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A method is developed to estimate the properties of a global hydrodynamic instability in turbulent flows from measurement data of the limit-cycle oscillations. For this purpose, the flow dynamics is separated into deterministic contributions representing the global mode and a stochastic contribution representing the intrinsic turbulent forcing. Stochastic models are developed that account for the interaction between the two and allow the determination of the dynamic properties of the flow from stationary data. The deterministic contributions are modelled by an amplitude equation, which describes the oscillatory dynamics of the instability, and in a second approach by a mean-field model, which additionally captures the interaction between the instability and the mean-flow corrections. The stochastic contributions are considered as coloured noise forcing, representing the spectral characteristics of the stochastic turbulent perturbations. The methodology is applied to a turbulent swirling jet with a dominant global mode. Particle image velocimetry measurements are conducted to ensure that the mode is the most dominant coherent structure, and further pressure measurements provide long time series for the model calibration. The supercritical Hopf bifurcation is identified from the linear growth rate of the global mode, and the excellent agreement between measured and estimated statistics suggest that the model captures the relevant dynamics. This work demonstrates that the sole observation of limit-cycle oscillations is not sufficient to determine the stability of turbulent flows, since the stochastic perturbations obscure the actual bifurcation point. However, the proposed separation of deterministic and stochastic contributions in the dynamical model allows the identification of the flow state from stationary measurements.

Three-dimensional (3D) instabilities on a (potentially turbulent) two-dimensional (2D) flow are still incompletely understood, despite recent progress. Here, based on known physical properties of such 3D instabilities, we propose a simple, energy-conserving model describing this situation. It consists of a regularized 2D point-vortex flow coupled to localized 3D perturbations ("ergophages"), such that ergophages can gain energy by altering vortex-vortex distances through an induced divergent velocity field, thus decreasing point-vortex energy. We investigate the model in three distinct stages of evolution: (i) The linear regime, where the amplitude of the ergophages grows or decays exponentially on average, with an instantaneous growth rate that fluctuates randomly in time. The instantaneous growth rate has a small auto-correlation time, and a probability distribution featuring a power-law tail with exponent between -2 and -5/3 (up to a cutoff) depending on the point-vortex base flow. Consequently, the logarithm of the ergophage amplitude performs a Levy flight. (ii) The passive-nonlinear regime of the model, where the 2D flow evolves independently of the ergophage amplitudes, which saturate by non-linear self-interactions without affecting the 2D flow. In this regime the system exhibits a new type of on-off intermittency that we name Levy on-off intermittency, which we define and study in a companion paper [van Kan et al., Phys. Rev. E 103, 052115 (2021)1063-651X10.1103/PhysRevE.103.052115]. We compute the bifurcation diagram for the mean and variance of the perturbation amplitude, as well as the probability density of the perturbation amplitude. (iii) Finally, we characterize the fully nonlinear regime, where ergophages feed back on the 2D flow, and study how the vortex temperature is altered by the interaction with ergophages. It is shown that when the amplitude of the ergophages is sufficiently large, the condensate is disrupted and the 2D flow saturates to a zero-temperature state. Given the limitations of existing theories, our model provides a new perspective on 3D instabilities growing on 2D flows, which will be useful in analyzing and understanding the much more complex results of DNS and potentially guide further theoretical developments.

Large eddy simulations are presented on the wake flow of a notional propeller (the INSEAN E1658), upstream of a NACA0020 hydrofoil of infinite spanwise extent, mimicking propeller–rudder interaction. Results show that the flow physics is dominated by the interaction between the coherent structures populating the wake of the propeller and the surface of the hydrofoil. The suction and pressure side branches of the tip vortices move towards inner and outer radii, respectively. The hub vortex is split into two branches at the leading edge of the hydrofoil. The two branches of the hub vortex shift in the opposite direction, compared to the tip vortices, towards the rudder suction sides. As a result, a contraction of the propeller wake on the suction sides occurs, leading to increased levels of shear stress and turbulence. At downstream locations along the hydrofoil the spanwise deflection of the suction side branches of the tip vortices affects the trajectory of the overall propeller wake, including also the smaller helical vortices across the span of the wake of each blade and the two branches of the hub vortex on the two sides of the hydrofoil. This cross-stream shift persists, producing a strong anti-symmetry of the overall wake.

