Vincent Jaunet
email: vincent.jaunet@univ-poitiers.fr

Erwan Collin

Joël Delville

POD-Galerkin advection model for convective flow Application to a flapping rectangular supersonic jet

Keywords: Supersonic jet, Proper Orthogonal Decomposition, Convection velocity, Advection equation, Low order dynamical systems Nomenclature, x Space derivative, t Time derivative

This article describes a model obtained by applying Proper Orthogonal Decomposition to the advection equation. The resulting set of equations links the POD modes, their temporal and spatial derivatives and the flow convection velocity. It provides a technique to calculate the convection velocity of coherent structures. It follows, from the model, that a priori knowledge of the convection velocity suffices to construct a dynamical model of the flow. This is demonstrated using experimental data.

Introduction

The convection velocity of organized structures is an important feature that, together with a length scale, permits in estimating the main frequency of a phenomenon. For example, it is one of the key paramaters used to predict screech tones of high speed jets (Powell, 1953b,a). In order to estimate this convection velocity many different methods can be employed, all of them requiring time and/or space separation. Two-point hotwire, LDA or unsteady pressure sensing can be used to discretize the space-time correlation function. The convection speed is obtained as the ratio between the space separation of the two sensors and the lag-time of the maximum correlation value between the two signals [START_REF] Morris | Velocity measurements in jets with application to noise source modeling[END_REF][START_REF] Kerherve | Two-point laser doppler velocimetry measurements in a mach 1.2 cold supersonic jet for statistical aeroacoustic source model[END_REF]. However, these methods only give access to a local representation of the convection velocity: lots of effort is then required to probe large area of flows. Particle Image Velocimetry (PIV) gives access to instantaneous velocity fields, it is thus particularly well suited to measure convection velocity. When time-resolved data can be obtained, the former method can still be used and the entire flow field can be probed. However, the sampling rate attainable with PIV is in many cases not yet sufficient to fully resolve the structures of interest. [START_REF] Shih | Investigation of flow at leading and trailing edges of pitching-up airfoil[END_REF] and [START_REF] Alkislar | Structure of a screeching rectangular jet : a stereoscopic particle image velocimetry study[END_REF] defined the convection velocity as the vorticity weighted average value within the area of flow in a coherent structure. A single PIV snapshot suffices to educe the speed of a structure. The results obtained with this definition may however be sensitive to the way the coherent structures are defined and identified in the data. Two-spark schlieren imaging has already been used years ago to obtain the velocity of coherent structures in highspeed mixing layers [START_REF] Papamoschou | A two-spark schlieren system for very-high velocity measurement[END_REF]. The identification of coherent structures in the images was left to the subjectivity of the experimentalists. Therefore, the aim of this paper is to describe a method that gives the speed of coherent structures.This method can be applied to any kind of data (e.g. velocity fields, schlieren images...) as long as it is representative of a convective motion. This method is based on the combination of the advection equation and a POD-Galerkin analysis and does not require the data to be time-resolved. As the method used by [START_REF] Papamoschou | A two-spark schlieren system for very-high velocity measurement[END_REF], two successive samples at short interval time also suffice with this technique. It is therefore particularly well suited to experimental data. Following the work of [START_REF] Perret | Polynomial identification of POD based low-order dynamical system[END_REF], it is shown that the use of this simple model permits the identification of the coefficients of low-order linear dynamical systems, even if the original data only contains independent realizations of a flow field. The first part of this paper comprises the mathematical description of the POD-Galerkin advection model. In the second part, we describe the experimental set-up and the main flow features. In the third part, the POD-Galerkin model is applied to dual-time Schlieren pictures of a screeching jet to educe the convective speed of large scale structures. Finally, we present the identification of a low order dynamical model, from independent PIV realizations of the same flow field.

POD based advection model

Overview of the Proper Orthogonal Decomposition

We provide here a brief introduction to the Proper Orthogonal Decomposition (POD). The reader can be referred to e.g. [START_REF] Sirovich | Turbulence and the dynamics of coherent strucutres. part 1 : coherent strucutres[END_REF] or [START_REF] Delville | La décomposition orthogonale aux valeurs propres et l'analyse de l'organisation tridimensionnelle des écoulement turbulents cisaillés libres[END_REF] for more precise discussions on POD.

