Collapse of a homogeneous fluid mass in a stratified fluid

1969 ◽  
pp. 321-330 ◽  
Author(s):  
C. C. Mei
Keyword(s):  
Stratified Fluid ◽  
Homogeneous Fluid ◽  
Fluid Mass

scholarly journals On the figure requisite to maintain the equilibrium of a homogeneous fluid mass that revolves upon an axis

1833 ◽  
Vol 2 ◽  
pp. 206-207
Keyword(s):  
Direct Analysis ◽  
Homogeneous Fluid ◽  
Mutual Attraction ◽  
Sir Isaac Newton ◽  
Isaac Newton ◽  
Fluid Mass

The author enumerates the various steps by which Sir Isaac Newton, M c Laurin, and Laplace have carried the theory of the equilibrium of a revolving fluid very near to perfection, but he observes that they have generally supposed the spheroid to differ but little from a sphere; and he proceeds in the present paper to investigate the figure “by a direct analysis, in which no arbitrary supposition is admitted.” Mr. Ivory thinks it necessary to distinguish carefully two separate cases; the first is when the particles of the fluid do not attract one another, and the second when the particles are endued with attractive powers. These, he says, are plainly two cases that are essentially different from one another; for in the first, a stratum added induces no other change than an increase of pressure caused by the action of the accelerating forces at the surface; but in the second, besides the pressure, a new force is introduced, arising from the mutual attraction between the matter of the stratum and the fluid mass to which it is added.

scholarly journals Steady boundary-layer solutions for a swirling stratified fluid in a rotating cone

1999 ◽  
Vol 384 ◽  
pp. 339-374 ◽  
Author(s):  
R. E. HEWITT ◽  
P. W. DUCK ◽  
M. R. FOSTER
Keyword(s):  
Boundary Layer ◽  
Stokes Equations ◽  
Swirling Flow ◽  
Rotating Disk ◽  
Stratified Fluid ◽  
Steady States ◽  
General Description ◽  
Homogeneous Fluid ◽  
Axisymmetric Solution ◽  
Steady Solutions

We consider a set of nonlinear boundary-layer equations that have been derived by Duck, Foster & Hewitt (1997a, DFH), for the swirling flow of a linearly stratified fluid in a conical container. In contrast to the unsteady analysis of DFH, we restrict attention to steady solutions and extend the previous discussion further by allowing the container to both co-rotate and counter-rotate relative to the contained swirling fluid. The system is governed by three parameters, which are essentially non-dimensional measures of the rotation, stratification and a Schmidt number. Some of the properties of this system are related (in some cases rather subtly) to those found in the swirling flow of a homogeneous fluid above an infinite rotating disk; however, the introduction of buoyancy effects with a sloping boundary leads to other (new) behaviours. A general description of the steady solutions to this system proves to be rather complicated and shows many interesting features, including non-uniqueness, singular solutions and bifurcation phenomena.We present a broad description of the steady states with particular emphasis on boundaries in parameter space beyond which steady states cannot be continued.A natural extension of this work (motivated by recent experimental results) is to investigate the possibility of solution branches corresponding to non-axisymmetric boundary-layer states appearing as bifurcations of the axisymmetric solutions. In an Appendix we give details of an exact, non-axisymmetric solution to the Navier–Stokes equations (with axisymmetric boundary conditions) corresponding to the flow of homogeneous fluid above a rotating disk.

Motion of an axisymmetric body in a rotating stratified fluid confined between two parallel planes

1969 ◽  
Vol 38 (4) ◽  
pp. 833-842 ◽  
Author(s):  
D. V. Krishna ◽  
L. V. Sarma
Keyword(s):  
Steady State ◽  
Oblate Spheroid ◽  
Vertical Axis ◽  
Vertical Gradient ◽  
Steady State Solution ◽  
Stratified Fluid ◽  
Axisymmetric Body ◽  
The Body ◽  
Homogeneous Fluid ◽  
Axis Of Rotation

