Waves and instabilities in rotating free surface flows

The stability properties of the rotating free surface flow in a cylindrical container is studied using a global stability approach, considering succesively three models. For the case of solid body rotation (Newton’s bucket), all eigenmodes are found to be stable, and are classified into three families : gravity waves, singular inertial modes, and Rossby waves. For the case of a potential flow, an instability is found. The mechanism is explained as a resonance between gravity waves and centrifugal waves, and is thought to be at the origin of the ”rotating polygon instability” observed in experiments where the flow is driven by rotation of the bottom plate (see [9]). Finally, we give some preliminary results concerning a third model : the Rankine vortex

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

Rotating Polygon Instability of a Swirling Free Surface Flow

We explain the rotating polygon instability on a swirling fluid surface [G. H. Vatistas, J. Fluid Mech. 217, 241 (1990) and Jansson et al., Phys. Rev. Lett. 96, 174502 (2006)] in terms of resonant interactions between gravity waves on the outer part of the surface and centrifugal waves on the inner part. Our model is based on potential flow theory, linearized around a potential vortex flow with a free surface for which we show that unstable resonant states appear. Limiting our attention to the lowest order mode of each type of wave and their interaction, we obtain an analytically soluble model, which, together with estimates of the circulation based on angular momentum balance, reproduces the main features of the experimental phase diagram. The generality of our arguments implies that the instability should not be limited to flows with a rotating bottom (implying singular behavior near the corners), and indeed we show that we can obtain the polygons transiently by violently stirring liquid nitrogen in a hot container

Generation of three-dimensional patterns through wave interaction in a model of free surface swirling flow

The free surface flow in a cylindrical tank over a rotating bottom is known to support spectacular three-dimensional patterns, including deformation of the inner free surface into the shape of rotating polygons, and sloshing behavior of the upper free surface (e.g. Iga et al., FDR 2014, same issue). Through a stability analysis of a simplified model of this flow, we show that such patterns can be explained as a resonance mechanism involving different families of waves. The approach extends a previous work (Tophoj et al., PRL 2013) which explained the rotating polygons as an interaction between gravity waves and centrifugal waves, under the assumption that the base flow can be modeled as a potential vortex. We show that this previous model is justified for strong rotation rates (Dry-Potential case), and that for weaker rotations it can be improved by introducing an inner vortex core in solid-body rotation, which either extends to the center of the plate (Wet case) or surrounds a dry central region (Dry-Composite case). The study of this improved model predicts two new kind of instabilities. The first occurs at low rotations (Wet case) and results from an interaction between gravity waves and the Kelvin-Kirchhoff wave (namely, oscillation of the boundary of the vortex core). This instability is proposed to be at the origin of the sloshing phenomenon. The second new instability occurs, for moderate rotations, (Dry-Composite case) as an interaction between gravity waves and a "Kelvin-Centrifugal" wave characterized by deformation of the inner surface and the vortex core boundary in opposite directions. This instability exists for all azimuthal wave numbers starting from m = 1, this case corresponding to a "monogon" pattern

Waves and instabilities in rotating free surface flows

The stability properties of the rotating free surface flow in a cylindrical container is studied using a global stability approach, considering successively three models. For the case of solid body rotation (Newton’s bucket), all eigenmodes are found to be stable, and are classified into three families: gravity waves, singular inertial modes, and Rossby waves. For the case of a potential flow, an instability is found. The mechanism is explained as a resonance between gravity waves and centrifugal waves, and is thought to be at the origin of the “rotating polygon instability” observed in experiments where the flow is driven by rotation of the bottom plate (see L. Tophøj, J. Mougel, T. Bohr, D. Fabre, The Rotating Polygon Instability of a Swirling Free Surface Flow, Phys. Rev. Lett. 110 (2013) 194502). Finally, in the case of the Rankine vortex which in fact consists in the combination of the two first cases, we report a new instability mechanism involving Rossby and gravity waves

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

Waves in Newton's bucket

The motion of a liquid in an open cylindrical tank rotating at a constant rate around its vertical axis of symmetry, a configuration called Newton’s bucket, is investigated using a linear stability approach. This flow is shown to be affected by several families of waves, all weakly damped by viscosity. The wave families encountered correspond to: surface waves which can be driven either by gravity or centrifugal acceleration, inertial waves due to Coriolis acceleration which are singular in the inviscid limit, and Rossby waves due to height variations of the fluid layer. These waves are described in the inviscid and viscous cases by means of mathematical considerations, global stability analysis and various asymptotic methods; and their properties are investigated over a large range of parameters (a, Fr), with a the aspect ratio and Fr the Froude number

Unexpected effect of copper ions on electrochemical impedance behaviour of self-assembled alkylaminethiol monolayer

Effect of copper ions on the electrochemical behaviour of an alkylaminethiol monolayer has been studied by electrochemical impedance spectrosocpy. RAMAN experiment shows the effective adsorption of receptor onto the gold surfaces. The study of Nyquist plot shows that the gold/monolayer/electrolyte interface can be described by a serial combination of two R, CPE electrical circuits. In the presence of increasing amounts of copper, the Nyquist plots at low frequencies were modified showing an increase of the resistance of the second R, CPE electrical circuit. Moreover, this increase of resistance varies linearly with the amounts of copper ions added in solution from 10−8 mol·L−1 to 10−5 mol·L−1