The POD was introduced by [START_REF] Lumley | The structure of inhomogeneous turbulent flows[END_REF] and consists in considering that the coherent structures are the one having the maximum projection onto the velocity field in a least mean square sense. This leads to an eigenvalue problem whose kernel is the spatial (classical POD) or temporal (Snapshot POD) correlation tensor. Thus, for a spatio-temporal field g(x, t) the decomposition is written :

g(x, t) = ∞ n=1 a (n) (t)Φ (n) (x), (1)
where Φ (n) (x) represents the n th eigenfunction (also referred to as spatial mode) and a (n) (t) the n th eigenvector (also referred to as temporal coefficient).

If the database is constituted of uncorrelated samples, the POD decomposition is calculated using the Snapshot POD [START_REF] Sirovich | Turbulence and the dynamics of coherent strucutres. part 1 : coherent strucutres[END_REF]. The Snapshot POD eigenvalue problem is defined as:

T C(t, t )a(t )dt = λa(t), (2)
where λ represents the eigenvalue, T the temporal length of the data base, and C(t, t ) the temporal correlation matrix calculated on a given spatial domain Ω as:

C(t, t ) = 1 T Ω Ω g(x, t)g(x, t )dx, (3)
The spatial modes are then obtained by projecting the original fields onto the temporal modes :

Φ (n) (x) = 1 T λ (n) T g(x, t)a (n) (t)dt. ( 4
)
By definition, the eigenfunction basis Φ (n) (x) is orthonormal:

Φ (n) (x), Φ (m) (x) = δ nm , (5)
where (, ) is the scalar product and δ nm is equal to 1 only if n = m. The temporal coefficients a (n) (t) are orthogonal to each other, so that:

a (n) a (m) = λ (n) δ nm , (6)
where • represents the ensemble average.

Convection velocity estimation

Consider the advection of a fluctuating spatio-temporal field g(x, t) at velocity U c :

∂ ∂t g(x, t) + U c ∂ ∂x g(x, t) = 0, (7)
The POD decomposition of g(x, t) is then applied to this equation, leading to:

∂ ∂t Ns n=1 a (n) (t)Φ (n) (x) + U c ∂ ∂x Ns m=1 a (m) (t)Φ (m) (x) = 0. ( 8
)
where N s being the number of available samples (i.e. the number of POD modes). This can be written using the linearity of the POD:

Ns n=1 a (n) ,t (t)Φ (n) (x) + U c Ns m=1 a (m) (t)Φ (m) ,x (x) = 0. ( 9
)
For reasons of readability, time and space derivatives are referred by the , t and , x underscripts respectively. A Galerkin projection is then performed on a chosen spatial function Φ (p) (x):

Ns n=1 a (n) ,t (t) Φ (n) (x), Φ (p) (x) + U c Ns m=1 a (m) (t) Φ (m) ,x (x), Φ (p) (x) = 0. ( 10
)
As the eigenfunctions Φ are orthonormal, we get:

a (p) ,t (t) + U c Ns m=1 a (m) (t) Φ (m) ,x (x), Φ (p) (x) = 0, ( 11
)
The last step is to multiply this equation by a (n) and take its ensemble average < • >:

a (n) a (p) ,t + U c λ (n) Φ (n) ,x (x), Φ (p) (x) = 0. ( 12
)
This equation states that, in the case of purely advective flow, there should be a correlation between the scalar product of the spatial POD modes and their space derivatives Φ (n)

,x (x), Φ (p) (x) on one hand, and the correlation weighted by the energy of the temporal POD modes and their temporal derivatives a (n) a (p) ,t /λ (n) on the other. The ratio between the two terms is the convective velocity :

