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.

This paper presents an investigation of the stability of a trailing vortex using mean flow profiles given by an approximate solution of the Navier-Stokes equations. The axial and tangential velocity profiles obtained from this solution, deduced by Moore and Saffman [“Axial flow in laminar trailing vortices,” Proc. R. Soc. London, Ser. A 333, 491–508 (1973)], agree well with experiments involving wings at slight angles of attack. In particular, the Moore-Saffman profiles better describe the jet-like and wake-like axial flows near the center of the vortex than does the much-studied Batchelor vortex. We determine solutions numerically for these profiles and find that they are well suited to describe the flow at short and intermediate distances behind the wingtip. Growth rates for unstable perturbations are presented for different values of n, the wingtip loading parameter. These growth rates are shown to be somewhat larger than those obtained for the Batchelor vortex, and instability persists for larger values of the swirl. The largest amplification rates were found to occur near n = 0.5, the value corresponding to elliptic loading. This is within the range 0.44 < n < 1.0, where the core axial flow is wake-like. For n < 0.44, the flow in the vortex core is jet-like and the growth rates of unstable perturbations become progressively smaller, with all modes damped for n ≈ 0.25.

We investigate the eigenvalue problem governing the propagation of long nonlinear surface waves when there is a currentū(y) beneath the surface, y being the vertical coordinate. The amplitude of such waves evolves according to the KdV equation and it was proved by Burns [1] that their speed of propagation c is such that there is no critical layer (i.e., c lies outside the range ofū(y)). If, however, the critical layer is nonlinear, the result of Burns does not necessarily apply because the phase change of linear theory then vanishes.In this paper, we consider specific velocity profiles and determine c as a function of Froude number for modes with nonlinear critical layers. Such modes do not always exist, the case of the asymptotic suction profileū = 1 − e −y being a notable example. We find, however, that singular modes can be obtained for boundary layer profiles of the Falkner-Skan similarity type, including the Blasius case. These and other examples are treated and we examine singular solutions of the Rayleigh equation to gain insight about the long wave limit of such solutions.