Waves generated in rotating fluids by travelling forcing effects

1973 ◽  
Vol 61 (1) ◽  
pp. 129-158 ◽  
Author(s):  
G. V. Prabhakara Rao
Keyword(s):  
Wave Pattern ◽  
Far Field ◽  
Rotating Fluid ◽  
Rotating Fluids ◽  
Fixed Frequency ◽  
Arbitrary Angle ◽  
Upstream Side ◽  
Axis Of Rotation ◽  
The Right ◽  
Taylor Column

The two-dimensional wave pattern produced in a homogeneous rotating fluid by a forcing effect oscillating with a frequency σ′0 and travelling with a uniform speed U along a line inclined to the axis of rotation at an arbitrary angle α is studied following Lighthill's technique. It is shown how the far field changes with α and σ′0.For all σ′0 < 2Ω, except for σ′0 = 2Ω sin α (Ω being the angular velocity of the fluid), the forcing effect excites two systems of waves. When σ′0 → 2Ω sin α one of these systems spreads out, influencing the upstream side while the other shrinks in the downstream direction. This upstream influence is to the left or to the right of the line of motion of the forcing effect (the forcing line) according as σ′0 − 2Ω sin α[lg ] 0 and increases as σ′0 − 2Ω sin α decreases. For σ′0 > 2Ω there is only a single system propagating downstream. As α varies these systems undergo a kind of rotation retaining the main features. α ≠ 0 or ½π makes the pattern asymmetric about the forcing line while a non-zero σ′0 splits the steady-case identical wave systems into two, which are otherwise coincident.When σ′0 = 2Ω sin α the forcing effect excites straight unattenuated waves of fixed frequency travelling both ahead and behind in a ‘column’ parallel to the forcing line and enclosing it. Also there are two other systems, which propagate without penetrating into an upstream wedge. It is shown that this ‘column’ is the counterpart of the ‘Taylor column’.

Detached shear layers in a rotating fluid

1967 ◽  
Vol 29 (1) ◽  
pp. 39-60 ◽  
Author(s):  
R. Hide ◽  
C. W. Titman
Keyword(s):  
Empirical Relationship ◽  
Axisymmetric Flow ◽  
Axial Flow ◽  
Vertical Axis ◽  
Wave Pattern ◽  
Shear Layers ◽  
Standard Errors ◽  
Regular Pattern ◽  
Rotating Fluid ◽  
Axis Of Rotation

The occurrence of detached shear layers should, according to straightforward theoretical arguments, often characterize hydrodynamical motions in a rapidly rotating fluid. Such layers have been produced and studied in a very simple system, namely a homogeneous liquid of kinematical viscosity v filling an upright, rigid, cylindrical container mounted coaxially on a turn-table rotating at Ω0 rad/s about a vertical axis, and stirred by rotating about the same axis at Ω1 rad/s a disk of radius a cm and thickness b’ cm immersed in the liquid with its plane faces parallel to the top and bottom end walls of the container. By varying Ω0, Ω1 and a, ranges of Rossby number, the modulus of ε ≡ (Ω1 + Ω0)/½ (Ω1 + Ω0), from 0·01 to 0·3, and Ekman number, E ≡ 2v/a2(Ω1 + Ω0), from 10−5 to 5 × 10−4 were attained. Although the apparatus was axisymmetric, only when |ε| did not exceed a certain critical value, |εT|, was the flow characterized by the same property of symmetry about the axis of rotation. Otherwise, when |ε| > |εT|, non-axisymmetric flow occurred, having the form in planes perpendicular to the axis of rotation of a regular pattern of waves, M in number, when ε was positive, and of a blunt ellipse when ε was negative.The axial flow in the axisymmetric detached shear layer, and the uniform rate of drift of the wave pattern characterizing the non-axisymmetric flow when ε is positive, depend in relatively simple ways on ε and E. The dependence of|εT| on E can be expressed by the empirical relationship |εT| = AEn, where A = 16·8 ± 2·2 and n = 0·568 ± 0·013 (= (4/7) × (1·000 − (0·005 ± 0·023))!), standard errors, 25 determinations. M does not depend strongly on E but generally decreases with increasing ε.