U c = λ (n) Φ (n) ,x (x), Φ (p) (x) a (n) a (p) ,t (13)
Note that in the specific case of n = p, the first term of equation 12 becomes:

a (n) a (n) ,t = 1 2 ∂ ∂t (a (n) ) 2 = 1 2 ∂λ (n) ∂t = 0, (14)
Assuming that the convective velocity is not zero, this means that, in a purely advective flow, the spatial POD modes are orthogonal to their space derivatives:

Φ (n) ,x (x), Φ (n) (x) = 0, (15)
In other words, a scalar product Φ (n)

,x (x), Φ (n) (x) differing from zero would indicate that the n th mode is not representative of convective motion. This can therefore be used as a criterion to verify if a a POD mode can be used in the POD-Galerkin advection model.

Dynamical system identification

We would like to emphasize one asset of the former calculations. Consider the system of N s equations of a linear dynamical system formed with the temporal POD modes:

a (i) ,t = L ij a (j) + D i , (16)
where L ij is the linear coefficients of the system and Di constants that represent linear trends. Neglecting the D i terms, which can almost always be done for stationnary signals, multiplying equation 16 by a (j) and taking its ensemble average, we can write:

a (i) ,t a (j) = L ij a (i) a (j) a (i) ,t a (j) = L ij λ (j) , (17)
Hence, the L ij coefficients of the linear dynamical system can be obtained from the temporal POD modes and their time derivatives. This implies time information being obtained (see [START_REF] Perret | Polynomial identification of POD based low-order dynamical system[END_REF]), which is not always feasible specially with experimental data.

In the specific case of a known convection velocity, the POD-Galerkin advection model 12 allows us to rewrite the equation 17:

U c Φ (i) ,x (x), Φ (j) (x) = -L ij , (18)
showing that in the case of an advective motion of known convective speed, the coefficients of a linear dynamical system of POD modes can be retrieved without the need for temporal information about the flow. Thus, independent realizations of the flow suffice to obtain spatial POD modes, their space derivatives and thus to identify a linear dynamical system.

We have shown that the POD-Galerkin advection model (equation 12) can be used in two different ways:

-Retrieve linear coefficients of a dynamical system, from independent realizations of a purely convective flow (equation 18) -Obtain the convection velocity, from data containing temporal information (equation 13)

In the following we demonstrate the use of this model onto experimental data obtained from a supersonic rectangular jet.

3 Experimental set-up and flow description

Wind tunnel

Experiments were conducted in the S150 wind-tunnel of the PPRIME Institute of Poitiers. They consisted of a

Mach number M = 1.45 supersonic rectangular supersonic jet of size h j = 30 mm by L j = 150 mm. The geometry of the convergent-divergent nozzle is modified by the addition of two divergents, initially designed in order to study fluidic thrust vectoring [START_REF] Jaunet | 3d effects in a supersonic rectangular jet vectored by flow separation control, a numerical and experimental study[END_REF]. These supplementary divergents consist in 10˚slopes of the wall with respect to the main flow direction, on the small sides of the rectangular nozzle.

Their length is equal to h j . As the air flows at supersonic speed, an expansion fan is generated on the corner. Therefore, the pressure of the flow at the exit section of the nozzle is not uniformly distributed and the fully adapted regime can never be reached. The stagnation pressure is set as the minimum pressure necessary to ensure full flowing conditions. It was checked, with oil flow visualizations, that the flow does not separate from the wall before the exit section. The jet flows into a 500mm square test section. The subsonic surrounding flow, obtained by entrainment of ambient air, flows at a Mach number M e = 0.2. A reference coordinate system is chosen so that x axis is aligned with the main flow stream and the y axis is parallel to the long sides of the nozzle. The origin of the coordinate system is located in the center of the nozzle exit section (see figure 1).

A sketch of the wind-tunnel together with a detailed view of the nozzle is given in figure 1, and the stagnation conditions and main characteristics of the flow are summarized in table 1.

