In this thesis I present an analysis of the effects induced by
the presence of a linear friction force and by the addition of
small quantities of polymers on the statistical properties of
two-dimensional turbulence.
The first part of the thesis deals with the effects of a linear friction force on the direct enstrophy cascade. It is shown that friction produces a steepening of the energy spectrum, and a strong intermittency in the statistics of small-scale fluctuations of vorticity.
These effects can be predicted and quantified by means of Lagrangian statistics. In particular I will show that a linear friction force reduces the statistical properties of vorticity at small scale to coincide with those of a passive scalar field with finite lifetime transported by the same flow.
Theoretical predictions obtained within the Lagrangian approach
are validated by the results of numerical simulations of
Navier-Stoker equation and the advection-reaction equation
for the passive scalar.
In the second part the study of two-dimensional dilute polymer solutions is addressed by means of a linear viscoelastic model (Oldroyd-B).
In the passive case, i.e. neglecting the feedback of polymers on the flow, it is shown that the probability distribution function of polymer elongation has a power law tail, whose slope can be predicted in terms of Lagrangian exit-time statistics.
In the active case I will show that the kinetic energy of the fluid is drastically reduced by the polymer back-reaction. This phenomenon should be contrasted with the three-dimensional case where, on the opposite, kinetic energy of the mean flow is enhanced by the presence of polymers.
The viscoelastic model adopted provides a clear explanation for the origin of energy suppression which has been observed also in soap-film experiments.
Moreover I will show that the presence of polymers causes
a strong reduction of Lagrangian chaos, and influences
the decay of two-dimensional turbulence as well as the
inverse energy cascade, which can be completely depleted
for large enough polymer elasticity.
The two parts of the thesis are strictly connected by the methods used
for studying the two different physical situations, which are both reduced
to the general problem of transport of active scalar or tensorial fields.
The Lagrangian description of turbulence transport, which has revealed
as the most powerful tool for the comprehension of the passive problem
is here used to study two cases where the transported quantities are active,
i.e. they have a feedback on the velocity field.
The thesis is organized in four chapter. The first chapter is intended as a short introduction to fully developed turbulence in two dimensions. I will summarize the basic phenomenology, introducing the concepts and terminology that will be used in the thesis, and stressing the main differences with the three-dimensional case.
The second chapter deals with the effects of friction in two-dimensional turbulence. I will show the analogy with the problem of transport of a passive scalar field with finite lifetime, giving a short review of recent results for this problem, and showing how and under which condition they can be applied to the case of vorticity in two-dimensional turbulence with friction. Then I will present a comparison between the results obtained by means of numerical simulations and the prediction based on the Lagrangian statistics.
In the third chapter I provide a brief introduction to the physics of dilute polymer solution, describing some of the phenomena which have been observed in experiments and introducing the viscoelastic model adopted in this thesis.
The fourth chapter contains the results of the theoretical and numerical study of two-dimensional dilute polymers solutions.