We consider here the flow due to the oscillation of a slender oblate spheroid in a non-homogeneous, rotating fluid confined between two parallel planes which are perpendicular to the (vertical) axis of rotation. The direction of oscillation of the spheroid is perpendicular to the axis of rotation. By solving a set of dual integrals the steady-state solution is obtained in the two cases when the plates are at an infinite distance from the body and when they are at a large but finite distance. The singular or discontinuous surfaces observed in the case of homogeneous fluid are absent here. Also, the steady-state velocity is no longer independent of the distance along the axis of rotation. The velocity has now a vertical gradient, an important feature in the case of stratified fluid. It is also found that the presence of the plane boundaries increases the force on the body.

scholarly journals Three-dimensional stability of a horizontally sheared flow in a stably stratified fluid

2007 ◽  
Vol 570 ◽  
pp. 297-305 ◽  
Author(s):  
AXEL DELONCLE ◽  
JEAN-MARC CHOMAZ ◽  
PAUL BILLANT
Keyword(s):  
Froude Number ◽  
Dimensional Stability ◽  
Numerical Study ◽  
Three Dimensional ◽  
Stratified Fluid ◽  
Two Dimensional ◽  
Horizontal Shear ◽  
Homogeneous Fluid ◽  
Sheared Flow ◽  
Vertical Scales

This paper investigates the three-dimensional stability of a horizontal flow sheared horizontally, the hyperbolic tangent velocity profile, in a stably stratified fluid. In an homogeneous fluid, the Squire theorem states that the most unstable perturbation is two-dimensional. When the flow is stably stratified, this theorem does not apply and we have performed a numerical study to investigate the three-dimensional stability characteristics of the flow. When the Froude number, Fh, is varied from ∞ to 0.05, the most unstable mode remains two-dimensional. However, the range of unstable vertical wavenumbers widens proportionally to the inverse of the Froude number for Fh ≪ 1. This means that the stronger the stratification, the smaller the vertical scales that can be destabilized. This loss of selectivity of the two-dimensional mode in horizontal shear flows stratified vertically may explain the layering observed numerically and experimentally.

Some aspects of the initial-value problem for the inviscid motion of a contained, rotating, weakly-stratified fluid

1971 ◽  
Vol 46 (1) ◽  
pp. 1-23 ◽  
Author(s):  
J. S. Allen
Keyword(s):  
Initial Value Problem ◽  
Initial Conditions ◽  
Stratified Fluid ◽  
Rotational Frequency ◽  
Rotating Fluid ◽  
Initial Value ◽  
Homogeneous Fluid ◽  
Relationship Of ◽  
The Relationship ◽  
Rotating Stratified Fluid

The initial-value problem for the linear, inviscid motion of a contained, rotating stratified fluid is considered in the limit of weak stratification, that is, for small values of the stratification parameter S = N2/Ω2, where N is the Brunt–Väisälä frequency and Ω is the rotational frequency. The limiting flow is of interest because, although the initial-value problem has been studied, both for the case of a homogeneous, rotating fluid and for the case of a stratified, rotating fluid, the exact relationship of the two flows, in the limit of vanishing stratification, is not straightforward. For example, the method of determining, from the initial conditions, the steady geostrophic component of the flow of a rotating, stratified fluid does not in general give a motion that reduces, in the limit S → 0, to the steady component of the flow of a homogeneous fluid. By including a consideration of slow unsteady motions that vary on a time scale dependent on the stratification parameter, the relationship of the limiting flow to the flow of a homogeneous fluid is established.

scholarly journals Experimental evidence for a new instability of a vertical columnar vortex pair in a strongly stratified fluid

2000 ◽  
Vol 418 ◽  
pp. 167-188 ◽  
Author(s):  
PAUL BILLANT ◽  
JEAN-MARC CHOMAZ
Keyword(s):  
Turbulent Flows ◽  
Three Dimensional ◽  
Vortex Pair ◽  
Reynolds Numbers ◽  
Stratified Fluid ◽  
Viscous Effects ◽  
Homogeneous Fluid ◽  
Cross Sectional ◽  
Columnar Vortex ◽  
New Type

