PROGRAM IDEI – CERCETARI EXPLORATORII

Cod Proiect ID_1104 / 2008

Director Proiect: Madalina VLAD

Nonlinear evolution, quasi-coherence 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 quasi-coherent 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 advection-diffusion 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 shock-turbulence interactions and the wave-wave 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 Burgers-Hopf 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, quasi-coherent 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 advection-diffusion 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 shok-turbulence and the wave-wave interactions;

C) To determine the evolution class associated to models of turbulence based on Burgers-Hopf 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 quasi-coherent 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 two-dimensional self-similar 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 two-dimensional 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 1-st order on the n-dimensional 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 Hamilton-Jacobi 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 1-st order on a space-time manifold of arbitrary dimension .

The second one concerns the inviscid n-dimensional 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 Hamilton-Jacobi equation and to characterize it in a projective invariant manner among the solutions of the inviscid Burgers system.

First Report