Cette thèse est constituée de deux parties, dont l'axe commun est la modélisation de phénomènes de petites échelles pour des écoulements turbulents.Dans une première partie on s'intéresse à l'influence des rotons sur la dynamique d'un modèle d'hélium superfluide. On commence par une calibration d'un modèle non-local d'interaction dans le but de reproduire la relation de dispersion expérimentale de l'hélium, mesurée par diffraction de neutrons. On utilise ensuite ce modèle calibré pour réaliser des simulations numériques directes (DNS) de l'équation de Gross-Pitaevskii, afin de sonder le phénomène de reconnexion des tourbillons quantiques. Ce phénomène est étudié en détail via une analyse géométrique et énergétique des résultats des DNS. On compare alors systématiquement ces résultats à ceux du modèle local afin d'étudier l'influence des rotons sur l'écoulement aux échelles de l'ordre de l'Angstrom.Dans un second temps on cherche à décrire la structure spatio-temporelle de la turbulence homogène et isotrope. Pour cela on commence par une analyse des propriétés statistiques du champ eulérien de vitesse, basée sur l'évaluation de ses incréments spatio-temporels. On utilise les données issues d'une DNS des équations de Navier-Stokes mises à disposition par l'Université Johns Hopkins. On propose ensuite un champ aléatoire spatio-temporel pour la vitesse eulérienne, en caractérisant d'abord la structure de ses corrélations par une approximation gaussienne. On modifie ensuite cette approximation par une mesure multi-fractale afin de reproduire les aspects non-gaussiens observés dans la DNS, tels que les hauts niveaux des coefficients d'asymétrie et d'aplatissement. / This thesis consists of two parts that share a common theme : the modeling of small-scale phenomena in turbulent flows.In a first part we focus on the influence of rotons on the dynamics of a model of superfluid helium. We begin by a calibration of a nonlocal model of the interaction, aiming at reproducing the experimental dispersion relation of helium, as measured by neutron scattering methods. This model is then used to perform Direct Numerical Simulations (DNS) of the Gross-Pitaevskii equation, in order to probe the reconnection of quantum vortices. This phenomenon is studied quantitatively through a geometrical and energetical analysis of the results of the DNS. We then systematically compare these results with those of the local model, so as to study the influence of rotons on flow scales of the order of the Angtstrom.The goal of the second part is to describe the spatio-temporal structure of homogeneous and isotropic turbulence. To achieve it we start by a standard analysis of the statistical properties of the eulerian velocity field, by computing its spatio-temporal increments. We use the data from a DNS of the Navier-Stokes equations, hosted and made available by the Johns Hopkins University. We then propose a random, spatio-temporal eulerian velocity field, by first characterizing the structure of its correlations through a gaussian approximation. This approximation is then modified by a multifractal measure in order to reproduce the non-gaussian features, as they are demanded by the observed high level of skewness and flatness of increments.

Steady state and decay of quantum turbulence generated in channel flows and detected by second sound attenuation Bc. Emil Varga Abstract Quantum turbulence is studied in superfluid 4 He under classical flow condi- tions. Turbulence is generated by a flow through a 7 × 7 mm square channel with a flow conditioner either with an additional grid or without it. The flow is generated mechanically by squeezing a stainless steel bellows. Vortex line den- sity is measured by attenuation of second sound in both steady state and decay for a range of temperatures 1.17 - 2.16 K. In the steady state, temperature- independent scaling of the vortex line density with flow velocity of the form L ∝ V 3/2 is observed. In the decay the expected late-time behaviour L ∝ t−3/2 is observed. Explanation for both of these observations is based on a quasi- classical model of quantum turbulence, that allows the extraction of the effective kinematic viscosity νeff, which approximately agree with the values available in the literature. Two models based on counterflow theory are also explored and the effect of inhomogeneous vortex line distribution on the measurement tech- nique is studied.

We have performed a study of quantum turbulence generated in oscillatory counterflow as a continuation of previous experiments on various channel flows of superfluid helium, in the form of coflow, thermal DC counterflow and pure superflow. We have investigated its development, steady state properties and temporal decay, as well as the effect of the resonant mode used to generate the turbulence at three different temperatures, 1.45 K, 1.65 K and 1.83 K. The attenuation of low amplitude second sound, orientated perpendicularly to the long axis of the resonator, was used to determine the amount of quantized vortices created. One of the main goals of this work was to characterize the critical parameters for the onset of instabilities in oscillatory counterflow and to determine their values. Decay measurements of the vortex line density allowed us to distinguish between Vinen-type and Kolmogorov- type decays of quantum turbulence.

