Stuart vortices on a hyperbolic sphere

2020 ◽  
Vol 61 (2) ◽  
pp. 023103
Author(s):  
Jongbin Yoon ◽  
Habin Yim ◽  
Sun-Chul Kim
Keyword(s):  
Stuart Vortices

Effects of the Coriolis force on the stability of Stuart vortices

1998 ◽  
Vol 356 ◽  
pp. 353-379 ◽  
Author(s):  
STÉPHANE LEBLANC ◽  
CLAUDE CAMBON
Keyword(s):  
Coriolis Force ◽  
Three Dimensional ◽  
Basic Mechanism ◽  
Short Wave ◽  
Stagnation Points ◽  
Floquet Modes ◽  
The Stability ◽  
Simple Flows ◽  
Stuart Vortices ◽  
Wave Breakdown

A detailed investigation of the effects of the Coriolis force on the three-dimensional linear instabilities of Stuart vortices is proposed. This exact inviscid solution describes an array of co-rotating vortices embedded in a shear flow. When the axis of rotation is perpendicular to the plane of the basic flow, the stability analysis consists of an eigenvalue problem for non-parallel versions of the coupled Orr–Sommerfeld and Squire equations, which is solved numerically by a spectral method. The Coriolis force acts on instabilities as a ‘tuner’, when compared to the non-rotating case. A weak anticyclonic rotation is destabilizing: three-dimensional Floquet modes are promoted, and at large spanwise wavenumber their behaviour is predicted by a ‘pressureless’ analysis. This latter analysis, which has been extensively discussed for simple flows in a recent paper (Leblanc & Cambon 1997) is shown to be relevant to the present study. The basic mechanism of short-wave breakdown is a competition between instabilities generated by the elliptical cores of the vortices and by the hyperbolic stagnation points in the braids, in accordance with predictions from the ‘geometrical optics’ stability theory. On the other hand, cyclonic or stronger anticyclonic rotation kills three-dimensional instabilities by a cut-off in the spanwise wavenumber. Under rapid rotation, the Stuart vortices are stabilized, whereas inertial waves propagate.

Stuart-type vortices on a rotating sphere

2019 ◽  
Vol 865 ◽  
pp. 1072-1084 ◽  
Author(s):  
A. Constantin ◽  
V. S. Krishnamurthy
Keyword(s):  
Large Scale ◽  
Inviscid Flow ◽  
Explicit Solutions ◽  
Perturbative Approach ◽  
Rotating Sphere ◽  
Water Flows ◽  
Shallow Water Flows ◽  
Ocean Gyres ◽  
Stuart Vortices ◽  
Insight Into

Stuart vortices are among the few known smooth explicit solutions of the planar Euler equations with a nonlinear vorticity, and they have a counterpart for inviscid flow on the surface of a fixed sphere. By means of a perturbative approach we adapt the method used to investigate Stuart vortices on a fixed sphere to provide insight into some large-scale shallow-water flows on a rotating sphere that model the dynamics of ocean gyres.

On Kelvin–Stuart vortices in a viscous fluid

2008 ◽  
Vol 366 (1876) ◽  
pp. 2717-2728 ◽  
Author(s):  
L.E Fraenkel
Keyword(s):  
Reynolds Number ◽  
Viscous Fluid ◽  
Small Time ◽  
Navier Stokes ◽  
Initial Condition ◽  
Small Reynolds Number ◽  
The One ◽  
Stuart Vortices ◽  
Reversed Time

When one contemplates the one-parameter family of steady inviscid shear flows discovered by J. T. Stuart in 1967, an obvious thought is that these flows resemble a row of vortices diffusing in a viscous fluid, with the parameter playing the role of a reversed time. In this paper, we ask how close this resemblance is. Accordingly, the paper begins to explore Navier–Stokes solutions having as initial condition the classical, irrotational flow due to a row of point vortices. However, since we seek explicit answers, such exploration seems possible only in two relatively easy cases: that of small time and arbitrary Reynolds number and that of small Reynolds number and arbitrary time.

Three-dimensional destabilization of Stuart vortices: the influence of rotation and ellipticity

1999 ◽  
Vol 387 ◽  
pp. 205-226 ◽  
Author(s):  
P. G. POTYLITSIN ◽  
W. R. PELTIER
Keyword(s):  
Cross Sections ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Steady State Solution ◽  
Vorticity Distribution ◽  
Columnar Vortex ◽  
Elliptical Instability ◽  
The Stability ◽  
Dimensional Instability ◽  
Stuart Vortices

We investigate the influence of the ellipticity of a columnar vortex in a rotating environment on its linear stability to three-dimensional perturbations. As a model of the basic-state vorticity distribution, we employ the Stuart steady-state solution of the Euler equations. In the presence of background rotation, an anticyclonic vortex column is shown to be strongly destabilized to three-dimensional perturbations when background rotation is weak, while rapid rotation strongly stabilizes both anticyclonic and cyclonic columns, as might be expected on the basis of the Taylor–Proudman theorem. We demonstrate that there exist three distinct forms of three-dimensional instability to which strong anticyclonic vortices are subject. One form consists of a Coriolis force modified form of the ‘elliptical’ instability, which is dominant for vortex columns whose cross-sections are strongly elliptical. This mode was recently discussed by Potylitsin & Peltier (1998) and Leblanc & Cambon (1998). The second form of instability may be understood to constitute a three-dimensional inertial (centrifugal) mode, which becomes the dominant mechanism of instability as the ellipticity of the vortex column decreases. Also evident in the Stuart model of the vorticity distribution is a third ‘hyperbolic’ mode of instability that is focused on the stagnation point that exists between adjacent vortex cores. Although this short-wavelength cross-stream mode is much less important in the spectrum of the Stuart model than it is in the case of a true homogeneous mixing layer, it nevertheless does exist even though its presence has remained undetected in most previous analyses of the stability of the Stuart solution.

