Stochastic forcing of the Lamb–Oseen vortex

The aim of the present paper is to analyse the dynamics of the Lamb–Oseen vortex when continuously forced by a random excitation. Stochastic forcing is classically used to mimic external perturbations in realistic configurations, such as variations of atmospheric conditions, weak compressibility effects, wing-generated turbulence injected in aircraft wake, or free-stream turbulence in wind tunnel experiments. The linear response of the Lamb–Oseen vortex to stochastic forcing can be decomposed in relation to the azimuthal symmetry of the perturbation given by the azimuthal wavenumber m. In the axisymmetric case m = 0, we find that the response is characterised by the generation of vortex rings at the outer periphery of the vortex core. This result is consistent with recurrent observations of such dynamics in the study of vortex-turbulence interaction. When considering helical perturbations m = 1, the response at large axial wavelengths consists of a global translation of the vortex, a feature very similar to the phenomenon of vortex meandering (or wandering) observed experimentally, corresponding to an erratic displacement of the vortex core. At smaller wavelengths, we find that stochastic forcing can excite specific oscillating modes of the Lamb–Oseen vortex. More precisely, damped critical-layer modes can emerge via a resonance mechanism. For perturbations with higher azimuthal wavenumber m > 2, we find no structure that clearly dominates the response of the vortex

Linear and nonlinear dynamics of axisymmetric waves in the hollow core vortex

The dynamics of trailing vortices are under constant investigation during last decades since it is of considerable interest to reduce aircraft wakes and associated hazards to forthcoming planes. The isolated axisymmetric vortex is the commonly used simplest elementary model when considering such issue. Although asymptotically stable, recent studies have revealed its sensitiveness to specific perturbations, leading in some cases to considerable gains of energy. Albeit of evident interest, the underlying mechanisms of energy growth are considered in the linear regime. The nonlinear dynamics of such vortices need also to be considered in order to complete the picture. Rather than performing direct numerical simulations3, an interesting way to investigate it is to consider the nonlinear interactions of waves. This approach is motivated by the possible existence of resonance between wave components. For this purpose, the base flow model is simplified by considering the hollow core vortex. Arising naturally when a tank is drained (bath-tube vortex), it presents simpler dynamics than the Lamb-Oseen vortex as it only possesses two families of waves. This point is of crucial importance for the tractability of the problem. In this work, the nonlinear temporal evolution of axisymmetric waves are investigated through numerical integration when the flow is submitted to various initial conditions (travelling or standing wave, pinching of the free surface, wave trains). We focus on wave trains as important energy exchanges between the main component and its sideband waves are observed. This phenomenon is related to the Benjamin-Feir instability4 (triadic resonance) occurring for wave trains on deep water

On the existence and evolution of a spanwise vortex in laminar shallow water dipoles

The present work investigates the existence and evolution of a spanwise vortex at the front of shallow dipolar vortices. The vortex dipoles are experimentally generated using a double flap apparatus. Particle image velocimetry measurements are performed in a horizontal plane and in the vertical symmetry plane of the flow. The dynamics of such vortical structures is investigated through a parametric study in which both the Reynolds number Re=U0D0/ν∈[90,470] and the aspect ratio α = h/D0∈[0.075,0.7],associated with the shallowness of the flow, are varied, where U0 is the initial velocity of the vortex dipole, D0 is the initial diameter, h is the water depth, and v is the kinematic viscosity of the fluid. The present experiments confirm the numerical results obtained in a companion paper by Duran-Matute et al. [Phys. Fluids 22, 116606 (2010)], namely that the flow remains quasi parallel with negligible vertical motions below a critical value of the parameter α2Re. By contrast, for large values of α2Re and α≲0.6, a three-dimensional regime is observed in the shape of an intense spanwise vortex generated at the front of the dipole. The present study reveals that the early-time motion and dynamics of the spanwise vortex do not scale on the unique parameter α2Re but is strongly influenced by both the aspect ratio and the Reynolds number. A mechanism for the generation of the spanwise vortex is proposed. For α≳0.6, a third regime is observed, where the spanwise vortex is replaced by a vorticity tongu

On vortex rings around vortices: an optimal mechanism

Stable columnar vortices subject to hydrodynamic noise (\eg turbulence) present some recurrent behaviours like the systematic development of vortex rings at the periphery of the vortex core. This phenomenon still lacks a comprehensive explanation, partly because it is not associated to an instability \textit{stricto sensu}. The aim of the present paper is to identify the physical mechanism triggering this intrinsic feature of vortices using an optimal perturbation analysis as a tool of investigation. We found that the generation of vortex rings is linked to the intense and rapid amplification of specific disturbances in the form of azimuthal velocity streaks that eventually evolve into azimuthal vorticity rolls generated by the rotational part of the local Coriolis force. This evolution thus appears to follow a scenario opposite to the classical lift-up view, where rolls give rise to streaks

Transient energy growth for the Lamb-Oseen vortex

The transient evolution of infinitesimal flow disturbances whichoptimally induce algebraic growth in the Lamb-Oseen (gaussian) vortex is studied using a direct-adjoint technique. This optimal perturbation analysis reveals that the Lamb-Oseen vortex allows for intense amplification of kinetic energy for 2D and 3D perturbations of azimuthal wavenumber $m = 1$. In both cases, the disturbances experiencing the most growth initially take the form of concentrated spirals at the outer periphery of the vortex which rapidly excite bending waves within the vortex core. In the limit of large wavelengths, the optimal perturbation leads to arbitrary large growths via an original scenario combining the Orr mechanism with vortex induction

Topology and dynamics of the A-pillar vortex

The topology and dynamics of the flow bypassing an automobile A-pillar modeled by a 30° dihedron are investigated experimentally. The various components of the A-pillar flow are identified by means of low and high frequency particle image velocimetry. For each component, the time evolution of the position, displacement, vorticity magnitude, circulation, and fluctuating kinetic energy are analyzed along the A-pillar. More precisely, the flow by passing the A-pillar is composed of two vortex structures with different behaviors. The major structure grows in size, circulation magnitude, and total amount of fluctuating kinetic energy along the A-pillar, whereas the minor structure does not vary significantly. Finally, the displacement of the major structure is identified as a movement of precession and the influence of the A-pillar geometry is emphasized

Absolute and convective secondary instabilities in spatially periodic shear flows

The generic problem of the spatiotemporal instability of a periodic basic flow (Stuart vortices) is considered in order to interpret the sequence of bifurcations observed in open shear flows. Using a novel numerical technique, we show that the more concentrated the vortices, the smaller the backflow needed to trigger absolute instability. These results allow us to propose an alternative interpretation for the subharmonic resonance observed in forced shear flows, which is classically attributed to an acoustic feedbac

Self-sustained oscillations of a confined impinging jet

The present paper investigates the dynamics of a laminar plane jet impinging on a flat plate in a channel. An experimental parametric study is carried out to determine the flow regimes at different levels of confinement and Reynolds numbers. For very confined jets, the flow is steady whatever the Reynolds number. The overall structure of the flow is symmetric with respect to the jet axis and is characterized by the presence of recirculation zones at the channel walls. The dynamics is radically different for less confined jets. Above a critical Reynolds number, the flow bifurcates in the form of an oscillating flapping mode of the impinging jet. Analyses of the experimental results provide with a quantitative characterization of this regime in terms of amplitude, wavelength and frequency. This self-oscillating bifurcated flow induces strong sweepings of the target plate by the jet and intense vortex dipole ejections from the impacted wall. Such a regime is expected to be particularly useful in the enhancement of the local heat transfer at relatively low cost in terms of flow rate

Forçage stochastique du tourbillon de Lamb-Oseen

We analyse the dynamics of the Lamb-Oseen vortex when continuously excited by a stochastic forcing. This maintained forcing can thus model any perturbation such as variations of atmospheric conditions or turbulence generated by aircraft wings. The mechanisms of transient growth identified by Antkowiak & Brancher (2004, 2007) are retrieved thanks to this approach. In the axisymmetric case, vortex rings appear at the vortex outskirts via an "anti-lift-up" mechanism. For the other azimuthal wavenumbers, the distribution that emerges at large times is a structure which has been excited by left-handed spiraling vorticity sheets. This response corresponds most often to a Kelvin wave. In the helical case, this analysis enables to propose a good candidate for the vortex meandering phenomenon evidenced by experimentalists

A bypass transition in the Lamb-Oseen vortex

Transient energy growth in the short-time linear dynamics of a Lamb-Oseen monopole is a potential mechanism for nonlinear bypass transition, a phenomenon already observed in both experiments and numerical simulations. In the present study, we investigate this scenario by means of a nonlinear optimal perturbation approach, i.e. by looking for the initial perturbation whose evolution satisfies the fully nonlinear Navier-Stokes equations and maximizes the energy gain at a given time horizon. Preliminary two-dimensional results show that, for small initial amplitudes, the optimal perturbation and growth mechanisms observed in the linear regime are recovered. More particularly, the time evolution of the m = 2 optimal perturbation leads to an elliptical core deformation of the monopole, which suggests a potential bypass scenario driven by the non-linear dynamics. This is confirmed by computations for larger initial perturbation amplitudes: the optimal perturbation is similar to that of the linear regime but a subcritical bifurcation to a quasi-steady, high-energy, rotating tripole is observed