Note on the method of matched-asymptotic expansions for determining the force acting on a particle

This paper is an addendum to the article by Candelier, Mehaddi & Vauquelin (2013) where the motion of a particle in a stratified fluid is investigated theoretically, at small Reynolds and P\'eclet numbers. We review briefly the method of matched asymptotic expansions which is generally used in order to determine the force acting on a particle embedded in a given flow, in order to account for small, but finite, inertia effects. As part of this method, we present an alternative matching procedure, which is based on a series expansion of the far-field solution of the problem, performed in the sense of generalized functions. The way to perform such a series is presented succinctly and a simple example is provided.Comment: 8 page

The history force on a small particle in a linearly stratified fluid

The hydrodynamic force experienced by a small spherical particle undergoing an arbitrary time-dependent motion in a density-stratified fluid is investigated theoretically. The study is carried out under the Oberbeck-Boussinesq approximation, and in the limit of small Reynolds and small P\'eclet numbers. The force acting on the particle is obtained by using matched asymptotic expansions in which the small parameter is given by a/l where a is the particle radius and l is the stratification length defined by Ardekani & Stocker (2010), which depends on the Brunt-Vaisala frequency, on the fluid kinematic viscosity and on the thermal or the concentration diffusivity (depending on the case considered). The matching procedure used here, which is based on series expansions of generalized functions, slightly differs from that generally used in similar problems. In addition to the classical Stokes drag, it is found the particle experiences a memory force given by two convolution products, one of which involves, as usual, the particle acceleration and the other one, the particle velocity. Owing to the stratification, the transient behaviour of this memory force, in response to an abrupt motion, consists of an initial fast decrease followed by a damped oscillation with an angular-frequency corresponding to the Brunt-Vaisala frequency. The perturbation force eventually tends to a constant which provides us with correction terms that should be added to the Stokes drag to accurately predict the settling time of a particle in a diffusive stratified-fluid.Comment: 16 page

Hauteur de fontaine retombantes

Une fontaine est un jet dont la flottabilité agit dans le sens inverse de sa quantité de mouvement. Ce type d’écoulement se rencontre dans divers processus industriels (dispersion atmosphérique, rejets de polluant dans la mer, etc) et environnementaux (cumulonimbus, fumeur noir, éruptions volcaniques, etc). La configuration à laquelle nous nous intéressons est celle d’un rejet lourd vertical dans un milieu de masse volumique homogène. La hauteur du rejet évolue durant sa phase transitoire d’une hauteur maximale vers une hauteur stabilisée atteinte en régime établi. Turner (JFM 1966) a montré que pour de grandes valeurs du nombre de Froude à l’injection, le rapport entre la hauteur finale et la hauteur stabilisée est constant et vaut lambda = 1,43. Des travaux récents contredisent cette valeur pour de faibles nombres de Froude. Dans ce travail, nous établissons à partir du ”confined top-hat model” de Carazzo, Kaminski et Tait (JFM 2010) une équation différentielle régissant la hauteur stabilisée de la fontaine. Une solution rapprochée de cette équation est proposée et permet d’obtenir la dépendance du rapport avec le nombre de Froude. Enfin, une comparaison est faite avec les résultats expérimentaux de Burridge et Hunt (JFM 2012). Un bon accord est constaté, en tout cas pour Fr > 1

Turbulent miscible fountains

Une fontaine peut se créer quand la flottabilité d'un rejet vertical s'oppose à sa quantité de mouvement. Ce type d'écoulement connaît beaucoup d'applications que ce soit dans la nature (panaches issus des éruptions volcaniques) ainsi que dans l'industrie du bâtiment (chauffage et refroidissement) ou dans le domaine des risques (rejets accidentel de gaz lourd). Dans cette thèse, nous nous focalisons sur l'étude des fontaines turbulentes miscibles. Dans le premier chapitre nous reformulons le modèle théorique de Morton et al. (1956) pour traiter le cas des fontaines en milieu linéairement stratifié. La résolution de ce modèle permet d'obtenir des relations analytiques pour la hauteur de la fontaine et sa hauteur d'étalement. Ce modèle est, par la suite, étendu au cas des panaches et des jets turbulents en milieu linéairement stratifié. Dans le second chapitre, nous proposons un modèle théorique permettant d'étudier une fontaine turbulente miscible en régime établi. Pour calibrer ce modèle, des simulations numériques aux grandes échelles (LES) sont utilisées pour obtenir une estimation des valeurs des constantes associées aux phénomènes d'échanges turbulents entre les parties ascendante et descendante de la fontaine. L'objectif du dernier chapitre est d'apporter, à partir d'expérimentations en laboratoire, des informations quantitatives sur l'influence de forts écarts de masses volumiques dans les écoulements de type fontaine. Les expériences sont réalisées pour des fontaines gazeuses (mélange air/hélium) en régime établi.A fountain can occur when the buoyancy of a vertically released fluid opposes its momentum. Such flows have many applications in nature (plumes issuing from volcanic eruption), building industry (cooling or heating) or in the area of risk management (accidental release of heavy dangerous gas). In this thesis, we focus on the study of miscible turbulent fountains. In the first chapter, we revisit the theoretical model of Morton et al. (1956) to handle the case of fountains in linearly stratified fluid. The resolution of this model allows us to obtain analytical relations for the fountain height as well as the spreading height of its horizontal layer. This model is subsequently extended to the case of turbulent jets and plumes in linearly stratified fluid. In the second chapter, we propose a theoretical model for the study of a turbulent miscible fountain in a steady state. To calibrate this model, large eddy simulations (LES) are used to obtain an estimate of the values of the constants associated with the additional terms appearing in the equations. The objective of the final chapter is to provide, from laboratory experiments, quantitative information on the influence of strong density differences on the behaviour of a turbulent fountain. These experiments shows that all the classical relations valid for the Boussinesq case can be extended to the non-Boussinesq case by using an appropriate definition of the Froude number