We study freely decaying quantum turbulence by performing high resolution numerical simulations of the Gross-Pitaevskii equation (GPE) in the Taylor-Green geometry. We use resolutions ranging from $1024^3$ to $4096^3$ grid points. The energy spectrum confirms the presence of both a Kolmogorov scaling range for scales larger than the intervortex scale $\ell$, and a second inertial range for scales smaller than $\ell$. Vortex line visualizations show the existence of substructures formed by a myriad of small-scale knotted vortices. Next, we study finite temperature effects in the decay of quantum turbulence by using the stochastic Ginzburg-Landau equation to generate thermal states, and then by evolving a combination of these thermal states with the Taylor-Green initial conditions using the GPE. We extract the mean free path out of these simulations by measuring the spectral broadening in the Bogoliubov dispersion relation obtained from spatio-temporal spectra, and use it to quantify the effective viscosity as a function of the temperature. Finally, in order to compare the decay of high temperature quantum and that of classical flows, and to further calibrate the estimations of viscosity from the mean free path in the GPE simulations, we perform low Reynolds number simulations of the Navier-Stokes equations., Comment: 15 figures, Supplemental material: Video M1 and M2

We investigate theoretically and experimentally the stability of two interlaced helical vortices with respect to displacement perturbations having wavelengths that are large compared to the size of the vortex cores. First, existing theoretical results are recalled and applied to the present configuration. Various modes of unstable perturbations, involving different phase relationships between the two vortices, are identified and their growth rates are calculated. They lead to a local pairing of neighbouring helix loops, or to a global pairing with one helix expanding and the other one contracting. A relation is established between this instability and the three-dimensional pairing of arrays of straight parallel vortices, and a striking quantitative agreement concerning the growth rates and frequencies is found. This shows that the local pairing of vortices is the driving mechanism behind the instability of the helix system. Second, an experimental study designed to observe these instabilities in a real flow is presented. Two helical vortices are generated by a two-bladed rotor in a water channel and characterised through dye visualisations and particle image velocimetry measurements. Unstable displacement modes are triggered individually, either by varying the rotation frequency of the rotor, or by imposing a small rotor eccentricity. The observed unstable mode structure, and the corresponding growth rates obtained from advanced processing of visualisation sequences, are in good agreement with theoretical predictions. The nonlinear late stages of the instability are also documented experimentally. Whereas local pairing leads to strong deformations and subsequent breakup of the vortices, global pairing results in a leapfrogging phenomenon, which temporarily restores the initial double-helix geometry, in agreement with recent observations from numerical simulations.

The evolution towards a finite-time singularity of the Navier–Stokes equations for flow of an incompressible fluid of kinematic viscosity$\unicode[STIX]{x1D708}$is studied, starting from a finite-energy configuration of two vortex rings of circulation$\pm \unicode[STIX]{x1D6E4}$and radius$R$, symmetrically placed on two planes at angles$\pm \unicode[STIX]{x1D6FC}$to a plane of symmetry$x=0$. The minimum separation of the vortices,$2s$, and the scale of the core cross-section,$\unicode[STIX]{x1D6FF}$, are supposed to satisfy the initial inequalities$\unicode[STIX]{x1D6FF}\ll s\ll R$, and the vortex Reynolds number$R_{\unicode[STIX]{x1D6E4}}=\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D708}$is supposed very large. It is argued that in the subsequent evolution, the behaviour near the points of closest approach of the vortices (the ‘tipping points’) is determined solely by the curvature$\unicode[STIX]{x1D705}(\unicode[STIX]{x1D70F})$at the tipping points and by$s(\unicode[STIX]{x1D70F})$and$\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})$, where$\unicode[STIX]{x1D70F}=(\unicode[STIX]{x1D6E4}/R^{2})t$is a dimensionless time variable. The Biot–Savart law is used to obtain analytical expressions for the rate of change of these three variables, and a nonlinear dynamical system relating them is thereby obtained. The solution shows a finite-time singularity, but the Biot–Savart law breaks down just before this singularity is realised, when$\unicode[STIX]{x1D705}s$and$\unicode[STIX]{x1D6FF}/\!s$become of order unity. The dynamical system admits ‘partial Leray scaling’ of just$s$and$\unicode[STIX]{x1D705}$, and ultimately full Leray scaling of$s,\unicode[STIX]{x1D705}$and$\unicode[STIX]{x1D6FF}$, conditions for which are obtained. The tipping point trajectories are determined; these meet at the singularity point at a finite angle. An alternative model is briefly considered, in which the initial vortices are ovoidal in shape, approximately hyperbolic near the tipping points, for which there is no restriction on the initial value of the parameter$\unicode[STIX]{x1D705}$; however, it is still the circles of curvature at the tipping points that determine the local evolution, so the same dynamical system is obtained, with breakdown again of the Biot–Savart approach just before the incipient singularity is realised. The Euler flow situation ($\unicode[STIX]{x1D708}=0$) is considered, and it is conjectured on the basis of the above dynamical system that a finite-time singularity can indeed occur in this case.