Nonlinear dispersive partial differential equations occur in a variety of areas within mathematical physics and engineering. We study several classes of such equations, including scalar complex partial differential equations, vector partial differential equations, and finally non-local integro-differential equations. For physically interesting families of these equations, we demonstrate the existence (and, when possible, stability) of specific solutions which are relevant for applications. While multiple application areas are considered, the primary application that runs through the work would be the nonlinear dynamics of vortex filaments under a variety of physical models. For instance, we are able to determine the structure and time evolution of several physical solutions, including the planar, helical, self-similar and soliton vortex filament solutions in a quantum fluid. Properties of such solutions are determined analytically and numerically through a variety of approaches. Starting with complex scalar equations (often useful for studying two-dimensional motion), we progress through more complicated models involving vector partial differential equations and non-local equations (which permit motion in three dimensions). In many of the examples considered, the qualitative analytical results are used to verify behaviors previously observed only numerically or experimentally.

A nonlinear second order differential equation related to vortex motion is derived. This equation is analyzed using various numerical and analytical techniques including finding approximate solutions using a perturbative approach.

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Daniel Duda Visualization of selected flows of superfluid helium using solid hydrogen tracer particles 5 Thesis title: Visualization of selected flows of superfluid helium using solid hydrogen tracer particles Author: Bc. Daniel Duda Department: Department of Low Temperature Physics, Supervisor: prof. RNDr. Ladislav Skrbek, DrSc, Department of Low Temperature Physics, Faculty of Mathematics and Physics, Charles University in Prague. Consultant: Dr. Marco La Mantia, PhD. Abstract: Quantum turbulence generated in thermal counterflow of He II is studied experimentally by visualization. The statistical properties of the motion of micron size solid deuterium particles are studied by using the particle tracking velocimetry technique at length scales comparable to the mean distance between quantized vorti- ces. The probability density function (PDF) of the longitudinal velocity displays two peaks that correspond to two velocity fields of the two-fluid description of He II. The PDF of the transversal velocity displays a classical-like Gaussian core with non- classical power-law tails, confirming the quantum nature of turbulence in counter- flowing He II. The distribution of the particle acceleration is found to be similar in shape to the classical one, in the range of investigated parameters. The observed de-...

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Cette thèse de doctorat à été réalisée au sein du Laboratoire des Écoulements Géophysiques et Industriels (LEGI) et du Service des Basses Températures du CEA de Grenoble. Ce travail expérimental a porté sur l'étude comparative de la turbulence classique et quantique à très grand nombre de Reynolds d'un écoulement turbulent de jet d'hélium liquide normal (HeI) et superfluide (HeII) entre 2.3K et 1.78K. Le travail s’est en premier lieu concentré sur le développement des moyens d'essais (une soufflerie cryogénique à boucle fermée pressurisée et régulée en température) et sur l'adaptation de la technique de l'anémomètrie à fil chaud aux basses températures. L’étude s’est poursuivie par l'analyse statistique du champ de vitesse en He I et, plus particulièrement, des incréments spatiaux de vitesse en fluide normal montrant un bon accord avec la littérature et fournissant un véritable point de départ pour la mise en évidence de comportements différents en HeII. Les résultats obtenus en superfluide montrent d'une part un comportement classique à grande échelle et, d’autre part, des écarts aux petites échelles qui dépendent de la température du fluide (i.e. de la fraction variable de superfluide). L'effet le plus évident se manifeste par un changement du signe de la fonction de structure d'ordre 3 des incréments de vitesse. / This experimental work is focused on the the statistical study of the high Reynolds number turbulent velocity field in an inertially driven liquid helium axis-symmetric round jet at temperatures above and below the lambda transition (between 2.3 K and 1.78 K) in a cryogenic wind tunnel. The possibility to finely tune the fluid temperature allows us to perform a comparative study of the quantum HeII turbulence within the classical framework of the Kolmogorov turbulent cascade in order to have a better understanding of the energy cascade process in a superfluid. In particular we focused our attention on the intermittency phenomena, in both He I and He II phases, by measuring the high order statistics of the longitudinal velocity increments by means of the flatness and the skewness statistical estimators.A first phase consisted in developing the cryogenic facility, a closed loop pressurized and temperature regulated wind tunnel, and adapting the classic hot-wire anemometry technique in order to be able to work in such a challenging low temperature environment. A detailed calibration procedure of the fully developed turbulent flow was the carried out at 2.3 K at Reynolds numbers based on the Taylor length scale up to 2600 in order to qualify our testing set-up and to identify possible facility-related spurious phenomena. This procedure showed that the statistical properties of the longitudinal velocity increments are in good agreement with respect to previous results.By further reducing the temperature of the working fluid (at a constant pressure) below the lambda point down to 1.78K local velocity measurements were performed at different superfluid density fractions. The results show a classic behaviour of the He II energy cascade at large scales while, at smaller scales, a deviation has been observed. The occurence of this phenomenon, which requires further investigation and modelling, is highlighted by the observed changing sign of the third order structure function of the longitudinal velocity increments. The intermittency phenomena is also observed and a quantitative analysis is carried out by measuring the scaling behaviour of the velocity increments flatness which is consistent with results obtained in Navier-Stokes fluids. This Ph.D. thesis has been carried out at the LEGI (Laboratoire des Écoulement Géophysiques et Industriels) laboratory in Grenoble and at the CEA Low Temperature Department (SBT) in Grenoble.

