875 research outputs found
Averaging versus Chaos in Turbulent Transport?
In this paper we analyze the transport of passive tracers by deterministic
stationary incompressible flows which can be decomposed over an infinite number
of spatial scales without separation between them. It appears that a low order
dynamical system related to local Peclet numbers can be extracted from these
flows and it controls their transport properties. Its analysis shows that these
flows are strongly self-averaging and super-diffusive: the delay for
any finite number of passive tracers initially close to separate till a
distance is almost surely anomalously fast (, with
). This strong self-averaging property is such that the dissipative
power of the flow compensates its convective power at every scale. However as
the circulation increase in the eddies the transport behavior of the flow may
(discontinuously) bifurcate and become ruled by deterministic chaos: the
self-averaging property collapses and advection dominates dissipation. When the
flow is anisotropic a new formula describing turbulent conductivity is
identified.Comment: Presented at Oberwolfach (October 2002), CIRM (March 2003), Lisbonne
(XIV international congress on mathematical physics: July 2003). Submitted on
October 2002, to appear in Communications in Mathematical Physics. 45 pages,
7 figure
Renormalization group analysis of the Reynolds stress transport equation
The pressure velocity correlation and return to isotropy term in the Reynolds stress transport equation are analyzed using the Yakhot-Orszag renormalization group. The perturbation series for the relevant correlations, evaluated to lowest order in the epsilon-expansion of the Yakhot-Orszag theory, are infinite series in tensor product powers of the mean velocity gradient and its transpose. Formal lowest order Pade approximations to the sums of these series produce a fast pressure strain model of the form proposed by Launder, Reece, and Rodi, and a return to isotropy model of the form proposed by Rotta. In both cases, the model constant are computed theoretically. The predicted Reynolds stress ratios in simple shear flows are evaluated and compared with experimental data. The possibility is discussed of driving higher order nonlinear models by approximating the sums more accurately
Phase Diagram for Turbulent Transport: Sampling Drift, Eddy Diffusivity and Variational Principles
We study the long-time, large scale transport in a three-parameter family of
isotropic, incompressible velocity fields with power-law spectra. Scaling law
for transport is characterized by the scaling exponent and the Hurst
exponent , as functions of the parameters. The parameter space is divided
into regimes of scaling laws of different {\em functional forms} of the scaling
exponent and the Hurst exponent. We present the full three-dimensional phase
diagram.
The limiting process is one of three kinds: Brownian motion (),
persistent fractional Brownian motions () and regular (or smooth)
motion (H=1).
We discover that a critical wave number divides the infrared cutoffs into
three categories, critical, subcritical and supercritical; they give rise to
different scaling laws and phase diagrams. We introduce the notions of sampling
drift and eddy diffusivity, and formulate variational principles to estimate
the eddy diffusivity. We show that fractional Brownian motions result from a
dominant sampling drift
Universality in Turbulence: an Exactly Soluble Model
The present note contains the text of lectures discussing the problem of
universality in fully developed turbulence. After a brief description of
Kolmogorov's 1941 scaling theory of turbulence and a comparison between the
statistical approach to turbulence and field theory, we discuss a simple model
of turbulent advection which is exactly soluble but whose exact solution is
still difficult to analyze. The model exhibits a restricted universality. Its
correlation functions contain terms with universal but anomalous scaling but
with non-universal amplitudes typically diverging with the growing size of the
system. Strict universality applies only after such terms have been removed
leaving renormalized correlators with normal scaling. We expect that the
necessity of such an infrared renormalization is a characteristic feature of
universality in turbulence.Comment: 31 pages, late
Multiple-scale analysis and renormalization for pre-asymptotic scalar transport
Pre-asymptotic transport of a scalar quantity passively advected by a
velocity field formed by a large-scale component superimposed to a small-scale
fluctuation is investigated both analytically and by means of numerical
simulations. Exploiting the multiple-scale expansion one arrives at a
Fokker--Planck equation which describes the pre-asymptotic scalar dynamics.