12 pages, 3 figures, tables, 1 appendix, supplementary movies are available at https://doi.org/10.1017/jfm.2018.892, Exact solutions for multipolar azimuthal-mode vortices in two-dimensional Euler flows are presented. Flow solutions with non-vanishing far-field velocity are provided for any set of azimuthal wavenumbers m and arbitrary number n of vorticity shells. For azimuthal wavenumbers m=0 and m=1, the far-field velocity is a rigid motion and unsteady flow solutions with vanishing far-field velocity are obtained by means of a time-dependent change of reference frame. Addition of these first two modes, in the case of n = 1, results in a particular Chaplygin-Lamb (C-L) dipole, with continuous and vanishing vorticity at the vortex boundary. Numerical simulations suggest that this particular C-L dipole is stable, Partial support for this study was obtained through project CTM2014-56987-P (Spanish Ministry of Science and Innovation)

International audience; All the methods which estimate the unperturbed fluid flow velocity relying on particle suspensions address the same question: How can the fluid velocity be computed measuring the particles trajectory and/or their velocities? The tracking of a few large density-mismatched particles is here used to efficiently and accurately reconstruct the background fluid flow. Approximating the particulate phase space and taking the limit of vanishing Stokes number St -> 0, we retrieve the background flow for three test cases: a shear flow near a wall, a rigid-body vortex, and a strained vortex. The major advantages and the potentials of this approach are discussed in the end, highlighting how to overcome the classic shortcomings of experimental measurements faced for near-boundaries particle tracking.

It is well known that the use of SPH models in simulating flow at high Reynolds numbers is limited because of the tensile instability inception in the fluid region characterized by high vorticity and negative pressure. In order to overcome this issue, the δ + -SPH scheme is modified by implementing a Tensile Instability Control (TIC). The latter consists of switching the momentum equation to a non-conservative formulation in the unstable flow regions. The loss of conservation properties is shown to induce small errors, provided that the particle distribution is regular. The latter condition can be ensured thanks to the implementation of a Particle Shifting Technique (PST). The novel variant of the δ + -SPH is proved to be effective in preventing the onset of tensile instability. Several challenging benchmark tests involving flows past bodies at large Reynolds numbers have been used. Within this a simulation characterized by a deforming foil that resembles a fish-like swimming body is used as a practical application of the δ + -SPH model in biological fluid mechanics.

We study, using 2D direct numerical simulations, the effects of particle inertia on the condensation process in a vortical flow. We find that particle inertia has a small effect on both the overall condensation rate as well as the size distribution of particles. These particles represent water droplets in the turbulent flows that are convecting clouds. The simulations presented here are indicative, and should serve as proof of concept. More detailed simulations are being performed.

