ON ONE APPROACH TO DETERMINING THE VORTEX SOUND SOURCE

The work consists of five sections and a bibliographic list. The first section provides answers to questions about the relevance, the applied value of the study, as well as the need to develop new approaches that allow modeling vortex structures in engineering practice. In the second section, some mathematical models and approaches used to solve problems of vortex dynamics are considered. The third section is devoted to solving the problem of determining the main parameters of the flow in the core of a vortex ring for given geometric dimensions. It is shown that a turbulent vortex ring is obtained as a result of the interaction of two vortex columns. The fourth section is devoted to methods for characterizing a concentrated vortex as a source of acoustic vibrations. As an object of research, the flow in the core of a turbulent vortex ring is considered. It is assumed that the core of the vortex ring has the shape of a torus. An approach is proposed that makes it possible to establish a strict link between the main flow parameters and the shape of the vortex ring. The aim of this work is to obtain the flow parameters in the core of a vortex ring with their subsequent substitution into the acoustic-vortex equation to analyze the source of acoustic oscillations. It is also necessary to show the presence of a structure in the vortex ring corresponding to some point symmetry and, thus, to abandon the concept of the circular symmetry of the core of the vortex ring. The proposed approach is based on the assertion that a vortex ring can be represented as a set formed according to a “rule” that determines a spatial geometric shape. As a result, an approach was proposed for analyzing the vortex ring as a source of acoustic oscillations, and it was also formulated and theoretically substantiated that the core of a turbulent vortex ring having the shape of a torus can be considered as a result of the interaction of two vortex columns.

A generalized vortex ring model

A conventional laminar vortex ring model is generalized by assuming that the time dependence of the vortex ring thickness ℓ is given by the relation ℓ = atb, where a is a positive number and 1/4 ≤ b ≤ 1/2. In the case in which $a=\sqrt{2\nu}$, where ν is the laminar kinematic viscosity, and b = 1/2, the predictions of the generalized model are identical with the predictions of the conventional laminar model. In the case of b = 1/4 some of its predictions are similar to the turbulent vortex ring models, assuming that the time-dependent effective turbulent viscosity ν∗ is equal to ℓℓ′. This generalization is performed both in the case of a fixed vortex ring radius R0 and increasing vortex ring radius. In the latter case, the so-called second Saffman's formula is modified. In the case of fixed R0, the predicted vorticity distribution for short times shows a close agreement with a Gaussian form for all b and compares favourably with available experimental data. The time evolution of the location of the region of maximal vorticity and the region in which the velocity of the fluid in the frame of reference moving with the vortex ring centroid is equal to zero is analysed. It is noted that the locations of both regions depend upon b, the latter region being always further away from the vortex axis than the first one. It is shown that the axial velocities of the fluid in the first region are always greater than the axial velocities in the second region. Both velocities depend strongly upon b. Although the radial component of velocity in both of these regions is equal to zero, the location of both of these regions changes with time. This leads to the introduction of an effective radial velocity component; the latter case depends upon b. The predictions of the model are compared with the results of experimental measurements of vortex ring parameters reported in the literature.