M Grandemange

M Gohlke

V Parezanović

O Cadot

On experimental sensitivity analysis of an axisymmetric turbulent wake

The sensitivity to local disturbances of the turbulent wake over a 3D blunt body with an axisymmetric detachment is investigated at Re = 2.1•10 4 . The flow presents a favored m = 2 azimuthal symmetry set by two wings. The instantaneous wake is measured either above or below the plane containing the wings but leads a statistical symmetric wake. Topology shifts are random but occur mostly after a large number of global mode periods. The statistical symmetry is highly sensitive to any asymmetric disturbance. As a consequence, depending on its position a small control cylinder in the close wake fixes the wake to one asymmetric topology affecting shedding activity and drag. The effect of an axisymmetric perturbation (m = 0) on flow topology and dynamics is also studied ; it induces significant drag reductions and global mode modifications when acting on mixing layers. Whatever the disturbance, the sensitivity of the wake seems concentrated into the mixing layers and may depend more on their local turbulent characteristics than on the inviscid dynamics of vorticity. A = ⟨a⟩ average value of a Std (a) standard deviation of a a * non-dimensional value of a |⃗ a| modulus of a c D drag coefficient c p pressure coefficient D diameter of the body base d R diameter of the control ring H = δ * /θ boundary layer shape factor L b recirculation length p 0 inlet flow pressure P (#A) probability of state #A P (#A|#B) probability of state #A, given #B

I. INTRODUCTION

New theoretical efforts have allowed to obtain the structural sensitivity of global mode wakes in laminar regimes [START_REF] Hill | A theoretical approach for analyzing the restabilization of wakes[END_REF][START_REF] Marquet | Sensitivity analysis and passive control of cylinder flow[END_REF][START_REF] Luchini | Structural sensitivity of the finite-amplitude vortex shedding behind a circular cylinder[END_REF] . Over bidimensional geometries, this approach is able to recover qualitatively the sensitivity obtained experimentally by Strykowski and Sceenivasan 4 . Recently, Meliga et al. [START_REF] Meliga | Unsteadiness in the wake of disks and spheres: Instability, receptivity and control using direct and adjoint global stability analyses[END_REF][START_REF] Meliga | Open-loop control of compressible afterbody flows using adjoint methods[END_REF] performed similar studies in the wake of axisymmetric bodies ; zones of high sensitivity have been found in the recirculating bubble, particularly around the separatrix. Although the theory is still limited to basic flows and laminar regimes, this approach might provide a useful tool in the frame work of control strategy since it predicts the placement of a disturbance to efficiently affect global properties of the flow. For practical and industrial interests, the method should be extended to large Reynolds number flows over complex geometries. So far, no theory is available in turbulent regimes however, over cylinders, some qualitative similitudes persist between laminar theoretical analyses and experiments at moderate Reynolds numbers [START_REF] Sakamoto | An optimum suppression of fluid forces by controlling a shear layer separated from a square prism[END_REF][START_REF] Parezanović | The impact of a local perturbation on global properties of a turbulent wake[END_REF][START_REF] Parezanović | Experimental sensitivity analysis of the global properties of a 2D turbulent wake[END_REF] . The empirical results of Parezanović and Cadot 9 point out that the distrubance of the close wake leads to variations of global mode Over axisymmetric bodies, the dynamic of the separated flow is more complicated than the bidimensional von-Kármán street. Indeed, the wake undergoes several transitions associated to different wake regimes as the Reynolds number increases [START_REF] Sakamoto | A study on vortex shedding from spheres in a uniform flow[END_REF] . At very low Reynolds numbers, the flow is steady and axisymmetric. A first bifurcation is reported (for Re ≈ 210 over spheres) : a steady azimuthal mode m = 1 appears. A lift force is generated and a pair of counter-rotating vortices develops downstream moving the wake off streamwise axis so the flow loses its axisymmetry but keeps a planar symmetry. A second bifurcation occurs at a higher Reynolds number (Re ≈ 277 over spheres) : the wake starts oscillating but preserves the same planar symmetry. Becoming turbulent, the preference toward this azimuthal plane of symmetry vanishes and the flow becomes statistically axisymmetric [START_REF] Mittal | Symmetry properties of the transitional sphere wake[END_REF] . At moderate Reynolds numbers, the high frequency mixing layer instabilities in the close wake degenerate into large scale vortex loops developing from the end of the recirculation bubble ; the wake oscillates randomly [START_REF] Sakamoto | A study on vortex shedding from spheres in a uniform flow[END_REF][START_REF] Taneda | Visual observations of the flow past a sphere at Reynolds numbers between 104 and 106[END_REF] and may get a helical structure highly coherent in space [START_REF] Pao | Vortex structure in the wake of a sphere[END_REF][START_REF] Berger | Coherent vortex structures in thewake of a sphere and a circular disk at rest and under forced vibrations[END_REF][START_REF] Yun | Vortical structures behind a sphere at subcritical Reynolds numbers[END_REF] . This unsteady global mode (dominant mode m = 1) is reported at 0.1 < St < 0.2 depending on geometry and Reynolds number.

