Separation and the Taylor-column problem for a hemisphere

1974 ◽  
Vol 66 (4) ◽  
pp. 767-789 ◽  
Author(s):  
J. D. A. Walker ◽  
K. Stewartson
Keyword(s):  
Angular Velocity ◽  
Incompressible Fluid ◽  
Viscous Incompressible Fluid ◽  
Shear Layers ◽  
Constant Angular Velocity ◽  
Vertical Shear ◽  
Rossby Number ◽  
Ekman Number ◽  
Free Shear Layers ◽  
Taylor Column

A layer of viscous incompressible fluid is confined between two horizontal plates which rotate rapidly in their own plane with a constant angular velocity. A hemisphere has its plane face joined to the lower plate and when a uniform flow is forced past such an obstacle, a Taylor column bounded by thin detached vertical shear layers forms. The linear theory for this problem, wherein the Rossby number ε is set equal to zero on the assumption that the flow is slow, is examined in detail. The nonlinear modifications of the shear layers are then investigated for the case when ε ∼ E½, where E is the Ekman number. In particular, it is shown that provided that the Rossby number is large enough separation occurs in the free shear layers. The extension of the theory to flow past arbitrary spheroids is indicated.

Stability of a Rotor Partially Filled With a Viscous Incompressible Fluid

1979 ◽  
Vol 46 (4) ◽  
pp. 913-918 ◽  
Author(s):  
S. L. Hendricks ◽  
J. B. Morton
Keyword(s):  
Angular Velocity ◽  
Circular Cylinder ◽  
Incompressible Fluid ◽  
Parametric Study ◽  
Viscous Incompressible Fluid ◽  
Constant Angular Velocity ◽  
Unstable Region ◽  
Stability Boundaries ◽  
System Parameters ◽  
Linear Spring

A hollow circular cylinder rotating with constant angular velocity and partially filled with a viscous incompressible fluid has been analyzed for stability. The analysis can be extended to apply to many different rotor geometries. The results of this analysis predict that over a range of operating speeds, the system is unstable. The extent of this unstable region is determined by the system parameters. The interplay between viscosity of the fluid and damping on the rotor is especially important in determining stability boundaries. A parametric study is presented for a rotor modeled as a cup in the middle of a symmetrically supported massless shaft. The rotor is subject to a linear spring and a linear damper. Rotor unbalance, gravity, and axial effects are considered negligible.

On nonlinear Ekman and Stewartson layers in a rotating fluid

1973 ◽  
Vol 333 (1595) ◽  
pp. 469-489 ◽  
Keyword(s):  
Angular Velocity ◽  
Shear Layers ◽  
Rossby Number ◽  
Inertial Effects ◽  
Rotating Fluid ◽  
Rotating Disks ◽  
Ekman Number ◽  
Axis Of Rotation ◽  
Ekman Layers ◽  
Source Sink

Boundary and shear layers parallel to the axis of rotation in a rapidly rotating fluid are modified by inertial effects when the Rossby number R is not small enough to be neglected. If E is the Ekman number, inertial modifications to Stewartson E ¼ layers become important when the Rossby number is comparable with, or larger than, E ¼ and changes in the thickness and the structure of the layer then occur. The nature of the change depends on whether the total component of vorticity parallel to the axis of rotation is increased or diminished by the contribution from the layer. When inertial effects are important in these layers, the Ekman layers are also modified by inertia and a nonlinear Ekman condition must be used. Such a condition is obtained and it is used to discuss the flow between rotating disks when there is a discontinuity in their angular velocity and source-sink flow in an annulus.

Stability of circular Couette flow with variable inner cylinder speed

1982 ◽  
Vol 123 ◽  
pp. 43-57 ◽  
Author(s):  
G. P. Neitzel
Keyword(s):  
Angular Velocity ◽  
Incompressible Fluid ◽  
Viscous Incompressible Fluid ◽  
Outer Cylinder ◽  
Finite Amplitude ◽  
Reverse Direction ◽  
Stability Bound ◽  
Concentric Cylinders ◽  
Theory Result ◽  
Stability Enhancement

Energy & ability theory is employed to study the finite-amplitude stability of a viscous incompressible fluid occupying the space between a pair of concentric cylinders when the inner-cylinder angular velocity varies linearly with time. For the case with a fixed outer cylinder and increasing inner-cylinder speed, we find an enhancement of stability, consistent with a linear-theory result due to Eagles. When the inner-cylinder speed decreases, we find an initially decreased stability bound, indicating the possibility of hysteresis, while, if the inner cylinder is allowed to reverse direction and linearly increase in speed, we find significant stability enhancement.

