Nadia Mkhinini

Thomas Dubos

Philippe Drobinski

Instabilities and transition in three-dimensional flows with rotation SECONDARY INSTABILITY OF STRATIFIED EKMAN LAYER ROLL VORTICES

We study the transition to turbulence of the Ekman ow in a linear density stratication, an exact solution of the Boussinesq equations occurring near boundaries in rotating uids. The control parameters are the Richardson number Ri = δ 2 N 2 G 2 , the Reynolds number Re = Gδ ν and the Prandtl number P r = ν κ where δ = 2ν/f is the Ekman length, ν is the kinematic viscosity, f is the Coriolis parameter, G is the geostrophic velocity far from the lower boundary and κ is the thermal diusivity. We consider Re = 500 and Re = 1000. In order to assess separately the eect the diusion of momentum and heat, we consider two values of the Prandtl number : P r = 1 and P r = 4.

Nonlinear primary instability

The Ekman ow is subject to linear instability for Ri < Ri c (Re, P r) where Ri c depends very little on P r [START_REF] Brown | On the inection point instability of a stratied Ekman boundary layer[END_REF]. This instability develops into traveling Kelvin-Helmholtz roll vortices which we compute as a function of stratication as in [START_REF] Dubos | Emergence and secondary instability of ekman layer rolls[END_REF]. Fig. 1 presents the amplitude of the equilibrated vortices as measured by their kinetic, potential and total energy. While for P r = 1 the roll amplitude vanishes at Ri = Ri c this is not the case at P r = 4. So far only supercritical bifurcations had been identied in the Ekman layer stability diagram [START_REF] Haeusser | Pattern formation in the marginally unstable Ekman layer[END_REF][START_REF] Dubos | Emergence and secondary instability of ekman layer rolls[END_REF]. The subcritical bifurcation observed at P r = 4 allows us to compute equilibrated vortices at Ri > Ri c .

2 Linear secondary instability Fig. 2 presents the growth rate σ 2 of inntesimal three-dimensional perturbations to the travelling rolls as a function of their horizontal wavenumber k y . For Ri = 0 we recover at k y > 1 the unstable modes identied by [START_REF] Dubos | Emergence and secondary instability of ekman layer rolls[END_REF]. New, less unstable modes branches are obtained for 0 < k y < 1. However as Ri increases the modes for k y > 1 become less unstable, become dominated by the unstable modes for k y < 1 and eventually disappear. For P r = 1 the overall maximum growth rate decreases as Ri increases, but not for P r = 4. This must be related to the fact that the roll vortices weaken as Ri approaches Ri c if P r = 1, but not if P r = 4.

Conclusion

We have studied the inuence of an ambient linear stratication on the secondary instability of the Ekman ow. This inuence depends markedly on the Prandtl number. Our results reveal that the bifurcation at Ri = Ri c is subcritical P r = 4. In this regime, nite amplitude vortices exist beyond Ri = Ri c and the growth rate of the secondary instability increases as the ambient stratication increases, instead of decreasing as occurs for P r = 1 and as intuition suggests. A complementary study of the nonlinear development of the secondary instabilty is under way.

Figure 1 :

 1 Figure 1: Energy of roll vortices as a function of ambient stratication. Arrows point to Ri = Ri c

Figure 2 :

 2 Figure 2: Growth rate σ 2 of secondary instability of Ekman roll vortices as a function of wave vector k y (right : zoom)


Ayah Lazar

Alexandre Stegner

Eyal Heifetz

Instabilities and transition in three-dimensional flows with rotation A STUDY ON THE 3D INERTIAL INSTABILITY MECHANISM IN THE SUB-MESOSCALE OCEAN

. Inertial instability (hereafter II), which is a centrifugal instability mechanism in the presence of the Coriolis force, is a destructive mechanism that acts only on anti-cyclones, and therefore is hypothesized to cause this asymmetry. Furthermore, since II is in fact the growth of the overturning vorticity, thus creating vertical mixing, it is expected to contribute to nutrient enrichment from the deep, affecting primary production and the oceanic carbon cycle.

. Axis-symmetry seems to be a proper simplification, as it was shown by [1] that the growth rates of axis-symmetric disturbances are larger than non axis-symmetric ones, and the latter are completely stable above a relatively low cut-off azimuthal wave number. Recent fully non-linear numerical simulations show that the area of the instability is breached [5] into the anticyclone core from the outside [3] in the nonlinear phase.

) with large-scale laboratory experiments studies, performed at the LEGI-Coriolis platform.

A good case study for II is expected in Island wakes, where the forcing that generates the anticyclones is strong. Then they are created, but destructed far more rapidly than their cyclonic counterparts in the wake. For that reason, we conducted experiments of island-wake vortices instability on the 14-meter diameter rotating platform at the LEGI-Coriolis in Grenoble with quantitative PIV measurements of the wake flow.

