89 research outputs found
Euler equations and turbulence: analytical approach to intermittency
Physical models of intermittency in fully developed turbulence employ many
phenomenological concepts such as active volume, region, eddy, energy
accumulation set, etc, used to describe non-uniformity of the energy cascade.
In this paper we give those notions a precise mathematical meaning in the
language of the Littlewood-Paley analysis. We further use our definitions to
recover scaling laws for the energy spectrum and second order structure
function with proper intermittency correction.Comment: This is an updated version. Published in SIMA, 201
An inviscid dyadic model of turbulence: the fixed point and Onsager's conjecture
Properties of an infinite system of nonlinearly coupled ordinary differential
equations are discussed. This system models some properties present in the
equations of motion for an inviscid fluid such as the skew symmetry and the
3-dimensional scaling of the quadratic nonlinearity. It is proved that the
system with forcing has a unique equilibrium and that every solution blows up
in finite time in -norm. Onsager's conjecture is confirmed for the
model system
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
Coherent vortex structures and 3D enstrophy cascade
Existence of 2D enstrophy cascade in a suitable mathematical setting, and
under suitable conditions compatible with 2D turbulence phenomenology, is known
both in the Fourier and in the physical scales. The goal of this paper is to
show that the same geometric condition preventing the formation of
singularities - 1/2-H\"older coherence of the vorticity direction - coupled
with a suitable condition on a modified Kraichnan scale, and under a certain
modulation assumption on evolution of the vorticity, leads to existence of 3D
enstrophy cascade in physical scales of the flow.Comment: 15 pp; final version -- to appear in CM
Leray and LANS- modeling of turbulent mixing
Mathematical regularisation of the nonlinear terms in the Navier-Stokes
equations provides a systematic approach to deriving subgrid closures for
numerical simulations of turbulent flow. By construction, these subgrid
closures imply existence and uniqueness of strong solutions to the
corresponding modelled system of equations. We will consider the large eddy
interpretation of two such mathematical regularisation principles, i.e., Leray
and LANS regularisation. The Leray principle introduces a {\bfi
smoothed transport velocity} as part of the regularised convective
nonlinearity. The LANS principle extends the Leray formulation in a
natural way in which a {\bfi filtered Kelvin circulation theorem},
incorporating the smoothed transport velocity, is explicitly satisfied. These
regularisation principles give rise to implied subgrid closures which will be
applied in large eddy simulation of turbulent mixing. Comparison with filtered
direct numerical simulation data, and with predictions obtained from popular
dynamic eddy-viscosity modelling, shows that these mathematical regularisation
models are considerably more accurate, at a lower computational cost.Comment: 42 pages, 12 figure
- …