Higher-order approximation for free shear layers in almost rigid rotations

1975 ◽  
Vol 69 (1) ◽  
pp. 191-195 ◽  
Author(s):  
L. Pamela Cook ◽  
G. S. S. Ludford
Keyword(s):  
Angular Velocity ◽  
Order Approximation ◽  
Shear Layers ◽  
Higher Order ◽  
Rigid Rotation ◽  
Free Shear Layer ◽  
Curvature Effects ◽  
Parallel Disks ◽  
Free Shear Layers ◽  
Theory And Experiment

The free shear layer stemming from a discontinuity in angular velocity at either of two parallel disks in almost rigid rotation with fluid between is re-examined. The sole discrepancy between theory and experiment is unaffected by higher-order approximation unless curvature effects are included, when it is reduced.

scholarly journals Stability of an isolated pancake vortex in continuously stratified-rotating fluids

2016 ◽  
Vol 801 ◽  
pp. 508-553 ◽  
Author(s):  
Eunok Yim ◽  
Paul Billant ◽  
Claire Ménesguen
Keyword(s):  
Angular Velocity ◽  
Baroclinic Instability ◽  
Base Flow ◽  
Shear Instability ◽  
Vertical Shear ◽  
Rossby Number ◽  
Centrifugal Instability ◽  
Pancake Vortex ◽  
To Come ◽  
The Stability

This paper investigates the stability of an axisymmetric pancake vortex with Gaussian angular velocity in radial and vertical directions in a continuously stratified-rotating fluid. The different instabilities are determined as a function of the Rossby number $Ro$, Froude number $F_{h}$, Reynolds number $Re$ and aspect ratio ${\it\alpha}$. Centrifugal instability is not significantly different from the case of a columnar vortex due to its short-wavelength nature: it is dominant when the absolute Rossby number $|Ro|$ is large and is stabilized for small and moderate $|Ro|$ when the generalized Rayleigh discriminant is positive everywhere. The Gent–McWilliams instability, also known as internal instability, is then dominant for the azimuthal wavenumber $m=1$ when the Burger number $Bu={\it\alpha}^{2}Ro^{2}/(4F_{h}^{2})$ is larger than unity. When $Bu\lesssim 0.7Ro+0.1$, the Gent–McWilliams instability changes into a mixed baroclinic–Gent–McWilliams instability. Shear instability for $m=2$ exists when $F_{h}/{\it\alpha}$ is below a threshold depending on $Ro$. This condition is shown to come from confinement effects along the vertical. Shear instability transforms into a mixed baroclinic–shear instability for small $Bu$. The main energy source for both baroclinic–shear and baroclinic–Gent–McWilliams instabilities is the potential energy of the base flow instead of the kinetic energy for shear and Gent–McWilliams instabilities. The growth rates of these four instabilities depend mostly on $F_{h}/{\it\alpha}$ and $Ro$. Baroclinic instability develops when $F_{h}/{\it\alpha}|1+1/Ro|\gtrsim 1.46$ in qualitative agreement with the analytical predictions for a bounded vortex with angular velocity slowly varying along the vertical.

The theory of trailing Taylor columns

1970 ◽  
Vol 68 (2) ◽  
pp. 485-491 ◽  
Author(s):  
M. J. Lighthill
Keyword(s):  
Flow Field ◽  
Experimental Evidence ◽  
Small Angle ◽  
Large Body ◽  
Physical Significance ◽  
The Body ◽  
Rossby Number ◽  
Ekman Number ◽  
Axis Of Rotation ◽  
Taylor Column

AbstractWhen Rossby number is small but Ekman number is very much smaller, study of the flow field far from a body moving at right angles to the axis of rotation of a large body of fluid indicates that the region of influence should not be a Taylor column parallel to the axis, but a trailing Taylor column, bent backwards on both sides of the body at a small angle (proportional to Rossby number) to the axis. The paper reviews the physical significance of, and experimental evidence for, this conclusion.