On the left, vorticity field of experiment C50 006 calculated from the PIV. The cylinder and its direction is shown schematically. Small scale patterns that can easily be seen with dye cannot be detected by 2D PIV measurements, therefore the signature of the II is extracted by the mesoscale evolution of individual eddies, in this case the velocity profiles of anticyclones taken when the vortex is at a distance of eight times the diameter of the cylindrical island model (in experiment C50 006 the profile is taken where the line A is indicated). Panel (A) and (B) show the velocity profiles for experiments C50 006 and C50 001 respectively. Experiment C50 001 is predicted by its place on the stability diagram (1b) to be stable and indeed, the anticyclone in this experiment is unperturbed. Experiment C50 006 is predicted to be unstable and indeed the anticyclone is perturbed at the edges, where the parameter χ < 0 (the cyclones, not shown, in all cases are unperturbed as the instability is asymmetric).

EUROMECHFigure 1 :Figure 2 :

 12 Figure 1: Panel (a) -First (top) and second (middle) horizontal modes for a sample Rankine vortex (Ro = -3). This is the solution of a modified bessel equation of order unity inside the vortex and of imaginary order outside. On the bottom is the χ parameter, and it is evident that the instability, in both modes, is localized to the area where χ < 0. Panel (b) -Marginal stability curves (σ = 0) -on the (1/E k , |Ro|) plane, for three stratification values S = 0, 8, 20 and δ = 0.2. Not shown are curves for different types of non-isolated vortices, which on this parameter space, are very close to the ones shown for the Rankine vortex. The parameters of experiments C50 001 and C50 006 from the experiments performed at the LEGI-Coriolis platform are both marked with red dots. Both are with δ = 0.2 and S 8, which means that C50 001 should be stable and C50 006 should be unstable as they are on either side of the marginal stability curve. See figure 2 for the results.


H Bolnot

S Le Dizès

T Leweke

Instabilities and transition in three-dimensional flows with rotation SPATIO-TEMPORAL DEVELOPMENT OF INSTABILITIES IN HELICAL VORTICES

In aeronautics, the dynamics of helical vortices are of interest for many applications. For instance, helicopter wakes exhibit a global transition to the Vortex Ring State (VRS) in steep descent regimes which can lead to a loss of control and crash the apparatus. This transition could be associated with an instability phenomenon and we will argue that it is due to the change of nature from convective to absolute of the pairing instability. For wind turbines, instabilities are also known to be responsible for the turbulence in the far wake. Understanding the spatio-temporal development of the different instabilities is therefore also important in this context. In this work, we consider the particular instability that leads to vortex pairing and address our results in both context of wind turbines and helicopter.

Numerical simulations have shown that several instabilities can occur in helical wakes. Both the threedimensional elliptic instability and the two-dimensional pairing instability were found to play an important role in helicopter as well as wind turbine wakes (Leishman et al. [START_REF] Bhagwat | The vortex ring state as a spatially and temporally developing wake instability[END_REF], Walther et al. [START_REF] Walther | A numerical study of the stability of helical vortices using vortex methods[END_REF]). However, the theoretical results known about the stability properties of a helical vortex wake were established for only few model cases: a helical filament (Widnall [START_REF] Widnall | The stability of a helical vortex filament[END_REF]) or in the limit of large wavenumbers (Okulov [4], Okulov and Sørensen [START_REF] Okulov | Stability of helical tip vortices in a rotor far wake[END_REF]). For the pairing instability, which requires to take into account the geometry of the vortex core, a systematic analysis of the absolute/convective nature of the instability was performed numerically in the case of an array of straight vortices (Brancher and Chomaz [START_REF] Brancher | Absolute and convective secondary instabilities in spatially periodic shear flows[END_REF]) but has still to be done for a helical vortex wake.

In order to focus on the development of the pairing instability and filter out all other instabilities, the helical wake is represented by an array of axisymmetric vortex rings. For small helix pitches, this provides a good approximation of the azimuthal vorticity field, as shown in Figure 1. The instabilities developing along the array of vortex rings are characterized in detail. The linearized Navier-Stokes equations for the perturbations are discretized by means of a spectral method, in order to describe the spatio-temporal evolution of the wave packet generated by a localized initial perturbation. The temporal growth rate of the pairing instability is obtained as a function of the perturbation propagation speed for the different parameters ( a0 R , a0 h , Re) of the system. A conclusion is eventually made on the different flow configurations that could lead to the Vortex Ring State, which arises when unstable perturbations (with a positive growth rate) propagate upstream faster than the advection speed of vortices. For wind turbine configurations, the distance (behind the rotor plane) at which the pairing instability initiates will be determined as a function of the parameters, which represents a major issue in the design of wind turbine farms. The initial condition in both cases is a Gaussian vortex of core radius a 0 and with no axial velocity. Simulations were run for a0 R = 1 16 and a0 h = 1 8 at Re = Γ 2 π ν = 500. The helical vortex is computed by forcing the flow to have a helical symmetry (see Delbende [START_REF] Delbende | DNS of vortices with helical symmetry[END_REF]).