scholarly journals On a model for internal waves in rotating fluids

2018 ◽  
Vol 3 (2) ◽  
pp. 627-648 ◽  
Author(s):  
A. Durán
Keyword(s):  
Internal Waves ◽  
Fluid Model ◽  
Well Posedness ◽  
Solitary Wave Solutions ◽  
Rotating Fluid ◽  
Rotating Fluids ◽  
Mathematical Properties ◽  
Gravity Effects ◽  
Two Fluid Model ◽  
Two Fluid

AbstractIn this paper a rotating two-fluid model for the propagation of internal waves is introduced. The model can be derived from a rotating-fluid problem by including gravity effects or from a nonrotating one by adding rotational forces in the dispersion balance. The physical regime of validation is discussed and mathematical properties of the new system, concerning well-posedness, conservation laws and existence of solitary-wave solutions, are analyzed.

Slow motion of a paraboloid of revolution in a rotating fluid

1958 ◽  
Vol 3 (4) ◽  
pp. 404-410 ◽  
Author(s):  
L. V. K. Viswanadha Sarma
Keyword(s):  
Rigid Body ◽  
Small Amplitude ◽  
Radial Displacement ◽  
Direct Consequence ◽  
Slow Motion ◽  
Uniform Motion ◽  
Rotating Fluid ◽  
Body Rotation ◽  
Axis Of Rotation ◽  
Paraboloid Of Revolution

The slow uniform motion, after an impulsive start from relative rest, of a paraboloid of revolution along the axis of a rotating fluid is investigated by using a perturbation method. The principal purpose of the note is to illustrate the mechanism by which the fluid is not subjected to any substantial radial displacement, which is a direct consequence of the requirement that the circulation round material circuits should be constant when the perturbation velocities due to the motion of the paraboloid remain small. It appears that the mechanism is an oscillatory one in which the distance between any fluid particle and the axis of rotation oscillates sinusoidally in time with small amplitude. As time progresses, the amplitude of the oscillation decays to zero everywhere except on the paraboloid. The ultimate motion is then a rigid body rotation everywhere except on the paraboloid and the axis of rotation, where the perturbation velocities continue to oscillate indefinitely with small amplitude.

The motion of a rising disk in a rotating axially bounded fluid for large Taylor number

1995 ◽  
Vol 291 ◽  
pp. 1-32 ◽  
Author(s):  
Marius Ungarish ◽  
Dmitry Vedensky
Keyword(s):  
Exact Solution ◽  
Boundary Value Problem ◽  
Equations Of Motion ◽  
Taylor Number ◽  
Rossby Number ◽  
Rotating Fluid ◽  
Asymptotic Results ◽  
Dual Integral Equations ◽  
Wall Distance ◽  
Taylor Column

The motion of a disk rising steadily along the axis in a rotating fluid between two infinite plates is considered. In the limit of zero Rossby number and with the disk in the middle position, the boundary value problem based on the linear, viscous equations of motion is reduced to a system of dual-integral equations which renders ‘exact’ solutions for arbitrary values of the Taylor number, Ta, and disk-to-wall distance, H (scaled by the radius of the disk). The investigation is focused on the drag and on the flow field when Ta is large (but finite) for various H. Comparisons with previous asymptotic results for ‘short’ and ‘long’ containers, and with the preceding unbounded-configuration ‘exact’ solution, provide both confirmation and novel insights.In particular, it is shown that the ‘free’ Taylor column on the particle appears for H > 0.08 Ta and attains its fully developed features when H > 0.25 Ta (approximately). The present drag calculations improve the compatibility of the linear theory with Maxworthy's (1968) experiments in short containers, but for the long container the claimed discrepancy with experiments remains unexplained.

