The motion of an axisymmetric body falling in a tube at moderate Reynolds numbers

This study concerns the rectilinear and periodic paths of an axisymmetric solid body (short-length cylinder and disk of diameter d and thickness h) falling in a vertical tube of diameter D. We investigated experimentally the influence of the confinement ratio (S=d/D<0.8) on the motion of the body, for different aspect ratios (χ=d/h=3, 6 and 10), Reynolds numbers (80<Re<320) and a density ratio between the fluid and the body close to unity. For a given body, the Reynolds number based on its mean vertical velocity is observed to decrease when S increases. The critical Reynolds number for the onset of the periodic motion decreases with S in the case of thin bodies (χ=10), whereas it appears unaffected by S for thicker bodies (χ=3 and 6). The characteristics of the periodic motion are also strongly modified by the confinement ratio. A thick body (χ=3) tends to go back to a rectilinear path when S increases, while a thin body (χ=10) displays oscillations of growing amplitude with S until it touches the tube (at about S=0.5). For a given aspect ratio, however, the amplitudes of the oscillations follow a unique curve for all S, which depends only on the relative distance of the Reynolds number to the threshold of path instability. In parallel, numerical simulations of the wake of a body held fixed in a uniform confined flow were carried out. The simulations allowed us to determine in this configuration the effect of the confinement ratio on the thresholds for wake instability (loss of axial symmetry at Rec₁ and loss of stationarity at Rec₂) and on the maximal velocity Vw in the recirculating region of the stationary axisymmetric wake. The evolution with χ and S of Vw at Rec₁ was used to define a Reynolds number Re*. Remarkably, for a freely moving body, Re* remains almost constant when S varies, regardless of the nature of the path

Sampling from a log-concave distribution with compact support with proximal Langevin Monte Carlo

This paper presents a detailed theoretical analysis of the Langevin Monte Carlo sampling algorithm recently introduced in Durmus et al. (Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau, 2016) when applied to log-concave probability distributions that are restricted to a convex body $\mathsf{K}$. This method relies on a regularisation procedure involving the Moreau-Yosida envelope of the indicator function associated with $\mathsf{K}$. Explicit convergence bounds in total variation norm and in Wasserstein distance of order $1$ are established. In particular, we show that the complexity of this algorithm given a first order oracle is polynomial in the dimension of the state space. Finally, some numerical experiments are presented to compare our method with competing MCMC approaches from the literature

The periodic motion of a disk freely falling in a tube

This study is devoted to the rectilinear and periodic paths of an axisymmetric solid body (short-length cylinder and disk of diameter d, thickness h and aspect ratio X= d/h varying between 3 and 10) falling in a vertical tube of diameter D. We investigated the influence of the confinement ratio (S = d/D) on the characteristics of the body motion, for different aspect ratios X and Reynolds numbers (80 < Re < 320), and a density ratio between the fluid and the body close to unity

Interaction of two axisymmetric bodies falling in tandem at moderate Reynolds numbers

This study considers the interaction of two identical solid axisymmetric bodies (of diameter d and thickness h) freely falling in a fluid at rest. We determine the domains of existence of the different interaction behaviour of the two bodies (i.e. attraction, repulsion and indifference) as a function of their initial relative position. We then investigate in detail the case of bodies falling in tandem, for both rectilinear and periodic paths, and the associated attraction behaviour. For all the Reynolds numbers and aspect ratios of the bodies (χ=d/h) investigated, the trailing body catches up with the leading body. We provide a quantitative description of the kinematics leading to the regrouping of the bodies and analyse its relationship with the wake of the leading body. In the case of rectilinear paths, a dynamical model that takes into account the axial evolution of the wake of the leading body is proposed to reproduce the acceleration observed for the trailing body until a vertical separation distance between the bodies of 1.5 diameters. In parallel, direct numerical simulations (DNS) of the flow about two fixed bodies in tandem in an oncoming flow are carried out, providing a good estimation of the motion of the bodies for separation distances larger than 5 diameters. For periodic paths, the kinematics leading to the regrouping of the bodies is slower than for rectilinear paths. However, in this case, the interaction also leads to significant changes in the characteristics of the oscillatory motion and is strongly dependent on the aspect ratio of the bodies. To explain the observed differences, we consider the effect of the transverse inhomogeneity of the wake of the leading body on the oscillatory motion of the trailing disk

Interaction of two axisymmetric bodies falling side by side at moderate Reynolds numbers

