Numerical realization of helical vortices: application to vortex instability

Abstract

The need to numerically represent a free vortex system arises frequently in fundamental and applied research. Many possible techniques for realizing this vortex system exist but most tend to prioritize accuracy either inside or outside of the vortex core, which therefore makes them unsuitable for a stability analysis considering the entire flow field. In this article, a simple method is presented that is shown to yield an accurate representation of the flow inside and outside of the vortex core. The method is readily implemented in any incompressible Navier-Stokes solver using primitive variables and Cartesian coordinates. It can potentially be used to model a wide range of vortices but is here applied to the case of two helices, which is of renewed interest due to its relevance for wind turbines and helicopters. Three-dimensional stability analysis is performed in both a rotating and a translating frame of reference, which yield eigenvalue spectra that feature both mutual inductance and elliptic instabilities. Comparison of these spectra with available theoretical predictions is used to validate the proposed baseflow model, and new insights into the elliptic instability of curved Batchelor vortices are presented. Furthermore, it is shown that the instabilities in the rotating and the translating reference frames have the same structure and growth rate, but different frequency. A relation between these frequencies is provided.

Nonlinear dynamics of two helical vortices: A dynamical system approach

“…When the radial displacement is small i.e. |η| ≪ 1, relation (7) reduces to η ′ = −η: the symmetry between displacements of the two vortices prevailing in section II is recovered. For larger amplitudes, however, figure 6 indicates that this top-bottom symmetry (that is outward versus inward vortex) is nearly preserved for L = 0.1 (figure 6a), but clearly broken for larger values of L (figures 6b and c).…”

“…For the vortex ring alley, this situation is seen to hold for very small values of the pitch, typically L ≤ 0.1. For larger values, amplitudes ± are asymmetric since equation (7) implies…”

“…These structures interact in a non trivial way, which eventually leads to wake destructuration. In order to understand such complex flows, it is pertinent to study the dynamics once the tip vortices are formed: indeed, the near wake made of N identical tip vortices can be used as a base flow [7,16,21,23]. The dynamics can then be analyzed in terms of linear and nonlinear evolution of perturbations evolving on top of this base flow.…”

Structure and stability of Joukowski's rotor wake model

“…They were able to calculate the spatial growth rate of individual modes through a spectral analysis of the nonlinear flow. A temporal linear stability analysis was recently performed by Selçuk, Delbende & Rossi (2018) and Brynjell-Rahkola & Henningson (2020), obtaining the growth rate of different unstable modes using time-stepping methods on the linearized Navier-Stokes equations.…”

Local and global pairing instabilities of two interlaced helical vortices

“…These authors performed a spectral analysis of the resulting non-linear flow and determined the spatial growth rates of the individual modes. The temporal linear stability problem was also recently revisited by Brynjell-Rahkola & Henningson (2017) and Selçuk et al (2018), who determined the growth rates and corresponding unstable modes of a helix configuration via time-stepping methods applied to the linearised Navier-Stokes equations.…”