Paul Billant

Experimental study of vortex breakdown in swirling jets

The goal of this study is to characterize the various breakdown states taking place in a swirling water jet as the swirl ratio S and Reynolds number Re are varied. A pressure-driven water jet discharges into a large tank, swirl being imparted by means of a motor which sets into rotation a honeycomb within a settling chamber. The experiments are conducted for two distinct jet diameters by varying the swirl ratio S while maintaining the Reynolds number Re fixed in the range 300 Re <1200. Breakdown is observed to occur when S reaches a well defined threshold Sc ≈1.3–1.4 which is independent of Re and nozzle diameter used. This critical value is found to be in good agreement with a simple criterion derived in the same spirit as the first stage of Escudier & Kellers (1983) theory. Four distinct forms of vortex breakdown are identified: the well documented bubble state, a new cone configuration in which the vortex takes the form of an open conical sheet, and two associated asymmetric bubble and asymmetric cone states, which are only observed at large Reynolds numbers. The two latter configurations differ from the former by the precession of the stagnation point around the jet axis in a co-rotating direction with respect to the upstream vortex flow. The two flow configurations, bubble or cone, are observed to coexist above the threshold Sc at the same values of the Reynolds number Re and swirl parameter S . The selection of breakdown state is extremely sensitive to small temperature inhomogeneities present in the apparatus. When S reaches Sc , breakdown gradually sets in, a stagnation point appearing in the downstream turbulent region of the flow and slowly moving upstream until it reaches an equilibrium location. In an intermediate range of Reynolds numbers, the breakdown threshold displays hysteresis lying in the ability of the breakdown state to remain stable for S Sc once it has taken place. Below the onset of breakdown, i.e. when 0 S Sc , the swirling jet is highly asymmetric and takes the shape of a steady helix. By contrast above breakdown onset, cross-section visualizations indicate that the cone and the bubble are axisymmetric. The cone is observed to undergo slow oscillations induced by secondary recirculating motions that are independent of confinement effects.

Scaling analysis and simulation of strongly stratified turbulent flows

Direct numerical simulations of stably and strongly stratified turbulent flows with Reynolds number Re >> 1 and horizontal Froude number F-h > 1, viscous forces are unimportant and l(v) scales as l ...

Self-similarity of strongly stratified inviscid flows

It is well-known that strongly stratified flows are organized into a layered pancake structure in which motions are mostly horizontal but highly variable in the vertical direction. However, what determines the vertical scale of the motion remains an open question. In this paper, we propose a scaling law for this vertical scale Lv when no vertical lengthscales are imposed by initial or boundary conditions and when the fluid is strongly stratified, i.e., when the horizontal Froude number is small: Fh=U/NLh≪1, where U is the magnitude of the horizontal velocity, N the Brunt–Vaisala frequency and Lh the horizontal lengthscale. Specifically, we show that the vertical scale of the motion is Lv=U/N by demonstrating that the inviscid governing equations in the limit Fh→0, without any a priori assumption on the magnitude of Lv, are self-similar with respect to the variable zN/U, where z is the vertical coordinate. This self-similarity fully accounts for the layer characteristics observed in recent studies reportin...

Experimental evidence for a new instability of a vertical columnar vortex pair in a strongly stratified fluid

This paper shows that a long vertical columnar vortex pair created by a double flap apparatus in a strongly stratified fluid is subjected to an instability distinct from the Crow and short-wavelength instabilities known to occur in homogeneous fluid. This new instability, which we name zigzag instability, is antisymmetric with respect to the plane separating the vortices. It is characterized by a vertically modulated twisting and bending of the whole vortex pair with almost no change of the dipoles cross- sectional structure. No saturation is observed and, ultimately, the vortex pair is sliced into thin horizontal layers of independent pancake dipoles. For the largest Brunt–Vaisala frequency N = 1.75 rad s −1 that may be achieved in the experiments, the zigzag instability is observed only in the range of Froude numbers: 0.13 F h 0 F h 0 = U 0 / NR , where U 0 and R are the initial dipole travelling velocity and radius). When F h 0 > 0.21, the elliptic instability develops resulting in three-dimensional motions which eventually collapse into a relaminarized vortex pair. Irregular zigzags are then also observed to grow. The threshold for the inhibition of the elliptic instability F h 0 = 0.2±0.01 is independent of N and in good agreement with the theoretical study of Miyazaki & Fukumoto (1992). Complete stabilization for F h 0 < 0.13 is probably due to viscous effects since the associated Reynolds number is low, Re 0 < 260. In geophysical flows characterized by low Froude numbers and large Reynolds numbers, we conjecture that this viscous stabilization will occur at much lower Froude number. It is tentatively argued that this new type of instability may explain the layering widely observed in stratified turbulent flows.

Resistance to Colletotrichum lindemuthianum in Phaseolus vulgaris: a case study for mapping two independent genes.

Anthracnose, caused by the hemibiotrophic fungal pathogen Colletotrichum lindemuthianum is a devastating disease of common bean. Resistant cultivars are economical means for defense against this pathogen. In the present study, we mapped resistance specificities against 7 C. lindemuthianum strains of various geographical origins revealing differential reactions on BAT93 and JaloEEP558, two parents of a recombinant inbred lines (RILs) population, of Meso-american and Andean origin, respectively. Six strains revealed the segregation of two independent resistance genes. A specific numerical code calculating the LOD score in the case of two independent segregating genes (i.e. genes with duplicate effects) in a RILs population was developed in order to provide a recombination value (r) between each of the two resistance genes and the tested marker. We mapped two closely linked Andean resistance genes (Co-x, Co-w) at the end of linkage group (LG) B1 and mapped one Meso-american resistance genes (Co-u) at the end of LG B2. We also confirmed the complexity of the previously identified B4 resistance gene cluster, because four of the seven tested strains revealed a resistance specificity near Co-y from JaloEEP558 and two strains identified a resistance specificity near Co-9 from BAT93. Resistance genes found within the same cluster confer resistance to different strains of a single pathogen such as the two anthracnose specificities Co-x and Co-w clustered at the end of LG B1. Clustering of resistance specificities to multiple pathogens such as fungi (Co-u) and viruses (I) was also observed at the end of LG B2.