Previous experiments of passive and active control of the turbulent wake show connections between parameters of the control set-up, drag and global mode intensity [START_REF] Berger | Coherent vortex structures in thewake of a sphere and a circular disk at rest and under forced vibrations[END_REF][START_REF] Mair | The effect of a rear-mounted disc on the drag of a blunt-based body of revolution(Drag of body of revolution with blunt base substantially reduced by mounting disk behind body with smaller diameter than body)[END_REF][START_REF] Weickgenannt | Control of vortex shedding in an axisymmetric bluff body wake[END_REF][START_REF] Sevilla | Vortex shedding in high reynolds number axisymmetric bluff-body wakes: Local linear instability and global bleed control[END_REF][START_REF] Higuchi | Passive and Active Controls of Three-Dimensional Wake of Bluff-Body[END_REF][START_REF] Morrison | Control of an axisymmetric turbulent wake by a pulsed jet[END_REF] . In particular, Berger et al. [START_REF] Berger | Coherent vortex structures in thewake of a sphere and a circular disk at rest and under forced vibrations[END_REF] proves that disk forced to oscillate near the helical mode frequency stabilizes this mode in space and time at a coherence level close to 1. Weickgenannt et al. [START_REF] Weickgenannt | Control of vortex shedding in an axisymmetric bluff body wake[END_REF] Eventually, the effects of these disturbances on the wake orientation and global mode frequency are discussed ; past tridimensional bodies, the turbulent characteristic of the wake might play a greater role than the inviscid dynamics of vorticity.

II. EXPERIMENTAL SET-UP

The Eiffel type wind tunnel is an open loop air flow facility. The turbulent intensity is less than 0.3% and the homogeneity of the velocity over the 400 mm × 400 mm blowing section is 0.4%. The wake is generated by a 3D symmetric blunt body ; its geometry is based on the "D" shape cylinder (see Fig. 1). Most experiment are performed with the blunt body aligned to incoming flow, however when specified a small pitching angle ϵ is set to add an antisymmetric disturbance.

The main flow velocity is U 0 = 6.5 m.s -1 and the Reynolds number

(Re = U 0 D/ν) is 2.1 • 10 4 .
The velocities are defined as The separated flow is first controlled by a 3 mm diameter cylinder oriented along z direction and moved in the wake using a robot. x C and y C refer to the cylinder position (see Fig. 1(a)).

⃗ u = u x . ⃗ e x + u y . ⃗ e y + u z . ⃗ e z ; u ij = √ u 2 i + u 2 j is
Except when located on the streamwise axis, the cylinder is considered a m = 1 disturbance.

The other control devices are flat rings with a length of 6 mm and a thickness of 1 mm mounted on a 3 mm diameter rod (see Fig. 1(b)). x R represents the gap between the base of the body and the ring center. The point R remains on the streamwise axis so the symmetry of the body is preserved and the rings are associated to m = 0 perturbations. Nine ring diameters d R between 0.7 and 1.1D are used to disturb the close wake.

A wake analysis is made from PIV and stereo PIV measurements respectively in planes z * = 0 and x * = 1.2. Statistics from 2000 instantaneous velocity field estimate the fluctuating velocity energy. To get unsteady characteristics of the flow, a 1D hot wire probe mounted on a three-axis robot records the velocity in the wake at a sampling frequency of 2 kHz. Power spectra are then calculated up to 1 kHz with a resolution of 0.5 Hz.

In addition, the pressure on the body is measured in the plane z * = 0. Thirteen taps are located on the half sphere of the forebody every 15 • ; the seven others give the pressure distribution on the base, the space between base taps is 6 mm. Pressure measurements are performed at 1 Hz over 120 s; the sampling frequency of 1 Hz is limited by the acquisition device : the record is a mean over at least 1 s to preserve precision.

The drag is estimated by integration of the interpolated pressure on the surface projected on x direction ; pressure distributions in each region y * > 0 and y * < 0 are assumed axisymmetric : (

C D = 1 1 2 ρπ D 2 4 V 0 2 [ ∫ ∫
For the uncontrolled flow, the so-evaluated drag is C D 0 = 0.261±0.002. Experimenting the method presented by Weickgenannt et al. [START_REF] Weickgenannt | Control of vortex shedding in an axisymmetric bluff body wake[END_REF] based on momentum deficiency in the far wake, a higher value of 0.29 is measured, partially due to friction effects. When the control device is in the wake, its contribution to the drag is neglected.

III. RESULTS

A. Natural flow

The boundary layer at the trailing edge is fully turbulent and depends on the azimuth due to the presence of the wings in the plane y * = 0. The boundary layers at detachment in planes In the plane y * = 0, the wake of the wings leads to a velocity deficiency and a turbulent boundary layer at the trailing edge at y * ≈ 0 (see empty and filled squares on Fig. 2(a)). As the wake of the wings is present at y * = 0, the fluctuations of velocity remains large even outside the boundary layer. The characteristics obtained from these velocity profiles are presented in table II. The high shape factor of the turbulent boundary layer is certainly due to the adverse pressure gradient along the side of the body.