Lentink & Dickinson (2009J. Exp. Biol.212, 2705–2719. (doi:10.1242/jeb.022269)) showed that rotational acceleration stabilized the leading-edge vortex on revolving, low aspect ratio (AR) wings and hypothesized that a Rossby number of around 3, which is achieved during each half-stroke for a variety of hovering insects, seeds and birds, represents a convergent high-lift solution across a range of scales in nature. Subsequent work has verified that, in particular, the Coriolis acceleration plays a key role in LEV stabilization. Implicit in these results is that there exists an optimal AR for wings revolving about their root, because it is otherwise unclear why, apart from possible morphological reasons, the convergent solution would not occur for an even lower Rossby number. We perform direct numerical simulations of the flow past revolving wings where we vary the AR and Rossby numbers independently by displacing the wing root from the axis of rotation. We show that the optimal lift coefficient represents a compromise between competing trends with competing time scales where the coefficient of lift increases monotonically with AR, holding Rossby number constant, but decreases monotonically with Rossby number, when holding AR constant. For wings revolving about their root, this favours wings of AR between 3 and 4.

When a vortex refracts surface waves, the momentum flux carried by the waves changes direction and the waves induce a reaction force on the vortex. We study experimentally the resulting vortex distortion. Incoming surface gravity waves impinge on a steady vortex of velocity $U_0$ driven magneto-hydrodynamically at the bottom of a fluid layer. The waves induce a shift of the vortex center in the direction transverse to wave propagation, together with a decrease in surface vorticity. We interpret these two phenomena in the framework introduced by Craik and Leibovich (1976): we identify the dimensionless Stokes drift $S=U_s/U_0$ as the relevant control parameter, $U_s$ being the Stokes drift velocity of the waves. We propose a simple vortex line model which indicates that the shift of the vortex center originates from a balance between vorticity advection by the Stokes drift and self-advection of the vortex. The decrease in surface vorticity is interpreted as a consequence of vorticity expulsion by the fast Stokes drift, which confines it at depth. This purely hydrodynamic process is analogous to the magnetohydrodynamic expulsion of magnetic field by a rapidly moving conductor through the electromagnetic skin effect. We study vorticity expulsion in the limit of fast Stokes drift and deduce that the surface vorticity decreases as $1/S$, a prediction which is compatible with the experimental data. Such wave-induced vortex distortions have important consequences for the nonlinear regime of wave refraction: the refraction angle rapidly decreases with wave intensity., Comment: 13 pages, 4 figures

Swirling jets and vortices can both be unstable to the centrifugal instability but with a different wavenumber selection: the most unstable perturbations for swirling jets in inviscid fluids have an infinite azimuthal wavenumber, whereas, for vortices, they are axisymmetric but with an infinite axial wavenumber. Accordingly, sufficient condition for instability in inviscid fluids have been derived asymptotically in the limits of large azimuthal wavenumber $m$ for swirling jets (Leibovich and Stewartson, J. Fluid Mech., vol. 126, 1983, pp. 335–356) and large dimensionless axial wavenumber $k$ for vortices (Billant and Gallaire, J. Fluid Mech., vol. 542, 2005, pp. 365–379). In this paper, we derive a unified criterion valid whatever the magnitude of the axial flow by performing an asymptotic analysis for large total wavenumber $ \sqrt{{k}^{2} + {m}^{2} } $. The new criterion recovers the criterion of Billant and Gallaire when the axial flow is small and the Leibovich and Stewartson criterion when the axial flow is finite and its profile sufficiently different from the angular velocity profile. When the latter condition is not satisfied, it is shown that the accuracy of the Leibovich and Stewartson asymptotics is strongly reduced. The unified criterion is validated by comparisons with numerical stability analyses of various classes of swirling jet profiles. In the case of the Batchelor vortex, it provides accurate predictions over a wider range of axial wavenumbers than the Leibovich–Stewartson criterion. The criterion shows also that a whole range of azimuthal wavenumbers are destabilized as soon as a small axial velocity component is present in a centrifugally unstable vortex.

Vortical structures and behaviour associated with vortex-ring collisions upon round cylinders with different cylinder-to-vortex-ring diameter ratios were studied using laser-induced fluorescence and time-resolved particle-image velocimetry techniques. Circular vortex rings of Reynolds number 4000 and three diameter ratios of $D/d=1$, 2 and 4 were considered in the present investigation. Results reveal that the collision behaviour is very different from that associated with flat surfaces, in which vortex disconnection and reconnection processes caused by the strong interactions between primary and secondary vortex rings produce small-scale vortex ringlets that eject away from the cylinders. For the cylinder with the largest diameter ratio used here, these vortex ringlets move towards each other along the collision axis, where they eventually collide to produce a vortex dipole that propagates upstream. However, as the diameter ratio decreases, these vortex ringlets are produced further away from the collision axis, which results in them ejecting away from the cylinder at increasingly larger angles relative to the collision axis. Trajectories of key vortex cores were extracted from the experimental results to demonstrate quantitatively the strong sensitivity of these vortical motions upon the diameter ratio. Furthermore, significant differences in the primary vortex-ring circulation along convex surfaces and straight edges after the collisions are observed. In particular, vortex flow models are presented here to better illustrate the highly three-dimensional flow dynamics of the collision behaviour, as well as highlighting the strong dependency of the secondary vortex-ring formation, vortex disconnection/reconnection processes, and ejection of the resulting vortex ringlets upon the diameter ratio. As such, these results are expected to shed more light on the more general scenario of vortex-ring collisions upon arbitrarily contoured solid boundaries.

Inviscid, unsteady simulations of the roll up of the start-up vortex issuing from a semi-infinite plate are compared with previous simulations of the viscous flow. The inviscid equations were solved by a lumped-vortex method, the two-dimensional, incompressible Navier–Stokes equations in the vorticity–streamfunction formulation modelled the viscous problem. The purpose is to verify whether the irregular behaviour found by the inviscid solution well approximates the unstable evolution of the viscous spiral vortex in the limit of infinitely large time (or equivalently Reynolds number).

It is known that rapidly rotating turbulent flows are characterized by the emergence of simultaneous upscale and downscale energy transfer. Indeed, both numerics and experiments show the formation of large-scale anisotropic vortices together with the development of small-scale dissipative structures. However the organization of interactions leading to this complex dynamics remains unclear. Two different mechanisms are known to be able to transfer energy upscale in a turbulent flow. The first is characterized by two-dimensional interactions among triads lying on the two-dimensional, three-component (2D3C) manifold, namely on the Fourier plane perpendicular to the rotation axis. The second mechanism is three-dimensional and consists of interactions between triads with the same sign of helicity (homochiral). Here, we present a detailed numerical study of rotating flows using a suite of high Reynolds number direct numerical simulations within different parameter regimes to analyze both upscale and downscale cascade ranges. We find that the upscale cascade at wave numbers close to the forcing scale is generated by increasingly dominant homochiral interactions which couple the three-dimensional bulk and the 2D3C plane. This coupling produces an accumulation of energy in the 2D3C plane, which then transfers energy to smaller wave numbers thanks to the two-dimensional mechanism. In the forward cascade range, we find that the energy transfer is dominated by heterochiral triads and is dominated primarily by interaction within the fast manifold where $k_z\ne0$. We further analyze the energy transfer in different regions in the real-space domain. In particular, we distinguish high-strain from high-vorticity regions and we uncover that while the mean transfer is produced inside regions of strain, the rare but extreme events of energy transfer occur primarily inside the large-scale column vortices.

The vortex–body interaction problem, which characterizes the flow field of a rudder placed downstream of a single-blade marine rotor, is investigated by numerical simulations. The particular topology of the propeller wake, consisting of a helicoidal vortex detached from the blade tips (tip vortex) and a longitudinal, streamwise oriented vortex originating at the hub (hub vortex), embraces two representative mechanisms of vortex–body collisions: the tip vortices impact almost orthogonally to the mean plane, whereas the hub vortex travels in the mean plane of the wing (rudder), perpendicularly to its leading edge. The two vortices evolve independently only during the approaching and collision phases. The passage along the body is instead characterized by strong interaction with the boundary layer on the rudder and is followed by reconnection and merging in the middle and far wake. The features of the wake were investigated by the $\unicode[STIX]{x1D706}_{2}$-criterion (Jeong & Hussain, J. Fluid Mech., vol. 285, 1995, pp. 69–94) and typical flow variables (pressure, velocity and vorticity) of the instantaneous flow field; wall pressure spectra were analysed and related to the tip and hub vortices evolution, revealing a non-obvious behaviour of the loading on the rudder that can be related to undesired unsteady loads.

