Patrice Meunier

A merging criterion for two-dimensional co-rotating vortices

We propose a quantitative criterion for the merging of a pair of equal two-dimensional co-rotating vortices. A cross-validation between experimental and theoretical analyses is performed. Experimental vortices are generated by the roll-up of a vortex sheet originating from the identical and impulsive rotation of two plates. The phenomenon is then followed up in time until a rapid pairing transition occurs for which critical parameters are measured. In the theoretical approach, the nonlinear Euler solution representing a pair of equal vortices is computed for various nonuniform vorticity distributions. The stability analysis of such a configuration then provides critical values for the onset of merging. From this data set, a criterion depending on global impulse quantities is extracted for different shapes of the vorticity distribution. This theoretical statement agrees well with our experimentally based criterion.

Three-dimensional instability during vortex merging

The interaction of two parallel vortices of equal circulation is observed experimentally. For low Reynolds numbers (Re), the vortices remain two dimensional and merge into a single one, when their time-dependent core size exceeds approximately 30% of the vortex separation distance. At higher Re, a three-dimensional (3-D) instability is discovered, showing the characteristics of an elliptic instability of the vortex cores. The instability rapidly generates small-scale turbulent motion, which initiates merging for smaller core sizes and produces a bigger final vortex than for laminar 2-D flow.

How vortices mix

The advection of a passive scalar blob in the deformation field of an axisymmetric vortex is a simple mixing protocol for which the advection–diffusion problem is amenable to a near-exact description. The blob rolls up in a spiral which ultimately fades away in the diluting medium. The complete transient concentration field in the spiral is accessible from the Fourier equations in a properly chosen frame. The concentration histogram of the scalar wrapped in the spiral presents unexpected singular transient features and its long time properties are discussed in connection with real mixtures.

Elliptic instability of a co-rotating vortex pair

In this paper, we report experimental results concerning a three-dimensional short-wave instability observed in a pair of equal co-rotating vortices. The pair is generated in water by impulsively started plates, and is analysed through dye visualizations and detailed quantitative measurements using particle image velocimetry. The instability mode, which is found to be stationary in the rotating frame of reference of the two-vortex system, consists of internal deformations of the vortex cores, which are characteristic of the elliptic instability occurring in strained vortical flows. Measurements of the spatial structure, wavelengths and growth rates are presented, as functions of Reynolds number and non-dimensional core size. The self-induced rotation of the vortex pair, which is not a background rotation of the entire flow, is found to lead to a shift of the unstable wavelength band to higher values, as well as to higher growth rates. In addition, a dramatic increase in the width of the unstable bands for large values of the rescaled core radius is found. Comparisons with recent theoretical results by Le Dizes & Laporte (2002) concerning elliptic instability of co-rotating vortices show very good agreement. At later stages of the flow, when the perturbation amplitude becomes sufficiently large, the two vortices merge into a single structure. This happens for smaller cores sizes than in the case of two-dimensional merging. The three-dimensional merging leads to a final vortex characterized by turbulent small-scale motion, whose size appears to be larger than it would have been without instability. The vorticity profile of the final vortex is non-Gaussian after both two-dimensional and three-dimensional merging. The profile contains more vorticity outside the inner core than a Gaussian vortex, resulting from the ejection of vorticity filaments during the merging stage.

A rotating fluid cylinder subject to weak precession

In this paper, we report experimental and theoretical results on the flow inside a precessing and rotating cylinder. Particle image velocimetry measurements have revealed the instantaneous structure of the flow and confirmed that it is the sum of forced inertial (Kelvin) modes, as predicted by the classical linear inviscid theory. But this theory predicts also that the amplitude of a mode diverges when its natural frequency equals the precession frequency. A viscous and weakly nonlinear theory has therefore been developed at the resonance. This theory has been compared to experimental results and shows a good quantitative agreement. For low Reynolds numbers, the mode amplitude scales as the square root of the Reynolds number owing to the presence of Ekman layers on the cylinder walls. When the Reynolds number is increased, the amplitude saturates at a value which scales as the precession angle to the power one-third for a given resonance. The nonlinear theory also predicts the forcing of a geostrophic (axisymmetric) mode which has been observed and measured in the experiments. These results allow the flow inside a precessing cylinder to be fully characterized in all regimes as long as there is no instability.

