scholarly journals The stably stratified Taylor–Couette flow is always unstable except for solid-body rotation

2013 ◽  
Vol 725 ◽  
pp. 262-280 ◽  
Author(s):  
Junho Park ◽  
Paul Billant
Keyword(s):  
Solid Body ◽  
Outer Cylinder ◽  
Inviscid Limit ◽  
Rotational Instability ◽  
Body Rotation ◽  
Viscous Effects ◽  
Solid Body Rotation ◽  
Concentric Cylinders ◽  
The Stability ◽  
The Waves

AbstractThe stability of the flow between two concentric cylinders is studied numerically and analytically when the fluid is stably stratified. We show that such flow is unstable when the angular velocity $\Omega (r)$ increases along the radial direction, a regime never explored before. The instability is highly non-axisymmetric and involves the resonance of two families of inertia–gravity waves like for the strato-rotational instability. The growth rate is maximum when only the outer cylinder is rotating and goes to zero when $\Omega (r)$ is constant. The sufficient condition for linear, inviscid instability derived previously, $\mathrm{d} {\Omega }^{2} / \mathrm{d} r\lt 0$, is therefore extended to $\mathrm{d} {\Omega }^{2} / \mathrm{d} r\not = 0$, meaning that only the regime of solid-body rotation is stable in stratified fluids. A Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) analysis in the inviscid limit, confirmed by the numerical results, shows that the instability occurs only when the Froude number is below a critical value and only for a particular band of azimuthal wavenumbers. It is also demonstrated that the instability originates from a reversal of the radial group velocity of the waves, or equivalently from a wave over-reflection phenomenon. The instability persists in the presence of viscous effects.

scholarly journals Inertial waves in a differentially rotating spherical shell

2013 ◽  
Vol 719 ◽  
pp. 47-81 ◽  
Author(s):  
C. Baruteau ◽  
M. Rieutord
Keyword(s):  
Spherical Shell ◽  
Solid Body ◽  
Shear Layers ◽  
Inviscid Limit ◽  
Order Partial Differential Equation ◽  
Body Rotation ◽  
Inertial Waves ◽  
Visible Absorption ◽  
Solid Body Rotation ◽  
Short Period

AbstractWe investigate the properties of small-amplitude inertial waves propagating in a differentially rotating incompressible fluid contained in a spherical shell. For cylindrical and shellular rotation profiles and in the inviscid limit, inertial waves obey a second-order partial differential equation of mixed type. Two kinds of inertial modes therefore exist, depending on whether the hyperbolic domain where characteristics propagate covers the whole shell or not. The occurrence of these two kinds of inertial modes is examined, and we show that the range of frequencies at which inertial waves may propagate is broader than with solid-body rotation. Using high-resolution calculations based on a spectral method, we show that, as with solid-body rotation, singular modes with thin shear layers following short-period attractors still exist with differential rotation. They exist even in the case of a full sphere. In the limit of vanishing viscosities, the width of the shear layers seems to weakly depend on the global background shear, showing a scaling in ${E}^{1/ 3} $ with the Ekman number $E$, as in the solid-body rotation case. There also exist modes with thin detached layers of width scaling with ${E}^{1/ 2} $ as Ekman boundary layers. The behaviour of inertial waves with a corotation resonance within the shell is also considered. For cylindrical rotation, waves get dramatically absorbed at corotation. In contrast, for shellular rotation, waves may cross a critical layer without visible absorption, and such modes can be unstable for small enough Ekman numbers.

scholarly journals Stability of stationary endwall boundary layers during spin-down

1996 ◽  
Vol 326 ◽  
pp. 373-398 ◽  
Author(s):  
J. M. Lopez ◽  
P. D. Weidman
Keyword(s):  
Boundary Layers ◽  
Solid Body ◽  
Experimental Studies ◽  
Steady Flows ◽  
Body Rotation ◽  
Solid Body Rotation ◽  
Centrifugal Instability ◽  
Self Similar ◽  
Layer Flow ◽  
The Stability

Since Bödewadt's (1940) seminal work on the boundary layer flow produced by a fluid in solid-body rotation over a stationary disk of infinite radius there has been much interest in determining the stability of such flows. To date, it appears that there is no theoretical study of the stability of Bödewadt's self-similar solution to perturbations that are not self-similar. Experimental studies have been compromised due to the difficulty in establishing these steady flows in the laboratory. Savaç (1983, 1987) has studied the endwall boundary layers of flow in a circular cylinder following impulsive spin-down. During the first few radians of rotation, the endwall boundary layers have a structure very similar to Bödewadt layers. For certain conditions, SavaÇ has observed a series of axisymmetric waves travelling radially inwards in the endwall boundary layers. The conjecture is that these waves represent a mode of instability of the Bödewadt layer. Within a few radians of rotation however, the centrifugal instability of the sidewall layer dominates the spin-down process and the endwall waves are difficult to examine further.Here, the impulsive spin-down problem is examined numerically for Savaç’ (1983, 1987) conditions and good agreement with his experiments is achieved. New experimental results are also presented, which include quantitative space-time information regarding the axisymmetric waves. These agree well with both the numerics and the earlier experimental work. Further, a related problem is considered numerically. This flow is also initially in solid-body rotation, but only the endwalls are impulsively stopped, keeping the sidewall rotating. This results in a flow virtually identical to the usual spin-down flow for the first few radians of rotation, except in the immediate vicinity of the sidewall. The sidewall layer is no longer centrifugally unstable and the circular waves on the endwalls are observed without the influence of the sidewall instability.

