39 research outputs found
Probability Densities in Strong Turbulence
According to modern developments in turbulence theory, the "dissipation"
scales (u.v. cut-offs) form a random field related to velocity
increments . In this work we, using Mellin's transform combined
with the Gaussain large -scale boundary condition, calculate probability
densities (PDFs) of velocity increments and the PDF of the
dissipation scales , where is the large-scale Reynolds
number. The resulting expressions strongly deviate from the Log-normal PDF
often quoted in the literature. It is shown that the
probability density of the small-scale velocity fluctuations includes
information about the large (integral) scale dynamics which is responsible for
deviation of from . A framework for
evaluation of the PDFs of various turbulence characteristics involving spatial
derivatives is developed. The exact relation, free of spurious Logarithms
recently discussed in Frisch et al (J. Fluid Mech. {\bf 542}, 97 (2005)), for
the multifractal probability density of velocity increments, not based on the
steepest descent evaluation of the integrals is obtained and the calculated
function is close to experimental data. A novel derivation (Polyakov,
2005), of a well-known result of the multi-fractal theory [Frisch, "Turbulence.
{\it Legacy of A.N.Kolmogorov}", Cambridge University Press, 1995)), based on
the concepts described in this paper, is also presented.Comment: 25 pages and 9 figure
Development of turbulence models for shear flows by a double expansion technique
Turbulence models are developed by supplementing the renormalization group (RNG) approach of Yakhot and Orszag with scale expansions for the Reynolds stress and production of dissipation terms. The additional expansion parameter (eta) is the ratio of the turbulent to mean strain time scale. While low-order expansions appear to provide an adequate description of the Reynolds stress, no finite truncation of the expansion for the production of dissipation term in powers of eta suffices - terms of all orders must be retained. Based on these ideas, a new two-equation model and Reynolds stress transport model are developed for turbulent shear flows. The models are tested for homogeneous shear flow and flow over a backward facing step. Comparisons between the model predictions and experimental data are excellent
Mean- Field Approximation and a Small Parameter in Turbulence Theory
Numerical and physical experiments on two-dimensional (2d) turbulence show
that the differences of transverse components of velocity field are well
described by a gaussian statistics and Kolmogorov scaling exponents. In this
case the dissipation fluctuations are irrelevant in the limit of small
viscosity. In general, one can assume existence of critical
space-dimensionality , at which the energy flux and all odd-order
moments of velocity difference change sign and the dissipation fluctuations
become dynamically unimportant. At the flow can be described by the
``mean-field theory'', leading to the observed gaussian statistics and
Kolmogorov scaling of transverse velocity differences. It is shown that in the
vicinity of the ratio of the relaxation and translation
characteristic times decreases to zero, thus giving rise to a small parameter
of the theory. The expressions for pressure and dissipation contributions to
the exact equation for the generating function of transverse velocity
differences are derived in the vicinity of . The resulting equation
describes experimental data on two-dimensional turbulence and demonstrate onset
of intermittency as and in three-dimensional flows in
close agreement with experimental data. In addition, some new exact relations
between correlation functions of velocity differences are derived. It is also
predicted that the single-point pdf of transverse velocity difference in
developing as well as in the large-scale stabilized two-dimensional turbulence
is a gaussian.Comment: 25 pages, 1 figur
Large eddy simulation of two-dimensional isotropic turbulence
Large eddy simulation (LES) of forced, homogeneous, isotropic,
two-dimensional (2D) turbulence in the energy transfer subrange is the subject
of this paper. A difficulty specific to this LES and its subgrid scale (SGS)
representation is in that the energy source resides in high wave number modes
excluded in simulations. Therefore, the SGS scheme in this case should assume
the function of the energy source. In addition, the controversial requirements
to ensure direct enstrophy transfer and inverse energy transfer make the
conventional scheme of positive and dissipative eddy viscosity inapplicable to
2D turbulence. It is shown that these requirements can be reconciled by
utilizing a two-parametric viscosity introduced by Kraichnan (1976) that
accounts for the energy and enstrophy exchange between the resolved and subgrid
scale modes in a way consistent with the dynamics of 2D turbulence; it is
negative on large scales, positive on small scales and complies with the basic
conservation laws for energy and enstrophy. Different implementations of the
two-parametric viscosity for LES of 2D turbulence were considered. It was found
that if kept constant, this viscosity results in unstable numerical scheme.