Three-dimensional stability of a vortex pair

This paper investigates the three-dimensional stability of the Lamb-Chaplygin vortex pair. Short-wavelength instabilities, both symmetric and antisymmetric, are found. The antisymmetric mode possesses the largest growth rate and is indeed the one reported in a recent experimental study [J. Fluid Mech. 360, 85 1998]. The growth rates, wave numbers of maximum amplification, and spatial eigenmodes of these short-wavelength instabilities are in good agreement with the predictions from elliptic instability theory. A long-wavelength symmetric instability similar to the Crow instability of a pair of vortex filaments is also recovered. Oscillatory bulging instabilities, both symmetric and antisymmetric, are identified albeit their growth rates are lower than for the short-wavelength instabilities. Their behavior and eigenmodes resemble those of the oscillatory bulging instability occurring in the mixing layer.

Three-dimensional stability of a vertical columnar vortex pair in a stratified fluid

This paper investigates the three-dimensional stability of a Lamb–Chaplygin columnar vertical vortex pair as a function of the vertical wavenumber k z , horizontal Froude number F h , Reynolds number Re and Schmidt number Sc . The horizontal Froude number F h ( F h = U / NR , where U is the dipole travelling velocity, R the dipole radius and N the Brunt–Vaisala frequency) is varied in the range [0.033, ∞[ and three set of Reynolds-Schmidt numbers are investigated: { Re = 10 000, Sc = 1}, Re = 1000, Sc = 1}, { Re = 200, Sc = 637}. In the whole range of F h and Re , the dominant mode is always antisymmetric with respect to the middle plane between the vortices but its physical nature and properties change when F h is varied. An elliptic instability prevails for F h > 0.25, independently of the Reynolds number. It manifests itself by the bending of each vortex core in the opposite direction to the vortex periphery. The growth rate of the elliptic instability is reduced by stratification effects but its spatial structure is almost unaffected. In the range 0.2 F h < 0.25, a continuous transition occurs from the elliptic instability to a different instability called zigzag instability. The transitional range F hc = 0.2–0.25 is in good agreement with the value F h = 0.22 at which the elliptic instability of an infinite uniform vortex is suppressed by the stratification. The zigzag instability dominates for F h [les ] 0.2 and corresponds to a vertically modulated bending and twisting of the whole vortex pair. The experimental evidence for this zigzag instability in a strongly stratified fluid reported in the first part of this study (Billant & Chomaz 2000 a ) are therefore confirmed and extended. The numerically calculated wavelength and growth rate for low Reynolds number compare well with experimental measurements. The present numerical stability analysis fully agrees with the inviscid asymptotic analysis carried out in the second part of this investigation (Billant & Chomaz 2000 b ) for small Froude number F h and long wavelength. This confirms that the zigzag instability is related to the breaking of translational and rotational invariances. As predicted, the growth rate of the zigzag instability is observed to be self-similar with respect to the variable F h k z , implying that the maximum growth rate is independent of F h while the most amplified dimensional wavenumber varies with N / U . The numerically computed eigenmode and dispersion relation are in striking agreement with the analytical results.

Radiative instability in stratified vortices

This paper investigates the stability of a columnar vortex in an inviscid stratified fluid. By means of a WKBJ analysis for large axial wave number, we demonstrate that the normal modes can be stable or unstable owing to the emission of internal waves from the vortex. This phenomenon is shown to be analogous to the radioactive decay of nuclei in quantum mechanics. The destabilized character of the wave emission is shown to be associated with the presence of a critical point in the radial structure of the normal mode. The theoretical predictions for the frequency and growth rate of the normal modes are shown to be in good agreement with numerical results for two examples.

Nonlinear evolution of the zigzag instability in stratified fluids : a shortcut on the route to dissipation

We present high-resolution direct numerical simulations of the nonlinear evolution of a pair of counter-rotating vertical vortices in a stratified fluid for various high Reynolds numbers Re and low Froude numbers F h . The vortices are bent by the zigzag instability producing high vertical shear. There is no nonlinear saturation so that the exponential growth is stopped only when the viscous dissipation by vertical shear is of the same order as the horizontal transport, i.e. when Z h max /Re = O(1) where Z h max is the maximum horizontal enstrophy non-dimensionalized by the vortex turnover frequency. The zigzag instability therefore directly transfers the energy from large scales to the small dissipative vertical scales. However, for high Reynolds number, the vertical shear created by the zigzag instability is so intense that the minimum local Richardson number Ri decreases below a threshold of around 1/4 and small-scale Kelvin-Helmholtz instabilities develop. We show that this can only occur when ReF 2 h is above a threshold estimated as 340. Movies are available with the online version of the paper.

Elliptic and zigzag instabilities on co-rotating vertical vortices in a stratified fluid

We present a three-dimensional linear stability analysis of a couple of co-rotating vertical vortices in a stratified fluid. When the fluid is non-stratified, the two vortices are unstable to the elliptic instability owing to the elliptic deformation of their core. These elliptic instability modes persist for weakly stratified flow: