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Steady axisymmetric flow in an open cylindrical with a partially rotating bottom wall
Abstract and Figures
The steady motion of a viscous fluid in a cylindrical container with a partially rotating bottom wall and a free surface is investigated by means of axisymmetric Navier–Stokes simulations. The flow above the spinning disk at the center of the bottom wall is dominated by an Ekman boundary layer that drives the fluid radially outward. In contrast, an inward flow ensues along the outer, stationary part of the bottom wall, where the radially increasing pressure distribution set up by the rotating fluid motion near the free surface is not balanced by a corresponding centrifugal force. As a result, flow separation occurs at an intermediate radial location close to the outer edge of the rotating disk. Thus a flow configuration results that is dominated by a meridional vortex above the spinning disk, and a counterrotating vortex above the stationary part of the bottom wall. Simulations are conducted for various aspect ratios and Reynolds numbers, in order to evaluate the resulting changes in the vortex breakdown configurations. As the ratio of container radius to disk radius increases above a value of about 2.3, the influence of the lateral container wall on the features of the central flow in the neighborhood of the spinning disk becomes insignificant. By means of a simplified model problem, it is demonstrated that this rapid loss of influence is due to the exponential decay of the azimuthal surface velocity beyond the edge of the disk. This exponential decay is confirmed by the numerical data, and it reflects the fact that as the lateral wall moves outward, the stationary part of the end wall becomes the main sink for the azimuthal momentum of the fluid. © 2005 American Institute of Physics.
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... Several subsequent studies were devoted to the effect of numerous operational parameters such as disk rotational speed and diameter on these flows (Jansson, Haspang, Jesen, Hersen, & Bohr, 2006;Suzuki, Iima, & Hayase, 2006;Vatistas, Wang, & Lin, 1992). Piva and Meiburg (2005) proposed a first numerical approximation to detect the free surface deflection but this is limited to small deformations. Kahouadji and Martin Witkowski (2014) performed a numerical study that takes into account the axisymmetric interfacial deformation using curvilinear coordinates. ...
... It is clear from figure 2(a) that both resolutions lead to accurate predictions of the interface deflection for both the DNS and LES approaches. It is also well known through the previous works of Piva and Meiburg (2005) and Kahouadji (2011) that the physical quantity that plays a major role in interface deflection is the azimuthal velocity distribution along the interface, u ∼ r, close to the shaft. Therefore, even for turbulent regimes, the interface can be accurately captured via the DNS or LES approaches. ...
... This spiralling motion reaches the fixed vessel wall inducing the formation of two Stewartson boundary layers (Kahouadji & Martin Witkowski, 2014;Poncet, 2005;Stewartson, 1953) at the top and bottom peripheries. The fluid motion separates and reaches both the free surface and the bottom of the tank, it then decelerates by a centripetal spiral motion toward the rotating shaft (rotation axis) above (below) the impeller; this is analogous to the behaviour reported previously in the literature for rotating disks (Daube, 1991;Kahouadji & Martin Witkowski, 2014;Piva & Meiburg, 2005;Spohn, 1991;Spohn et al., 1993Spohn et al., , 1998. The flow is not axisymmetric, and one can see in figures 3(a) and 3(b) that the position of the rotating blades matter in terms of understanding the reasons underlying the flow patterns. ...
We consider the mixing dynamics of an air–liquid system driven by the rotation of a pitched blade turbine (PBT) inside an open, cylindrical tank. To examine the flow and interfacial dynamics, we use a highly parallelised implementation of a hybrid front-tracking/level-set method that employs a domain-decomposition parallelisation strategy. Our numerical technique is designed to capture faithfully complex interfacial deformation, and changes of topology, including interface rupture and dispersed phase coalescence. As shown via transient, a three-dimensional (3-D) LES (large eddy simulation) using a Smagorinsky–Lilly turbulence model, the impeller induces the formation of primary vortices that arise in many idealised rotating flows as well as several secondary vortical structures resembling Kelvin–Helmholtz, vortex breakdown, blade tip vortices and end-wall corner vortices. As the rotation rate increases, a transition to ‘aeration’ is observed when the interface reaches the rotating blades leading to the entrainment of air bubbles into the viscous fluid and the creation of a bubbly, rotating, free surface flow. The mechanisms underlying the aeration transition are probed as are the routes leading to it, which are shown to exhibit a strong dependence on flow history.
... Much more recently, E. Serre has numerically investigated vortex breakdown in a cylinder of aspect ratio G=4, with a free surface ( [13] and [14]). And in 2005, M. Piva introduced a new parameter into the experiment, with the variation of the rotating disc diameter [15]. Figure extracted from [15]. ...
... And in 2005, M. Piva introduced a new parameter into the experiment, with the variation of the rotating disc diameter [15]. Figure extracted from [15]. ...
... Approaches with surface deformation were realized in [42], [51] and [15]. Without entering too deeply in surface deformation analysis, such as in the two first references, one can consider the leading order of the free surface deflection, ∆h. ...
Our study focuses on the first instability that appears in a flow generated by a rotating disc placed at the bottom of a fixed cylindrical cavity, with a flat liquid-gas interface. Owing to the dual numerical and experimental approach, we highlight a robust mismatch on the threshold of this instability. After having assumed and later ruled out that mechanical vibrations could be responsible for these mismatches, this investigation has focused on the boundary condition used in simulations at the interface air-water and its influence on the flow. We show that the free slip condition does not allow to simulate the same flow as in the experiments. Several models of this interface are explored in order to reduce the mismatches. In particular, taking into account variations in surface tension depending on the concentration of a pollutant at the interface results in a considerable reduction of the error on the critical Reynolds number, and to a more accurate reproduction of the flow in the simulations.
... To validate the Sunfluidh code, we have chosen a well known configuration, G = 1 and Re = 900 with flat surface, studied before by [Piva and Meiburg, 2005] and later on by [Bouffanais and Lo Jacono, 2009a]. The configuration is calculated with Sunfluidh on a regular grid 96 × 96, modified ROSE (see Appendix A.1) on a regular curvilinear grid 151 × 151 and Gerris on a regular grid 128 × 128. ...
... solid body rotation), the deviation h(r)−G is proportional to the Froude number. Using numerical simulations with an undeformed free surface, [Piva and Meiburg, 2005] proposed a first order approximation for the surface elevation h(r) based on the normal stress balance, yielding a scaling proportional to F r. [Kahouadji and Martin Witkowski, 2014] observed and interpreted such a scaling for weaker deformations and moderate Reynolds numbers. Figure 4.3 shows the rescaled surface deformation (h(r) − G)/F r for the set of parameters of figure 4.2: this scaling remains here a fair approximation, even though at F r = 1 the surface deformation is of the same order of magnitude as the fluid layer thickness. ...
... Moreover, the graph reveals a flow reversal (V z < 0) similar to that observed in vortex-breakdown bubbles for rotating-lid experiments [Escudier, 1984]. This analogy has already been pointed out by [Iwatsu, 2004, Piva andMeiburg, 2005] or [Herrada et al., 2013] and is sometimes referred to as "off-axis vortex breakdown". ...
This work is a contribution to the understanding of instabilities in rotating flows with deformable free surface. The configuration under study is a cylindrical container partially filled with water or water-glycerin mixture and the rotation is driven by a disk located at the bottom. In such flow, several families of instability patterns can be triggered depending on the flow parameters (Froude and Reynolds numbers and aspect ratio). As a mean of investigation, steady and unsteady simulations are conducted using the two in-house codes ROSE and Sunfluidh, in conjunction with experiments. The novelty lies in two-phase numerical simulations with strongly deformed interface and large density ratio between the phases, in accurate surface height and Laser-Doppler Velocimetry measurements, and in cross validations between numerical and experimental techniques. The base flow is first carefully described. Axisymmetric steady solutions show the structure of the rotating flow: a central solid body rotation and a toroidal flow cell located at the periphery, the intensity and geometry of which are characterized. Focusing on Reynolds numbers large enough so that boundary layers form at solid walls, two additional layers are described: an internal layer at the edge of the central region and a layer under the free surface never reported before. This flow structure is shown to be robust as Froude number is varied. Three-dimensional unsteady simulations reveal that the time and space averaged flow shows many similarities with steady axisymmetric solutions and that the surface remains almost flat even though three-dimensional large structure instabilities have developed in the bulk. Through this full numerical approach corroborated by velocity measurements, the flow structure is thus clarified and compared to the simplified models used in the literature. The transient spin-up dynamics of a fluid from rest are also investigated for two configurations, namely Newton's bucket and the bottom-driven flow. Experimental surface height and velocity evolution are found in very good agreement with axisymmetric unsteady simulations. These results are meant to serve as a benchmark for unsteady rotating two-phase flow simulations. Large amplitude three-dimensional surface deformation waves known as rotating polygons are eventually studied experimentally. These patterns are usually observed at very large Reynolds number. Conducting investigations in more viscous flow regimes, the link between bulk and surface polygon instabilities is tentatively elucidated. Through a spectral analysis of surface deformation and velocity measurements, the phase-velocity to disk speed ratio is found to be close to 1/3 for rotating polygons whereas it is 2/3 for the bulk instability.