BL at y * = 0.5 and z * = 0 BL at y * = 0 and z * = 0.5 After detachment, the mixing layers grows from the trailing edge to the far wake. α = 0.1, 0.5 and 0.9) are plot on Figure 2(c). The growth of δ M is approximatively linear between

δ
x * = 0.1 and x * = 0.7 and dδ M * /dx * is measured respectively at 0.17 and 0.18 for the upper and lower mixing layer. The expansion occurs toward the recirculation bubble where U x * ∼ 0, the contours U x * = 0.9 being roughly parallel to the streamwise axis. This particularly high growth rate in comparison to the results presented by of Champagne et al. [START_REF] Champagne | On the two-dimensional mixing region[END_REF] may be due to the axisymmetry of the wake but also to the oscillations induced by the presence of a global mode. Indeed, the power spectrum analysis of hot wire probe signals at (x * , y * , z * ) = (4, 0.5, 0) reports a wake oscillation at frequency f S = 25.8 Hz, i.e. St S = 0.199 (see Fig. 3). The two peaks at St ≈ 0.05 and 0.07 correspond to wind tunnel signature.

Figure 4(a) displays the streamlines of the time-averaged PIV measurements in the plane Theses fluctuations are partially associated to the development of the unsteady global mode. As expected, it is equally present at the top and the bottom of the end of the recirculation bubble.

In agreement with remarks present in literature [START_REF] Weickgenannt | Control of vortex shedding in an axisymmetric bluff body wake[END_REF][START_REF] Delery | Topologie des écoulements tridimensionnels décollés stationnaires : points singuliers, séparatrices et structures tourbillonnaires[END_REF] , the symmetry of the wake is highly sensitive to the set-up configuration. An ϵ incidence moves the whole wake up or down. just above a single zone of intense fluctuations. Therefore, as soon as the symmetry is lost, the unsteady m = 1 mode develops mostly from the lower part of the wake. It is associated to the formation of parallel hairpin shaped vortex loops at the end of the recirculation bubble [START_REF] Sakamoto | A study on vortex shedding from spheres in a uniform flow[END_REF][START_REF] Achenbach | Vortex shedding from spheres[END_REF] . When the symmetry is preserved, these vortex loops must develop statistically from both sides of the wake.

The observed balanced wake for ϵ = 0 (see Fig. an artificial velocity field can be computed:

U x (x * , y * ) = 1 2 (U xϵ (x * , y * ) + U xϵ (x * , -y * )) and U y (x * , y * ) = 1 2 (U yϵ (x * , y * ) -U yϵ (x * , -y * ))
. This artificial wake presented on Figure 6 has exactly the same properties as the measured one over a lined-up body : two collar points associated to two zones of intense fluctuations of velocity at y * ≈ ±0.4.

Two different wake positions are then expected for the flow over a lined-up configuration i.e. ϵ = 0. When the wake follows the configuration #1, the upper recirculation structure is bigger and closer to the base than the lower one. This asymmetry of the recirculation bubble leads to an average diagonal recirculating flow and a non-uniform pressure distribution on the base ; the base pressure gradient dC p /dy * is negative in state #1 (see Fig 7(a)). The slope of the linear fit of the base pressure distribution gives the average pressure gradient and indicates whether the instantaneous flow follows the state #1 or #2. Hence, each record of base pressure distribution gives the dominant topology over the second of measurement.

The time evolution of the base pressure signal during 5 • 10 3 s is studied. The pressure gradient, i.e. the dominant topology, is shown on Figure 7(b) over 1000 s . Two preferred positions are visible at dC p /dy * ≈ ±0.1. This point is confirmed by the probability density function presented on Figure 7(c). The two topologies #1 and #2 respectively associated to negative and positive gradients (-0.7 and 0.9) are clearly visible. The presence of the minimum between these peaks suggests that a sampling period of 1 s remains below the mean shift time estimated at 4.6 s, i.e. more than 100 global mode periods. The standard deviation of the base pressure gradient is 0.084. This value is equivalent to the absolute values of pressure gradients associated to states #1 and #2 presented on the histogram Figure 7(b). Such a high value is then an indicator of the coexistence of the two wake topologies. Eventually, the spectrum analysis of this signal over the 5 • 10 3 s does not present any characteristic frequency : the process seems random.

The two asymmetric topologies are then discriminated against the sign of the base pressure gradient. The probability of state #1 or #2 at a moment t (resp. named S t = #1 and S t = #2) depends not only on geometrical parameters like incidence but also on the past events (see Tab. III). P (S t = #1) < P (S t = #1 | S t-1 = #1) < P (S t = #1 | S t-1 = S t-2 = #1) (idem for topology #2), P (#A|#B) referring to the conditional probability of #A, given #B. As a result, a configuration is more likely to appear at t if it was already there the seconds before.

P (S t = #1) =0.53 P (S t = #2) = 0.47 P (S t = #1 | S t-1 = #1) = 0.79 P (S t = #2 | S t-1 = #2) = 0.77 P (S t = #1 | S t-1 = S t-2 = #1 ) = 0.85 P (S t = #2 | S t-1 = S t-2 = #2 ) = 0.80
Table III: Probabilities of states #1 and #2 depending on the previous states over lined-up uncontrolled configuration (precision better than 0.02), P (#A|#B) is the conditional probability of #A, given #B This points out that both topologies #1 and #2 are present in the wake, the shift characteristic time being implicitly between 1 s and 60 s. Further downstream, for x C * > 1, the flow is less influenced by the control cylinder : the effect of the disturbance is not important enough to really ) but remains at the same order of magnitude. In addition, the two collar points are still observed in the wake. This tends indicate that the flow is not stabilized but that the two states #1 and #2 may still be present with a shifting time smaller than 1 s.

