Asymptotic theory for a bathtub vortex in a rotating tank

AbstractFluid entering the periphery of a cylindrical tank mounted on a rotating table is pumped inwards toward a central, floor drain by a potential vortex that is established in the fluid interior. We present here an asymptotic theory for small Rossby and Ekman numbers, including detailed solutions in the vortex core. Results for azimuthal velocity variation with radius agree quite well with the experiments of Andersen et al. (J. Fluid Mech., vol. 556, 2006, pp. 121–146), in spite of their free upper boundary. Modifications of the flow are presented in the instance that a short cylinder is place on the tank axis as in the work of Chen et al. (J. Fluid Mech., vol. 733, 2013, pp. 134–157). The overall flow structure found here is exactly that noted by both Andersen et al. and Chen et al.

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Structure of a steady drain-hole vortex in a viscous fluid

Journal of Fluid Mechanics ◽

10.1017/s0022112010001473 ◽

2010 ◽

Vol 656 ◽

pp. 177-188 ◽

Cited By ~ 9

Author(s):

L. BØHLING ◽

A. ANDERSEN ◽

D. FABRE

Keyword(s):

Reynolds Number ◽

Vortex Core ◽

Integrable Model ◽

Azimuthal Velocity ◽

Meridional Flow ◽

Cylindrical Tank ◽

Drain Hole ◽

Height Dependence ◽

Qualitative Structure ◽

Bathtub Vortex

We use direct numerical simulations to study a steady bathtub vortex in a cylindrical tank with a central drain-hole, a flat stress-free surface and velocity prescribed at the inlet. We find that the qualitative structure of the meridional flow does not depend on the radial Reynolds number, whereas we observe a weak overall rotation at a low radial Reynolds number and a concentrated vortex above the drain-hole at a high radial Reynolds number. We introduce a simple analytically integrable model that shows the same qualitative dependence on the radial Reynolds number as the simulations and compares favourably with the results for the radial velocity and the azimuthal velocity at the surface. Finally, we describe the height dependence of the radius of the vortex core and the maximum of the azimuthal velocity at a high radial Reynolds number, and we show that the data on the radius of the vortex core and the maximum of the azimuthal velocity as functions of height collapse on single curves by appropriate scaling.

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Tracer Stirring around a Meddy: The Formation of Layering

Journal of Physical Oceanography ◽

10.1175/jpo-d-14-0061.1 ◽

2015 ◽

Vol 45 (2) ◽

pp. 407-423 ◽

Cited By ~ 22

Author(s):

Thomas Meunier ◽

Claire Ménesguen ◽

Richard Schopp ◽

Sylvie Le Gentil

Keyword(s):

Vortex Core ◽

Azimuthal Velocity ◽

Thin Layers ◽

Vertical Shear ◽

In Situ Data ◽

Simple Scaling ◽

Diffusive Processes ◽

Tracer Experiments ◽

Good Agreement

AbstractThe dynamics of the formation of layering surrounding meddy-like vortex lenses is investigated using primitive equation (PE), quasigeostrophic (QG), and tracer advection models. Recent in situ data inside a meddy confirmed the formation of highly density-compensated layers in temperature and salinity at the periphery of the vortex core. Very high-resolution PE modeling of an idealized meddy showed the formation of realistic layers even in the absence of double-diffusive processes. The strong density compensation observed in the PE model, in good agreement with in situ data, suggests that stirring might be a leading process in the generation of layering. Passive tracer experiments confirmed that the vertical variance cascade in the periphery of the vortex core is triggered by the vertical shear of the azimuthal velocity, resulting in the generation of thin layers. The time evolution of this process down to scales of O(10) m is quantified, and a simple scaling is proposed and shown to describe precisely the thinning down of the layers as a function of the initial tracer column’s horizontal width and the vertical shear of the azimuthal velocity. Nonlinear QG simulations were performed and analyzed for comparison with the work of Hua et al. A step-by-step interpretation of these results on the evolution of layering is proposed in the context of tracer stirring.