This paper shows that a long vertical columnar vortex pair created by a double flap apparatus in a strongly stratified fluid is subjected to an instability distinct from the Crow and short-wavelength instabilities known to occur in homogeneous fluid. This new instability, which we name zigzag instability, is antisymmetric with respect to the plane separating the vortices. It is characterized by a vertically modulated twisting and bending of the whole vortex pair with almost no change of the dipole's cross- sectional structure. No saturation is observed and, ultimately, the vortex pair is sliced into thin horizontal layers of independent pancake dipoles. For the largest Brunt–Väisälä frequency N = 1.75 rad s−1 that may be achieved in the experiments, the zigzag instability is observed only in the range of Froude numbers: 0.13 < Fh0 < 0.21 (Fh0 = U0/NR, where U0 and R are the initial dipole travelling velocity and radius). When Fh0 > 0.21, the elliptic instability develops resulting in three-dimensional motions which eventually collapse into a relaminarized vortex pair. Irregular zigzags are then also observed to grow. The threshold for the inhibition of the elliptic instability Fh0 = 0.2±0.01 is independent of N and in good agreement with the theoretical study of Miyazaki & Fukumoto (1992). Complete stabilization for Fh0 < 0.13 is probably due to viscous effects since the associated Reynolds number is low, Re0 < 260. In geophysical flows characterized by low Froude numbers and large Reynolds numbers, we conjecture that this viscous stabilization will occur at much lower Froude number.It is tentatively argued that this new type of instability may explain the layering widely observed in stratified turbulent flows.

scholarly journals Dynamics of a Turbulent Buoyant Plume in a Stratified Fluid: An Idealized Model of Subglacial Discharge in Greenland Fjords

2017 ◽  
Vol 47 (10) ◽  
pp. 2611-2630 ◽  
Author(s):  
Ekaterina Ezhova ◽  
Claudia Cenedese ◽  
Luca Brandt
Keyword(s):  
Internal Waves ◽  
Lower Layer ◽  
Subsurface Layer ◽  
Turbulent Jets ◽  
Stratified Fluid ◽  
Buoyant Plume ◽  
Homogeneous Fluid ◽  
Idealized Model ◽  
Turbulent Entrainment ◽  
Large Eddy

AbstractThis study reports the results of large-eddy simulations of an axisymmetric turbulent buoyant plume in a stratified fluid. The configuration used is an idealized model of the plume generated by a subglacial discharge at the base of a tidewater glacier with an ambient stratification typical of Greenland fjords. The plume is discharged from a round source of various diameters and characteristic stratifications for summer and winter are considered. The classical theory for the integral parameters of a turbulent plume in a homogeneous fluid gives accurate predictions in the weakly stratified lower layer up to the pycnocline, and the plume dynamics are not sensitive to changes in the source diameter. In winter, when the stratification is similar to an idealized two-layer case, turbulent entrainment and generation of internal waves by the plume top are in agreement with the theoretical and numerical results obtained for turbulent jets in a two-layer stratification. In summer, instead, the stratification is more complex and turbulent entrainment by the plume top is significantly reduced. The subsurface layer in summer is characterized by a strong density gradient and the oscillating plume generates internal waves that might serve as an indicator of submerged plumes not penetrating to the surface.

Unstable jets generated by a sphere descending in a very strongly stratified fluid

2019 ◽  
Vol 867 ◽  
pp. 26-44 ◽  
Author(s):  
Shinsaku Akiyama ◽  
Yusuke Waki ◽  
Shinya Okino ◽  
Hideshi Hanazaki
Keyword(s):  
Froude Number ◽  
Particle Image ◽  
Stratified Fluid ◽  
Turbulent Jet ◽  
Homogeneous Fluid ◽  
Image Velocimetry ◽  
Low Froude Number ◽  
Long Time ◽  
Short Time ◽  
Unstable States

