On the Stability of Stationary Rotation of a Point Vortices System in a Circular Domain

2015 ◽  
pp. 68-73
Author(s):  
Leonid Gennadievich Kurakin ◽  
◽  
Andrei Petrovich Melekhov ◽  
Irina Vladimirovna Ostrovskaya ◽  
◽  
...  
Keyword(s):  
Circular Domain ◽  
Point Vortices ◽  
Stationary Rotation ◽  
The Stability

scholarly journals On the equilibrium and stability of a row of point vortices

1995 ◽  
Vol 290 ◽  
pp. 167-181 ◽  
Author(s):  
Hassan Aref
Keyword(s):  
Vortex Dynamics ◽  
Stability Problem ◽  
Regular Polygon ◽  
Symmetric Matrix ◽  
Classical Problem ◽  
Point Vortices ◽  
Instability Modes ◽  
Systematic Enumeration ◽  
The Stability ◽  
Accessible Form

The equilibrium and stability of a single row of equidistantly spaced identical point vortices is a classical problem in vortex dynamics, which has been addressed by several investigators in different ways for at least a century. Aspects of the history and the essence of these treatments are traced, stating some in more accessible form, and pointing out interesting and apparently new connections between them. For example, it is shown that the stability problem for vortices in an infinite row and the stability problem for vortices arranged in a regular polygon are solved by the same eigenvalue problem for a certain symmetric matrix. This result also provides a more systematic enumeration of the basic instability modes. The less familiar theory of equilibria of a finite number of vortices situated on a line is also recalled.

scholarly journals On the action of viscosity in increasing the spacing ratio of a vortex street

1936 ◽  
Vol 154 (881) ◽  
pp. 67-89 ◽  
Keyword(s):  
Viscous Fluid ◽  
Point Vortices ◽  
Vortex Street ◽  
Von Karman ◽  
Spacing Ratio ◽  
The Subject ◽  
The Stability ◽  
Von Kármán ◽  
The Ideal ◽  
Stable Vortex

1—von Kármán, by considering two parallel rows (of indefinite extent) of isolated, equal, point-vortices existing in a non-viscous fluid, has shown that the only stable vortex arrangement is the asymmetrical staggered one; and then only provided that the geometry of the system is such that h / a = 0·281, where h = width between the rows, and a = distance between consecutive vortices in one row. Since von Kármán’s investigation was published, writers on the subject have attempted to connect up the street with an obstacle producing it; and to investigate the effect of channel walls upon the stability and spacing ratio of the ideal street. At the same time efforts have been made to verify von Kármán’s spacing prediction by experiment, and to check the theoretical conclusions concerning the effect of parallel walls ;§ but the results have been far from satisfactory.

scholarly journals THE RESEARCH OF STABILITY OF STATIONARY ROTATION A ROTOR SYSTEM WITH A LIQUID, THE AXLE OF WHICH IS LOCATED IN ANISOTROPIC FIXINGS

2021 ◽  
Vol 83 (3) ◽  
pp. 324-334
Author(s):  
D.N. Derendyaev ◽  
N.V. Derendyaev
Keyword(s):  
Liquid Surface ◽  
Original Method ◽  
Rotor System ◽  
Stability Regions ◽  
Stationary Rotation ◽  
Rotation Mode ◽  
The Stability ◽  
Rotation Constant ◽  
Degree Of Instability ◽  
Rotor Axis

Earlier, one of the authors proposed and developed (together with coworkers) an original method to study the stability of stationary rotation of rotary systems containing a viscous liquid and having a drive that maintains the angular velocity of rotation constant. It was assumed that the rotor has axial symmetry, the anchors of its axis are isotropic. The method is based on two theorems, according to which a change in the degree of instability is associated with the possibility of a perturbed motion of the circular precession type. This motion has a remarkable property: the velocity field and the shape of the liquid surface do not depend on time in a specially selected non-inertial reference frame associated with the line of centers. Finding the conditions for the feasibility of circular precession makes it possible to effectively construct the boundaries of the stability regions of the stationary rotation regime in the space of problem parameters. In addition, the study of the occurrence of circular precession allows us to find the conditions under which a subcritical (supercritical) Andronov-Hopf bifurcation takes place in the rotor system and to identify "dangerous" (“safe”) sections of the boundaries of the stability regions. In this paper, the previously proposed method of stability research applies to systems in which the rotor axis is located in anisotropic Laval type anchors. In the study of rotary systems of this type, it is possible to link the change in the degree of instability with the feasibility of perturbed movements of the elliptical precession type. It can be shown that the imaginary characteristic numbers of the equations in deviations from the stationary rotation mode are possible only in the case when there is a perturbed motion in the form of an elliptical precession. An example of a study of the stability of stationary rotation of a typical rotary system is given. Mechanical effects caused by the fact that gyroscopic stabilization becomes impossible with anisotropic fixing of the rotor axis are noted.