We consider the interaction of two identical disks freely falling side by side in a fluid at rest for Reynolds numbers ranging from 100 to 300, corresponding to rectilinear and oscillatory paths. For the three aspect ratios of the disks investigated, we observed that the bodies always repel one another when the horizontal distance between their centres of gravity is less than 4.5 diameters. They never come closer for distances spanning between 4.5 and 6 diameters. Beyond the latter distance, the disks appear indifferent to each other. For both rectilinear and periodic paths, the repulsion effect is weak, leading to an overall horizontal drift lower than 3 % of the vertical displacement. We propose a model for the repulsion coefficient Cr, which decreases with the separation distance between the bodies and is inversely proportional to the aspect ratio of the bodies, Cr thus being stronger for the thicker ones. Furthermore, in the case of the oscillatory paths, we show that the effect of the interaction reduces to the repulsion effect, since the characteristics of the oscillatory motion of each disk appear unaffected by the presence of the companion disk and no synchronization is observed between the paths, nor between the wakes, of the two disks

Trajectoire et sillage d'un corps en chute libre en interaction avec un autre corps ou en présence d'un confinement

Un corps en chute libre dans un fluide sous l'effet de la gravité peut être soumis à des perturbations. De façon générale, celles-ci peuvent être dues aux mouvements propres du fluide porteur, à la présence d'autres corps mobiles ou encore à la présence de parois. Dans un premier temps, nous avons choisi de nous intéresser à l'interaction de deux disques identiques chutant dans un fluide de densité proche de celle du corps. Différents comportements d'interaction sont observés pour des disques de rapports de forme variable (diamètre sur épaisseur) et des nombres de Reynolds (effets inertiel sur effets visqueux) couvrant des trajectoires rectilignes et périodiques oscillantes. Lorsque les disques sont lâchés en tandem, ils s'attirent et se rencontrent. Après le contact, le comportement dépend du rapport de forme : les disques épais se séparent et tombent côte à côte, tandis que les disques minces continuent leur évolution ensemble dans une configuration relative stable. Lorsque les corps sont lâchés côte à côte, on observe une répulsion des corps qui se traduit par un éloignement horizontal. Une modélisation pour chacun de ces comportements (attraction, répulsion) a été proposée. Dans un deuxième temps, nous avons étudié l’effet de parois fixes sur le mouvement d'un disque isolé en chute libre. Les disques sont lâchés dans des tubes cylindriques créant différents rapports de confinement (diamètre du disque sur diamètre du tube). Nous avons mis en évidence que le comportement du corps dépendait du rapport de forme : la trajectoire d'un disque mince est déstabilisée par le confinement, alors que celle d'un disque épais est stabilisée. Des visualisations des sillages à l'aide de colorants ainsi que des simulations numériques de l'écoulement autour de disques fixes ont été réalisées et ont permis de mieux comprendre le rôle du sillage sur les interactions. ABSTRACT : A body falling in a fluid under the effect of gravity may be perturbed by the presence of other bodies or fixed boundaries. We first focused our attention on the interaction of two identical disks in freefall. We investigated the kinematics of disks with different aspect ratios (ratio of diameter to thickness) and with different initial relative positions, for a range of Reynolds numbers (ratio of inertial effects to viscous effects) covering both rectilinear trajectories and periodic, oscillating trajectories. When the disks are falling in tandem, the trailing body accelerates until it catches up the leading one. After the contact, the behaviour depends on the aspect ratio. Thicker disks separate and fall side by side and separated while thinner disks continue their fall together in Y or T reversed position. A model of the different types of interaction (entrainment by the wake and horizontal repulsion) is proposed. The second part of the study is devoted to the effect of fixed walls on the kinematics of freely falling bodies. The behaviour again depends strongly on the aspect ratio : a thin disk is destabilized when the blockage ratio (ratio of disk diameter to tube diameter) increases, whereas thicks disks are stabilized. In addition, visualizations of the wakes using dyes and numerical simulations for fixed disks were performed and provide a better understanding of the role of the wake in the interaction

Interaction de deux corps identiques chutant librement l’un à la suite de l’autre de façon rectiligne

Cette étude expérimentale porte sur l’interaction de deux corps identiques en chute libre rectiligne sous l’effet de la flottabilité. Lorsqu’un corps suit un autre, un phénomène d’entraînement gouverné par le sillage du corps de tête se produit. Le corps suiveur a alors tendance à accélérer et à rattraper le premier corps. Des résultats sur la cinématique de disques pour ce type d’interaction ont été obtenus pour un nombre de Reynolds de l’ordre de 100 et différents rapports de forme

Di-tert-butyl 2-benzoyl­hydrazine-1,1-dicarboxyl­ate

The crystal structure of the title compound, C17H24N2O5, was determined in the course of our studies on the preparation of two families of pseudopeptides, viz. hydrazino- and N-amino- peptides. The most significant inter­action in the crystal structure is a bifurcated inter­molecular N—H⋯O hydrogen bond

Depolymerization of fractionated wood by hydrothermal liquefaction

Direct thermochemical conversion of lignocellulosic biomass produces a mixture of compounds that have to be separated to produce purified building blocks. Moreover, lignin derived products have a detrimental effect on further biological conversion processes, such as fermentation. For all these reasons, it is important to develop an integrated approach for a better fractionation and valorisation of macromolecules (carbohydrates and lignin) in bio-refineries. Please click Additional Files below to see the full abstract