Oval Flow Mode Between Two Corotating Disks With Stationary Shroud

2014 ◽  
Vol 137 (3) ◽  
Author(s):  
Ching Min Hsu ◽  
Jia-Kun Chen ◽  
Min Kai Hsieh ◽  
Rong Fung Huang
Keyword(s):  
Flow Structure ◽  
Solid Body ◽  
Circumferential Velocity ◽  
Body Rotation ◽  
Flow Core ◽  
Core Flow ◽  
Node Point ◽  
Solid Body Rotation ◽  
Characteristic Flow ◽  
Buffer Region

The characteristic flow behavior, time-averaged velocity distributions, phase-resolved ensemble-averaged velocity profiles, and turbulence properties of the flow in the interdisk midplane between shrouded two corotating disks at the interdisk spacing to disk radius aspect ratio 0.2 and rotation Reynolds number 3.01 × 105 were experimentally studied by flow visualization method and particle image velocimetry (PIV). An oval core flow structure rotating at a frequency 60% of the disks rotating frequency was observed. Based on the analysis of relative velocities, the flow in the region outside the oval core flow structure consisted of two large vortex rings, which move circumferentially with the rotation motion of the oval flow core. Four characteristic flow regions—solid-body-rotation-like region, buffer region, vortex region, and shroud-influenced region—were identified in the flow field. The solid-body-rotation-like region, which was featured by its linear distribution of circumferential velocity and negligibly small radial velocity, was located within the inscribing radius of the oval flow core. The vortex region was located outside the circumscribing radius of the oval flow core. The buffer region existed between the solid-body-rotation-like region and the vortex region. In the buffer region, there existed a “node” point that the propagating circumferential velocity waves diminished. The circumferential random fluctuation intensity presented minimum values at the node point and high values in the solid-body-rotation-like region and shroud-influenced region due to the shear effect induced by the wall.

scholarly journals Azimuthal velocity profiles in Rayleigh-stable Taylor–Couette flow and implied axial angular momentum transport

2015 ◽  
Vol 774 ◽  
pp. 342-362 ◽  
Author(s):  
Freja Nordsiek ◽  
Sander G. Huisman ◽  
Roeland C. A. van der Veen ◽  
Chao Sun ◽  
Detlef Lohse ◽  
...  
Keyword(s):  
Angular Momentum ◽  
Couette Flow ◽  
Outer Cylinder ◽  
Azimuthal Velocity ◽  
Velocity Profiles ◽  
Body Rotation ◽  
Solid Body Rotation ◽  
Angular Momentum Transport ◽  
Taylor Couette ◽  
Taylor Couette Flow

We present azimuthal velocity profiles measured in a Taylor–Couette apparatus, which has been used as a model of stellar and planetary accretion disks. The apparatus has a cylinder radius ratio of ${\it\eta}=0.716$, an aspect ratio of ${\it\Gamma}=11.74$, and the plates closing the cylinders in the axial direction are attached to the outer cylinder. We investigate angular momentum transport and Ekman pumping in the Rayleigh-stable regime. This regime is linearly stable and is characterized by radially increasing specific angular momentum. We present several Rayleigh-stable profiles for shear Reynolds numbers $\mathit{Re}_{S}\sim O(10^{5})$, for both ${\it\Omega}_{i}>{\it\Omega}_{o}>0$ (quasi-Keplerian regime) and ${\it\Omega}_{o}>{\it\Omega}_{i}>0$ (sub-rotating regime), where ${\it\Omega}_{i,o}$ is the inner/outer cylinder rotation rate. None of the velocity profiles match the non-vortical laminar Taylor–Couette profile. The deviation from that profile increases as solid-body rotation is approached at fixed $\mathit{Re}_{S}$. Flow super-rotation, an angular velocity greater than those of both cylinders, is observed in the sub-rotating regime. The velocity profiles give lower bounds for the torques required to rotate the inner cylinder that are larger than the torques for the case of laminar Taylor–Couette flow. The quasi-Keplerian profiles are composed of a well-mixed inner region, having approximately constant angular momentum, connected to an outer region in solid-body rotation with the outer cylinder and attached axial boundaries. These regions suggest that the angular momentum is transported axially to the axial boundaries. Therefore, Taylor–Couette flow with closing plates attached to the outer cylinder is an imperfect model for accretion disk flows, especially with regard to their stability.

