokes flows excited by a unidirectional external force and gives an explicit example of Hopf bifurcation phenomenon occurring in the NavierâStokes problem. Complete rigorous analysis on the existence of this instability behavior is provided