The diffusive strip method for scalar mixing in two dimensions

We introduce a new numerical method for the study of scalar mixing in two-dimensional advection fields. The position of an advected material strip is computed kinematically, and the associated convection–diffusion problem is solved using the computed local stretching rate along the strip, assuming that the diffusing strip thickness is smaller than its local radius of curvature. This widely legitimate assumption reduces the numerical problem to the computation of a single variable along the strip, thus making the method extremely fast and applicable to any large Peclet number. The method is then used to document the mixing properties of a chaotic sine flow, for which we relate the global quantities (spectra, concentration probability distribution functions (PDFs), increments) to the distributed stretching of the strip convoluted by the flow, possibly overlapping with itself. The numerical results indicate that the PDF of the strip elongation is log normal, a signature of random multiplicative processes. This property leads to exact analytical predictions for the spectrum of the field and for the PDF of the scalar concentration of a solitary strip. The present simulations offer a unique way of discovering the interaction rule for building complex mixtures which are made of a random superposition of overlapping strips leading to concentration PDFs stable by self-convolution.

Self-preservation in stratified momentum wakes

A general model is described for drag wakes in a linearly stratified fluid, based on the self-preservation of the flow. It is assumed that the buoyancy-controlled self-similar wake expands in the horizontal direction due to turbulent diffusion and in the vertical direction due to viscous diffusion. The mean characteristics of the wake (height, width and velocity defect) are analytically derived and show good agreement with existing data from experimental and numerical results. Moreover, the three regimes previously found in the literature that characterize different dynamical phases of the wake evolution are recovered, and two new regimes are found. The model allows for prediction of characteristic length and velocity scales at the high Reynolds numbers of large-scale applications of geophysical and naval origin.

Precessional instability of a fluid cylinder

In this paper, the instability of a fluid inside a precessing cylinder is addressed theoretically and experimentally. The precessional motion forces Kelvin modes in the cylinder, which can become resonant for given precessional frequencies and cylinder aspect ratios. When the Reynolds number is large enough, these forced resonant Kelvin modes eventually become unstable. A linear stability analysis based on a triadic resonance between a forced Kelvin mode and two additional free Kelvin modes is carried out. This analysis allows us to predict the spatial structure of the instability and its threshold. These predictions are compared to the vorticity field measured by particle image velocimetry with an excellent agreement. When the Reynolds number is further increased, nonlinear effects appear. A weakly nonlinear theory is developed semi-empirically by introducing a geostrophic mode, which is triggered by the nonlinear interaction of a free Kelvin mode with itself in the presence of viscosity. Amplitude equations are obtained coupling the forced Kelvin mode, the two free Kelvin modes and the geostrophic mode. They show that the instability saturates to a fixed point just above threshold. Increasing the Reynolds number leads to a transition from a steady saturated regime to an intermittent flow in good agreement with experiments. Surprisingly, this weakly nonlinear model still gives a correct estimate of the mean flow inside the cylinder even far from the threshold when the flow is turbulent.

Tilt-induced instability of a stratified vortex

This experimental and theoretical study considers the dynamics and the instability of a Lamb-Oseen vortex in a stably stratified fluid. In a companion paper, it was shown that tilting the vortex axis with respect to the direction of stratification induces the formation of a rim of strong axial flow near a critical radius when the Froude number of the vortex is larger than one. Here, we demonstrate that this tilt-induced flow is responsible for a three-dimensional instability. We show that the instability results from a shear instability of the basic axial flow in the critical-layer region. The theoretical predictions for the wavelength and the growth rate obtained by a local stability analysis of the theoretical critical-layer profile are compared to experimental measurements and a good agreement is observed. The late stages of the instability are also analysed experimentally. In particular, we show that the tilt-induced instability does not lead to the destruction of the vortex, but to a sudden decrease of its Froude number, through the turbulent diffusion of its core size, when the initial Froude number is close to 1. A movie is available with the online version of the paper.

Viscous stability properties of a Lamb-Oseen vortex in a stratified fluid

In this work, we analyse the linear stability of a frozen Lamb―Oseen vortex in a fluid linearly stratified along the vortex axis. The temporal stability properties of three-dimensional normal modes are obtained under the Boussinesq approximation with a Chebychev collocation spectral code for large ranges of Froude numbers and Reynolds numbers (the Schmidt number being fixed to 700). A specific integration technique in the complex plane is used in order to apply the condition of radiation at infinity. For large Reynolds numbers and small Froude numbers, we show that the vortex is unstable with respect to all non-axisymmetrical waves. The most unstable mode is however always a helical radiative mode (m =1) which resembles either a displacement mode or a ring mode. The displacement mode is found to be unstable for all Reynolds numbers and for moderate Froude numbers (F ∼ 1). The radiative ring mode is by contrast unstable only for large Reynolds numbers above 10 4 and is the most unstable mode for large Froude numbers (F > 2). The destabilization of this mode for large Froude numbers is shown to be associated with a resonance mechanism which is analysed in detail. Analyses of the scaling and of the spatial structure of the different unstable modes are also provided.