System Of Vortex Rings Research Articles

On the Topology of the Atmosphere Advected by a Periodic Array of Axisymmetric Thin-cored Vortex Rings

The fluid motion produced by a periodic array of identical, axisymmetric, thin-cored vortex rings is investigated. It is well known that such an array moves uniformly without change of shape or form in the direction of the central axis of symmetry, and is therefore an equilibrium solution of Euler's equations. In a frame of reference moving with the system of vortex rings, the motion of passive fluid particles is investigated as a function of the two non-dimensional parameters that define this system: $\varepsilon = a/R$, the ratio of minor radius to major radius of the torus-shaped vortex rings, and $\lambda = L/R$, the separation of the vortex rings normalized by their radii. Two bifurcations in the streamline topology are found that depend significantly on $\varepsilon$ and $\lambda$; these bifurcations delineate three distinct shapes of the 'atmosphere' of fluid particles that move together with the vortex ring for all time. Analogous to the case of an isolated vortex ring, the atmospheres can be 'thin-bodied' or 'thick-bodied'. Additionally, we find the occurrence of a 'connected' system, in which the atmospheres of neighboring rings touch at an invariant ring of fluid particles that is stationary in a frame of reference moving with the rings.

Aerodynamic Load Estimation of Helicopter Rotor in Hovering Flight

The aerodynamic characteristics of a helicopter rotor blade are highly dependent on the wake induced flow. The rotor blades, which are rotating wings, shed vortices which trail in helical path along the axis of rotation forming a wake. The velocity field induced by the vortex system influences the blade loading. The free wake model analysis is computationally intensive and numerically unstable in very low speed applications. For the aerodynamic analysis of a hovering rotor, Miller and Reddy have proposed simple wake models in which the spiral vortex wake is replaced by a system of vortex rings or vortex line elements in conjunction with lifting line theory to overcome some of the problems encountered in free-wake models. In the Proposed model of the rotor, the same lifting line model of the rotor is retained, but a further simplification of the vortex wake is proposed. This method requires relatively very small computing time compared to the above models, but without necessarily sacrificing the accuracy in estimating the aerodynamic blade loading.

Problem of the motion of annular vortices in an ideal fluid

The study discusses problems in the numerical modeling of axisymmetric flows of an incompressible fluid induced by a continuous vortex region or vortex sheet by means of a discrete system of vortex rings.

Hamiltonian description of axisymmetric vortex flows and the system of vortex rings

The Hamiltonian equations for the trajectories of fluid particles in axisymmetric vortex flows are obtained. On the basis of these equations, the system of thin vortex rings is considered. The problem of the interaction of two vortex rings has a general analytic solution. The problems of stochastization of the system of three and more vortex rings as well as stochastization of trajectories of fluid particles in the system of two and more vortex rings (with appropriate initial conditions) are discussed.

Thrust on a Piston Driven Half-Open Tube

The motion of the gas in and near a tube open at one end and closed at the other by a piston driven in nearly sinusoidal motion is investigated. Experiments disclose that when the frequency of the piston motion is near the fundamental acoustic resonance frequency of the gas column an intermittent air jet associated with a system of vortex rings is expelled from the tube's open end. This results in an appreciable average thrust on the tube which, however, is much less than might be expected from purely acoustic considerations. A theoretical analysis has been made which takes account of the intermittent vortex and jet formation, as well as of the acoustic radiation field. The analysis leads to formulas for the average thrust and power dissipation and to a relation between the driving frequency and the amplitude of the gas motion at the open end. An empirical parameter κ which represents the fraction of energy dissipated in vortex motion appears in these formulas. It is found possible to evaluate this parameter in terms of the observed thrust at resonance. From this datum the curve of thrust versus driving frequency is calculated and is found to agree well with experiment.

The stability of an infinite system of circular vortices

A system of equal circular vortex filaments have their centres evenly spaced along a straight line, and their planes at right angles to this line. The present investigation is concerned with the stability or instability of such an arrangement. The corresponding problem in two dimensions has been dealt with by Kármán who considered the case of two infinite trails of parallel rectilinear vortices, with the object of applying his results to the resistance of an infinite cylinder moving in a fluid and to the state of motion in the rear of the cylinder. The infinite system of circular vortex filaments, on the other hand, may be supposed, in certain circumstances to be discussed in a later paper, to be generated in the rear of a three-dimensional body in motion in a fluid. The present investigation may therefore be regarded as a first step towards an examination of the three-dimensional problem analogous to that treated by Kármán for two dimensions. A special difficulty arises in an investigation of this type from the fact that the system of vortex rings, possessing an infinite number of degrees of freedom, are capable of adopting an infinite number of possible configurations about the position of equilibrium.