The flow around a sphere descending at constant speed in a very strongly stratified fluid ($Fr\lesssim 0.2$) is investigated by the shadowgraph method and particle image velocimetry. Unlike the flow under moderately strong stratification ($Fr\gtrsim 0.2$), which supports a thin cylindrical jet, the flow generates an unstable jet, which often develops into turbulence. The transition from a stable jet to an unstable jet occurs for a sufficiently low Froude number $Fr$ that satisfies $Fr/Re<1.57\times 10^{-3}$. The Froude number $Fr$ here is in the range of $0.0157<Fr<0.157$ or lower, while the Reynolds number $Re$ is in the range of $10\lesssim Re\lesssim 100$ for which the homogeneous fluid shows steady and axisymmetric flows. Since the radius of the jet can be estimated by the primitive length scale of the stratified fluid, i.e. $l_{\unicode[STIX]{x1D708}}^{\ast }=\sqrt{\unicode[STIX]{x1D708}^{\ast }/N^{\ast }}$ or $l_{\unicode[STIX]{x1D708}}^{\ast }/2a^{\ast }=\sqrt{Fr/2Re}$, this predicts that the jet becomes unstable when it becomes thinner than approximately $l_{\unicode[STIX]{x1D708}}^{\ast }/2a^{\ast }=0.028$, where $N^{\ast }$ is the Brunt–Väisälä frequency, $a^{\ast }$ the radius of the sphere and $\unicode[STIX]{x1D708}^{\ast }$ the kinematic viscosity of the fluid. The instability begins when the boundary-layer thickness becomes thin, and the disturbances generated by shear instabilities would be transferred into the jet. When the flow is marginally unstable, two unstable states, i.e. a meandering jet and a turbulent jet, can appear. The meandering jet is thin with a high vertical velocity, while the turbulent jet is broad with a much smaller velocity. The meandering jet may persist for a long time, or develop into a turbulent jet in a short time. When the instability is sufficiently strong, only the turbulent jet could be observed.

The spin down of rotating stratified fluids

1971 ◽  
Vol 47 (4) ◽  
pp. 689-711 ◽  
Author(s):  
W. L. Siegmann
Keyword(s):  
Rotation Axis ◽  
Vertical Wall ◽  
Steady Motion ◽  
Density Perturbation ◽  
Stratified Fluid ◽  
Homogeneous Fluid ◽  
Ekman Layers ◽  
Two Factors ◽  
Small Change ◽  
Spherical Container

The transient process by which an incompressible dissipative rotating stratified fluid adjusts to a small change in the rotation rate of its container is examined theoretically. The aim is to clarify the effects of the imposed density stratification and of the boundary condition specified for the density perturbation on the behaviour of the fluid, particularly during the time span when the adjustment is performed in a homogeneous fluid. For a weakly stratified fluid in a cylinder, it is shown how these two factors govern the nature and intensity of boundary layers on the vertical wall which close the secondary meridional circulation generated by Ekman layers along the horizontal boundaries. For a more strongly stratified fluid, the usefulness and importance of potential vorticity conservation in determining the quasi-steady motion is verified, and a calculation for a spherical container demonstrates some new features that arise only when the container boundaries are not normal or parallel to the rotation axis. It is shown that experimental results of Holton (1965) are in less good agreement with predictions of the linear theory than had been previously indicated.

Spin-up of a stratified fluid: theory and experiment

1971 ◽  
Vol 50 (3) ◽  
pp. 579-608 ◽  
Author(s):  
George Buzyna ◽  
George Veronis
Keyword(s):  
Experimental Study ◽  
Vertical Plane ◽  
Limited Range ◽  
Stratified Fluid ◽  
Theoretical Treatment ◽  
Homogeneous Fluid ◽  
Fluid Theory ◽  
Side Walls ◽  
Range Of Values ◽  
Normal Laboratory

Stratified spin-up, the process of adjustment of a uniformly rotating stratified fluid to an abrupt change in the rotation of the container, is important in many geophysical contexts. An experimental study of this process is presented here for the case where a linearly stratified salt solution is enclosed in a cylindrical container whose rotation rate is changed by a small amount. Results are presented for a limited range of values of B, the internal Froude number, which measures the ratio of the frequencies due to buoyancy and rotation. The experimental study is augmented by a theoretical treatment of idealized models which clarify the more fundamental physical processes that occur. The response of a stratified fluid is faster than that of a homogeneous fluid but the adjustment is limited to layers near the bottom and top boundaries the thickness of which is determined by the value of B. A comparison of the experimental results with the theories of Holton, Walin and Sakurai is also made and it is shown that for the present physical arrangement (insulated side walls) the theories of the latter two authors agree much more closely with experiment than does the theory of Holton. However, all three theories tend to over-estimate the azimuthal displacement in the regions near the upper and lower boundaries where the spin-up is most rapid. The Sweet-Eddington circulation, which accompanies the ideal state of rigid-body rotation, can be significant under normal laboratory conditions and it was necessary to correct some of the spin-up results for this effect. The circulation in the vertical plane is described qualitatively.

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