595 research outputs found
Geometry of inertial manifolds probed via a Lyapunov projection method
A method for determining the dimension and state space geometry of inertial
manifolds of dissipative extended dynamical systems is presented. It works by
projecting vector differences between reference states and recurrent states
onto local linear subspaces spanned by the Lyapunov vectors. A sharp
characteristic transition of the projection error occurs as soon as the number
of basis vectors is increased beyond the inertial manifold dimension. Since the
method can be applied using standard orthogonal Lyapunov vectors, it provides a
simple way to determine also experimentally inertial manifolds and their
geometric characteristics
On the interaction of small and large eddies in two dimensional turbulent flows
Some results concerning the interaction of small and large eddies to two dimensional turbulent flows are presented. It is shown that the amplitude of small structures decays exponentially to a small value, and from this is inferred a simplified interaction law of small and large eddies. Beside their intrinsic interest for the understanding of the physics of turbulence, these results lead to new numerical schemes to be studied in a separate work
Variational bounds on the energy dissipation rate in body-forced shear flow
A new variational problem for upper bounds on the rate of energy dissipation
in body-forced shear flows is formulated by including a balance parameter in
the derivation from the Navier-Stokes equations. The resulting min-max problem
is investigated computationally, producing new estimates that quantitatively
improve previously obtained rigorous bounds. The results are compared with data
from direct numerical simulations.Comment: 15 pages, 7 figure
Confluent operator algebras and the closability property
Certain operator algebras A on a Hilbert space have the property that every
densely defined linear transformation commuting with A is closable. Such
algebras are said to have the closability property. They are important in the
study of the transitive algebra problem. More precisely, if A is a
two-transitive algebra with the closability property, then A is dense in the
algebra of all bounded operators, in the weak operator topology. In this paper
we focus on algebras generated by a completely nonunitary contraction, and
produce several new classes of algebras with the closability property. We show
that this property follows from a certain strict cyclicity property, and we
give very detailed information on the class of completely nonunitary
contractions satisfying this property, as well as a stronger property which we
call confluence.Comment: Preliminary versio
Estimates for the two-dimensional Navier-Stokes equations in terms of the Reynolds number
The tradition in Navier-Stokes analysis of finding estimates in terms of the
Grashof number \bG, whose character depends on the ratio of the forcing to
the viscosity , means that it is difficult to make comparisons with other
results expressed in terms of Reynolds number \Rey, 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 \cite{DF} to the two-dimensional
Navier-Stokes equations on a periodic domain by estimating
quantities of physical relevance, particularly long-time averages
\left, in terms of the Reynolds number \Rey = U\ell/\nu, where
U^{2}= L^{-2}\left and is the forcing scale. In
particular, the Constantin-Foias-Temam upper bound \cite{CFT} on the attractor
dimension converts to a_{\ell}^{2}\Rey(1 + \ln\Rey)^{1/3}, while the estimate
for the inverse Kraichnan length is (a_{\ell}^{2}\Rey)^{1/2}, where
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.Comment: 21 pages, 1 figure, accepted for publication from J. Math. Phys. for
the special issue on mathematical fluid mechanic
Optimal solutions to matrix-valued Nehari problems and related limit theorems
In a 1990 paper Helton and Young showed that under certain conditions the
optimal solution of the Nehari problem corresponding to a finite rank Hankel
operator with scalar entries can be efficiently approximated by certain
functions defined in terms of finite dimensional restrictions of the Hankel
operator. In this paper it is shown that these approximants appear as optimal
solutions to restricted Nehari problems. The latter problems can be solved
using relaxed commutant lifting theory. This observation is used to extent the
Helton and Young approximation result to a matrix-valued setting. As in the
Helton and Young paper the rate of convergence depends on the choice of the
initial space in the approximation scheme.Comment: 22 page
On the Clark-alpha model of turbulence: global regularity and long--time dynamics
In this paper we study a well-known three--dimensional turbulence model, the
filtered Clark model, or Clark-alpha model. This is Large Eddy Simulation (LES)
tensor-diffusivity model of turbulent flows with an additional spatial filter
of width alpha (). We show the global well-posedness of this model with
constant Navier-Stokes (eddy) viscosity. Moreover, we establish the existence
of a finite dimensional global attractor for this dissipative evolution system,
and we provide an anaytical estimate for its fractal and Hausdorff dimensions.
Our estimate is proportional to , where is the integral spatial
scale and is the viscous dissipation length scale. This explicit bound is
consistent with the physical estimate for the number of degrees of freedom
based on heuristic arguments. Using semi-rigorous physical arguments we show
that the inertial range of the energy spectrum for the Clark- model has
the usual Kolmogorov power law for wave numbers and
decay power law for This is evidence that the
Clark model parameterizes efficiently the large wave numbers within
the inertial range, , so that they contain much less translational
kinetic energy than their counterparts in the Navier-Stokes equations.Comment: 11 pages, no figures, submitted to J of Turbulenc
On the behavior of the Lorenz equation backward in time
AbstractThe sets of solutions to the Lorenz equations that exist backward in time and are bounded at an exponential rate determined by the eigenvalues of the linear part of the equation are examined. The set associated with the middle eigenvalue is shown to project surjectively onto a plane, thereby providing a lower estimate for its dimension. Specific bounds are also found for a cone containing this set
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