... The oil flow then remains in a configuration where the original meridional flow is confined to a thin layer in the vicinity of interface while the counter one above dominates in the oil phase (see figure 2l). On the other hand, flow separation (figure 2g) occurs in the water close to the edge of the rotating impeller (as described in the work of Piva & Meiburg, 2005). The water flow is characterised by a high level of turbulence involving the development of a multitude of vortices of varying scales. ...
The mixing of immiscible oil and water by a pitched blade turbine in a cylindrical vessel is studied numerically. Three-dimensional simulations combined with a hybrid front-tracking/level-set method are employed to capture the complex flow and interfacial dynamics. A large eddy simulation approach, with a Lilly–Smagorinsky model, is employed to simulate the turbulent two-phase dynamics at large Reynolds numbers $Re=1802{-}18\ 026$ . The numerical predictions are validated against previous experimental work involving single-drop breakup in a stirred vessel. For small $Re$ , the interface is deformed but does not reach the impeller hub, assuming instead the shape of a Newton's Bucket. As the rotating speed increases, the deforming interface attaches to the impeller hub which leads to the formation of long ligaments that subsequently break up into small droplets. For the largest $Re$ studied, the system dynamics becomes extremely complex wherein the creation of ligaments, their breakup and the coalescence of drops occur simultaneously. The simulation outcomes are presented in terms of spatio-temporal evolution of the interface shape and vortical structures. The results of a drop size analysis in terms of the evolution of the number of drops, and their size distribution, is also presented as a parametric function of $Re$ .
... The water film grows with the increase of rotational speed growth; the rotating liquid surface is divided into two parts: (1) Rotary paraboloid in the middle of the container and (2) the water film at the edge of the container on the inclined wall; the critical angular velocity in which the water film forms can be obtained by the deformation of eq 5: ω cri = (gK/x max ) 0.5 . At the same time, due to g, K is constant; as the container angular velocity increases, x max is constantly decreasing, that is, the range of the rotation parabolic is continuously reduced and the water film is larger. ...
Research shows that the surface shape of rotary liquid depends on the rotation mode. Mode A is that when the container wall rotates the liquid, the rotating liquid surface is paraboloid. Mode B is that when the rotor in the center of the container rotates the liquid, the rotating liquid surface is vortex. Based on the paraboloid formed by the mode A, the identity between the liquid level parameter and the wall slope K (K ≠ 0) is derived. When K → ∞, with the increase of the container angular spin rate, the liquid level parameter changes are infinite, the liquid level change and volume relationship are fixed. When K > 0, the container is a cylinder with a large upper part and a small lower part and the liquid level parameter changes are limited, and the limit ratio between the liquid level parameters is + 1. In addition, through the vortex experiment by the mode B, it is concluded that the vortex curve can be regarded as composed of three parabolas: the center triggering part, the rising part, and the edge attenuation part. Different from the mode A, the liquid level change and volume relationship caused by the vortex formed by the mode B are both variables. According to the experimental results, the influences of container inner diameter, initial liquid level, rotor size, and rotor speed on the vortex characteristics are discussed in detail. At the same time, based on the experiment, the liquid level change and volume relationship caused by the formation of the vortex are deduced under the ideal condition when a stable liquid surface is formed by the vortex.
... 8 The role of azimuthal vorticity gradient self-sustaining the radial expansion was further analyzed by Kurosaka et al. 9 Several investigations have been made by modifying the boundary conditions of the Vogel-Escudier flow. They include having a free surface, [10][11][12] an open cylinder with a partially rotating end wall, 13 a closed cylinder with partially rotating end wall, 14 both end walls rotating at the same rate, 15,16 replacing the disk with a rotating cone, 17 using a spin up or spin down process with a small rotation ratio of the top and bottom end walls, 18 adding a straight and sloped cylinder to the center of the flow 19 using two fluids of different densities, 20,21 and using different turbulence models. 22 A review paper published by Lucca-Negro and O'doherty 2 gives a comprehensive guide to the experimental, numerical, and theoretical work undertaken in the previous forty-five years. ...