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Self-similar behaviour of a rotor wake vortex core

Journal of Fluid Mechanics ◽

10.1017/jfm.2013.636 ◽

2014 ◽

Vol 740 ◽

Cited By ~ 9

Author(s):

Mohamed Ali ◽

Malek Abid

Keyword(s):

Vortex Core ◽

Scaling Laws ◽

Stokes Equations ◽

Three Dimensional ◽

Azimuthal Velocity ◽

Reynolds Numbers ◽

Navier Stokes ◽

Navier Stokes Equations ◽

Rotating Blade ◽

Self Similar

AbstractWe report a self-similar behaviour of solutions (obtained numerically) of the Navier–Stokes equations behind a single-blade rotor. That is, the helical vortex core in the wake of a rotating blade is self-similar as a function of its age. Profiles of vorticity and azimuthal velocity in the vortex core are characterized, their similarity variables are identified and scaling laws of these variables are given. Solutions of incompressible three-dimensional Navier–Stokes equations for Reynolds numbers up to $Re= 2000$ are considered.

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Axisymmetric rotating flow with free surface in a cylindrical tank

Journal of Fluid Mechanics ◽

10.1017/jfm.2018.929 ◽

2018 ◽

Vol 861 ◽

pp. 796-814 ◽

Cited By ~ 3

Author(s):

Wen Yang ◽

Ivan Delbende ◽

Yann Fraigneau ◽

Laurent Martin Witkowski

Keyword(s):

Reynolds Number ◽

Free Surface ◽

Surface Deformation ◽

Fluid Layer ◽

Three Dimensional ◽

Azimuthal Velocity ◽

Meridional Plane ◽

Cylindrical Tank ◽

Aspect Ratios ◽

Steady Solutions

The flow induced by a disk rotating at the bottom of a cylindrical tank is characterised using numerical techniques – computation of steady solutions or time-averaged two-dimensional and three-dimensional direct simulations – as well as laser-Doppler velocimetry measurements. Axisymmetric steady solutions reveal the structure of the toroidal flow located at the periphery of the central solid body rotation region. When viewed in a meridional plane, this flow cell is found to be bordered by four layers, two at the solid boundaries, one at the free surface and one located at the edge of the central region, which possesses a sinuous shape. The cell intensity and geometry are determined for several fluid-layer aspect ratios; the flow is shown to depend very weakly on Froude number (associated with surface deformation) or on Reynolds number if sufficiently large. The paper then focuses on the high Reynolds number regime for which the flow has become unsteady and three-dimensional while the surface is still almost flat. Direct numerical simulations show that the averaged flow shares many similarities with the above steady axisymmetric solutions. Experimental measurements corroborate most of the numerical results and also allow for the spatio-temporal characterisation of the fluctuations, in particular the azimuthal structure and frequency spectrum. Mean azimuthal velocity profiles obtained in this transitional regime are eventually compared to existing theoretical models.

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Design Rules for the Velocity Field of Vortex Breakdown Swirl Burners

Volume 1: Combustion and Fuels, Education ◽

10.1115/gt2006-90495 ◽

2006 ◽

Cited By ~ 6

Author(s):

Stephan Burmberger ◽

Christoph Hirsch ◽

Thomas Sattelmayer

Keyword(s):

Velocity Field ◽

Combustion Chamber ◽

Pressure Distribution ◽

Vortex Core ◽

Total Pressure ◽

Vortex Breakdown ◽

Azimuthal Velocity ◽

Velocity Profiles ◽

Flame Stabilization ◽

Stable Vortex

Most gas turbine premix burners without centrebody employ the breakdown of a swirling flow at the transition between the mixing section and the combustor for aerodynamic flame stabilization. As the formation of the desired vortex breakdown pattern depends very sensibly on the shape of the axial and azimuthal velocity profiles in the mixing section, the design of suitable swirlers is typically a cumbersome process and requires an iterative approach consisting of numerical as well as experimental development steps to be iteratively applied until a geometry is found, that provides a spatially as well as temporarily stable vortex breakdown in the primary zone of the combustion chamber without backflow on the centerline of the vortex into the swirler. These difficulties stem from the lack of generally applicable aerodynamic design criteria. The paper attempts to contribute to the development of such design guidelines, which lead quickly to successful swirler designs without need for an excessive number of iterations. For this purpose a family of swirl profiles was generated and the corresponding axial velocity profiles were calculated assuming several radial total pressure distributions. In the next step, the flows were calculated using CFD in order to find out, which velocity profiles produce stable vortex breakdown bubbles at the burner exit. This study reveals that the stable breakdown of the vortex can be achieved for a wide range of velocity distributions, if the radial total pressure distribution is properly selected. However, the radial total pressure distribution in the vortex core is essential for the robustness of the design. Interestingly, velocity profiles with constant total pressure do not show a stable transition of the velocity field at the cross-sectional area change at the entrance of the combustion chamber. In addition, theoretical considerations reveal that an increase of the azimuthal velocity in the vortex core in streamwise direction avoids backflow on the centreline as well as flame flashback. This increase can be achieved using a slightly conical nozzle and introducing a swirl free jet on the centreline upstream of the mixing zone. All effects are explained using the vorticity transport equation.