scholarly journals DESIGN OF THE BOUNDARIES OF THE STABILITY REGIONS THE STATIONARY ROTATION MODE OF A ROTARY SYSTEM WITH A LIQUID, THE AXLE OF WHICH IS LOCATED IN ANISOTROPIC FIXINGS

2021 ◽  
Vol 83 (3) ◽  
pp. 335-343
Author(s):  
D.N. Derendyaev ◽  
N.V. Derendyaev
Keyword(s):  
Original Method ◽  
Dimensionless Frequency ◽  
Negative Effects ◽  
Stability Regions ◽  
Rotor Systems ◽  
Stationary Rotation ◽  
Opposite Situation ◽  
The Stability ◽  
Permissible Deviation ◽  
Rotor Axis

Earlier, the authors generalized the original method for studying the stability of stationary rotation of rotor systems containing a viscous incompressible fluid, the axis of which is located in isotropic anchors, in the case when the viscoelastic anchors of the axis of the rotor system are anisotropic. The generalization is based on two theorems that say that finding the stability conditions of such systems is associated with the possibility of elliptical precession-type motion, and with such motion there is a special non-inertial reference frame in which the hydrodynamic elements of the system periodically change in time. The study of such movements allows us to construct the boundaries of regions with different degrees of instability, in particular, the boundaries of the stability regions of the stationary rotation regime in the parameter space of the problem. The boundaries of the stability regions are constructed for cases when the anchoring of the rotor axis is anisotropic. In the space of the anchorage parameters, a parametrically defined D-curve is obtained as a function of the dimensionless frequency of the rotor precession. The two most interesting cases are considered – anisotropic stiffness of anchors (damping is isotropic in this case) and the opposite situation: isotropic stiffness of anchors with anisotropic damping. The obtained results are compared with the known results for the case of isotropic anchoring of the rotor axis. It is shown that the anisotropy of anchors, which is always present in real rotary systems due to the imperfection of technologies for the production of anchors, does not lead to negative effects. Moreover, using the obtained D-curves, it is possible to obtain technological tolerances for the production of fasteners, using what is known as the permissible deviation of the stiffness or damping value along the axes.

scholarly journals The finite-dipole dynamical system

2012 ◽  
Vol 468 (2146) ◽  
pp. 3006-3026 ◽  
Author(s):  
Andrew A. Tchieu ◽  
Eva Kanso ◽  
Paul K. Newton
Keyword(s):  
Equations Of Motion ◽  
Point Vortices ◽  
Inviscid Fluid ◽  
Straight Line ◽  
Single Dipole ◽  
Gradient Based ◽  
Finite Distance ◽  
The Difference ◽  
The Stability ◽  
Induced Velocity

The notion of a finite dipole is introduced as a pair of equal and opposite strength point vortices (i.e. a vortex dipole) separated by a finite distance. Equations of motion for N finite dipoles interacting in an unbounded inviscid fluid are derived from the modified interaction of 2 N independent vortices subject to the constraint that the inter-vortex spacing of each constrained dipole, ℓ, remains constant. In the absence of all other dipoles and background flow, a single dipole moves in a straight line along the perpendicular bisector of the line segment joining the two point vortices comprising the dipole, with a self-induced velocity inversely proportional to ℓ. When more than one dipole is present, the velocity of the dipole centre is the sum of the self-induced velocity and the average of the induced velocities on each vortex comprising the pair due to all the other dipoles. Each dipole orients in the direction of shear gradient based on the difference in velocities on each of the two vortices in the pair. Several numerical experiments are shown to illustrate the interactions between two and three dipoles in abreast and tandem configurations. We also show that equilibria (multi-poles) can form as a result of the interactions, and we study the stability of polygonal equilibria, showing that the N =3 case is linearly stable, whereas the N >3 case is linearly unstable.

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