PROGRAM
IDEI – CERCETARI EXPLORATORII
Cod Proiect
ID_1104 / 2008

Director
Proiect: Madalina VLAD

Nonlinear evolution, quasicoherence and transport
in turbulent fluids 
PROJECT SUMMARY

This
project will bring an important contribution to the physical understanding of
turbulence and to its theoretical description. The aim is to develop new
theoretical approaches for the study of strong turbulence and of
quasicoherent structures. The project is based on recent results obtained in
our research group. We propose fundamental theoretical topics with
interdisciplinary character. The
nonlinear effects in the advectiondiffusion processes due to trajectory
trapping in turbulent incompressible fluids will be determined. Trapping
determines memory and coherence in particle motion, which are estimated to
have a strong influence on the advected fields. This
Project extends [and goes deep into the details of] some original research
concerning the nonlinear admissible evolution of certain significant
gasdynamic models by identifying some
new concepts and techniques. Extension takes into account 3 classes of
multidimensional evolutions. At first,
we consider 2 classes of evolution through gasdynamic interactions: the
shockturbulence interactions and the wavewave interactions. For these 2
classes: the steps of the mentioned
extension consider more and more complex versions [in a complexity hierarchy
]; for the objects contributing to an
interaction, the evolutions are [multidimensionally] characterized as
admissible and, essentially, as endowed with coherent structures, of a
classifying importance. For a third class of evolution, associated to some
BurgersHopf systems [significant in turbulence modelling], we aim to provide
multidimensional extensions of the admissibility conditions. The results
obtained would be put in parallel with significant comparable results
available in the literature. 
PROJECT OBJECTIVES

The
aim of this project is to bring important contributions to the understanding
of the nonlinear evolution, quasicoherent structure formation and transport
in fluid turbulence. We propose fundamental theoretical topics with
interdisciplinary character and applications in several fields. The
objectives of the project are: A) To determine the nonlinear effects in
the advectiondiffusion processes by developing a new statistical Lagrangian
method, that includes trajectory trapping in the structure of velocity field; B) To determine the nonlinear evolution
of some gasdynamic models by studyng the shokturbulence and the wavewave
interactions; C) To determine the evolution class
associated to models of turbulence based on BurgersHopf equations and to provide multidimensional
extensions of the admissibility conditions. Obiective
A In the first part of the project we
will develop the methods for the study of the statistics of trajectories in stochastic
fields, the decorrelation trajectory method (DTM) and the nested subensemble
approach (NSM). The DTM will be improved by a better description of
trajectory fluctuations using the path functional integral technique of
Feynman. We will also find a method of introducing in the statistical
description the constraints determined by the existance of statistical
invariant quantities, as the distribution of the Lagrangian velocity. The DTM
method does not have this property. The systematic expantion developed in the
NSM describes this statistical invariance only in the limit of large orders.
The results concerning the diffusion coefficients are not much influenced by
the absence of the invariance of the distribution of the Lagragian velocity,
but we expect that this property is important in the study of field
advection. The influence of the special
statistical properties of the trajectories on the advection of passive fields
will be studied in the second half of the project. We expect a strong connection
between the memory and coherence effects that we have found in the statistics
of trajectories and the quasicoherent structure formation in the advected
field. The effect of the diffusion on the evolution of the advected fields
will be determined. The effects of a weak compressibility on particle and
fields stochastic transport will be studied in the last year of the project. Obiective
B In the present project
we aim to obtain some important extensions in the study of ST
interactions1pt: details concerning
the persistence of a factorizing character, and of
the coherence associated with it in a ST interaction, to more
complex descriptions of a gasdynamic type1pt; construction of a general model of a ST interaction 
via an extension of the modal incidence considered in [13], [16]; completion of the
subcritical details of our anterior construction ([13], [16]) with some
supercritical details; extension of the
nature of the shock discontinuity which contribute in a ST interaction to a more complex gasdynamic type. For the study of WW interactions we aim to provide a multidimensional extension of the parallel between the
algebraic and differential approaches. This would require a previous adaptation of Martin's approach to a
multidimensional context. Finally, starting
from the class of the twodimensional selfsimilar exact isentropic solutions
constructed in [17] we aim to complete a parallel between a Burnat regular interaction structure and a Zhang  Zheng irregular interaction structure associated to the
qualitative study of a class of solutions [with the simplest structure] of
the twodimensional Riemann problem for an isentropic system of equations. Obiective
C In a first
stage we aim to characterize the class of scalar
partial differential equations of 1st order on the ndimensional Euclidean space whose
Cauchy characteristics are straight lines.
Next, on taking into account our recent results on the inviscid
Burgers equation, we would look for an unified statement
of the entropy conditions, invariant under the group of projective
transformations, both for those found by Hopf, Lax, Oleinik, Volpert and
Kruzhkov for dicontinuous solutions of quasilinear equations and for the
conditions found by Kruzhkov, Crandall and P.L. Lions for the continuous
solutions with discontinuous gradient of the HamiltonJacobi equations. We
should try to understand the role of the differentiation [along the Cauchy
characteristics] operator for the variational formulation of entropic
solutions; this operator is naturally implied in our variational formulation
for the inviscid Burgers equation. We should also try to identify, in this
general projective invariant context, an analogue of the barycentric
concentration of the singularities with respect to an entropy measure. Such
an analogue would support an extension of the
entropy conditions  phenomenon that we pointed out in the case of the
inviscid Burgers equation. We have here to look for the connection between
the entropy measure and the elliptic operator that defines the viscous
approximation of the entropy solution. We set aside 2 semesters for this
work. There
are next two directions of study that open and we aim to follow them in
parallel: The first one
concerns an invariant under diffeomorphisms definition of entropy solutions
for scalar partial differential equations of the 1st order on a spacetime
manifold of arbitrary dimension . The second one
concerns the inviscid ndimensional
Burgers system (where n denotes the number of space [nontemporal]
dimensions, of the unknown functions and of the equations in the system). We
aim to identify the solution obtained by the transport of the viscosity
solution of Crandall and P.L. Lions for this HamiltonJacobi equation and to
characterize it in a projective invariant manner among the solutions of the
inviscid Burgers system. 
