Colliding Dipolar Vortices in a Stratified Fluid

International audienceThe photographs illustrate experiments that have been performed on the collapse of a three-dimensional turbulent patch in a linearly stratified fluid. The turbulence was gener­ated by horizontal injection of a small volume of fluid during a short time interval. A transition to two-dimensional flow occurs when the turbulent patch collapses under gravity, as can be observed from the spectral flux of kinetic energy to larger scales. The collapsed fluid eventually gets organized in a dipolar flow structure that moves slowly forward along a straight line. The robustness of this dipolar coherent struc­ture is demonstrated in experiments on head-on collisions of two dipoles with approximately identical characteristics. Consecutive stages of a head-on collision are shown by the (plan view) photographs, taken (a) 42 sec, (b) 70 sec, (c) 120 sec, and (d) 225 sec after the injections were stopped. The asymmetry in the observed flow patterns is due to a slight misalignment of the initial dipoles. Nevertheless, the experiment shows nicely that the original dipoles exchange partners, and that two new dipoles emerge, moving along straight lines away from the collision area. Further experi­mental details are described elsewhere

The role of Stewartson and Ekman layers in turbulent rotating Rayleigh-B\'enard convection

When the classical Rayleigh-B\'enard (RB) system is rotated about its vertical axis roughly three regimes can be identified. In regime I (weak rotation) the large scale circulation (LSC) is the dominant feature of the flow. In regime II (moderate rotation) the LSC is replaced by vertically aligned vortices. Regime III (strong rotation) is characterized by suppression of the vertical velocity fluctuations. Using results from experiments and direct numerical simulations of RB convection for a cell with a diameter-to-height aspect ratio equal to one at $Ra \sim 10^8-10^9$ ($Pr=4-6$) and $0 \lesssim 1/Ro \lesssim 25$ we identified the characteristics of the azimuthal temperature profiles at the sidewall in the different regimes. In regime I the azimuthal wall temperature profile shows a cosine shape and a vertical temperature gradient due to plumes that travel with the LSC close to the sidewall. In regime II and III this cosine profile disappears, but the vertical wall temperature gradient is still observed. It turns out that the vertical wall temperature gradient in regimes II and III has a different origin than that observed in regime I. It is caused by boundary layer dynamics characteristic for rotating flows, which drives a secondary flow that transports hot fluid up the sidewall in the lower part of the container and cold fluid downwards along the sidewall in the top part.Comment: 21 pages, 12 figure

Effects of rotation and stratification: an introduction

Large-scale flows in the natural environment can be influenced by the planetary rotation and also by density differences. This chapter aims to provide an informal introduction into the effects of background rotation and stratification