Analytical solutions for turbulent Boussinesq fountains in a linearly stratified environment

International audienceThis paper theoretically investigates the initial up-flow of a vertical turbulent fountain (round or plane) in a linearly stratified environment. Conservation equations (volume, momentum and buoyancy) are written under the Boussinesq approximation assuming an entrainment proportional to the vertical velocity of the fountain. Analytical integration leads to exact values of both density and flow rate at the maximal height reached by the fountain. This maximal height is expressed as a function of the release conditions and the stratification strength and plotted from a numerical integration in order to exhibit overall behaviour. Then, analytical expressions for the maximal height are derived from asymptotic analysis and compared to experimental correlations available for forced fountains. For weak fountains, these analytical expressions constitute a new theoretical model. Finally, modified expressions are also proposed in the singular case of an initially non-buoyant vertical release

Naturally bounded plumes

International audienceThis paper investigates theoretically the vertical evolution of a turbulent plume into a linearly stratified ambient fluid, by regarding it as composed of two distinct regions. In the first region, called the positive buoyant region, the plume buoyancy and the plume momentum act in the same upward direction, whereas in the second region, called the negative buoyant region, they act in opposite directions. In a first step, analytical expressions for the plume variables at the transition height (i.e. between the two regions) are obtained from one-dimensional conservation equations, using the plume entrainment model and under the Boussinesq approximation. In a second step, these variables are used in order to determine analytically the buoyancy and volume fluxes as well as the density deficit of the plume at its top. In this investigation, the transition height (denoted z(t)) and the total plume height (denoted z(p)) are obtained in the form of two integrals. These integrals are evaluated asymptotically in three different cases associated with particular flow regimes. Finally, the limit of the Boussinesq assumption for such flows is discussed

Experimental non-Boussinesq fountains

International audienceLaboratory experiments involving downward air-helium fountains are presented. The large density differences between these releases and the ambient allow us to investigate how non-Boussinesq effects modify fountain heights and fountain fluctuations in comparison with the Boussinesq case (i.e. marginal density differences). In these experiments, the source Froude number is varied over a wide range covering (i) the very weak, (ii) the weak and (iii) the forced fountain regimes. It is shown that the classical Boussinesq correlations can be extended to the non-Boussinesq case provided that the Froude number is multiplied by the square root of the ratio between the released fluid density and that of the ambient. In the range investigated, no influence of the source Reynolds number is observed

Second-order inertial forces and torques on a sphere in a viscous steady linear flow

39 pages, 3 figuresWe compute the full set of second-order inertial corrections to the instantaneous force and torque acting on a small spherical rigid particle moving unsteadily in a general steady linear flow. This is achieved by using matched asymptotic expansions and formulating the problem in a coordinate system co-moving with the background flow. Effects of the fluid-velocity gradients are assumed to be small, but to dominate over those of the velocity difference between the body and fluid, which makes the results essentially relevant to nearly neutrally buoyant particles. The outer solution (which at first order is responsible for the Basset-Boussinesq history force at short time and for shear-induced forces such as the Saffman lift force at long time) is expressed via a flow-dependent tensorial kernel. The second-order inner solution brings a number of different contributions to the force and torque. Some are proportional to the relative translational or angular acceleration between the particle and fluid, while others take the form of products of the rotation/strain rate of the background flow and the relative translational or angular velocity between the particle and fluid. Adding the outer and inner contributions, the known added-mass force or the spin-induced lift force are recovered, and new effects involving the rotation/strain rate of the background flow are revealed. The resulting force and torque equations provide a rational extension of the classical Basset-Boussinesq-Oseen equation incorporating all first- and second-order fluid inertia effects resulting from both unsteadiness and velocity gradients of the carrying flow

Unstable modes of laminar round fountains on inclined wall

International audienceExperiments were carried out on reversed weak laminar inclined fountains to asses that unstable modes of round fountains are disturbed by the inclination. Indeed, compared to fountains developing on horizontal wall, some modes disappeared while others are split in several modes. This paper aims at describing and mapping these new modes regarding to the inclination and the inlet velocity. Explanations about what made the unstable modes evolve are also proposed. (C) 2011 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved