Elliptical instability of the Moore–Saffman model for a trailing wingtip vortex

Abstract

In this paper, we investigate the elliptical instability exhibited by two counter-rotating trailing vortices. This type of instability can be viewed as a resonance between two normal modes of a vortex and an external strain field. Recent numerical investigations have extended earlier results that ignored axial flow to include models with a simple wake-like axial flow such as the similarity solution found by Batchelor (J. Fluid Mech., vol. 20, 1964, pp. 645–658). We present herein growth rates of elliptical instability for a family of velocity profiles found by Moore & Saffman (Proc. R. Soc. Lond. A, vol. 333, 1973, pp. 491–508). These profiles have a parameter $n$ that depends on the wing loading. As a result, unlike the Batchelor vortex, they are capable of modelling both the jet-like and the wake-like axial flow present in a trailing vortex at short and intermediate distances behind a wingtip. Direct numerical simulations of the linearized Navier–Stokes equations are performed using an efficient spectral method in cylindrical coordinates developed by Matsushima & Marcus (J. Comput. Phys., vol. 53, 1997, pp. 321–345). We compare our results with those for the Batchelor vortex, whose velocity profiles are closely approximated as the wing loading parameter $n$ approaches 1. An important conclusion of our investigation is that the stability characteristics vary considerably with $n$ and $W_{0}$, a parameter measuring the strength of the mean axial velocity component. In the case of an elliptically loaded wing ($n=0.50$), we find that the instability growth rates are up to 50 % greater than those for the Batchelor vortex. Our results demonstrate the significant effect of the distribution and intensity of the axial flow on the elliptical instability of a trailing vortex.

Curvature instability of a curved Batchelor vortex

“…In the very large Reynolds number context of aeronautics, other models have been introduced to describe the vortices generated by wing tips (Moore & Saffman 1973;Spalart 1998). It would be interesting to analyse the occurrence of the curvature instability in these models as well as the competition with the elliptic instability (Fabre & Jacquin 2004a;Feys & Maslowe 2016 …”

Reconnection of vortex tubes with axial flow

“…Traditionally, most studies of anti-parallel vortex reconnection used vortex tubes in which the vortex lines are untwisted, being oriented parallel to the tube axes. However, it is also of physical interest to study the interaction of vortex tubes that individually have non-zero helicity -since for example wing-tip vortices are known to exhibit non-zero axial flow [13,14]. Introducing twist to the vortex tubes in the anti-parallel configuration, there are two options we can consider: both tubes having the same sign of twist, or twists of opposite signs.…”

Spectral/hp element methods: Recent developments, applications, and perspectives

“…Simulating and understanding vortical flows are vital in hydrodynamics and aerodynamics. In many applications, manipulation of vortices in the vicinity of flow boundaries is crucial for improving performance in engineering practice [7,85,86,87,88,89]. Developing a better understanding of the near wake of the vortex, lying within one chord length of the trailing edge, is essential in understanding the complex flow structure interactions [7], which is also crucial in understanding cavitation in vortical structures generated by hydro propellers [86].…”