Measurements and Visualizations

Unsteady pressure measurements

Two walls of the test section are equipped with Kulite TM sensors, placed on the y = 0 plane at a streamwise distance of x = 8 × h. A sketch of the sensor locations with respect to the flow is given in figure 2. The pressure signals were acquired simultaneously at a sample rate of 80kHz after being low pass filtered with a cut-off frequency of 40kHz to avoid any aliasing, and consisted of 2 × 10 6 samples each, acquired in a single experimental run. In the following, when pressure data is invoked the subscript i refers to the signal of sensor number i. Then, S ii refers to the auto-spectrum of signal i.

Dual-spark schlieren photographs

Pairs of instantaneous schlieren visualizations are acquired using a double spark light source together with a PIV camera. apparatus. The light source comprises a HSPS TM twin nano flash and two spark lamps. Hence, pairs of schlieren visualizations can be performed with a time delay between succesive images that can be adjusted up to a minimum of ∆t = 1µs. The 12 bit CCD sensor of the camera is composed of 1200 × 1600 pixels covering a field of view of about 200mm × 200mm, so that the resolution is sufficient to study the large scale coherent features of the flow. Synchronization between the light source and the camera is controlled by a computer.

The two successive sparks of the light source are obtained with two different pairs of electrodes. Consequently, the two successive images may not possess the same exposure and/or contrast and to avoid problems during the post-processing, the energy of each image is normalized and a histogram specification is applied. Histogram specification is performed using the histogram of the average image as a target. Figure 3 presents the histograms of 5 schlieren pictures taken from the data base before and after application of the histogram specification. As can be seen, the post-processed images have now the same exposure, ensuring that future analysis will less be corrupted by illumination fluctuations. Schlieren flow visualizations are performed either with a light source axis parallel to y (side view) or z (top view).

Grayscale

Particle Image Velocimetry

Particle image velocimetry (PIV) is performed in the plane of symmetry of the jet (z = 0), as well as in the z = 0.5h j plane i.e. along the lipline. The PIV device consists of a 1600x1200 pixel CCD camera equipped with a 28 mm lens placed perpendicularly to the measurement plane, giving a field of view of 350 × 270mm 2 . A sketch of the set-up is given in figure 5. The PIV was calibrated by measuring the aspect ratio of images along the diagonal of the field of view. Care was taken during the alignment of the camera to the laser plane so that the aspect ratio remains constant along the field of view, avoiding paralax effects. The lens aperture was set at f#4 to ensure a correct sharpness of the particle image across the entire field of view and that sufficient light reached the camera's sensor. The optical set-up gives a diffraction-limited particle image size of 5.3 µm [START_REF] Adrian | Particle imaging techniques for experimental fluid mechanics[END_REF] slightly smaller than the camera pixel size (i.e. 7.4 µm). Since a 190 mJ Nd-YAG laser was used to illuminate the particles, sufficient energy was provided for the particle image to be larger than a pixel size, avoiding pixel-locking. This is illustrated in figure 4 where a typical velocity histogram of a PIV snapshot is plotted. The absence of discrete peaks, expected around the equivalent velocity given by a single pixel displacement ( 30m/s), shows that no sign of peak-locking is visible in the data. The flow is seeded with SiO 2 particles. The particle mean diameter was estimated to 0.3µm and their relaxation time was measured to 0.019ms [START_REF] Lammari | Mesures par vélocimétrie laser doppler dans une couche de mélange turbulente supersonique : quelques aspects du processus de mesure[END_REF]. This gives an equivalent frequency response of more than 50kHz, which is sufficient for the time scales of interest in this study.

The time separation between each laser illumination for a given particle image pair is set to 7 µs, which corresponds to a maximum particle displacement of 7 pixels, and a set of 200 pairs of images were acquired at each measurement position. The PIV images are processed using a multi-pass iterative correlation technique (Willert and Gharib, 1991;[START_REF] Soria | An investigation of the near wake of a circular cylinder using a video-based digital crosscorrelation particle image velocimetry technique[END_REF] with image deformation to account for local velocity gradients [START_REF] Scarano | Iterative image deformation methods in piv[END_REF]. The process started with interrogation windows of 64 × 64 pixels and finished with interrogation areas of 16 × 16. The overlap between two successive interrogation areas was set at 50%. At each pass, a peak-ratio validation criterion was used; vectors were rejected if the ratio between the first and the second highest correlation peaks was less than 1.2 ensuring the reliability of a given vector. This gives a final PIV resolution of 1 vector every 2.0mm. Finally a spatial UOD filter [START_REF] Westerweel | Universal outlier detection for piv data[END_REF] together with a temporal statistical 3-sigma filter were employed to detect spurious vectors from the instantaneous PIV fields. Less than 3% of vectors were rejected and the spurious vectors were replaced using an iterative Gappy POD procedure [START_REF] Murray | An application of gappy pod[END_REF].