scholarly journals On a possible corrugation of the galactic plane

1970 ◽  
Vol 38 ◽  
pp. 147-150 ◽  
Author(s):  
C. M. Varsavsky ◽  
R. J. Quiroga
Keyword(s):  
Brightness Temperature ◽  
Solid Body ◽  
Rotation Curve ◽  
Galactic Plane ◽  
Maximum Temperature ◽  
Body Rotation ◽  
Maximum Brightness ◽  
Solid Body Rotation ◽  
The Galaxy ◽  
Sinusoidal Pattern

We have studied the rotation curve of the Galaxy at different heights below and above the equator. In the course of this work we noticed that the maximum brightness temperature of hydrogen oscillates around the galactic plane following a fairly sinusoidal pattern. It is further noticed that the maximum temperature of hydrogen occurs right on the plane in the regions where the rotation curve has a form indicating solid body rotation. A rotation curve based on points of maximum hydrogen temperature does not differ appreciably from a rotation curve measured on the galactic plane.

Fragmentation of elongated cylindrical clouds. IV - Clouds with solid-body rotation about an arbitrary axis

1992 ◽  
Vol 400 ◽  
pp. 579 ◽  
Author(s):  
Ian Bonnell ◽  
Jean-Pierre Arcoragi ◽  
Hugo Martel ◽  
Pierre Bastien
Keyword(s):  
Solid Body ◽  
Body Rotation ◽  
Solid Body Rotation ◽  
Arbitrary Axis

scholarly journals Observations of an Inertial Peak in the Intrinsic Wind Spectrum Shifted by Rotation in the Antarctic Vortex

2012 ◽  
Vol 69 (12) ◽  
pp. 3800-3811 ◽  
Author(s):  
L. J. Gelinas ◽  
R. L. Walterscheid ◽  
C. R. Mechoso ◽  
G. Schubert
Keyword(s):  
Solid Body ◽  
Wave Spectrum ◽  
Polar Vortex ◽  
Relative Vorticity ◽  
Body Rotation ◽  
Background Flow ◽  
Absolute Vorticity ◽  
Solid Body Rotation ◽  
The Absolute ◽  
The Antarctic

Abstract Spectral analyses of time series of zonal winds derived from locations of balloons drifting in the Southern Hemisphere polar vortex during the Vorcore campaign of the Stratéole program reveal a peak with a frequency near 0.10 h−1, more than 25% higher than the inertial frequency at locations along the trajectories. Using balloon data and values of relative vorticity evaluated from the Modern Era Retrospective-Analyses for Research and Applications (MERRA), the authors find that the spectral peak near 0.10 h−1 can be interpreted as being due to inertial waves propagating inside the Antarctic polar vortex. In support of this claim, the authors examine the way in which the low-frequency part of the gravity wave spectrum sampled by the balloons is shifted because of effects of the background flow vorticity. Locally, the background flow can be expressed as the sum of solid-body rotation and shear. This study demonstrates that while pure solid-body rotation gives an effective inertial frequency equal to the absolute vorticity, the latter gives an effective inertial frequency that varies, depending on the direction of wave propagation, between limits defined by the absolute vorticity plus or minus half of the background relative vorticity.

Waves in a gas in solid-body rotation

1972 ◽  
Vol 56 (2) ◽  
pp. 277-286 ◽  
Author(s):  
J. B. Morton ◽  
E. J. Shaughnessy
Keyword(s):  
Eigenvalue Problem ◽  
Solid Body ◽  
Transverse Wave ◽  
Speed Of Sound ◽  
Acoustic Waves ◽  
Body Rotation ◽  
Perfect Gas ◽  
Solid Body Rotation ◽  
Rotational Waves

The axial and transverse wave motions of an inviscid perfect gas in isothermal solid-body rotation in a cylinder are investigated. Solutions of the resulting eigenvalue problem are shown to correspond to two types of waves. The acoustic waves are the rotational counterparts of the well-known Rayleigh solutions for a gas at rest in a cylinder. The rotational waves, whose amplitudes and frequencies go to zero in the non-rotating limit, exhibit phase speeds both larger and smaller than the speed of sound. The effect of rotation on the frequency and structure of these waves is discussed.

The Elastic Distortion of Rollers Under Combined Radial and Thrust Loads

1983 ◽  
Vol 105 (2) ◽  
pp. 189-197 ◽  
Author(s):  
H. So ◽  
R. Gohar
Keyword(s):  
Pressure Distribution ◽  
Solid Body ◽  
Axial Load ◽  
Axial Loading ◽  
Radial Pressure ◽  
Approximate Analysis ◽  
Body Rotation ◽  
Solid Body Rotation ◽  
Elastic Distortion ◽  
Radial Loading

This paper describes an approximate analysis for finding the elastostatic radial and end face distortion, radial pressure distribution, and solid body rotation of a flat ended axially profiled bearing roller under combined radial and axial loading through the ribs. It is found that a small but significant end face bulge occurs at each roller end when there is radial loading only. Upon the addition of an axial load, this bulge becomes a small depression. The altered geometry there may become significant during bearing operation, as it affects roller skew, wear, and lubrication between the ribs and roller and faces.

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