Volume 9, Number 4

Volume 9, Number 4, 2004
Effective computations in modern dynamics

Riccardi G.
The two-dimensional motion at infinite Reynolds number of an elliptical, uniform vortex outside a circular cylinder is investigated. The analysis is performed under the main assumption that the vortex generation at the wall can be neglected: the boundary layer reduces to a vortex sheet lying on the body wall. The flow is invariant under rotations around the body, so that the system possesses three degrees of freedom and two first integrals of the motion are known. A qualitative analysis of the motion is carried out in a suitable phase plain and comparisons with numerical simulations are also discussed.
Citation: Riccardi G., Motion of an elliptical uniform vortex outside a circular cylinder, Regular and Chaotic Dynamics, 2004, vol. 9, no. 4, pp. 399-415
Sokolovskiy M. A.,  Verron J.
The problem of studying the motion of three vortex lines with arbitrary intensities in an unbounded two-dimensional finite-thickness layer of a homogeneous fluid is known [25], [9], [28], [1] to belong to the class of integrable problems. However, a complete classification of possible motions was constructed only recently [10], [28], [41]. In [40], [39], [20] a generalization is given for two-layer rotating fluid in the particular case determined by the conditions of (i) zero total circulation of vortices, and (ii) the equality of the intensities of two vortices. Here, the first of these restrictions is lifted.
Citation: Sokolovskiy M. A.,  Verron J., Dynamics of three vortices in a two-layer rotating fluid, Regular and Chaotic Dynamics, 2004, vol. 9, no. 4, pp. 417-438
Koshel' K. V.,  Stepanov D. V.
In this paper we consider chaotic advection in a barotropic inviscid unidirectional pulsating flow over a point topographic vortex located near a rectilinear boundary. The process of passive markers' transport from the vortical area into the flow-through region is examined. In particular, the evolution of the corresponding Poincaré sections as a function of the frequency of oscillations is studied. The optimal frequency for chaotic mixing caused by an external excitation is calculated. An approach to studying the chaotic advection mechanism and parameters for open systems with finite residence time trajectories is proposed. This approach is based on the distribution of time intervals necessary to carry markers into the flow-through region, and on the Lyapunov exponent, which can be calculated on a finite time interval. A classification of behavior of fluid particle trajectories is obtained by comparing the named parameters. It has been established that as the boundary influence increases, a new type of trajectories with large Lyapunov exponent and residence time occurs.
Citation: Koshel' K. V.,  Stepanov D. V., Some specific features of chaotization and transport in pulsating barotropic flow over a topographic point vortex near boundary, Regular and Chaotic Dynamics, 2004, vol. 9, no. 4, pp. 439-450
Elhefnawy A. F.,  Zakaria K.,  Sirwah M. A.
The nonlinear stability of magnetized standing waves on the plane interface separating two immiscible inviscid magnetic fluids in a cylindrical container is investigated by means of the method of the multiple scales. The fluid system is stressed by both a uniform magnetic field and a periodic acceleration normal to the free surface, with the effect of the surface tension between the two fluids taken into account. The modulation of the amplitudes and phases of the tow modes is represented by a system of nonlinear first-order differential equations. The different cases of resonance are studied. Numerical applications are achieved to indicate the stability of the different types of the fixed points. The trivial motion, a periodically modulated sinusoid (limit cycle), and a chaotically modulated sinusoid are discussed.
Citation: Elhefnawy A. F.,  Zakaria K.,  Sirwah M. A., Bifurcations of magnetized surface waves between two immiscible fluids inside a vertical cylindrical container, Regular and Chaotic Dynamics, 2004, vol. 9, no. 4, pp. 451-475
Maze G.,  Carton X.,  Lapeyre G.
The influence of an external strain (or shear) field on the evolution of two identical vortices is investigated in a two-dimensional incompressible fluid. Using point vortex modeling, two regimes of the vortex doublet (co-rotation and irreversible separation) are determined; the critical intensity of the large scale flow separating these two regimes for a given initial separation of vortices, is calculated. Finite-area effects are then considered for the vortices. The steady states of piecewise constant vortices are computed algebraically and numerically; positive strain (or shear) favors vortex deformation. This deformation has a dominant elliptical component. An elliptical model of two vortices confirms the point vortex model results for centroid trajectories, and the steady state model results concerning the influence of positive strain on vortex deformation. It also provides an estimate of critical merger distance in the presence of large scale flow. Finally, the finite-time, nonlinear evolution of the vortex doublet is simulated with a numerical code of the 2D vorticity equation. The various regimes (stationarity, merger, co-rotation, ejection) are classified in the plane of initial vortex separation and of external deformation. These regimes are analyzed, and the critical merger distance is evaluated for negative and positive external strain; the results are in agreement with the elliptical model prediction. Merger efficiency, defined as the ratio of final to initial vortex circulation, is computed; for the same initial distance, it is smaller for negative strain. It also depends in a more complex way of the initial vortex distance.
Citation: Maze G.,  Carton X.,  Lapeyre G., Dynamics of a 2D vortex doublet under external deformation, Regular and Chaotic Dynamics, 2004, vol. 9, no. 4, pp. 477-497
Pavec M.,  Carton X.
The parametric instability of a wall jet, with time-varying potential vorticity or transport of the baroclinic mean flow, is studied in a two-layer quasi-geostrophic model. This wall jet is composed of two superimposed strips of uniform potential vorticity, and the layer thicknesses are equal. The steady flow is stable with respect to short waves and its domain of linear instability grows with stratification. The time-dependent flow evolution is governed by a Hill equation which allows parametric instability. This instability indeed appears in numerical flow calculations. It is favored near the marginal stability curve of the steady flow. Near that curve, the evolution equation of the flow is calculated with a multiple time-scale expansion. This equation shows that for zero baroclinic transport of the mean flow, subcritical steady flows can be destabilized by flow oscillation, and supercritical steady flows can be stabilized by medium frequency oscillations. For finite baroclinic transport, this parametric instability vanishes in the limit of short waves or of long waves and narrow potential vorticity strips. Consequences for coastal flows in the ocean are drawn.
Citation: Pavec M.,  Carton X., Parametric instability of a two-layer wall jet, Regular and Chaotic Dynamics, 2004, vol. 9, no. 4, pp. 499-507
Kizner Z.,  Khvoles R.
Departing from the classical circular dipole and combining analytical and numerical methods, two kinds of distributed vortical solutions with noncircular boundaries are constructed. These are the translating elliptical dipole with a continuously differentiable vorticity field, and the rotating multipoles (tripole, quadrupole, pentapole etc) with one azimuthal mode in the far field. The features of the new solutions and their correspondence to observations are discussed.
Citation: Kizner Z.,  Khvoles R., Two variations on the theme of Lamb–Chaplygin: supersmooth dipole and rotating multipoles, Regular and Chaotic Dynamics, 2004, vol. 9, no. 4, pp. 509-518
Schmidt D. S.
In 1882 J. J. Thomson had claimed in his Adams prize essay "The motion of vortex rings" that a ring of seven vortices would be unstable. It was shown later that linear analysis can not decide stability in this case. In 1999 Cabral and Schmidt proved stability by calculating the higher order terms in the normal form of the Hamiltonian with the help of POLYPACK, a personal algebraic processor. The work is repeated here with the help of the more readily available computer algebra system MATHEMATICA.
Citation: Schmidt D. S., The stability of the Thomson heptagon, Regular and Chaotic Dynamics, 2004, vol. 9, no. 4, pp. 519-528

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