Therefore, another scheme was advanced in which the two-parametric viscosity
depends on the flow field. In addition, to extend simulations beyond the limits
imposed by the finiteness of computational domain, a large scale drag was
introduced. The resulting LES exhibited remarkable and fast convergence to the
solution obtained in the preceding direct numerical simulations (DNS) by
Chekhlov et al. (1994) while the flow parameters were in good agreement with
their DNS counterparts. Also, good agreement with the Kolmogorov theory was
found. This LES could be continued virtually indefinitely. Then, a simplifiedComment: 34 pages plain tex + 18 postscript figures separately, uses auxilary
djnlx.tex fil
Large-Eddy Simulations of Fluid and Magnetohydrodynamic Turbulence Using Renormalized Parameters
In this paper a procedure for large-eddy simulation (LES) has been devised
for fluid and magnetohydrodynamic turbulence in Fourier space using the
renormalized parameters. The parameters calculated using field theory have been
taken from recent papers by Verma [Phys. Rev. E, 2001; Phys. Plasmas, 2001]. We
have carried out LES on grid. These results match quite well with direct
numerical simulations of . We show that proper choice of parameter is
necessary in LES.Comment: 12 pages, 4 figures: Proper figures inserte
Local shell-to-shell energy transfer via nonlocal Interactions in fluid turbulence
In this paper we analytically compute the strength of nonlinear interactions
in a triad, and the energy exchanges between wavenumber shells in
incompressible fluid turbulence. The computation has been done using
first-order perturbative field theory. In three dimension, magnitude of triad
interactions is large for nonlocal triads, and small for local triads. However,
the shell-to-shell energy transfer rate is found to be local and forward. This
result is due to the fact that the nonlocal triads occupy much less Fourier
space volume than the local ones. The analytical results on three-dimensional
shell-to-shell energy transfer match with their numerical counterparts. In
two-dimensional turbulence, the energy transfer rates to the near-by shells are
forward, but to the distant shells are backward; the cumulative effect is an
inverse cascade of energy.Comment: 10 pages, Revtex
Anisotropy in Turbulent Flows and in Turbulent Transport
We discuss the problem of anisotropy and intermittency in statistical theory
of high Reynolds-number turbulence (and turbulent transport). We present a
detailed description of the new tools that allow effective data analysis and
systematic theoretical studies such as to separate isotropic from anisotropic
aspects of turbulent statistical fluctuations. Employing the invariance of the
equations of fluid mechanics to all rotations, we show how to decompose the
(tensorial) statistical objects in terms of the irreducible representation of
the SO(3) symmetry group. For the case of turbulent advection of passive scalar
or vector fields, this decomposition allows rigorous statements to be made: (i)
the scaling exponents are universal, (ii) the isotropic scaling exponents are
always leading, (iii) the anisotropic scaling exponents form a discrete
spectrum which is strictly increasing as a function of the anisotropic degree.
Next we explain how to apply the SO(3) decomposition to the statistical
Navier-Stokes theory. We show how to extract information about the scaling
behavior in the isotropic sector. Doing so furnishes a systematic way to assess
the universality of the scaling exponents in this sector, clarifying the
anisotropic origin of the many measurements that claimed the opposite. A
systematic analysis of Direct Numerical Simulations and of experiments provides
a strong support to the proposition that also for the non-linear problem there
exists foliation of the statistical theory into sectors of the symmetry group.
The exponents appear universal in each sector, and again strictly increasing as
a function of the anisotropic degreee.Comment: 150 pages, 26 figures, submitted to Phys. Re
Statistical Properties of Turbulence: An Overview
We present an introductory overview of several challenging problems in the
statistical characterisation of turbulence. We provide examples from fluid
turbulence in three and two dimensions, from the turbulent advection of passive
scalars, turbulence in the one-dimensional Burgers equation, and fluid
turbulence in the presence of polymer additives.Comment: 34 pages, 31 figure
Statistical Theory of Magnetohydrodynamic Turbulence: Recent Results
In this review article we will describe recent developments in statistical
theory of magnetohydrodynamic (MHD) turbulence. Kraichnan and Iroshnikov first
proposed a phenomenology of MHD turbulence where Alfven time-scale dominates
the dynamics, and the energy spectrum E(k) is proportional to k^{-3/2}. In the
last decade, many numerical simulations show that spectral index is closer to
5/3, which is Kolmogorov's index for fluid turbulence. We review recent
theoretical results based on anisotropy and Renormalization Groups which
support Kolmogorov's scaling for MHD turbulence.
Energy transfer among Fourier modes, energy flux, and shell-to-shell energy
transfers are important quantities in MHD turbulence. We report recent
numerical and field-theoretic results in this area. Role of these quantities in
magnetic field amplification (dynamo) are also discussed. There are new
insights into the role of magnetic helicity in turbulence evolution. Recent
interesting results in intermittency, large-eddy simulations, and shell models
of magnetohydrodynamics are also covered.Comment: 116 pages, Submitted to Phys. Rep., Due to size restrictions, the
figures are not of high quality. Download from http://home.iitk.ac.in/~mk