In this paper, the Vogel–Escudier flow is studied at an aspect ratio of 2.5 and a Reynolds number range of 330–3000. An attempt is made to control the vortex breakdown bubbles using a thin rod in co- and counter-rotation cases. The flow is studied qualitatively using planar laser-induced fluorescence and quantitatively using particle image velocimetry measurements. In terms of swirl strength, defined as the ratio of Reynolds numbers of the disk and control rod, a complete suppression and subsequent reemergence of vortex breakdown bubbles is observed at 56 and 37, respectively. Counter-rotation shows a slight decrease until −150 and a periodic and aperiodic shedding of the vortices at −112 and −75. An attempt is made to explain the flow behavior by experimental and analytical means taking into consideration the azimuthal plane flow using the swirl decay mechanism. Further, experiments are performed for time varying co- and counter-rotations for various combinations of amplitude and frequency. The anomalous behavior at high frequencies of destabilization of vortices in co-rotation and a milder change in counter-rotation is observed.
... Several innovative designs of contact-less liquid metal stirrers and pumps use a rotating 22 permanent magnet as the key element. These designs are remarkably simple and efficient. ...
Four configurations of a rotating magnetic dipole-driven turbulent flow in an electrically conducting liquid cylinder are considered by spectral direct numerical simulation. These configurations differ by parallel or perpendicular orientation of the dipole rotation vector with respect to the nearest surface of the cylinder or its axis. The rotating dipole generates electromagnetic force in a thin outer liquid layer facing it. A concentrated vortex is driven when the dipole rotation vector is perpendicular to the nearest surface. This vortex closely resembles the rotating disk-driven flow. When the dipole rotation vector is parallel to the nearest surface, then a distributed vortex occurs akin of the translating wall-driven cavity flow. The characteristic velocity is comparably little influenced by dipole orientation despite the electromagnetic force magnitude varying by a factor of three. Perpendicular orientation of the magnetic dipole rotation vector with respect to the cylinder's axis causes secondary corner eddies increasing the overall turbulent fluctuation. The simulations are supplemented by an experiment featuring a deep and narrow funnel-shaped quasi-stationary free surface deformation above a concentrated vortex.
... The equations were coupled to the free surface deformation and solved by an iterative process. The free surface deformation results agreed with those reported in [47] for Re = 900, but there was disagreement with the results of [36] for Re = 900 and 1500. In addition, the free surface deformation reported in [4,8], was qualitatively compared in [46] for the case before the symmetry breaking and without reaching the rotating disk. ...
The present paper provides a physically sound numerical modeling of liquid flows experimentally observed inside a vertical circular cylinder with a stationary envelope, rotating bottom and open top. In these flows, the resulting vortex depth may be such that the rotating bottom disk becomes partially exposed, and rather peculiar polygon shapes appear. The parameters and features of this work are chosen based on a careful analysis of the literature. Accordingly, the cylinder inner radius is 145 mm and the initial water height is 60 mm. The experiments with bottom disk rotation frequencies of 3.0, 3.4, 4.0 and 4.6 Hz are simulated. The chosen frequency range encompasses the regions of ellipse and triangle shapes as observed in the experimental studies reported in the literature. The free surface flow is expected to be turbulent, with the Reynolds number of O(105). The Large Eddy Simulation (LES) is adopted as the numerical approach, with a localized dynamic Subgrid-Scale Stresses (SGS) model including an energy equation. Since the flow obviously requires a surface tracking or capturing method, a volume-of-fluid (VOF) approach has been chosen based on the findings, where this method provided stable shapes in the ranges of parameters found in the corresponding experiments. Expected ellipse and triangle shapes are revealed and analyzed. A detailed character of the numerical results allows for an in-depth discussion and analysis of the mechanisms and features which accompany the characteristic shapes and their alterations. As a result, a unique insight into the polygon flow structures is provided.
In an experimental context, the contamination of an air–liquid interface by ambient pollutants can strongly affect the dynamics and the stability of a given flow. In some configurations, the interfacial flow can even be blocked by surface tension effects. A cylindrical free-surface flow driven by a slow rotating disc is considered here as a generic example of such effects and is investigated both experimentally and numerically. We suggest here a simple numerical model, without any superficial transport of the pollutants, adaptable into any code for single-phase flows. For the stationary axisymmetric base flow, the radial velocity at the interface is set to zero whereas the usual stress-free boundary conditions are retained for the perturbations. The model does not feature any free parameter. For a geometrical aspect ratio of 1/4, known to display ambiguous behaviour regarding stability thresholds, the modal selection as well as a nonlinear stability island found in the experiments are well reproduced by the model, both qualitatively and quantitatively. The robustness of the model has also been validated by replacing the radial velocity profile by a more accurate experimental fit, with very little influence on the stability results.