scholarly journals Effects of an axial flow on the centrifugal, elliptic and hyperbolic instabilities in Stuart vortices

2014 ◽  
Vol 758 ◽  
pp. 565-585 ◽  
Author(s):  
Manikandan Mathur ◽  
Sabine Ortiz ◽  
Thomas Dubos ◽  
Jean-Marc Chomaz
Keyword(s):  
Axial Velocity ◽  
Local Stability ◽  
Numerical Solutions ◽  
Axial Flow ◽  
Axial Direction ◽  
Threshold Value ◽  
Base Flow ◽  
Centrifugal Instability ◽  
Axisymmetric Vortex ◽  
Stuart Vortices

AbstractLinear stability of the Stuart vortices in the presence of an axial flow is studied. The local stability equations derived by Lifschitz & Hameiri (Phys. Fluids A, vol. 3 (11), 1991, pp. 2644–2651) are rewritten for a three-component (3C) two-dimensional (2D) base flow represented by a 2D streamfunction and an axial velocity that is a function of the streamfunction. We show that the local perturbations that describe an eigenmode of the flow should have wavevectors that are periodic upon their evolution around helical flow trajectories that are themselves periodic once projected on a plane perpendicular to the axial direction. Integrating the amplitude equations around periodic trajectories for wavevectors that are also periodic, it is found that the elliptic and hyperbolic instabilities, which are present without the axial velocity, disappear beyond a threshold value for the axial velocity strength. Furthermore, a threshold axial velocity strength, above which a new centrifugal instability branch is present, is identified. A heuristic criterion, which reduces to the Leibovich & Stewartson criterion in the limit of an axisymmetric vortex, for centrifugal instability in a non-axisymmetric vortex with an axial flow is then proposed. The new criterion, upon comparison with the numerical solutions of the local stability equations, is shown to describe the onset of centrifugal instability (and the corresponding growth rate) very accurately.

A perturbation study of particle dynamics in a plane wake flow

1999 ◽  
Vol 384 ◽  
pp. 1-26 ◽  
Author(s):  
T. J. BURNS ◽  
R. W. DAVIS ◽  
E. F. MOORE
Keyword(s):  
Bluff Body ◽  
Reynolds Numbers ◽  
Stokes Number ◽  
Wake Flow ◽  
Spherical Particles ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Focusing Effect ◽  
Stuart Vortices ◽  
Far Wake

We analyse the dynamics of small, rigid, dilute spherical particles in the far wake of a bluff body under the assumption that the background flow field is approximated by a periodic array of Stuart vortices that can be considered to be a regularization of the von Kármán vortex street. Using geometric singular perturbation theory and numerical methods, we show that when inertia (measured by a dimensionless Stokes number) is not too large, there is a periodic attractor in the phase space of the dynamical system governing the particle motion. We argue that this provides a simple mechanism to explain the unexpected ‘focusing’ effect that has been observed both numerically and experimentally in the far-wake flow past a bluff body by Tang et al. (1992). Their results show that over a range of Reynolds numbers and intermediate values of the Stokes number, particles injected into the wake of a bluff body concentrate near the edges of the vortex structures downstream, thus tending to ‘demix’ rather than disperse homogeneously.

Stuart vortices on a beta-plane

1995 ◽  
Vol 22 (4) ◽  
pp. 213-238 ◽  
Author(s):  
Roland Mallier
Keyword(s):  
Beta Plane ◽  
Stuart Vortices

The role of transverse secondary instabilities in the evolution of free shear layers

1989 ◽  
Vol 202 ◽  
pp. 367-402 ◽  
Author(s):  
G. P. Klaassen ◽  
W. R. Peltier
Keyword(s):  
Linear Stability ◽  
Initial Conditions ◽  
Shear Layers ◽  
Nonlinear Flow ◽  
Stability Analyses ◽  
Vortex Merging ◽  
Theoretical Predictions ◽  
High Reynolds ◽  
Secondary Instabilities ◽  
Stuart Vortices

Linear stability analyses and nonlinear flow simulations reveal several important features of transverse secondary instabilities of two-dimensional Kelvin–Helmholtz billows and Stuart vortices. Vortex pairing is found to be the most rapidly amplified mode in a continuous spectrum of vortex merging instabilities. In certain not uncommon circumstances it is possible for more than two vortices to amalgamate in a single interaction, demonstrating that the phenomenon that has become known as the pairing resonance in fact has a rather low quality factor. Another form of merging instability in which a vortex is deformed and drained by its neighbours has been revealed by our linear stability analyses of nonlinear shear-layer disturbances. It appears, however, that this vortex draining instability may be important only in unstratified or very weakly stratified flows, since in moderately stratified Kelvin–Helmholtz flow, it is replaced by a highly localized instability which leads to a temporary distortion of the braids. Nonlinear simulations of vortex merging events in moderately stratified, high-Reynolds-number shear layers are compared to the theoretical predictions of our stability analyses. We investigate and quantify the sensitivity of merging events to variations in the initial conditions. The character of the flow after merging instability saturates and the nonlinear aspects of multiple merging events are also considered.

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