B. 3 mm control cylinder in the wake

In parallel to these different flow modifications, drag is estimated through the pressure distribution on the body. Drag reductions are mostly observed for x C * ≈ 0.5 and y C * ≈ 0 (see Fig. 10). The optimal position leads to a 8% decrease in drag mostly associated to a recovery of base pressure (see Fig. 11(c)) ; the forebody pressure distribution is found independent of the cylinder position. When the cylinder induces dC p /dy * ≈ 0 for y C * ̸ = 0, there is almost no effect on the drag. At these locations, the flow does not seem disturbed by the cylinder, the only difference is that the global mode frequency is slightly reduced. to be attenuated referring to the natural flow. In particular, the impact of the control ring on Std (dc p /dy * ) follows the inner frontier of the mixing layer. As for the control cylinder, these low values mean that either the wake is stabilized or the shift characteristic time is smaller than the measurement sampling frequency. For x R * > 0.8, the fluctuations of base pressure gradient return close to its natural values.

C. Rings acting on mixing layers

The modification of the global mode activity due to the presence the control rings is now considered. Figures 13(a)-(c) present the power spectrum of the hot wire probe signal at (x * , y * , z * ) = (4, 0.5, 0). For small ring diameters in the close wake (d R * < 0.85 and x R * < 0.2), the disturbance is not in the mixing layer, the global mode frequency and amplitude are close to the natural case. As the disturbance reaches the inner part of the mixing layer for x C * < 0.4, the shedding frequency is decreased by approximately 15%. While the ring diameter increases, the perturbation affects the middle of the mixing layer and the global mode is reported far less energetic and then poorly defined (grey zones on Fig. 13(c)). Reaching the outer part of the mixing layer, the global mode is measured again but at a higher frequency in comparison to the natural value.

Further downstream for 0.4 < x C * < 0.6, a different scheme is observed (see Fig. 13( For x R * > 0.6 whatever d R * is, there is no more peak reported in the power spectra. The attenuation of the global mode may be due to the presence of the 3 mm support. Indeed, Figure 9(b) points out that the control cylinder prevent the global mode development when placed on the streamwise axis for x C * > 0.7. Then, for these locations, the effect of the support may not be negligible.

These evolutions in the dynamic of the flow result in drag reductions and increases. Figure 14 presents the estimation of the drag depending on the ring diameter and position. Alike the global mode activity, the effect of the rings approximately follows the position of the mixing layer in the natural flow. The optimal drag reductions are observed when the control device acts on the inner part of the mixing layers. In opposition, drag is globally increased when the outer part of the mixing layer is disturbed. An exception is present for x C * ≈ 0.2 where drag is also decreased for the highest ring diameters. Thus, the drag evolutions do not directly correspond to the global mode frequency map presented on Figure 13(c). Eventually, as observed with the control cylinder, if x R * > 0.9 the effect is limited on drag indicating that the sensitivity of the flow is concentrated particularly in the close wake mixing layers.

Different flow topologies correspond to these variations of drag and shedding frequency ; three of them are presented on Figures 15(a but after the recirculation bubble ; thus they may not be associated to equivalents of states #1 and #2 but rather to the proper wake of the ring. The area where U x * < 0 is strongly shortened and the recirculation centers move near the body and the curvature of the recirculation separatrix increases reducing peripheral base pressure. As shown on Figure 15(d), a compression occurs at the center of the body due to an intensified recirculating flow in average. However this high pressure at the base center is not sufficient to offset the loss of peripheral pressure. Indeed, the assumed axisymmetry of the base pressure implies that the area associated to the pressure considered to calculate drag is proportional to |y * | (see ( 1)). The base pressure at y * ≈ 0 has then a smaller impact on drag than at the periphery. The drag is measured equal to the uncontrolled case despite this different topology corresponding to the low frequency global mode (St S ≈ 0.1).

A third topology associated to high drag case is presented on Figure 15(c). Alike for x R * = 0.6, the flow reattaches on the rings and the recirculating structures are moved closer to the base of the body inducing a shorter recirculation length. The drag is increased due to the loss of pressure reported on the entire base : in opposition to the x R * = 0.6 there is not intense backward flow to counter the loss peripheral base pressure. This position of the ring is associated to an absence of global mode activity. The flow of over tridimensional geometries seems less influenced by the inviscid dynamics of the vorticity in the wake. The global mode does not correspond to intense shedding and the refill of the bubble presented by Gerrard is no more relevant. However, the dynamics of the flow can still be analyzed considering that the turbulence characteristic of the wake is dominant. The growth of the mixing layer is associated to the entrainment of fluid from both the bubble and the freestream flow at a characteristic velocity v E following the relationship

v E dx ∼ U 0 [δ M (x + dx) -δ M (x)],
then

v E * ∼ dδ M * dx * .
Estimating the bubble shape area and volume roughly at L b D and L b D 2 with L b the length in x direction, a characteristic time of the dynamic of the bubble T b can be defined. T b corresponds to the time needed to empty the volume of the bubble with a outflow speed v E over the interface.

