Nonlinear stability of axisymmetric swirling flows

1988 ◽  
Vol 326 (1590) ◽  
pp. 327-354 ◽  
Keyword(s):  
Poisson Bracket ◽  
Linear Stability ◽  
Nonlinear Stability ◽  
Equations Of Motion ◽  
Sufficient Conditions ◽  
Swirling Flows ◽  
Two Dimensional ◽  
Stability Studies ◽  
Vortex Density

We establish sufficient conditions for the nonlinear stability of incompressible, inviscid, swirling flows using the Arnol’d energy-Casimir method. We derive an axisymmetric Lie-Poisson bracket and work with equations of motion in swirl-function-vortex-density form. The flows and perturbations we consider may have axial variations. The formulation is closely analogous to that of two-dimensional, stratified, Boussinesq flows considered by Abarbanel et al . [ Phil. Trans. R. Soc. Lond. A 318, 349-409 (1986)) and a high-wavenumber cut off is necessary to overcome indefiniteness, as in that case. We give several examples of columnar swirling flows and discuss the relation of our results to linear stability studies of swirling flows.

scholarly journals THE CONSERVED CHARGES AND INTEGRABILITY OF THE CONFORMAL AFFINE TODA MODELS

1994 ◽  
Vol 09 (30) ◽  
pp. 2783-2801 ◽  
Author(s):  
H. ARATYN ◽  
L. A. FERREIRA ◽  
J. F. GOMES ◽  
A. H. ZIMERMAN
Keyword(s):  
Poisson Bracket ◽  
Equations Of Motion ◽  
Curvature Form ◽  
Two Dimensional ◽  
Conserved Charges ◽  
Zero Curvature ◽  
Algebraic Properties ◽  
Toda Model ◽  
Infinite Sets ◽  
Poisson Commute

We construct infinite sets of local conserved charges for the conformal affine Toda model. The technique involves the abelianization of the two-dimensional gauge potentials satisfying the zero-curvature form of the equations of motion. We find two infinite sets of chiral charges and apart from two lowest spin charges, all the remaining ones do not possess chiral densities. Charges of different chiralities Poisson commute among themselves. We discuss the algebraic properties of these charges and use the fundamental Poisson bracket relation to show that the charges conserved in time are in involution. Connections to other Toda models are established by taking particular limits.

scholarly journals Dynamics of constrained many body problems in constant curvature two-dimensional manifolds

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170425 ◽  
Author(s):  
Andrzej J. Maciejewski ◽  
Maria Przybylska
Keyword(s):  
Constant Curvature ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Rigid Bodies ◽  
The Body ◽  
Two Dimensional ◽  
Many Body ◽  
Finite Dimensional ◽  
Necessary And Sufficient

In this paper, we investigate systems of several point masses moving in constant curvature two-dimensional manifolds and subjected to certain holonomic constraints. We show that in certain cases these systems can be considered as rigid bodies in Euclidean and pseudo-Euclidean three-dimensional spaces with points which can move along a curve fixed in the body. We derive the equations of motion which are Hamiltonian with respect to a certain degenerated Poisson bracket. Moreover, we have found several integrable cases of described models. For one of them, we give the necessary and sufficient conditions for the integrability. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

Two-dimensional leaps in near-critical flow over isolated orography

2005 ◽  
Vol 461 (2064) ◽  
pp. 3747-3763 ◽  
Author(s):  
E.R Johnson ◽  
G.G Vilenski
Keyword(s):  
Unsteady Flow ◽  
Equations Of Motion ◽  
Fold Increase ◽  
Flow Speed ◽  
Two Dimensional ◽  
Subcritical Flow ◽  
One Dimensional ◽  
Petviashvili Equation ◽  
Lee Wave ◽  
Supercritical Flows