An analytical method for determining the shape of hollow vortices in shear flows is presented in detail. In a non-dimensional formulation, it is shown that the problem has one degree of freedom represented by the free choice of the non-dimensionalized speed $\unicode[STIX]{x1D705}$ at the boundary of the vortex. The solutions form two families of shapes corresponding to vortex circulation and shear-flow vorticity having the opposite or same sign. When the signs are opposite, the shape family resembles that described by Llewellyn Smith & Crowdy (J. Fluid Mech., vol. 691, 2012, pp. 178–200) for hollow vortices in a potential flow with strain. As for that flow, there is a minimum value of $\unicode[STIX]{x1D705}$ below which there is no solution as the boundary of the vortex self-intersects, while, when the signs are the same, solutions exist for $0.

This paper investigates the stability of an axisymmetric pancake vortex with Gaussian angular velocity in radial and vertical directions in a continuously stratified-rotating fluid. The different instabilities are determined as a function of the Rossby number $Ro$, Froude number $F_{h}$, Reynolds number $Re$ and aspect ratio ${\it\alpha}$. Centrifugal instability is not significantly different from the case of a columnar vortex due to its short-wavelength nature: it is dominant when the absolute Rossby number $|Ro|$ is large and is stabilized for small and moderate $|Ro|$ when the generalized Rayleigh discriminant is positive everywhere. The Gent–McWilliams instability, also known as internal instability, is then dominant for the azimuthal wavenumber $m=1$ when the Burger number $Bu={\it\alpha}^{2}Ro^{2}/(4F_{h}^{2})$ is larger than unity. When $Bu\lesssim 0.7Ro+0.1$, the Gent–McWilliams instability changes into a mixed baroclinic–Gent–McWilliams instability. Shear instability for $m=2$ exists when $F_{h}/{\it\alpha}$ is below a threshold depending on $Ro$. This condition is shown to come from confinement effects along the vertical. Shear instability transforms into a mixed baroclinic–shear instability for small $Bu$. The main energy source for both baroclinic–shear and baroclinic–Gent–McWilliams instabilities is the potential energy of the base flow instead of the kinetic energy for shear and Gent–McWilliams instabilities. The growth rates of these four instabilities depend mostly on $F_{h}/{\it\alpha}$ and $Ro$. Baroclinic instability develops when $F_{h}/{\it\alpha}|1+1/Ro|\gtrsim 1.46$ in qualitative agreement with the analytical predictions for a bounded vortex with angular velocity slowly varying along the vertical.

We consider a pair of like-signed, initially elliptical vortices with uniform vorticity distribution embedded in an incompressible, inviscid fluid occupying a two-dimensional, infinite domain. We characterize this finite-time, aperiodic, dynamical system in terms of its fixed points and separatrices, which divide the flow into inner core, inner recirculation, outer recirculation regions and outer flow. We numerically simulate the time evolution of the vortex pair using a contour dynamics algorithm. The rotational and co-rotational motion of the vortices perturb the separatrices, which undergo to deformations, yielding a tangle whose complexity increases as the amplitude of the perturbation increases. We analyse the dynamics of the tangle and explain the transport of fluid between different regions. We use two diagnostics to quantify stirring: stretching of the interface and the mix-norm. These two diagnostics characterize stirring in contradicting ways and present different sensitivity to the parameters considered. We find that stretching is dominated by the chaotic advection induced within the inner core and inner recirculation regions, whereas the mix-norm is dominated by the laminar transport induced within the outer recirculation regions. For pairs of vortices of small aspect ratio, stretching is piecewise linear and the mix-norm does not decrease monotonically. We show that these two effects are strongly coupled and synchronized with the rotational motion of the vortices. Since the nominal domain is unbounded, we quantify stirring on three concentric, circular domains. One domain nearly encloses the outer separatrices of the vortex pair, one is smaller and one larger than the first one. We show that the mix-norm is very sensitive to the size of the domain, while stretching is not. To quantify the sensitivity of stirring to the geometry of the initial concentration field, we consider, as an initial scalar field, two concentrations delimited by a straight-line interface of adjustable orientation. We show that the interface passing through the centroids of the vortices is the one most efficiently stretched, while the initial concentration field with an orthogonal interface is the most efficiently stirred. Finally, we investigate the effects of the angular impulse on the stirring performance of the vortex pair. Stretching is very sensitive to the angular impulse, while the mix-norm is not. We show that there is a value of the angular impulse which maximizes stretching and argue that this is due to two competing mechanisms.