L b D v E T b ∼ L b D 2 , T b ∼ D U 0 v E * ∼ D U 0 • dδ M * /dx * .
The associated Strouhal number is

St b ∼ D U 0 T b ∼ dδ M * dx * . ( 2
)
If the turbulent characteristic of the flow dominates its vorticity dynamic, then the global mode frequency of the recirculation bubble may linearly depend on the growth of the mixing layers in turbulent flows. The value of dδ M * /dx * ≈ 0.17 is reached. At moderate Reynolds number, the mixing layer is laminar in the close wake and its growth is slow ; this state is then associated to a low mode frequency. As the Reynolds number increases, the mixing layer turns into turbulence before the end of the recirculation. The higher the Reynolds number is, the sooner the transition occurs. This results in an increasing flow entrainment into the mixing layers and an higher global mode frequency.

The effect of the control ring on the shedding frequency may be interpreted with this basic model (see Fig. 13(c)). When the ring disturb the inner part of the mixing layer, the fluctuations of velocity are reduced in the mixing layers. Their growth rate is then attenuated corresponding to an increase in the recirculation length (see Fig. 14(a)). The associated shedding frequency is measured slightly reduced. When the rings act on the outer mixing layer in the close wake, the mixing layer may become more unstable and enhance fluid entrainment increasing the Strouhal number of the mode.

Further downstream, a part of the mixing layer flow is oriented toward the streamwise axis and the rings have their own wake. Thus, the dynamic of the flow changes drastically and the mode frequency at St ≈ 0.1, when observed, cannot be compared to the natural value. Eventually, the influence of the control cylinder on the global mode frequency is not discussed here. Indeed, m = 1 perturbations mostly impact the planar symmetry and the orientation of the wake, these effects are analyzed in the following section. This analysis coincides with the effect of small pitching angles ϵ and positions of the control cylinder.

Indeed over a nose-up configuration the upper boundary layer faces a higher adverse pressure gradient than the lower one. This results in an increased height and a higher turbulence level at the trailing edge. Therefore, the spread of the mixing layer is more intense on the upper side of the recirculation bubble and the wake moves down, the vortex loops being generated from the lower side of the bubble (see Fig. 5). In the same way, the control cylinder can enhance or reduce the rate dδ M * /dx * . In the close wake, when located in the inner part of the mixing layers the flow may reattach on it, inhibiting the shear layers instability. Then the opposite mixing layer grows faster and the wake moves in the direction of the disturbance (see Fig. 8(a)). Further downstream or in the outer mixing layers, the control cylinder has its own wake generating important fluctuations of the velocity fields. The disturbance intensifies the growth of the mixing layer and the wake moves to the other side. Only cylinder positions in the mixing layers in plane z * = 0 are discussed here ; when located inside the recirculation bubble, the cylinder get through the entire wake, the disturbance is then much more complicated. The presented interpretation is no more pertinent.

V. CONCLUDING REMARKS

In conclusion, the natural flow over a body with an axisymmetric detachment is proved to be a mean of asymmetric topologies. Due to the presence of two wings, i.e. m = 2 azimuthal periodicity of the body, the wake is not axisymmetric but keep a statistical m = 2 symmetry. Instantaneous wake follows a m = 1 azimuthal topology and is oriented either above or below the streamwise axis (phase 0 or π), shifting randomly. The unsteady global mode then develops from on side of the bubble depending on the orientation of the instantaneous wake.

A m = 1 disturbance, small pitching angle or control cylinder, sets one of the two asymmetric topologies depending on its characteristics. This study highlights that the sensitivity of the flow over a body of revolution to a non-axisymmetric local disturbance may only be observable in the azimuthal plane of the perturbation. Any shift in the azimuthal position of the disturbance will be followed by an equal shift of the azimuthal orientation of the global mode.

The use of control rings to generate a m = 0 disturbance has a strong influence on drag and wake.

As observed by Meliga et al. [START_REF] Meliga | Open-loop control of compressible afterbody flows using adjoint methods[END_REF] , the effect follows the position of the mixing layers of the natural flow. In particular, when placed in the inner part of the mixing layer, the rings may prevent the development of the shear layer instability. It reduces the global mode frequency and moves the wake structures downstream inducing significant drag reductions.

The mean flow symmetry as well as the global mode development are then highly sensitive to any disturbance in the recirculation area, particularly when the mixing layers equilibrium or characteristics are altered. The growth rate of the turbulent mixing layer seems to be a critical factor for the flow dynamics, more than the inviscid dynamics induced by the vorticity.

