Optimal amplification of the Crow instability

Abstract

A mechanism for promoting the Crow instability in a counter-rotating vortex pair is presented within the framework of linear dynamics. It consists of ͑i͒ the creation of a periodic array of vortex rings along the length of the vortices by stretching of vorticity at the leading hyperbolic point of the dipole, and ͑ii͒ the deformation of the vortices by the vortex rings leading to the Crow instability. A reduction of the characteristic time of the Crow instability by a factor of roughly 2 can be obtained by this mechanism.Vortex hazard caused by aircraft trailing vortices has gained much attention during the past decades with the advent of jumbo jets. In certain situations, vortex wakes collapse through a chain process including deformation of columnar vortices by the Crow instability, vortex linking, and turbulence. 1 One way to alleviate vortex hazard is to accelerate the process by exciting the intrinsic instabilities of the wake. This is the idea developed in this article, in which we optimize the energy of the Crow perturbation by means of an appropriate initial perturbation. The growth rate of the Crow instability was first derived by Crow. 2 Vortices of the pair deform by mutual induction and oscillate in a plane inclined at approximately 45°about the horizontal. The oscillations grow exponentially in amplitude until the point when the two vortices touch, leading to final collapse. Several studies such as that of Crow and Bate 3 showed that exciting the vortex pair at the wavelength of the mutual induction instability could be efficient in accelerating the chain process. Other studies such as those of Crouch 4 and Fabre, Jacquin, and Loof 5 showed by a vortex filament method that systems of four vortices exhibit much larger amplification rates than the Crow instability. While these previous studies arbitrarily specify the structure of the perturbation as vortex filaments, in this article we use a global stability method based on a finite-element discretization with a high number of degrees of freedom that presupposes no particular shape for the initial perturbation. In the case of a single Lamb-Oseen vortex, Antkowiak and Brancher 6 and Pradeep and Hussain 7 have already reported that the optimal perturbation takes the form of spirals of vorticity outside the vortex core, which suggests that a similar mechanism of amplification can be expected in the case of the dipole as far as the two vortices are not too close to each other. Yet this suggestion is partially hindered by the presence of two hyperbolic stagnation points ͑hereaf-ter simply referred to as hyperbolic points, "hyperbolic" standing for the hyperbolicity of the streamlines in the vicinity of these points, see Fig. 1͒ in the flow that are also known 8 to behave as energy amplifiers. The main objective of the study is to understand the roles that these two dynamics ͑that of the vortex and that of the hyperbolic point͒ play in the optimal amplification of the Crow instability.Base flow. The basic flow is a two-dimensional pair of counter-rotating vor...

Crow instability: nonlinear response to the linear optimal perturbation

“…Farrell (1988) established that the linear optimal perturbation of a particular flow corresponds to the adjoint of the most unstable mode of the flow. Brion, Sipp & Jacquin (2007) applied this principle to a counter-rotating vortex pair. The dominant instability of this configuration is the Crow instability (Crow 1970) which induces symmetrical sinusoidal displacement of the vortex cores.…”

“…The dominant instability of this configuration is the Crow instability (Crow 1970) which induces symmetrical sinusoidal displacement of the vortex cores. The linear optimal perturbation corresponding to the Crow instability established by Brion et al (2007) takes the form of vorticity sheets concentrated close to the plane separating the two vortices and produces considerable transient growth before reaching the classic linear Crow instability behaviour.…”

“…The aim of this paper is to study the nonlinear response to the linear optimal perturbation of a counter-rotating vortex pair determined by Brion et al (2007) using DNS. The paper is organised in the following manner.…”

Vortex Filament and Global Instability Analysis of the Crow Mode

“…Other works as Brion et al [1] have also compared results from the two methods, but less details are given. In this case, the comparison is studied more extensively in the range and amount of parameters.…”

Active Management of Flap-Edge Trailing Vortices

“…Recently, a similar approach was pursued with a view to exploiting the multiple vortex growth mechanisms created by an airplane on approach with flaps-down (Crouch el al, 2001). Another approach exploits transient growth or bypass mechanisms of the vortex pair (Pradeep and Hussain, 2006; Brion et al, 2007). Despite their inherent appeal, active methods must address issues such as "ride quality, dynamicload effects on the structure, and the ability to maintain control authority during operation" (Crouch, 2001).…”