Such equation is associated to a Langevin equation involving a multiplicative
noise and an effective (compressible) drift. For the general case, no explicit
expression for both the effective drift and the effective diffusivity (actually
a tensorial field) can be obtained. We discuss an approximation under which an
explicit expression for the diffusivity (and thus for the drift) can be
obtained. Its expression permits to highlight the important fact that the
diffusivity explicitly depends on the large-scale advecting velocity. Finally,
the robustness of the aforementioned approximation is checked numerically by
means of direct numerical simulations.Comment: revtex4, 12 twocolumn pages, 3 eps figure
Renormalized transport of inertial particles in surface flows
Surface transport of inertial particles is investigated by means of the
perturbative approach, introduced by Maxey (J. Fluid Mech. 174, 441 (1987)),
which is valid in the case the deflections induced on the particle trajectories
by the fluid flow can be considered small. We consider a class of compressible
random velocity fields, in which the effect of recirculations is modelled by an
oscillatory component in the Eulerian time correlation profile. The main issue
we address here is whether fluid velocity fluctuations, in particular the
effect of recirculation, may produce nontrivial corrections to the streaming
particle velocity. Our result is that a small (large) degree of recirculation
is associated with a decrease (increase) of streaming with respect to a
quiescent fluid. The presence of this effect is confirmed numerically, away
from the perturbative limit. Our approach also allows us to calculate the
explicit expression for the eddy diffusivity, and to compare the efficiency of
diffusive and ballistic transport.Comment: 18 pages, 13 figures, submitted to JF
Some relations between Lagrangian models and synthetic random velocity fields
We propose an alternative interpretation of Markovian transport models based
on the well-mixedness condition, in terms of the properties of a random
velocity field with second order structure functions scaling linearly in the
space time increments. This interpretation allows direct association of the
drift and noise terms entering the model, with the geometry of the turbulent
fluctuations. In particular, the well known non-uniqueness problem in the
well-mixedness approach is solved in terms of the antisymmetric part of the
velocity correlations; its relation with the presence of non-zero mean helicity
and other geometrical properties of the flow is elucidated. The well-mixedness
condition appears to be a special case of the relation between conditional
velocity increments of the random field and the one-point Eulerian velocity
distribution, allowing generalization of the approach to the transport of
non-tracer quantities. Application to solid particle transport leads to a model
satisfying, in the homogeneous isotropic turbulence case, all the conditions on
the behaviour of the correlation times for the fluid velocity sampled by the
particles. In particular, correlation times in the gravity and in the inertia
dominated case, respectively, longer and shorter than in the passive tracer
case; in the gravity dominated case, correlation times longer for velocity
components along gravity, than for the perpendicular ones. The model produces,
in channel flow geometry, particle deposition rates in agreement with
experiments.Comment: 54 pages, 8 eps figures included; contains additional material on
SO(3) and on turbulent channel flows. Few typos correcte
Cascades and Dissipative Anomalies in Nearly Collisionless Plasma Turbulence
We develop first-principles theory of kinetic plasma turbulence governed by
the Vlasov-Maxwell-Landau equations in the limit of vanishing collision rates.
Following an exact renormalization-group approach pioneered by Onsager, we
demonstrate the existence of a "collisionless range" of scales (lengths and
velocities) in 1-particle phase space where the ideal Vlasov-Maxwell equations
are satisfied in a "coarse-grained sense". Entropy conservation may
nevertheless be violated in that range by a "dissipative anomaly" due to
nonlinear entropy cascade. We derive "4/5th-law" type expressions for the
entropy flux, which allow us to characterize the singularities
(structure-function scaling exponents) required for its non-vanishing.
Conservation laws of mass, momentum and energy are not afflicted with anomalous
transfers in the collisionless limit. In a subsequent limit of small gyroradii,
however, anomalous contributions to inertial-range energy balance may appear
due both to cascade of bulk energy and to turbulent redistribution of internal
energy in phase space. In that same limit the "generalized Ohm's law" derived
from the particle momentum balances reduces to an "ideal Ohm's law", but only
in a coarse-grained sense that does not imply magnetic flux-freezing and that
permits magnetic reconnection at all inertial-range scales. We compare our
results with prior theory based on the gyrokinetic (high gyro-frequency) limit,
with numerical simulations, and with spacecraft measurements of the solar wind
and terrestrial magnetosphere.Comment: Several additions have been made that were requested by the referees
of the PRX submission. In particular, discussion previously relegated to
Supplemental Materials are now included in the main text as appendice
- …