Eventually, these results might be associated to the disturbance of the reminiscent global modes observed in laminar regimes. The steady asymmetric wake reported after the first bifurcation may concentrate the sensitivity to m ≥ 1 disturbances whereas the oscillating mode from the second bifurcation (oriented along the steady asymmetric mode) seems more sensitive to m = 0 perturbations.

z velocity u ij = √ u 2 i + u 2 j

 22 Re = U 0 D/ν Reynolds number St = f D/U 0 Stouhal number St S Strouhal number of the global mode St b characteristic Stouhal number of the recirculation T b characteristic time of the recirculation ⃗ u = u x . ⃗ e x + u y . ⃗ e y + u z . ⃗ e modulus of velocity in plane (⃗ e i , ⃗ e j ) U 0 inlet flow velocity v E characteristic velocity of flow incorporation into the mixing layers x C position x of the control cylinder x R position x of the control ring y C position y of the control cylinder y α position y verifying U x * (x * , y * , z * = 0) = α δ 99 boundary layer thickness based on 99% of freestream velocity δ * boundary layer displacement thickness δ M mixing layer thickness base on 10% and 90% of freestream velocity Ω z mean vorticity in z direction ρ density of the fluid θ boundary layer momentum thickness Table I: Nomenclature frequency and base pressure. The associated physical mechanisms seem relied on the perturbation ability to change the size of the formation region of the von-Kármán street vortices and to interact with the shear layers.

  performed experiments displacing a control disk in the recirculation bubble pointing out different flow regimes depending on the disk position associated to important evolutions of drag and Strouhal number. However this control set-up may be too intrusive to be considered as a local perturbation.The present work explores the sensitivity of wake of a 3D blunt body with a fixed axisymmetric flow separation at Re = 2.1 • 10 4 . Experiments of passive control using reasonably small devices are performed to disturb the flow with m = 0 and m = 1 perturbations ; to our knowledge, such an approach has not been carried out over bodies of revolution so far. The impact of the control device is reported through sensitivity maps of global mode frequency, pressure distribution on the body and estimation of the drag.The article is organized as follows. First, the natural flow is analysed ; it is proved that two antisymmetric flows cohabit and generate a statistically symmetric natural flow. Then the sensitivity to different local perturbations is investigated. Any m = 1 disturbance change the statistical equilibrium of the mean flow moving it off axis. On the contrary, a m = 0 disturbance preserve the mean flow symmetry but has significant impacts global mode frequency and drag. The sensitivity of the flow seems concentrated in the close wake, particularly into the mixing layers.

  used in the experiments of Parezanović and Cadot 8 . The body is made up by a cylinder of diameter D = 50 mm and a half sphere forebody. The total length is L = 125 mm. The axisymmetric body is supported by two NACA 0021 profiles fixing the azimuthal planes of symmetry (xOy) and (xOz)

  the amplitude of velocity at the considered point in the plane (⃗ e i , ⃗ e j ). A and Std (a) are respectively average value and standard deviation of any quantity a. The diameter D, density ρ, inlet velocity U 0 and static pressure p 0 are used to obtain non-dimensional values marked with an asterisk.

Figure 1 :

 1 Figure 1: Experimental set-up of the body controlled by a 3 mm cylinder (a) and rings (b) ; O sets the origin of the coordinate system ; pitching angle ϵ = 0 in most experiments.

4 ∫ 0. 5 y * =-0. 5 C 4 ∫ 0. 5 y * =-0. 5 C

 455455 p nose (y * ) • |y * | dy * p base (y * ) • |y * | dy * .

y * = 0

 0 and z * = 0 are presented on Figure 2(a). A separation is observed on the upper and lower part of the the end of the forebody. The flow reattaches turbulent on the cylindrical part before the massive separation at the trailing edge. This boundary layer history leads to the mean and fluctuating velocity profiles in the plane z * = 0 (see empty and filled circles on Fig. 2(a)). The energy of the fluctuations of velocity diminishes when y * increases and tends to the free flow level of turbulence. This particularly high fluctuation level in the entire boundary layer is a result of the forebody detachment.

Figure 2

 2 (b) displays the streamwise velocity profiles in plane z * = 0 at different positions x * . The size of the mixing layer is defined as δ M (x) = y 0.9 (x) -y 0.1 (x) with y α (x) verifying U x * (x, y α (x)) = α. The experiments of Champagne et al.[START_REF] Champagne | On the two-dimensional mixing region[END_REF] report that plane turbulent mixing layers has a characteristic growth rate constant dδ M * /dx * ∼ 0.1. In parallel, the eddy viscosity model predicts that it expands toward the low velocity domain[START_REF] Schlichting | Boundary-layer theory[END_REF] . The contours of U x * = 0.1, 0.5 and 0.9 (positions of y α (x) for

Figure 2 :

 2 Figure 2: (a) Mean and fluctuating velocity profiles respectively empty and filled markers at the trailing edge in plane z * = 0 (blue ⃝) and y * = 0 (red ) ; the hot wire probe measures respectively U xy and U xz . (b) Streamwise velocity profiles in the upper mixing layer from PIV in plane z * = 0 at different x * positions: -, x * = 0.2 ; --, x * = 0.4 ; ••, x * = 0.6 ;-•-, x * = 0.8. (c) Contours of streamwise velocity U x * from PIV in plane z * = 0: ••, 0.1 ; -, 0.5 ; --, 0.9.

Figure 3 :

 3 Figure 3: Power spectrum analysis of hot wire probe signal at (x * , y * , z * ) = (4, 0.5, 0).

Figure 4 :

 4 Figure 4: Streamlines of the time-average vector field in the plane z * = 0 ; colormap of the mean vorticity Ω z (a) and fluctuating velocities (b) ; white cross, collar point.