Cette thèse a pour objectif l’étude comparée de la turbulence isotherme en hélium I (HeI, fluide classique) et en hélium II (HeII, dont une partie du fluide est inviscide) par une analyse lagrangienne. Les analogies et/ou différences du comportement dynamique en écoulement classique ou superfluide devraient ainsi nous renseigner sur les caractéristiques multi-échelles intrinsèques de la turbulence. Ce type de recherche, qui constituait une première mondiale en début de thèse, semble promis à un bel avenir (à titre d'exemple, des mesures similaires de suivi lagrangien en HeI/HeII, en aval d’un barreau oscillant, sont actuellement en cours à Charles Université, Prague).Nous avons opté pour l’écoulement canonique de grille oscillante car il est isotrope, sans vitesse moyenne (autorisant ainsi un temps d’observation long) et homogène par plan. Nous basons nos mesures sur la visualisation de microsphères creuses de verre, dont la densité est voisine de celle de l’hélium liquide. Cette expérience étant nouvelle, il a fallu procéder à sa conception et à son dimensionnement, à sa mise en place ainsi qu'à sa calibration. Le cryostat que l’on a conçu est entièrement en verre, permettant ainsi une observation multi-angles de l’écoulement. Le système de visualisation utilisé est basé sur l'imagerie haute cadence en éclairage "backlight" (diffusion avant). A partir de la reconstruction des trajectoires des particules, nous avons pu réaliser des analyses lagrangiennes à une particule (à un temps et à deux temps), mais aussi étudier la dispersion relative de paires de particules.La résolution spatiale et temporelle de nos mesures nous a permis de pleinement caractériser les échelles inertielles de la turbulence, tandis que les échelles sont plus marginalement résolues. Dans ces conditions, nous trouvons que les caractéristiques de la turbulence en HeI sont en accord avec les mesures de la littérature dans des écoulements similaires en fluide classique. Ce même comportement est également observé en HeII. / This thesis aims at comparing the isothermal turbulence in Helium I (HeI, classical fluid) and in Helium II (HeII, which has one inviscid component) through a Lagrangian analysis. Analogies and/or differences between the classical and superfluid behaviors are expected to shed new light on the intrinsic multi-scale properties of turbulence. This is a pioneering study, as no similar cryogenic experiment had been performed prior to this thesis, which opens a full range of possible future studies (as an example, similar measurements of Lagrangian tracking in the wake of an oscillating cylinder, are currently in progress at Charles University in Prague). We opted for an oscillating grid turbulence generation, which produces isotropic and homogeneous (per plane) turbulence, with no mean velocity (hence allowing longer observation times). As tracers we use hollow glass microspheres, which are almost neutrally buoyant in liquid Helium. This being a new facility, significant effort was dedicated to the design and the tailoring of the experiment, its implementation and the calibration of the flow. The cryostat hosting the experiment is fully transparent (made of glass), allowing visualization at multiple angles. Measurements were performed using high speed imaging in backlight illumination. Particle trajectories were reconstructed using Lagrangian tracking, from which we performed analysis of single particle statistics (single time and two times) as well as relative dispersion of pairs of particles. The spatial and temporal resolution of our measurements give us access to the dynamics of the flow at inertial scales, while dissipative scales are marginally resolved. In these conditions, we find that turbulence in HeI behaves in agreement with previous results obtained in similar flows using classical fluids. Interestingly, the same turbulence properties stands also for superfluid conditions.