A comparison of the PIV results with measurements made with Laser Doppler Anemometry, obtained in a related work [START_REF] Jaunet | Etude d'un jet rectangulaire supersonique à nombre de mach 1.45 vectorise par actionneur fluidique[END_REF] is presented in figure 6. An excellent agreement is obtained by the two measurement techniques for the mean flow, showing the ability of the PIV to recover the essential features of the flow, such as the complex network of shock waves and expansion waves existing in the jet. The fluctuation intensities obtained with the PIV set-up are slightly underestimated compared to the ones observed with LDA, this is attributed to low-pass filtering effect of interrogation areas. Nonetheless, both techniques agree well on the location of peak of RMS values. The results presented in figure 6 show the excellent reproducibility of the experiment as well as the reliability of the PIV data.

Main flow features

An example of top-view spark schlieren picture of the jet (optical axis parallel to the long axis of the rectangular nozzle) is given in figure 7. It is clear that the jet is undergoing a coherent flapping motion in the (x, z) plane. This motion seems to be coupled with the generation of coherent structures in the mixing layer, in the vicinity of the nozzle lips. These structures are visible on figure 7, they are highlighted by dashed white circles. This figure also shows that strong acoustic waves exist in the subsonic co-flow. These waves are propagating upstream and are supposed to trigger the creation of the coherent structures when they reach the nozzle lip. This description is very similar to a screech mechanism, but one major difference is the origin and direction of propagation of these acousitc waves. Indeed, in a classical screech phenomenon, the acoustic perturbations generally emanate spherically from the third or fourth shock cell [START_REF] Raman | Supersonic jet screech : half-century from powell to the present[END_REF][START_REF] Alkislar | Structure of a screeching rectangular jet : a stereoscopic particle image velocimetry study[END_REF]. Here, the acoustic source location could not be seen and seems to be located downstream of the field of view: the waves first reflect on the wall of the wind tunnel before impinging the nozzle lips.

In figure 8 we plot the power spectral density of the pressure signal acquired at the wall of the windtunnel. Even though the jet does not exhibit a standard screech resonance, the energy of the signal here is also dominated by a fundamental frequency f s = 2930Hz and harmonics. Hence, the main flow features comprise shock cells of various size and a strong flapping of the jet. The mechanism responsible for this very large scale unsteadiness might be found in an acoustic coupling.

Application to data from a flapping jet

The results of section 2 are now applied to data obtained from the experiment described previously. First we will show that it is possible to obtain relevant quantitative information (main frequency and convective velocity) on the flow solely based on dual-time schlieren visualization. Then, a low-order linear dynamical system is identified, as presented in section 2, from independent PIV acquisitions allowing us to assess some of the dynamics of the flow field.

Obtaining a convective velocity

The analysis presented in section 2 is applied in this section to Dual-Spark schlieren photographs. It should be noted that schlieren images contain information integrated along the line of sight. However, the phenomenon of interest in the study is two-dimensional in the (x, z) plane, as it can be seen in the figure 7, so that no direct bias from the line of sight integration is expected in this part of the study.