This paper describes steady two-dimensional disturbances forced on the interface of a two-layer fluid by flow over an isolated obstacle. The oncoming flow speed is close to the linear longwave speed and the layer densities, layer depths and obstacle height are chosen so that the equations of motion reduce to the forced two-dimensional Korteweg–de Vries equation with cubic nonlinearity, i.e. the forced extended Kadomtsev–Petviashvili equation. The distinctive feature noted here is the appearance in the far lee-wave wake behind obstacles in subcritical flow of a ‘table-top’ wave extending almost one-dimensionally for many obstacles widths across the flow. Numerical integrations show that the most important parameter determining whether this wave appears is the departure from criticality, with the wave appearing in slightly subcritical flows but being destroyed by two-dimensional effects behind even quite long ridges in even moderately subcritical flow. The wave appears after the flow has passed through a transition from subcritical to supercritical over the obstacle and its leading and trailing edges resemble dissipationless leaps standing in supercritical flow. Two-dimensional steady supercritical flows are related to one-dimensional unsteady flows with time in the unsteady flow associated with a slow cross-stream variable in the two-dimensional flows. Thus the wide cross-stream extent of the table-top wave appears to derive from the combination of its occurrence in a supercritical region embedded in the subcritical flow and the propagation without change of form of table-top waves in one-dimensional unsteady flow. The table-top wave here is associated with a resonant steepening of the transition above the obstacle and a consequent twelve-fold increase in drag. Remarkably, the table-top wave is generated equally strongly and extends laterally equally as far behind an axisymmetric obstacle as behind a ridge and so leads to subcritical flows differing significantly from linear predictions.

On the application of the Kelvin–Arnol'd energy principle to the stability of forced two-dimensional inviscid flows

1998 ◽  
Vol 356 ◽  
pp. 221-257 ◽  
Author(s):  
P. A. DAVIDSON
Keyword(s):  
Magnetic Field ◽  
Nonlinear Stability ◽  
Broad Class ◽  
Mhd Flow ◽  
Two Dimensional ◽  
Energy Principle ◽  
Inviscid Flows ◽  
Virtual Displacement ◽  
Euler Flows ◽  
Universal Procedure

Arnol'd developed two distinct yet closely related approaches to the linear stability of Euler flows. One is widely used for two-dimensional flows and involves constructing a conserved functional whose first variation vanishes and whose second variation determines the linear (and nonlinear) stability of the motion. The second method is a refinement of Kelvin's energy principle which states that stable steady Euler flows represent extremums in energy under a virtual displacement of the vorticity field. The conserved-functional (or energy-Casimir) method has been extended by several authors to more complex flows, such as planar MHD flow. In this paper we generalize the Kelvin–Arnol'd energy method to two-dimensional inviscid flows subject to a body force of the form −ϕ∇f. Here ϕ is a materially conserved quantity and f an arbitrary function of position and of ϕ. This encompasses a broad class of conservative flows, such as natural-convection planar and poloidal MHD flow with the magnetic field trapped in the plane of the motion, flows driven by electrostatic forces, swirling recirculating flow, self-gravitating flows and poloidal MHD flow subject to an azimuthal magnetic field. We show that stable steady motions represent extremums in energy under a virtual displacement of ϕ and of the vorticity field. That is, d1E=0 at equilibrium and whenever d2E is positive or negative definite the flow is (linearly) stable. We also show that unstable normal modes must have a spatial structure which satisfies d2E=0. This provides a single stability test for a broad class of flows, and we describe a simple universal procedure for implementing this test. In passing, a new test for linear stability is developed. That is, we demonstrate that stability is ensured (for flows of the type considered here) whenever the Lagrangian of the flow is a maximum under a virtual displacement of the particle trajectories, the displacement being of the type normally associated with Hamilton's principle. A simple universal procedure for applying this test is also given. We apply our general stability criteria to a range of flows and recover some familiar results. We also extend these ideas to flows which are subject to more than one type of body force. For example, a new stability criterion is obtained (without the use of Casimirs) for natural convection in the presence of a magnetic field. Nonlinear stability is also considered. Specifically, we develop a nonlinear stability criterion for planar MHD flows which are subject to isomagnetic perturbations. This differs from previous criteria in that we are able to extend the linear criterion into the nonlinear regime. We also show how to extend the Kelvin–Arnol'd method to finite-amplitude perturbations.

EQUIVALENCE OF TWO-DIMENSIONAL GRAVITIES

1990 ◽  
Vol 05 (16) ◽  
pp. 1251-1258 ◽  
Author(s):  
NOUREDDINE MOHAMMEDI
Keyword(s):  
Gauge Theory ◽  
Explicit Solution ◽  
Light Cone ◽  
Equations Of Motion ◽  
Two Dimensional ◽  
Induced Gravity ◽  
Light Cone Gauge ◽  
Dimensional Gravity ◽  
The Relationship ◽  
Chern Simons

We find the relationship between the Jackiw-Teitelboim model of two-dimensional gravity and the SL (2, R) induced gravity. These are shown to be related to a two-dimensional gauge theory obtained by dimensionally reducing the Chern-Simons action of the 2+1 dimensional gravity. We present an explicit solution to the equations of motion of the auxiliary field of the Jackiw-Teitelboim model in the light-cone gauge. A renormalization of the cosmological constant is also given.

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