View Video Presentation: https://doi.org/10.2514/6.2021-2860.vid The present work presents experimental results on the Vogel-Escudier flow. In this paper, the parameter space is extended to an aspect ratio of 2.5 and the Reynolds Number range of 1700-3000. An attempt is made to control the Vortex Breakdown Bubbles using a thin rod in co and counter-rotation. The flow is studied qualitatively using PLIF and quantitatively using PIV measurements. In terms of swirl strength, defined as the ratio of Reynolds numbers of the disk and control rod, a complete suppression and subsequent re-emergence of Vortex Breakdown Bubbles is observed at 56 and 37, respectively. Counter-rotation shows a slight decrease till a swirl strength of -150 and a periodic and aperiodic shedding of the vortices at -112 and -75. The variation from the base flow due to the addition of the rod is quantified along with an analysis of the possible sources of error in the PIV measurements. The present results show the method of addition of a near axis swirl to be an effective way to control vortex breakdown.
Patterns on the air–water interface of a swirling cylinder flow are produced via hydrodynamic symmetry-breaking instability of the bulk flow. The patterns are rotating waves breaking the axisymmetry of the system and are longitudinal at the free surface (i.e., not surface deforming). Qualitative observations and quantitative measurements of velocity and vorticity are provided. Three-dimensional Navier–Stokes computations identify the symmetry-breaking mode responsible for the waves. These waves are then used to pattern Langmuir monolayers at concentrations sufficiently below saturation.
A systematic experimental investigation of the flow in an open cylinder, driven by the constant rotation of the bottom endwall, shows that axisymmetry is spontaneously broken via a supercritical Hopf bifurcation to a rotating wave with azimuthal wave number 4. The physical mechanism responsible for the symmetry breaking is shown to be due to the instability of the shear layer that is produced by the boundary layer on the bottom rotating endwall being turned into the interior by the stationary sidewall. Comparison with other experiments and numerical studies (restricted to axisymmetric subspaces) sheds new light on disparate observations in the literature and helps distinguish between spontaneous and forced (via imperfections) symmetry breaking.
A numerical investigation of the multiple stable solutions found in confined swirling flows is presented. The flows consist of fluid in a completely filled cylinder driven by the constant corotation of the two end walls. When reflectional symmetry at the cylinder half-plane is imposed, the flow corresponds to that in a cylinder of half the height driven by the bottom end wall, with the top surface being flat and stress-free. Comparisons with available experiments in this case are made and the observed toroidal recirculation zones attached to the free surface are described in terms of secondary motions induced by the bending of vortex lines. Calculations are also presented where the reflectional symmetry is not imposed and the possibility of the flow breaking this symmetry is discussed.
Recent experiments on concentrated vortices produced by swirling flow into a constricted, cylindrical tube have resulted, we believe, in a new understanding of vortex flows in a variety of similar devices, the main purposes of which have been to simulate atmospheric vortices. As a major part of this review we present a synopsis of these new results and show their connection to various laboratory models of atmospheric vortices. The utility of such models in the study of large scale atmospheric flows is also discussed.
Three-dimensional and compressibility effects on swirling flows in pipes have been analyzed using full Navier-Stokes equations. Previously obtained axisymmetric numerical simulations of compressible and incompressible flows in pipes showed that for sufficiently large values of the Reynolds number, swirling flows presented a multiplicity of steady state axisymmetric solutions within a range of values of the swirl strength. In particular, some flow quantities when represented versus the swirl strength S lie on a curve with three branches connected by two folding points. The numerical results show that the only branch that is stable under small non-axisymmetric perturbations is that corresponding to columnar flows; columnar flows are obtained for values of the swirl parameter S smaller than a critical one S1(Mc) that depends on the characteristic Mach number of the flow Mc. For values of S larger than the critical, the initial axisymmetric flow evolves towards a 3D and unsteady flow. To complete the study, a local, spatial, nonparallel, stability linear analysis of axisymmetric incompressible flows which presents breakdown (bubble) has been also carried out. The stability results show that these axisymmetric flows are absolute unstable for non-axisymmetric small perturbations.
We report the results of a numerical study of the creation of stagnation points in a rotating cylinder of uid where both end-walls are rotated. Good agreement is found with previous results where the stagnation points are formed on the core of the pri-mary columnar vortex. Novel phenomena have been uncovered at small aspect ratios where stagnation occurs o-axis directly and the secondary vortex which is created forms a toroid. The case is then considered of a small cylinder placed along the centre of the ow and despite the qualitatively di erent bound-ary condition, the phenomena are found to be robust.