Figure 5 :

 5 Figure 5: Streamlines of the velocity field in the plane z * = 0 (a) and in the plane x * = 1.2 (b) for slightly nose-up configuration, state #1 ; colormap of the fluctuating velocities ; white cross, collar point.

Figure 6 :

 6 Figure 6: Flow computed from velocity field #1 in plane z * = 0 (see Fig. 5(a)) and the expected velocity field for state #2.

  Figures 5(a)-(b) present the asymmetric wake topology for a slightly nose-up configuration. Measurements in the plane x * = 1.2 show only one pair of counter-rotating vortices at y * ≈ -0.3 and z * ≈ ±0.2

  2(b)) is consistent with an average of two asymmetric topologies (named #1 and #2), each of them being respectively close to the flow presented on Figures 5(a)-(b) and its symmetric referring to the plane y * = 0. Indeed, averaging the velocity field ⃗ U ϵ for state #1 shown on Figure 5(a) with the expected flow for state #2,

Figure 7 :

 7 Figure 7: (a) Base pressure distribution of topology #1 characterized by the negative slope of the linear fit line), the corresponding bubble velocity field from Figure 4(b) is presented. (b) Time evolution of 1 s averaged base pressure gradient over 1000 s. (c) Probability density function of base pressure slope in y direction (precision ±0.01).

Figure 8 :

 8 Figure 8: Effect of the position of the 3 mm control cylinder on dC p /dy * (a) and on Std (dc p /dy * ) (b) ; contours of U x * for the natural flow: ••, 0.1 ; -, 0.5 ; --, 0.9.

Figure 9 :

 9 Figure 9: (a) Effect of the position of the control cylinder y C * at x C * = 0.5 on the power spectrum at (x * , y * , z * ) = (4, 0.5, 0) ; positions of U x * for the natural flow: ••, 0.1 ; -, 0.5 ; --, 0.9. (b) Strouhal of the global mode as a function of the cylinder position ; the grey areas correspond to the absence of peak in the measured power spectrum ; contours of U x * of the natural flow: ••, 0.1 ; -, 0.5 ; --, 0.9.

Figure 10 :

 10 Figure 10: Effect of the position of the control cylinder on drag ; contours of U x * for the natural flow: ••, 0.1 ; -, 0.5 ; --, 0.9.

Figure 11 :

 11 Figure 11: Streamlines in the plane z * = 0 for the controlled case x C * = 0.2 and y C * = -0.3 (a) and x C * = 0.5 and y C * = 0 (b) ; colormap of the fluctuating velocities ; white cross, collar point. (c) Base pressure distribution: -(×), natural flow ; --(⃝), x C * = 0.2 and y C * = -0.3 (see Fig. 11(a)) ; •• ( ), x C * = 0.5 and y C * = 0 (see Fig. 11(b)).

Figure 9 (

 9 Figure 9(b) report the global mode frequency at positions where a peak of energy is present in the power spectrum. This map is particularly correlated to the map of absolute values of base pressure gradient. The more the control cylinder induced an asymmetric topology, the higher the frequency is. On the contrary, St S tends to be reduced where the control cylinder sets dC p /dy * ∼ 0. An exception is observed when y C * ≈ 0 but the energy associated to the global mode activity is spreading over a large band of frequency and St S is poorly defined. For theses positions, the total energy of the fluctuations of velocity may be slightly lower compared to that of the natural flow (compare Fig. 11(b) to Fig. 4(b)) but remains at the same order of magnitude. In addition, the

Figure 12 :

 12 Figure 12: Effect of the position of the control rings on Std (dc p /dy * ) ; contours of U x * from uncontrolled flow are plot: ••, 0.1 ; -, 0.5 ; --, 0.9.

Figure 13 :

 13 Figure 13: (a) Effect of the ring diameter d R * at x R * = 0.5 on the power spectrum at (x * , y * , z * ) = (4, 0.5, 0) (see Fig 3 for comparison to natural flow) ; positions of U x * of the natural flow are plot: ••, 0.1 ; -, 0.5 ; --, 0.9.. (b) Effect of the position x R * of the d R * = 0.85 ring on the power spectrum at (x * , y * , z * ) = (4, 0.5, 0) (see Fig. 3 for comparison to natural flow and Fig. 13(a) for legend). (c) Strouhal number of the global mode as a function of the ring position ; the grey areas correspond to the absence of peak in the measured power spectrum ; contours of U x * for the natural flow: ••, 0.1 ; -, 0.5 ; --, 0.9.

Figure 14 :

 14 Figure 14: Effect of the position of the control ring on drag ; contours of U x * from uncontrolled flow are plot: ••, 0.1 ; -, 0.5 ; --, 0.9.

Figure 15 :

 15 Figure 15: (a) Streamlines in the plane z * = 0 for the controlled flow with the d R * = 0.85 diameter ring at x R * = 0.3 ; colormap of the fluctuating velocities ; white cross, collar point. (b) Idem Figure 15(a) at x R * = 0.6. (c) Idem Figure 15(a) at x R * = 0.8. (d) Base pressure distribution: -(×), natural flow ; --(⃝), x R * = 0.3 and d R * = 0.85 (see Fig. 15(a)) ; •• ( ), x R * = 0.6 and d R * = 0.85 (see Fig. 15(b)) ; -•-(△), x R * = 0.8 and d R * = 0.85 (see Fig. 15(c)).

  a)). As measured for x C * < 0.4, the global mode frequency is reduced for small ring diameters and the peak of energy in the spectra disappears increasing d R * . In the middle of the mixing layer, a new global mode regime is measured with a frequency of St S ≈ 0.1. Eventually, for d R * > 1, the perturbation of the outer part of the mixing layer leads to an increase in global mode frequency.

