Vortex Pair of Coaxial Helical Filaments
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VORTEX PAIR OF COAXIAL HELICAL FILAMENTS V. L. Okulov∗
UDC 532.5.01
Abstract: Possible existence of vortex pairs of coaxially rotating helical filaments with circulation in the opposite directions is theoretically studied. Such pairs are generated, e.g., by a rotating blade of a rotor with identical strengths of the tip and root vortices. Conditions that ensure uniform rotation of the vortex pair of helical filaments are found, which is needed for modeling vortex wakes behind rotor blades. Keywords: vortex dynamics, helical vortex, vortex pair, equilibrium rotation. DOI: 10.1134/S0021894420030049
INTRODUCTION Alekseenko and Shtork [1] published an important experimental result that confirms the existence of an equilibrium state for two helical filaments of identical strengths. The problem of equilibrium states of vortices is the classical problem of hydrodynamics; however, only the two-dimensional case has been considered for a long time [2], and the problem was reduced to determining equilibrium configurations, i.e., vortex polygons whose vertices contain point vortices with identical rotation and circulation strengths [3, 4]. This problem was initiated by the experiment where identical configurations for floating identical magnets were observed. These configurations are similar to equilibrium polygons of point vortices, which had to be studied for developing the vortex model of the atom [5]. Similar to the classical experiment on equilibrium of magnets, which formed the basis of the theory of equilibrium states of point vortices, the experimental study of Alekseenko and Shtork [1] initiated a series of theoretical and numerical investigations aimed at determining stable equilibrium configurations of helical vortices [6–9]. Stability of a multiplet of helical vortices with identical circulation strengths located uniformly on a cylindrical surface, forming a regular vortex polygon, and uniformly moving along the axis was theoretically studied [6]. Then the possibility of existence of an equilibrium configuration for two helical vortices with identical circulation strengths and different radii asymmetrically arranged with respect to the common central axis (asymmetric helical vortex doublet) was theoretically confirmed [9]. The existence of such an asymmetric doublet in addition to the symmetric doublet described in [1] was not confirmed experimentally. It should be noted that the classical formulation of problems of point vortices is not limited to the case of identical circulation strengths; another case of interest is the case of vortices with identical circulation strengths of the opposite signs. In the classical plane case, these two vortices are called the vortex pair; this term was introduced and described by Helmholtz earlier than the problem of vortex polygons. In [10], Helmholtz considered possible motions of a set of rectangular vortex filaments or two-dimensional point vortices. In particular, it follows from [10] that there is only one unique solution for point vortices, where the vortex pair with identical, but opp
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