A note on forward wakes in rotating fluids

1970 ◽  
Vol 42 (1) ◽  
pp. 219-223 ◽  
Author(s):  
K. Stewartson
Keyword(s):  
Angular Velocity ◽  
Azimuthal Angle ◽  
Rotating Fluid ◽  
Rotating Fluids ◽  
Uniform Velocity

The two studies by Professor Miles (1970a, b) on the motion of a rotating fluid past a body raise the important question of the determinancy of such flows, by theoretical arguments, which it seems worth while making more precise. Suppose we have a fluid which when undisturbed has a uniform velocityUin the directionOxand a uniform angular velocity Ω aboutOx. It is slightly disturbed, the resulting motion having velocity components (u+U,v, Ωr+w) relative to cylindrical polar axes (x,r, θ), centreOand in whichrmeasures distance fromOx, while θ is the azimuthal angle. Assuming thatu, v, ware sufficiently small for their squares and products to be neglected, and are independent of θ, the equations governing their behaviour reduce to

On hydromagnetic waves in a stratified rotating incompressible fluid

1969 ◽  
Vol 39 (2) ◽  
pp. 283-287 ◽  
Author(s):  
R. Hide
Keyword(s):  
Incompressible Fluid ◽  
Angular Speed ◽  
Density Stratification ◽  
Stratified Fluids ◽  
Rotating Fluid ◽  
Rotating Fluids ◽  
Hydrodynamic Behaviour ◽  
Dispersion Relationship ◽  
Hydromagnetic Waves ◽  
Made In

The dispersion relationship for plane hydromagnetic waves in a stratified rotating fluid (α) indicates that the well-known analogy between rotating fluids and stratified fluids in regard to their hydrodynamic behaviour does not extend to magnetohydrodynamic behaviour, and (b) lends credence to a certain conjecture made in a previous paper, namely that effects due to density stratification can be neglected when considering the dispersion relationship for free hydromagnetic oscillations of the Earth's core if the Brunt—Väisälä frequency is much less than twice the angular speed of the Earth's rotation.

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.

scholarly journals NUMERICAL CALCULATION OF SEICHE MOTIONS IN HARBOURS OF ARBITRARY SHAPE

1982 ◽  
Vol 1 (18) ◽  
pp. 11
Author(s):  
P. Gaillard
Keyword(s):  
Finite Element ◽  
Numerical Calculation ◽  
Arbitrary Shape ◽  
Wave Pattern ◽  
Offshore Structures ◽  
Experimental Investigations ◽  
Far Field ◽  
Method Of Calculation ◽  
Long Wave ◽  
Element Technique

A new method of calculation of wave diffraction around islands, offshore structures, and of long wave oscillations within offshore or shore-connected harbours is presented. The method is a combination of the finite element technique with an analytical representation of the wave pattern in the far field. Examples of application are given, and results are compared with other theoretical and experimental investigations.

scholarly journals Acoustic and inertial modes in planetary-like rotating ellipsoids

2020 ◽  
Vol 476 (2239) ◽  
pp. 20200131
Author(s):  
Jérémie Vidal ◽  
David Cébron
Keyword(s):  
Finite Element ◽  
Liquid Core ◽  
Normal Modes ◽  
Flow Dynamics ◽  
Rotating Fluid ◽  
Rotating Fluids ◽  
Dynamical Models ◽  
Planetary Interiors ◽  
Physical Insight ◽  
Insight Into

The bounded oscillations of rotating fluid-filled ellipsoids can provide physical insight into the flow dynamics of deformed planetary interiors. The inertial modes, sustained by the Coriolis force, are ubiquitous in rapidly rotating fluids and Vantieghem (2014, Proc. R. Soc. A , 470 , 20140093. doi:10.1098/rspa.2014.0093 ) pioneered a method to compute them in incompressible fluid ellipsoids. Yet, taking density (and pressure) variations into account is required for accurate planetary applications, which has hitherto been largely overlooked in ellipsoidal models. To go beyond the incompressible theory, we present a Galerkin method in rigid coreless ellipsoids, based on a global polynomial description. We apply the method to investigate the normal modes of fully compressible, rotating and diffusionless fluids. We consider an idealized model, which fairly reproduces the density variations in the Earth’s liquid core and Jupiter-like gaseous planets. We successfully benchmark the results against standard finite-element computations. Notably, we find that the quasi-geostrophic inertial modes can be significantly modified by compressibility, even in moderately compressible interiors. Finally, we discuss the use of the normal modes to build reduced dynamical models of planetary flows.

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