Structures de vortex cohérentes en fluide stratifié

International audienceTwo-dimensional turbulent flows have the characteristic property to organize into large, coherent vortex structures. Due to their slow dissipation such coherent structures dominate to a large extent the flow evolution. Although no flow is ever purely two-dimensional, flows that are close to it show similar features. Large scale geophysical flows are quasi two-dimensional due to back-ground rotation, stratification and the geometry of the flow domain. During the past decades it has been shown in particular by satellite imagery that both the Earth's atmosphere and oceans contain coherent vortices on a range of horizontal scales. Also in other fields of physical interest, such as plasma physics and astrophysics, the dynamics of two-dimensional flows and coherent vortex structures play an important role.Initially motivated by their relevance to geophysical flow systems, the characteristics of coherent vortices as well as the organization properties of flows in a stratified fluid were investigated experimentally by using flow visualization and digital image processing techniques. Additionally, certain flow features were simulated numerically by rise of a numerical code for purely two-dimensional flows.In stratified fluids, vertical motions are suppressed by the buoyancy force. In such fluids a local perturbation (such as e.g. a pulsed horizontal jet) leads to a small mixed region, that after a gravitational collapse of vertically mixed fluid, becomes approximately two-dimensional. As a result of the self-organizing properties .of such flows, eventually, a large (horizontal) vortex emerges that in the case of a stratified fluid is confined to a thin layer of fluid. The inviscid dynamics of such planar vortex structures is shown to be in good approximation two- dimensional. The decay of the structures is mainly due to vertical diffusion of vorticity, and thereby its decay is three-dimensional. The dynamics of dipolar vortices, monopolar vortex structures and some higher-mode structures, like the tripolar vortex and the triangular vortex, is discussed.The Lamb-Chaplygin dipole model is found to describe the observed dipolar structures quite well. By two analytical models, based on the Lamb-Chaplygin model, the viscous decay of dipolar structures is described. Both models showed good comparison with the experimental data. Interaction properties of a dipolar vortex with a solid circular cylinder showed that after colliding with the cylinder, two asymmetric dipoles form, both of which can be characterized by a linear relation between vorticity (w) and stream function.Monopolar vortices in stratified fluids show, in contrast to their counterpart in non- stratified rotating fluids (see Kloosterziel & van Heijst, 1991), a tendency to increase their stability: during their evolution, the vortices advect ambient (irrotational) fluid under the in- teraction with internal waves and obtain a less steep velocity profile which corresponds with a higher stability. The Leith (1984) minimum enstrophy vortices were found to give a reasonable description of these stable monopoles.In certain cases an unstable monopolar vortex may give rise to the formation of a tripolar vortex or a triangular vortex. It is found that planar multipoled structures with net angular momentum retransform eventually to a monopolar vortex. Due to vertical diffusion of vorticity, the thinner edges of the structure decay relatively fast, which generally leads to the deformation and disappearance of the satellite vortices. This effect is amplified by the lateral expansion of the structure due to entrainment. The triangular and the tripolar vortex structures were never observed before in a non-rotating stratified fluid and therefore their dynamical characteristics have been investigated into detail. Because for both vortices simple analytical model with a distributed vorticity are lacking, the vortex structures are compared with point-vortex models that are found to describe some of their characteristics nicely well.Besides vortex structures in isolation, the self-organization of a two-dimensional turbulent flows on a finite domain with solid lateral boundaries are studied. The flow evolution to a quasi-final equilibrium state is determined both by the initial vorticity distribution and the boundary geometry. On a rectangular domain a cellular vortex pattern was observed to form, with generally the number of cells equal to the aspect ratio of the domain. Interactions of the vortex cells with the wall and with neighbouring cell regions induced oscillatory drift motions of the individual cells. These drift motions caused the exchange of fluid between different cell regions as well as the chaotic mixing of the outer cell regions and thereby influenced the flow evolution to an ultimate state. Single vortices, as arising in square and circular tanks, were found to have different 'initial' vorticity distributions for two different types of forcings used here (i.e. a random-forcing or a well defined localized forcing by which a global flow was generated). When advection of vorticity that is generated along the walls was relatively large, the flow evolved to a. vortex with its core characterized by a linear w, '4'-relation. In the quasi- final equilibrium state, the individual vortex cells were in agreement with the isolated Leith (1984) vortex that was found for minimum enstrophy. The flows on a bounded domain, as well as the ultimate flow states of isolated coherent vortices, were in agreement with different available maximum-entropy theories. However, the measurements did not show a significantly better correspondance with one of the available theoretical descriptions in particular.xczTweedimensionale turbulente stromingen hebben de karakterisitieke eigenschap zich te orga- niseren in grote coherente wervelstructuren. Deze coherente structuren bepalen vanwege hun lange levensduur in grote mate de verdere evolutie van de stroming. Hoewel in de natuur geen enkele stroming puur tweedimensionaal is, zijn