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An experimental and numerical study of the secular spin-up of a thermally stratified rotating fluid

Journal of Fluid Mechanics ◽

10.1017/s002211207900183x ◽

1979 ◽

Vol 93 (1) ◽

pp. 161-184 ◽

Cited By ~ 8

Author(s):

Robert C. Beardsley ◽

Kim D. Saunders ◽

Alex C. Warn-Varnas ◽

John M. Harding

Keyword(s):

Angular Velocity ◽

Time Scale ◽

Numerical Experiments ◽

Asymptotic Theory ◽

Numerical Study ◽

Angular Acceleration ◽

Azimuthal Velocity ◽

Velocity Fields ◽

Rotating Fluid ◽

Good Agreement

Laboratory and numerical experiments have been conducted to study the secular spin-up of both a homogeneous and a thermally stratified rotating fluid in a right cylinder. In these experiments, the angular velocity of the container increases linearly in time from some initial rotation rate at t = 0. A simple quasi-geostrophic model is developed to describe the adjustment of the fluid over the characteristic spin-up time scale to the constant angular acceleration of the basin. Good agreement is found between the observed interior temperature and azimuthal velocity fields and the theory in both the homogeneous and stratified secular experiments. This result is in contrast to the much faster adjustment observed in stratified instantaneous spin-up experiments reported earlier. The main difference between these experimental cases is the inability of secular forcing to excite energetic inertial–gravity-wave transients during the initial phases of secular spin-up. Thus, the asymptotic theory which has filtered out these initial higher-frequency transients is accurate even though the inertial period is not much smaller than the characteristic spin-up time scale.

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Lectures on the Asymptotic Theory of Ideals

10.1017/cbo9780511525957 ◽

1988 ◽

Cited By ~ 13

Author(s):

D. Rees

Keyword(s):

Asymptotic Theory

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Absolute Pitch and Its Frequency Range

Archives of Acoustics ◽

10.2478/v10168-011-0020-1 ◽

2011 ◽

Vol 36 (2) ◽

pp. 251-266 ◽

Cited By ~ 4

Author(s):

Andrzej Rakowski ◽

Piotr Rogowski

Keyword(s):

Single Case ◽

Random Order ◽

Absolute Pitch ◽

Screening Tests ◽

Music Students ◽

Frequency Range ◽

Musical Scale ◽

Loudness Level ◽

Pure Tones ◽

Upper Boundary

AbstractThis paper has two distinct parts. Section 1 includes general discussion of the phenomenon of "absolute pitch" (AP), and presentation of various concepts concerning definitions of "full", "partial" and "pseudo" AP. Sections 2-4 include presentation of the experiment concerning frequency range in which absolute pitch appears, and discussion of the experimental results. The experiment was performed with participation of 9 AP experts selected from the population of 250 music students as best scoring in the pitch-naming piano-tone screening tests. Each subject had to recognize chromas of 108 pure tones representing the chromatic musical scale of nine octaves from E0 to D#9. The series of 108 tones was presented to each subject 60 times in random order, diotically, with loudness level about 65 phon. Percentage of correct recognitions (PC) for each tone was computed. The frequency range for the existence of absolute pitch in pure tones, perceived by sensitive AP possessors stretches usually over 5 octaves from about 130.6 Hz (C3) to about 3.951 Hz (B7). However, it was noted that in a single case, the upper boundary of AP was 9.397 Hz (D9). The split-halves method was applied to estimate the reliability of the obtained results.

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