257 research outputs found
Anomalous Scaling in a Model of Passive Scalar Advection: Exact Results
Kraichnan's model of passive scalar advection in which the driving velocity
field has fast temporal decorrelation is studied as a case model for
understanding the appearance of anomalous scaling in turbulent systems. We
demonstrate how the techniques of renormalized perturbation theory lead (after
exact resummations) to equations for the statistical quantities that reveal
also non perturbative effects. It is shown that ultraviolet divergences in the
diagrammatic expansion translate into anomalous scaling with the inner length
acting as the renormalization scale. In this paper we compute analytically the
infinite set of anomalous exponents that stem from the ultraviolet divergences.
Notwithstanding, non-perturbative effects furnish a possibility of anomalous
scaling based on the outer renormalization scale. The mechanism for this
intricate behavior is examined and explained in detail. We show that in the
language of L'vov, Procaccia and Fairhall [Phys. Rev. E {\bf 50}, 4684 (1994)]
the problem is ``critical" i.e. the anomalous exponent of the scalar primary
field . This is precisely the condition that allows for
anomalous scaling in the structure functions as well, and we prove that this
anomaly must be based on the outer renormalization scale. Finally, we derive
the scaling laws that were proposed by Kraichnan for this problem, and show
that his scaling exponents are consistent with our theory.Comment: 43 pages, revtex
Exact Resummations in the Theory of Hydrodynamic Turbulence: II A Ladder to Anomalous Scaling
In paper I of this series on fluid turbulence we showed that exact
resummations of the perturbative theory of the structure functions of velocity
differences result in a finite (order by order) theory. These findings exclude
any known perturbative mechanism for anomalous scaling of the velocity
structure functions. In this paper we continue to build the theory of
turbulence and commence the analysis of nonperturbative effects that form the
analytic basis of anomalous scaling. Starting from the Navier-Stokes equations
(at high Reynolds number Re) we discuss the simplest examples of the appearance
of anomalous exponents in fluid mechanics. These examples are the nonlinear
(four-point) Green's function and related quantities. We show that the
renormalized perturbation theory for these functions contains ``ladder``
diagrams with (convergent!) logarithmic terms that sum up to anomalous
exponents. Using a new sum rule which is derived here we calculate the leading
anomalous exponent and show that it is critical in a sense made precise below.
This result opens up the possibility of multiscaling of the structure functions
with the outer scale of turbulence as the renormalization length. This
possibility will be discussed in detail in the concluding paper III of this
series.Comment: PRE in press, 15 pages + 21 figures, REVTeX, The Eps files of figures
will be FTPed by request to [email protected]
Dynamics of the vortex line density in superfluid counterflow turbulence
Describing superfluid turbulence at intermediate scales between the
inter-vortex distance and the macroscale requires an acceptable equation of
motion for the density of quantized vortex lines . The closure of such
an equation for superfluid inhomogeneous flows requires additional inputs
besides and the normal and superfluid velocity fields. In this paper
we offer a minimal closure using one additional anisotropy parameter .
Using the example of counterflow superfluid turbulence we derive two coupled
closure equations for the vortex line density and the anisotropy parameter
with an input of the normal and superfluid velocity fields. The
various closure assumptions and the predictions of the resulting theory are
tested against numerical simulations.Comment: 7 pages, 5 figure
Saturation of Turbulent Drag Reduction in Dilute Polymer Solutions
Drag reduction by polymers in turbulent wall-bounded flows exhibits universal
and non-universal aspects. The universal maximal mean velocity profile was
explained in a recent theory. The saturation of this profile and the crossover
back to the Newtonian plug are non-universal, depending on Reynolds number Re,
concentration of polymer and the degree of polymerization . We
explain the mechanism of saturation stemming from the finiteness of
extensibility of the polymers, predict its dependence on and in the
limit of small and large Re, and present the excellent comparison of our
predictions to experiments on drag reduction by DNA.Comment: 4 pages, 4 figs., included, PRL, submitte
Exact Resummations in the Theory of Hydrodynamic Turbulence: III. Scenarios for Anomalous Scaling and Intermittency
Elements of the analytic structure of anomalous scaling and intermittency in
fully developed hydrodynamic turbulence are described. We focus here on the
structure functions of velocity differences that satisfy inertial range scaling
laws , and the correlation of energy dissipation
. The goal is to understand the
exponents and from first principles. In paper II of this series
it was shown that the existence of an ultraviolet scale (the dissipation scale
) is associated with a spectrum of anomalous exponents that characterize
the ultraviolet divergences of correlations of gradient fields. The leading
scaling exponent in this family was denoted . The exact resummation of
ladder diagrams resulted in the calculation of which satisfies the
scaling relation . In this paper we continue our analysis and
show that nonperturbative effects may introduce multiscaling (i.e.
not being linear in ) with the renormalization scale being the infrared
outer scale of turbulence . It is shown that deviations from K41 scaling of
() must appear if the correlation of dissipation is
mixing (i.e. ). We derive an exact scaling relation . We present analytic expressions for for all
and discuss their relation to experimental data. One surprising prediction is
that the time decay constant of scales
independently of : the dynamic scaling exponent is the same for all
-order quantities, .Comment: PRE submitted, 22 pages + 11 figures, REVTeX. The Eps files of
figures will be FTPed by request to [email protected]
Parametric Generation of Second Sound by First Sound in Superfluid Helium
We report the first experimental observation of parametric generation of
second sound (SS) by first sound (FS) in superfluid helium in a narrow
temperature range in the vicinity of . The temperature dependence
of the threshold FS amplitude is found to be in a good quantitative agreement
with the theory suggested long time ago and corrected for a finite geometry.
Strong amplitude fluctuations and two types of the SS spectra are observed
above the bifurcation. The latter effect is quantitatively explained by the
discreteness of the wave vector space and the strong temperature dependence of
the SS dissipation length.Comment: 4 pages, 4 postscript figures, REVTE
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