er veel stromingen die quasi-tweedimensionaal zijn doordat de bewegingen in een bepaalde richting onderdrukt worden. Zo zijn grootschalige geofysische stromingen in benadering tweedimensionaal onder invloed van de aardrotatie, stratificatie en de geometrie van de stroming. De afgelopen decennia hebben o.a. satellietbeelden een scala aan grootschalige coherente structuren in de atmosfeer en de oceanen getoond. Ook in andere takken van de fysica, zoals de plasma-fysica en de astrofysica, blijken coherente structuren en de dynamica van tweedimensionale stromingen een belangrijke rol te spelen.Dit promotie onderzoek is hoofdzakelijk gemotiveerd vanuit de geofysische sfeer. De karak- teristieken van coherente structuren alsmede de organisatie-eigenschappen van stromingen in een gestratificeerde vloeistof zijn experimented onderzocht. Er is gebruik gemaakt van visualisatie-technieken en digitale beeldverwerkingsmethoden. Ter vergelijking met de exper- imentele resultaten zijn enkele stromingen numeriek gesimuleerd.In gestratificeerde vloeistoffen worden verticale bewegingen onderdrukt door de Archimedes kracht. Een locale verstoring (bijvoorbeeld door een gepulseerde horizontale jet) leidt tot een klein doorgemengd vloeistofvolume, dat na enige tijd onder invloed van de zwaartekracht inklapt en resulteert in een quasi-tweedimensionale stroming. Als gevolg van dezelf-organisatie in dit soort stromingen ontstaat een grote (horizontale) wervelstructuur die zich in een dun vloeistoflaagje bevindt. Uit een schalingsanalyse blijkt dat de niet-visceuze dynamica van deze vlakke structuren in goede benadering tweedimensionaal is. Het verval van deze structuren heeft echter een driedimensionaal karakter en wordt hoofdzakelijk door verticale diffusie van vorticiteit bepaald. De dynamica van dipolaire wervelstructuren, monopolaire wervels en enkele hogere mode structuren zoals tripolaire wervels en driehoekswervels is onderzocht.Dipolen zijn vergeleken met het Lamb-Chaplygin dipool model dat een goede beschrijving van de waargenomen structuren blijkt te geven. Het visceuze verval van deze dipolen is beschreven aan de hand van twee analytische modellen die beide gebaseerd zijn op dit Lamb- Chaplygin dipoolmodel. De overeenkomst van de modellen met de experimentele data geeft aan dat diffusie van vorticiteit in verticale richting in belangrijke mate het verval van de dipolen bepaalt. Interacties van een dipool met een vaste cirkelvormige cilinder laten zien dat zich na de interactie twee nieuwe asymmetrische dipolen vormen die beide door een lineaire relatie tussen vorticiteit (omega) en stroomfunktie (psi) worden gekarakteriseerd.Monopolaire wervels in gestratificeerde vloeistoffen hebben -in tegenstelling tot de monopolen in roterende homogene vloeistoffen zoals waargenomen door Kloosterziel & van Heijst, 1991- de neiging steeds stabieler te worden. Onder invloed van interacties met interne golven vervormt de cirkelvormige wervelstructuur en vindt er advectie van (irrotationele) omgev- ingsvloeistof plaats. Daardoor wordt het verloop van de tangentiele snelheid met de straal minder steil en neemt de stabiliteit van de wervel toe. Het Leith minimum enstrofie model geeft een redelijke beschrijving van deze stabiele monopolen.In sommige gevallen leiden onstabiele monopolen tot de formatie van een tripolaire wervel of driehoekswervel. Uit de experimenten blijkt in het algemeen dat deze multipoolstructuren uiteindelijk overgaan in een monopool structuur. Als gevolg van verticale diffusie van vorticiteit vervallen de dunnere randen van deze multipoolstructuren eerder dan de kern, wat leidt tot de deformatie en het uiteindelijk verdwijnen van de satellietwervels. Dit effect wordt versterkt door de laterale expansie van de structuur als gevolg van entrainment.De tripolaire wervel en de driehoekswervel zijn niet eerder in een niet-roterende gestra- tificeerde vloeistof waargenomen en hun karakteristieken zijn uitvoerig onderzocht. Omdat tot nog toe geen eenvoudige modellen met een continue vorticiteitsverdeling bekend zijn voor deze wervelstructuren, zijn beide wervels met puntwervel modellen vergeleken. Deze puntwer- vel modellen beschrijven ondanks dat ze een sterke vereenvoudiging van de werkelijkheid zijn enkele karakteristieke eigenschappen van de wervels redelijk goed.Naast de evolutie van geisoleerde coherente structuren zijn quasi-tweedimensionale stro- mingen onderzocht die op het gehele domein zijn gegenereerd. De evolutie van deze stromingen naar een stationaire eindtoestand blijkt zowel door de soort forcering als door de geometric van het stromingsdomein te worden bepaald. Stromingen op een rechthoekig domein orga- niseren zich in een celvormig wervelpatroon, waarbij in het algemeen het aantal cellen gelijk is aan de lengte-breedte verhouding van het domein. De interacties van de wervels met de wand en tussen de wervelcellen onderling leiden tot oscillatie-achtige verschuivingen van de wervelcellen. Deze 'verschuivingen' veroorzaken uitwisseling van vloeistof tussen de cellen en chaotische menging van de buitenste celgebieden, en heinvloeden zo de evolutie naar een stationaire eindtoestand. Op een vierkant of cirkelvormig domein vormt zich na zelf-organisatie van de stroming een enkele wervelcel die een andere vorticiteitsverdeling heeft voor een ran- dom forcering dan voor een goed gedefinieerde locale forcering. Als gevolg van advectie van aan de wand gegenereerde vorticiteit verandert dit 'initiele' vorticiteitsprofiel. In alle gevallen waar advectie van vorticiteit relatief groot is, ontwikkelt de stroming zich naar een wervel waarvan de kern gekarakteriseerd wordt door een lineare w, ,P-relatic. In de stationaire eind- toestand komen de individuele wervel-cellen goed overeen met het Leith (1984) model voor geisoleerde minimum-enstrofie wervels. De meetresultaten van zowel deze stromingen op een eindig domein als die van de geisoleerde wervelstructuren kwamen overeen met de bestaande maximum-entropie theorieen maar waren niet significant genoeg om de voorkeur te kunnen geven aan een bepaalde maximum-entropie theorie