On some exact solutions in magnetohydrodynamics with astrophysical applications

1972 ◽  
Vol 51 (1) ◽  
pp. 33-38 ◽  
Author(s):  
C. Sozou
Keyword(s):  
Free Surface ◽  
Angular Velocity ◽  
Incompressible Fluid ◽  
Exact Solutions ◽  
Equilibrium Configuration ◽  
Constant Angular Velocity ◽  
Magnetohydrodynamic Equations ◽  
The Stability

Some exact solutions of the steady magnetohydrodynamic equations for a perfectly conducting inviscid self-gravitating incompressible fluid are discussed. It is shown that there exist solutions for which the free surface of the liquid is that of a planetary ellipsoid and rotates with constant angular velocity about its axis. The stability of the equilibrium configuration is not investigated.

scholarly journals The flow induced by the transverse motion of a thin disk in its own plane through a contained rapidly rotating viscous liquid

1969 ◽  
Vol 39 (4) ◽  
pp. 831-847 ◽  
Author(s):  
D. W. Moore ◽  
P. G. Saffman ◽  
T. Maxworthy
Keyword(s):  
Fat Body ◽  
Circular Disk ◽  
Shear Layers ◽  
Transverse Motion ◽  
Leading Order ◽  
Stream Surface ◽  
Free Shear Layers ◽  
Plane Transverse ◽  
Unbounded Fluid ◽  
Taylor Column

A thin circular disk translates slowly in its own plane transverse to the axis of rotation of parallel plane boundaries filled with viscous incompressible liquid. It is shown that the indeterminateness of the geostrophic flow is removed by constraints imposed by the dynamics of free shear layers (Stewartson layers), which surround a Taylor column whose boundary is not a stream surface. Fluid particles cross the Taylor column at the expense of deflexion through a finite angle. A comparison is made with the flow past a fat body (Jacobs 1964), where the geostrophie flow is determined without appeal to the dynamics of the shear layers. The problem is also considered for a disk in an unbounded fluid, and it is shown that to leading order there is no disturbance.

On the stability of the Stewartson layer

1976 ◽  
Vol 76 (2) ◽  
pp. 289-306 ◽  
Author(s):  
Kiyoshi Hashimoto
Keyword(s):  
Boundary Layer ◽  
Growth Rate ◽  
Incompressible Fluid ◽  
Layer Structure ◽  
Rossby Number ◽  
Governing Equations ◽  
Azimuthal Component ◽  
Ekman Number ◽  
The Stability ◽  
Good Agreement

The stability of the Stewartson layer in a rotating incompressible fluid is investigated within the framework of a linear theory. The boundary-layer structure of the shear layer is correctly taken into account and the effect of viscous dissipation on the disturbance is included in the governing equations. The growth rate ωi of the disturbance is given as a function of the unified parameter mRo/(γ½), where m, an integer, is the azimuthal component of the wavenumber vector, γ the radius of the layer, Ro the Rossby number and E the Ekman number. Instability occurs when m Ro/(γ½) > 9·5. The time evolution of a growing disturbance is given schematically. Comparison of our results with the experiments by Hide & Titman shows good agreement.

scholarly journals Синхронизация в турбулентном сферическом течении Куэтта под действием неравномерного вращения

2019 ◽  
Vol 89 (7) ◽  
pp. 992
Author(s):  
Д.Ю. Жиленко ◽  
О.Э. Кривоносова
Keyword(s):  
Experimental Data ◽  
Angular Velocity ◽  
Turbulent Flow ◽  
Incompressible Fluid ◽  
Turbulent Flows ◽  
Viscous Incompressible Fluid ◽  
Spherical Layer ◽  
Numerical Results ◽  
Spherical Boundary

Turbulent flows of viscous incompressible fluid in rotating spherical layer in the presence of synchronization are under consideration. Numerical results are presented. Synchronization of turbulent flow is due to the action of periodical modulation of the angular velocity of inner spherical boundary. The angular velocity of outer spherical boundary is constant. Obtained results were compared with experimental data. The interval of modulation amplitudes was determined where synchronization is followed by intermittency “chaos – chaos”.

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