  Figure 15(b) point out that the mixing layers reattach on the flat ring. The directions of the

  the global mode frequency An unsteady global mode is usually observed in the flow past bluff bodies at moderate Reynolds numbers : its frequency and energy depend on different parameters. Over cylinders, Gerrard 25 presents a dynamic of the bubble in the wake considering the von-Kármán vortices. The flow entrained from the bubble to the mixing layers is renewed by the flow injected at the end of the recirculation through the velocity induced by the shear layers rollings. This model is highly related to the vorticity dynamic following Biot & Savart law and leads to a dependence between the characteristics of the shear layers and the global mode activity. The experiments of Parezanović and Cadot 9 recently confirm that the disturbance of the distance between the two shear layers or their intensity impacts the global mode frequency. The presented results are in good agreement with this Gerrard's analysis : the Strouhal number of the global mode of the wake is governed by the shear layer interaction.

  is measured in our experiments for the natural flow where the Strouhal of the shedding is measured at 0.199 so St b ≈ α dδ M * /dx * with α ≈ 1. This dependence is consistent with the observations of Achenbach 24 on the Strouhal number of the wake oscillation past a sphere. The frequency of this mode increases with the Reynolds number from St = 0.12 at Re = 6 • 10 4 to St = 0.20 at Re = 3 • 10 5 just before the critical Reynolds number

B.

  On the azimuthal symmetry of the wake It has been observed that any slight perturbation of the set-up leads to an asymmetric wake.The work of Mittal et al.[START_REF] Mittal | Symmetry properties of the transitional sphere wake[END_REF] on shedding activity past spheres highlights that the Reynolds number impacts the azimuthal position of the oscillating global mode m = 1. It appears that after the unsteady bifurcation at Re ≈ 277 over a sphere, the global mode has a preferred azimuthal orientation which may evolve at very low frequency. This effect disappears as Reynold number increases and the wake gets statistically axisymmetric in turbulent flows. In our experiments even at moderate Reynolds number, the wings prevent the statistical wake axisymmetry. The shedding occurs mostly at the upper and lower part of the wake corresponding respectively to topology #2 and #1 but the mean flow keeps the top-bottom symmetry.Any non-axisymmetric disturbance leads to a preferred azimuthal orientation for the antisymmetric global mode. For example, as soon as the body has a small pitching angle, the shedding occurs exclusively at the upper part of the wake or lower part depending on the sign of ϵ and the wake loses this statistical symmetry. Therefore, if the disturbance has an azimuthal periodicity m = 1, the shedding occurs in the same azimuthal plane, at the same side or the opposite one depending on the position of the perturbation. This point was proved by Meliga et al.[START_REF] Meliga | Global mode interaction and pattern selection in the wake of a disk: a weakly nonlinear expansion[END_REF] at low Reynolds numbers through a theoretical analysis. Over blunt bodies of revolution, the forcing therm associated to the presence of a m = 1 disturbance selects the plane of symmetry of the first bifurcation. The different bifurcations observed in the wake of a disk degenerate into imperfect bifurcations and the flow remains tridimensional, its orientation based on the azimuthal position of the perturbation.When the disturbance has a higher azimuthal periodicity m ≥ 2, for example the two wings in our experimental set-up associated to m = 2, the wake follows a statistical m flow topology. The main coherent structure remains a m = 1 global mode shifting randomly between the m preferred locations and generating a m statistical wake.As a consequence, the experimental sensitivity analysis of the oscillating or helical mode m = 1 to an antisymmetric perturbation can only be studied in the azimuthal plane of the disturbance. The global mode follows any change of azimuthal position of the control leading to a mode sensitivity independent of the azimuth.The planar symmetry being set, the azimuthal phase of the wake is then 0 or π referring to the position of the disturbance. The rate dδ M * /dx * may play a dominant role for the selection of the phase, i.e. state #1 or #2. If one side of the axisymmetric mixing layer has a higher growth rate, as previously seen, the development of δ M toward the recirculation bubble is enhanced. Thus, the curvature of the streamlines is increased[START_REF] Simpson | Turbulent boundary-layer separation[END_REF] and the whole wake is shifted to the opposite side where the mixing layer remains thin. This concentration of vorticity leads to rolls-up and the development of the vortex loops presented by Sakamoto & Haniu 10 .

Table II :

 II Boundary layer characteristics at the trailing edge.

	99	1.57 mm ±0.05 mm	1.82 mm ±0.05 mm
	δ *	0.36 mm ±0.02 mm	0.40 mm ±0.02 mm
	θ	0.18 mm ±0.02 mm	0.19 mm ±0.02 mm
	H	2.0 ±0.3	2.1 ±0.3

Acknowledgements

The authors wish to thank P. Meliga for fruitful scientific discussions.