Influence of a wall on the three-dimensional dynamics of a vortex pair

In this paper, we are interested in perturbed vortices under the influence of a wall or ground plane. Such flows have relevance to aircraft wakes in ground effect, to ship hull junction flows, to fundamental studies of turbulent structures close to a ground plane and to vortex generator flows, among others. In particular, we study the vortex dynamics of a descending vortex pair, which is unstable to a long-wave instability (Crow, AIAA J., vol. 8 (12), 1970, pp. 2172–2179), as it interacts with a horizontal ground plane. Flow separation on the wall generates opposite-sign secondary vortices which in turn induce the ‘rebound’ effect, whereby the primary vortices rise up away from the wall. Even small perturbations in the vortices can cause significant topological changes in the flow, ultimately generating an array of vortex rings which rise up from the wall in a three-dimensional ‘rebound’ effect. The resulting vortex dynamics is almost unrecognizable when compared with the classical Crow instability. If the vortices are generated below a critical height over a horizontal ground plane, the long-wave instability is inhibited by the wall. We then observe two modes of vortex–wall interaction. For small initial heights, the primary vortices are close together, enabling the secondary vortices to interact with each other, forming vertically oriented vortex rings in what we call a ‘vertical rings mode’. In the ‘horizontal rings mode’, for larger initial heights, the Crow instability develops further before wall interaction; the peak locations are farther apart and the troughs closer together upon reaching the wall. The proximity of the troughs to each other and the wall increases vorticity cancellation, leading to a strong axial pressure gradient and axial flow. Ultimately, we find a series of small horizontal vortex rings which ‘rebound’ from the wall. Both modes comprise two small vortex rings in each instability wavelength, distinct from Crow instability vortex rings, only one of which is formed per wavelength. The phenomena observed here are not limited to the above perturbed vortex pairs. For example, remarkably similar phenomena are found where vortex rings impinge obliquely with a wall.

Crow instability: nonlinear response to the linear optimal perturbation

The potential for anticipated destruction of a counter-rotating vortex pair using the linear optimal perturbation of the Crow instability is assessed. Direct numerical simulation is used to study the development of the Crow instability and the subsequent evolution of the flow up to 30 characteristic times at a circulation-based Reynolds number of 1000. The conventional development of the instability leads to multiple contortions of the vortices including the linear growth of sinusoidal deformation, vortex linking and the formation of vortex rings. A new evolution stage is identified, succeeding this well-established sequence: the vortex rings undergo periodic oscillation. Two complete periods are simulated during which the vortical system is hardly altered, thereby demonstrating the extraordinary resilience of the vortices. The possibility of preventing these dynamics using the linear optimal perturbation of the Crow instability, the adjoint mode, is analysed. By appropriately setting the forcing amplitude, the lifetime of the vortices until their loss of coherence is reduced to approximately 13 characteristic times, which is less than half that of the natural Crow behaviour observed with infinitesimal forcing. The dynamics of the flow induced by the linear optimal perturbation that enable this result are connected to processes already known to efficiently alter vortical flows, in particular transient growth and four-vortex dynamics.

CROW INSTABILITY IN UNITARY FERMI GAS

In this paper, we investigate the initiation and subsequent evolution of Crow instability in an inhomogeneous unitary Fermi gas using zero-temperature Galilei-invariant nonlinear Schrödinger equation. Considering a cigar-shaped unitary Fermi gas, we generate the vortex–antivortex pair either by phase-imprinting or by moving a Gaussian obstacle potential. We observe that the Crow instability in a unitary Fermi gas leads to the decay of the vortex–antivortex pair into multiple vortex rings and ultimately into sound waves.

Stability characteristics of a counter-rotating unequal-strength Batchelor vortex pair

A Batchelor vortex represents the asymptotic solution of a trailing vortex in an aircraft wake. In this study, an unequal-strength, counter-rotating Batchelor vortex pair is employed as a model of the wake emanating from one side of an aircraft wing; this model is a direct extension of several prior investigations that have considered unequal-strength Lamb–Oseen vortices as representations of the aircraft wake problem. Both solution of the linearized Navier–Stokes equations and direct numerical simulations are employed to study the linear and nonlinear development of a vortex pair with a circulation ratio of$\Lambda = \ensuremath{-} 0. 5$. In contrast to prior investigations considering a Lamb–Oseen vortex pair, we note strong growth of the Kelvin mode$[\ensuremath{-} 2, 0] $coupled with an almost equal growth rate of the Crow instability. Three stages of nonlinear instability development are defined. In the initial stage, the Kelvin mode amplitude becomes sufficiently large that oscillations within the core of the weaker vortex are easily observable and significantly affect the profile of the weaker vortex. In the secondary stage, filaments of secondary vorticity emanate from the weaker vortex and are convected around the stronger vortex. In the tertiary stage, a transition in the dominant instability wavelength is observed from the short-wavelength Kelvin mode to the longer-wavelength Crow instability. Much of the instability growth is observed on the weaker vortex of the pair, although small perturbations in the stronger vortex are observed in the tertiary nonlinear growth phase.

The instability of a vortex ring impinging on a free surface

Direct numerical simulation is used to study the development of a single laminar vortex ring as it impinges on a free surface directly from below. We consider the limiting case in which the Froude number approaches zero and the surface can be modelled with a stress-free rigid and impermeable boundary. We find that as the ring expands in the radial direction close to the surface, the natural Tsai–Widnall–Moore–Saffman (TWMS) instability is superseded by the development of the Crow instability. The Crow instability is able to further amplify the residual perturbations left by the TWMS instability despite being of differing radial structure and alignment. This occurs through realignment of the instability structure and shedding of a portion of its outer vorticity profile. As a result, the dominant wavenumber of the Crow instability reflects that of the TWMS instability, and is dependent upon the initial slenderness ratio of the ring. At higher Reynolds number a short-wavelength instability develops on the long-wavelength Crow instability. The wavelength of the short waves is found to vary around the ring dependent on the local displacement of the long waves.