A method for predicting the outcome of vortex breakup in a rotating flow is introduced. The vortices dealt with here are subject to both centrifugal and barotropic instabilities. The prediction of the aftermath of the breakup relies on knowing how both centrifugal and barotropic instabilities would equilibrate separately. A theoretical model for non-linear equilibration in centrifugal instability is wedded to two-dimensional simulation of barotropic instability to predict the final vortices that emerge from the debris of the original vortex. This prediction method is tested against three-dimensional Navier–Stokes simulations. For vortices in which a rapid centrifugal instability triggers a slower barotropic instability, the method is successful both qualitatively and quantitatively. The skill of the prediction method decreases as the time scales of the two instabilities become comparable.

We present a study of polygons forming on the free surface of a water flow confined to a stationary cylinder and driven by a rotating bottom plate as described by Jansson et al. (Phys. Rev. Lett., vol. 96, 2006, 174502). In particular, we study the case of a triangular structure, either completely ‘wet’ or with a ‘dry’ centre. For the dry structures, we present measurements of the surface shapes and the process of formation. We show experimental evidence that the formation can take place as a two-stage process: first the system approaches an almost stable rotationally symmetric state and from there the symmetry breaking proceeds like a low-dimensional linear instability. We show that the circular state and the unstable manifold connecting it with the polygon solution are universal in the sense that very different initial conditions lead to the same circular state and unstable manifold. For a wet triangle, we measure the surface flows by particle image velocimetry (PIV) and show that there are three vortices present, but that the strength of these vortices is far too weak to account for the rotation velocity of the polygon. We show that partial blocking of the surface flow destroys the polygons and re-establishes the rotational symmetry. For the rotationally symmetric state our theoretical analysis of the surface flow shows that it consists of two distinct regions: an inner, rigidly rotating centre and an outer annulus, where the surface flow is that of a point vortex with a weak secondary flow. This prediction is consistent with the experimentally determined surface flow.

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.

The influence of longitudinal vortices on the high-lift behavior of a generic three-dimensional wing is presented. A grid convergence study is performed for the two-dimensional high-lift airfoil and different grid topologies are discussed. Numerical simulations are performed with the DLR TAU-Code at different angles of attack. For the simulations, the Menter-SST turbulence model is applied. A simplified vortex system originates at a spanwise slat cut-off. The vortex system passes along the suction side of the wing and influences the high-lift and stall behavior. The characteristics of the vortices are described and the influence on the stall mechanism is shown.

Using complementary numerical approaches at high resolution, we study the late-time behaviour of an inviscid, incompressible two-dimensional flow on the surface of a sphere. Starting from a random initial vorticity field comprised of a small set of intermediate wavenumber spherical harmonics, we find that -- contrary to the predictions of equilibrium statistical mechanics -- the flow does not evolve into a large-scale steady state. Instead, significant unsteadiness persists, characterised by a population of persistent small-scale vortices interacting with a large-scale oscillating quadrupolar vorticity field. Moreover, the vorticity develops a stepped, staircase distribution, consisting of nearly homogeneous regions separated by sharp gradients. The persistence of unsteadiness is explained by a simple point vortex model characterising the interactions between the four main vortices which emerge., Comment: 21 pages, 14 figures, submitted to JFM

