Numerical stability analysis of a vortex ring with swirl

Abstract

The linear instability of a vortex ring with swirl with Gaussian distributions of azimuthal vorticity and velocity in its core is studied by direct numerical simulation. The numerical study is carried out in two steps: first, an axisymmetric simulation of the Navier–Stokes equations is performed to obtain the quasi-steady state that forms a base flow; then, the equations are linearized around this base flow and integrated for a sufficiently long time to obtain the characteristics of the most unstable mode. It is shown that the vortex rings are subjected to curvature instability as predicted analytically by Blanco-Rodríguez & Le Dizès (J. Fluid Mech., vol. 814, 2017, pp. 397–415). Both the structure and the growth rate of the unstable modes obtained numerically are in good agreement with the analytical results. However, a small overestimation (e.g. 22 % for a curvature instability mode) by the theory of the numerical growth rate is found for some instability modes. This is most likely due to evaluation of the critical layer damping which is performed for the waves on axisymmetric line vortices in the analysis. The actual position of the critical layer is affected by deformation of the core due to the curvature effect; as a result, the damping rate changes since it is sensitive to the position of the critical layer. Competition between the curvature and elliptic instabilities is also investigated. Without swirl, only the elliptic instability is observed in agreement with previous numerical and experimental results. In the presence of swirl, sharp bands of both curvature and elliptic instabilities are obtained for $\unicode[STIX]{x1D700}=a/R=0.1$, where $a$ is the vortex core radius and $R$ the ring radius, while the elliptic instability dominates for $\unicode[STIX]{x1D700}=0.18$. New types of instability mode are also obtained: a special curvature mode composed of three waves is observed and spiral modes that do not seem to be related to any wave resonance. The curvature instability is also confirmed by direct numerical simulation of the full Navier–Stokes equations. Weakly nonlinear saturation and subsequent decay of the curvature instability are also observed.

Inviscid and viscous global stability of vortex rings

“…Note that these are typical eigenfunctions for vortex rings subjected to elliptic instability, which are also reported elsewhere (Gargan-Shingles et al 2016; Hattori et al 2019). A couple of eigenfunctions of the rotating modes that look qualitatively similar to the spiral modes of Hattori et al (2019) are shown in figure 8. We could track these modes only for the viscous Gaussian cases (see table 4), while neither for the corresponding inviscid cases nor for any of the equilibrated cases were these found.…”

“…Vortex rings are themselves unstable to azimuthal perturbations, whose inviscid asymptotic theories were developed in Widnall & Sullivan (1973) and Widnall, Bliss & Tsai (1974), for slender rings whose core radius , the ring radius. These were later extended to viscous flows (Eloy & Le Dizés 2001; Fukumoto & Hattori 2005), while viscous growth rates have also been extracted from direct numerical simulations (DNS) (Shariff, Verzicco & Orlandi 1994; Archer, Thomas & Coleman 2008; Hattori, Blanco-Rodríguez & Le Dizés 2019). In this work a detailed global, modal stability analysis of vortex rings for both inviscid and viscous flows is performed where one of the primary aims is to clarify some of the existing discrepancies between asymptotic theories and numerical simulations.…”

“…Gargan, Rudman & Ryan (2016) integrated the linearised Navier–Stokes equations beginning with an equilibrated velocity field due to a vortex ring, and extracted growth rates which were in between the DNS of Shariff et al (1994) and Archer et al (2008). Recently, Hattori et al (2019) considered linearised and full Navier–Stokes equations to study the evolution of base flow and perturbations for swirling vortex rings, where curvature instability was found to dominate at large and small values of and , respectively. Even though curvature instabilities are not explored in our work, a few of our unstable modes are similar to the spiral modes of Hattori et al (2019), as we show in § 3.3.…”

Structure and stability of Joukowski's rotor wake model

“…The effect of an axial flow in the vortex cores was analysed by Fukumoto & Miyazaki (1991). More recently, global temporal stability characteristics were also derived by Okulov (2004) and Okulov & Sørensen (2007) using Hardin's expressions. In all these works, the core size is assumed small and short-wavelength instabilities (Blanco-Rodríguez & Le Dizès 2016Hattori, Blanco-Rodríguez & Le Dizès 2019) that may develop in vortex cores are neglected.…”

Numerical realization of helical vortices: application to vortex instability

“…Such a procedure has also been employed in previous studies e.g. [42][43][44][45]. It should be noted that since no stabilization or restriction (aside from streamwise periodicity) is imposed on the flow, the baseflow may contain a low amplitude component of possibly unstable eigenmodes at the end of the adaptation stage.…”