Spin-up of a source-sink flow over a model continental shelf

Results of analytical and experimental models are presented in which the role of various forms of bottom topography on externally driven continental shelf currents has been investigated. The shelf currents are generated in a relating cylindrical geometry by means of a source-sink technique. A linear analytical model for a homogeneous fluid in this configuration predicts that the azimuthal (swirl) velocity above a flat bottom is inversely proportional to the radial distance from the origin. This velocity profile is shown to be altered if the bottom boundary consists of a model continental shelf and slope. Then a geometrical function has to be included to describe the azimuthal velocity profile above the sloping bottom. This function depends only on the slope angle alpha and differs only significantly from unity for large values of alpha (alpha &gt; 30 degrees). As a result, a free Stewartson layer is generated above the shelf break to account for the azimuthal velocity shear between the two interior regions. The net vertical transport in this shear layer is again only important for large slope angles.Some aspects of the analytical model were verified in laboratory experiments on source-sink driven hows in both homogeneous and weakly linearly stratified fluids. The results show that the stratification was sufficiently weak not to have a significant effect on the dynamics in the interior regions. Reference experiments were carried out to measure the azimuthal velocity profiles above a flat bottom. Then, a part of the bottom profile was replaced by a slope with a slope angle of 25 degrees or 55 degrees. Comparison of the azimuthal velocity profiles of the 25 degrees slope with its equivalent reference case reveals no measurable difference, as predicted by the analytial model. However, with the 55 degrees slope, the difference between the interior regions above the slope and the flat bottom is significant and in quantitative agreement with the results of the analytical model. In addition to the analytical description of the steady state flow, the experiments also provided information on the spin-up phase of the flow. The experimentally obtained spin-up times confirm the theoretical results of Greenspan and Howard (1963) when the local fluid depth is taken into account.</p

Near-field pollutant dispersion in an actual urban area: Analysis of the mass transport mechanism by high-resolution Large Eddy Simulations

Large-Eddy Simulation of near-field pollutant dispersion from stacks on the roof of a low-rise building in downtown Montreal is performed. Two wind directions are considered, with different wind-flow patterns and plume behavior. The computed mean concentration field is analyzed by means of the convective and turbulent (including subgrid-scale) mass fluxes. This decomposition provides insight into the dispersion process and allows an evaluation of common turbulent transport models used with the Reynolds-Averaged Navier-Stokes approach, such as the standard gradient-diffusion hypothesis. Despite the specific character of the flow and dispersion patterns due to the complex geometry of the urban area under study, some similarities are found with the generic case of dispersion around an isolated simple building. Moreover, the analysis of dispersion in downtown Montreal is facilitated by the physical insight gained by the study of the generic case. In this sense, the present study supports the use of generic, simplified cases to investigate and understand environmental processes as they occur in real and more complex situations. Reciprocally, the results of this applied study show the influence on the dispersion process of the rooftop structures and of the orientation of the emitting building with respect to the incoming wind flow, providing directions for further research on generic cases.status: publishe

Vortex interactions with flapping wings and fins can be unpredictable

As they fly or swim, many animals generate a wake of vortices with their flapping fins and wings that reveals the dynamics of their locomotion. Previous studies have shown that the dynamic interaction of vortices in the wake with fins and wings can increase propulsive force. Here, we explore whether the dynamics of the vortex interactions could affect the predictability of propulsive forces. We studied the dynamics of the interactions between a symmetrically and periodically pitching and heaving foil and the vortices in its wake, in a soap-film tunnel. The phase-locked movie sequences reveal that abundant chaotic vortex-wake interactions occur at high Strouhal numbers. These high numbers are representative for the fins and wings of near-hovering animals. The chaotic wake limits the forecast horizon of the corresponding force and moment integrals. By contrast, we find periodic vortex wakes with an unlimited forecast horizon for the lower Strouhal numbers (0.2–0.4) at which many animals cruise. These findings suggest that swimming and flying animals could control the predictability of vortex-wake interactions, and the corresponding propulsive forces with their fins and wings

Stereoscopic-PIV study of a dipole in a shallow fluid layer

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