For a stratified incompressible Euler fluid under gravity confined by rigid boundaries, sources of vorticity are classified with the aim of isolating those which are sensitive to the topological configurations of density isopycnals, for both layered and continuous density variations. The simplest case of a two-layer fluid is studied first. This shows explicitly that topological sources of vorticity are present whenever the interface intersects horizontal boundaries. Accordingly, the topological separation of the fluid domain due to the interface–boundary intersections can contribute additional terms to the vorticity balance equation. This phenomenon is reminiscent of Klein’s ‘Kaffeelöffel’ thought-experiment for a homogeneous fluid (Klein, Z. Math. Phys., vol. 59, 1910, pp. 259–262), and it is essentially independent of the vorticity generation induced by the baroclinic term in the bulk of the fluid. In fact, the two-layer case is generalized to show that for the continuously stratified case topological vorticity sources are generically present whenever density varies along horizontal boundaries. The topological sources are expressed explicitly in terms of local contour integrals of the pressure along the intersection curves of isopycnals with domain boundaries, and their effects on vorticity evolution are encoded by an appropriate vector, termed here the ‘topological vorticity’.

Journal of Fluid Mechanics, 731, ISSN:0022-1120, ISSN:1469-7645

The energy decay of vortices in viscous fluids governed by the compressible Navier–Stokes equations is investigated. It is shown that the main reason for the slow decay is that zero eigenvalues exist in the matrix related to the dissipative terms. The theoretical analysis is purely mathematical and based on the energy method. To check the validity of the theoretical result in practice, numerical solutions to the Navier–Stokes equations are computed using a stable high-order finite difference method. The numerical computations corroborate the theoretical conclusion.

International audience; This paper addresses the instability of a two-layer coastal current in a quasigeostrophic model; the potential vorticity (PV) structure of this current consists in two uniform cores, located at different depths, with finite width and horizontally shifted. This shift allows both barotropic and baroclinic instability for this current. The PV cores can be like-signed or opposite-signed, leading to their vertical alignment or to their hetonic coupling. These two aspects are novel compared to previous studies. For narrow vorticity cores, short waves dominate, associated with barotropic instability; for wider cores, longer waves are more unstable and are associated with baroclinic processes. Numerical experiments were performed on the f-plane with a finite-difference model. When both cores have like-signed PV, trapped instability develops during the nonlinear evolution: vertical alignment of the structures is observed. For narrow cores, short wave breaking occurs close to the coast; for wider cores, substantial turbulence results from the entrainment of ambient fluid into the coastal jet. When the two cores have opposite-signed PV, the nonlinear regimes range from short wave breaking to the ejection of dipoles or tripoles, via a regime of dipole oscillation near the wall. The Fourier analysis of the perturbed flow is appropriate to distinguish the regimes of short wave breaking, of dipole formation, and of turbulence, but not the differences between regimes involving only vortex pairs. To explain more precisely the regimes where two vortices (and their wall images) interact, a point vortex model is appropriate.

International audience; Numerical simulations of the Lighthill-Weis-Fogh mechanism are performed using a Fourier pseudo-spectral method with volume penalization. Single-winged and double-winged configurations are compared, and the vortex shedding patterns are related to the lift generated in both cases. The computations of the lift coefficient are validated against the results reported previously by Miller and Peskin (J Exp Biol 208:195-212, 2005).

The instability of the Kármán vortex street is revisited under a spatio-temporal perspective that allows the taking into account of the advection of the vortices by the external flow. We analyse a simplified point vortex model and show through numerical simulations of its linear impulse response that the system becomes convectively unstable above a certain critical advection velocity. This critical velocity decreases as the aspect ratio approaches its specific value for temporal stability, and increases with the confinement induced by lateral walls. In the limiting unconfined case, direct application of the Briggs–Bers criterion to the dispersion relation gives results in excellent agreement with the numerical simulations. Finally, a direct numerical simulation of the $Re=100$ flow past a confined cylinder is performed, and the actual advection velocity of the resulting vortex street is found to be much larger than the critical advection velocity for convective instability given by our model. The Kármán vortex street is therefore strongly convectively unstable.