Multipolar spherical and cylindrical vortices

Abstract

Multipolar spherical solutions to the three-dimensional steady vorticity equation are provided. These solutions are based on the separation of radial and angular contributions in terms of the spherical Bessel functions and vector spherical harmonics, respectively. In this set of multipolar vortex solutions, the Hicks–Moffatt swirling vortex is categorized as a vortex of degree ${\ell }=1$ and therefore as a vortex dipole. This swirling vortex is the three-dimensional dipole in spherical geometry equivalent to the two-dimensional Lamb–Chaplygin dipole in polar geometry. The three-dimensional dipole solution admits two linearly superposable solutions. The first one is a Trkalian flow and the second one is a cylindrical solid-body rotation with swirl. The higher ${\ell }>1$ multipolar vortices found are either vanishing-helicity vortices or Trkalian flow vortices. The multipolar Trkalian flows admit two circular polarizations given by the sign of the wavenumber $k$ . It is also found that piecewise vortex solutions, consisting of interior rotational and exterior potential flow domains, satisfying velocity continuity conditions at the vortex boundary, are possible in the general multipolar Trkalian spherical vortex. A particular polarized dipole solution in three-dimensional cylindrical geometry, consisting as well in the superposition of a Trkalian flow and a rigid motion, is also analysed. This swirling vortex may be interpreted as the three-dimensional dipole in cylindrical geometry equivalent to the two-dimensional Lamb–Chaplygin dipole in polar geometry.

Exact solutions of time-dependent oscillations in multipolar spherical vortices

“…These solutions may be interpreted as time and space oscillations, with spherical geometry, embedded in a cylindrical constant flow with swirl, and are characterized as inertial oscillations in background flow. These time-dependent azimuthal oscillating velocity solutions are a generalization of the steady three-dimensional multipolar vortex solutions given in Viúdez (2022), which are recovered in the case of vanishing time dependence (m𝔴 = 0). The necessary and sufficient condition for the existence of the inertial waves is a double condition: the flows experience inertial waves as long as they have a non-vanishing azimuthal wavenumber (m / = 0) in the presence of a background rotation (angular speed 𝔴 / = 0).…”

“…Also, the indices , m will generally be omitted from the symbols U , etc. The oscillating function (2.1) is divergence-free (∇ • U = 0), and when 𝔴 → 0, equals the steady multipolar vortex solutions given in Viúdez (2022). We notice that in the general time-dependent case (m𝔴 / = 0), the time-dependent function (2.1) is not a solution of the vorticity equation (1.1).…”

“…In three-dimensional flows, a similar decomposition of the velocity field into spherical Bessel functions for the radial () and angular () and spherical harmonics , in the particular case and , lead to the discovery of the Hicks–Moffatt spherical vortex (Hicks 1899; Moffatt 1969, 2017), whose limit for vanishing radial wavenumber is Hill's spherical vortex (Hill 1894). These steady solutions were generalized recently to multipolar spherical vortices with any degree and order (Viúdez 2022). These steady-state, or rigidly translating, vortex solutions are further generalized in this work.…”

Three-Dimensional Unsteady Axisymmetric Viscous Beltrami Vortex Solutions to the Navier–Stokes Equations

“…Steady axisymmetric inviscid Beltramian solutions in bounded or unbounded space have been found by focusing on the so-called Bragg-Hawthorne equation [17,22,23]. How the structure of the fundamental modes admitted by the linear equation varies by depending on the coordinate system chosen for solving the equation has been revealed in [22,24]. Non-axisymmetric solutions with vanishing helicity, i.e., inner products of velocity and vorticity, as well as the vortices with a constant proportionality coefficient, c hereafter, are presented in [24].…”

“…How the structure of the fundamental modes admitted by the linear equation varies by depending on the coordinate system chosen for solving the equation has been revealed in [22,24]. Non-axisymmetric solutions with vanishing helicity, i.e., inner products of velocity and vorticity, as well as the vortices with a constant proportionality coefficient, c hereafter, are presented in [24]. A Beltrami flow whose c is constant is called a Trkalian flow.…”