In parallel with this theoretical approach, experimental results concerning helical wakes will be presented. The characteristics of the wake behind a rotor for different tip speed ratios, defined as the tip speed over the external speed T SR = Vtip Vext , are analyzed qualitatively as well as quantitatively. The facility used here is a closed loop water channel in which an external field can be imposed to the rotor. This free surface channel has a test section of 38 * 50 * 150 cm 3 and generates a constant incoming velocity (less than 1 % of turbulence) up to 100 cm/s. A visual access is available on the 5 sides of the test section. The rotor diameter is 16 cm and its rotational speed can be varied up to 10 rps. The sense of rotation determines whether the rotor accelerates the fluid (helicopter climb regime) or if it produces a counter flow (helicopter descent regimes and wind turbines configuration). Qualitative observations of the wake are made through dye visualizations, either by injecting a line of fluorescein upstream of the rotor plane or by painting directly the surface of the blades at the tip location in order to visualize the center of the tip vortices which evolves in the wake downstream as shown in Figure 2(a). A quantitative analysis of the wake is also performed thanks to Stereo Particle Image Velocimetry (SPIV). The vortical structure developing in the wake is identified precisely by calculating the characteristics of the vortices (position, core size, circulation, axial velocity) as a function of their age. For instance, Figure 2 These experimental results are eventually compared with the numerical predictions concerning the threshold of the transition leading to VRS for helicopter configurations and the distance at which the pairing instability occurs in the case of wind turbine wakes.

This work is supported by EUROCOPTER S.A.S. through a doctoral grant (contract no. IPEC 0187N/2009).

Figure 1 :

 1 Figure 1: Sketches and azimuthal vorticity contours of a helical vortex (left) and of a vortex rings array (right).The initial condition in both cases is a Gaussian vortex of core radius a 0 and with no axial velocity. Simulations were run for a0 R = 1 16 and a0 h = 1 8 at Re = Γ 2 π ν = 500. The helical vortex is computed by forcing the flow to have a helical symmetry (see Delbende[START_REF] Delbende | DNS of vortices with helical symmetry[END_REF]).

Figure 2 :

 2 Figure 2: (Left) Dye visualization of the wake illuminated in volume in climb regime at T SR = 4.5 The incoming velocity as well as the induced velocity of the rotor are oriented downwards. (Right) Isocontour of vorticity of the 3D wake in climb regime at T SR = Vtip Vext = 10 (with V tip = 100 cm/s, V ext = 10 cm/s and Reynolds number based on the chord and the tip speed Re = 15000).


A unified

come

2 LFMI, EPFL, 1015 Lausanne, Switzerland.

It is well known that swirling jets can become centrifugally unstable like pure vortices but with a different azimuthal wavenumber selection. The Leibovich and Stewartson [START_REF] Leibovich | Stewartson A sufficient condition for the instability of columnar vortices[END_REF] criterion is a generalization of the Rayleigh criterion to swirling jets: it is a sufficient condition for instability with respect to perturbations with both large axial and azimuthal wavenumbers. We have relaxed the large azimuthal wavenumber assumption in this criterion and obtained a new criterion that is valid whatever the azimuthal wavenumber and whatever the magnitude of the axial flow: from zero (pure vortex) to finite values (swirling jets). The new criterion recovers the Leibovich-Stewartson criterion when the azimuthal wavenumber is large and the Rayleigh criterion when the azimuthal wavenumber is finite [START_REF] Billant | Gallaire Generalized rayleigh criterion for non-axisymmetric centrifugal instabilities[END_REF]. The criterion is confirmed by comparisons with numerical stability analyses of various classes of swirling jet profiles. In the case of the Batchelor vortex, it provides more accurate results for perturbations with finite azimuthal wavenumbers than the Leibovich-Stewartson criterion (figure 1). The criterion shows also that a whole range of azimuthal wavenumbers are destabilized as soon as a non-zero axial velocity component is present in a centrifugally unstable vortex.

EUROMECH

  Colloquium 525, 21-23 June 2011, Écully, France Instabilities and transition in three-dimensional flows with rotation A UNIFIED CRITERION FOR THE CENTRIFUGAL INSTABILITY OF VORTICES AND SWIRLING JETS Paul Billant 1 , Francois Gallaire 2 1 LadHyX, CNRS-Ecole Polytechnique, F-91128 Palaiseau, France.

Figure 1 :

 1 Figure 1: Growth rate as a function of the axial wavenumber k for the Batchelor vortex for the swirling parameter q = 0.8 and the azimuthal wavenumbers m = -6 and m = -8 (from left to right). Solid line: exact result, dashed line: present criterion, dashed-dotted line: Leibovich & Stewartson criterion.