Domain of study and POD analysis

As shown in figure 9, the schlieren images show a complex network of discontinuities (i.e. shock waves) near the nozzle. This makes a POD analysis difficult in this area: an infinite number of POD modes is necessary to represent the different locations of such discontinuities. Therefore, it has been decided to perform the POD analysis on a sub-domain of the original visualizations, slightly downstream of the nozzle, where no sign of discontinuity is clearly visible. The chosen domain of study, of size 5 ≤ x/h ≤ 9.5 and -1.6 ≤ z/h ≤ 1.6, is presented figure 9. This domain was chosen for two reasons. Firstly, it avoids POD analysis on the external flow field, where no feature can be observed, minimizing the computational cost. Secondly, the main objective of the study is to capture the convection speed of the large structures observed in the flow field, so the chosen area includes slightly more than a complete wavelength of the phenomenon, the relevant information of the flapping motion. The POD is performed on the fluctuating gray-scale level field g(x, t) and computed using a Snapshot POD [START_REF] Sirovich | Turbulence and the dynamics of coherent strucutres. part 1 : coherent strucutres[END_REF]:

g(x, t) = Ns n=1 a (n) (t)Φ (n) (x), ( 19
)
where N s represents the total number of POD modes given by the number of instantaneous visualizations (200 samples in our case).

As dual-time schlieren photographs were acquired, the time derivative of the temporal coefficients can be calculated with the following scheme:

a (n) ,t (t + ∆t 2 ) = a (n) (t + ∆t) -a (n) (t) ∆t . ( 20
)
To preserve the temporal coefficients at the same sampling time than their time derivatives, the temporal modes are interpolated:

a (n) (t + ∆t 2 ) = a (n) (t + ∆t) + a (n) (t) 2 . ( 21
)
Then, the new eigenfunctions are calculated by projecting the gray-scale images onto the interpolated temporal modes using equation 4.

The POD eigenspectrum is plotted in figure 10(a). The energy contained in the first n modes, defined as n = n i=1 λ i / Ns i=1 λ i , is also plotted in figure 10(b). The first 4 POD modes represent 60% of the total energy and are sufficient to produce partial reconstructions of the flow that well represent the flapping motion of the jet (see figure 11). The higher order POD modes (i.e. remaining 40% of the energy) describes the smaller gray-scale wavelength, visible in figure 10(a). These are small turbulent structures, living in the mixing layers, that are also convected downstream. However, the combination of their 3-dimensional motion and the line of sight integration renders the POD analysis unable to correctly capture their convective motion. It is therefore expected that the POD-Galerkin analysis, presented earlier, will only allow us to obtain the convective speed of the large structure of the flapping jet. The energy of mode 1 and mode 2 on one hand and mode 3 and mode 4 on the other hand are of the same order of magnitude. This results from the fact that the visualizations capture the convective motion of the flow. Indeed, the reproduction of a propagating flow pattern from eigenfunctions that are fixed in space needs pairs of eigenfunctions Φ (n) (x). Therefore, the temporal dynamics of mode 1 and 2 are very similar, the main difference being a simple phase shift of π/2. This leads to a simple dynamics in the phase space (a (1) -a (2) ) in which the temporal modes describe a circle (see figure 12(a)). The dynamics of mode 3 (and mode 4, not presented here) is the same as the one of mode 1 and mode 2, but occurs at twice the pulsation. This is shown in figure 12. Modes 3 and 4 thus represent the first harmonic of the jet motion. As can be observed in figure 12, the measured phases do not lie on a perfect circle and some scatter is visible. Some of the time derivatives of the temporal modes, represented as arrows in figure 12, also indicate slightly different directions than the average modes in the phase domain. This can arise for several reasons, either from the jet dynamics itself (screeching jets are well known for staging for example [START_REF] Raman | Supersonic jet screech : half-century from powell to the present[END_REF]), or from the schlieren image quality that is sometimes corrupted by slight changes in the illumination quality. Even though a lot of care was taken to account for illumination quality by processing the images, we assume the latter to be responsible for the scatter visible in figure 12. Nevertheless, the method described in the former section uses statistics : the convective velocity is obtained as the ratio of spatial mode scalar products over temporal mode averages (see equation 13), so that uncorrelated noise should not disturb the estimation of the convection velocity.

Flapping frequency estimation

Perret et al. (2006) succeeded in obtaining the main frequency of the flapping motion of a jet-mixing layer interaction using POD analysis and low-order dynamical system. In a very simple way, it is possible to consider a linear low-order dynamical subsystem composed of mode 1 and mode 2, that can be written as follows:

a (i) ,t = L ij a (j) + D i , (22)
where the subscript , t refers to the time derivative, with implicit summation notation. As the temporal coefficients and their time derivatives are zero-centered, the D i term can be neglected. Furthermore, in case of a strong correlation between the time derivatives a (i)

,t and the coefficients a (j) , the dynamical system (22) can be reduced to:

a (1) ,t (t) = L 12 a (2) a (2) ,t (t) = L 21 a (1) , (23)
that can easily be integrated giving:

a (1) (t) = K 1 cos(ωt + φ 1 ) a (2) (t) = K 2 cos(ωt + φ 2 ), (24)
with ω = √ -L 12 L 21 = 2πf . Thus, the characteristic frequency of the flow is directly related to L 12 and L 21 .

The phase portraits of a

,t versus a (2) , and a

(2)

,t versus a (1) are presented in figure 13. A clear correlation between a

(1) ,t and a (2) on the one hand, and between a

(2) ,t and a (1) on the other hand can be observed. A slight scatter of the data can be observed in this figure. This is an effect of the noise observed earlier in figure 12. Since modes 1 and 2 are of the same order of magnitude, the simple linear dynamical system composed of these two modes (equation 23) can be used to model the main flow dynamics. A linear regression on the phase portraits leads to L 12 = 1.88 × 10 4 s -1 ± 1.80% and L 21 = -1.89 × 10 4 s -1 ± 1.35%. We see that L 12 and L 12 have very close values with a small confidence interval, which shows that the noise transmitted from the original data have only a small influence on the results. This gives a flapping frequency f = 3000Hz obtained with an accuracy of ±1.575%. Hence, the observed flapping motion is periodic with a -20 -10

0 10 20 -1 -0.5 0 0.5 1 da (j) /dt [kHz] a (i) a (2) ,t = f(a ( 1) ) a (2)
,t = -18.9a

(1) a ( 1) ,t = f(a (2) ) a

(1) ,t = 18.8a

(2) Fig. 13 Frequency estimation of the motion of the 2 first modes dominant frequency corresponding to the narrow band acoustic waves propagating in the co-flow. This ensures that POD captures the main dynamics of the flow, and it also shows the relevance of performing the analysis downstream of the nozzle.

Convection velocity of large scale structures

In figure 14 are plotted the distributions of a (n) a (p) ,t /λ (n) and Φ (n) ,x (x), Φ (p) (x) in the (n -p) plane for the first 8 modes. Both distributions show a very similar shape which is, as expected from equation 12, signature of the convective feature of the flow. Moreover, one can see that for n = p both the correlation and the scalar product have low values as expected for purely advective flows.

The main differences between these distributions occur for the larger mode numbers. As mentionned earlier, this may be attributed to the fact that the information is integrated along the optical path, the threedimensionality of the flow at small scales disrupts the analysis. Moreover, only a small number of samples (100 (a (n) , a (p) ,t ) couples) were used to calculate the POD decomposition, thus higher order modes may have not fully converged. ' < a (n) a (p) ,t >/λ (n) , [kHz]

), It is possible to see that the points corresponding to the maximum values in the distributions (figure 14) are disposed along a line and, by linear regression, a convective speed of 222.7 m/s is retrieved ± + -5.16%. We see here that the etimation of the convective velocity is more prone to uncertainty than the frequency estimation. This is attributed to the low number of snapshot used for this study. It is likely that the spatial modes and their gradient have perfectly converged. Neverthe-less, it is a satisfactory result, noting that this analysis is performed on flow visualization. The estimated convective velocity represents 0.64×∆U j , where ∆U j is the velocity difference between the jet exit and the entrained subsonic flow. This result is in good agreement with the measured value of [START_REF] Alkislar | Structure of a screeching rectangular jet : a stereoscopic particle image velocimetry study[END_REF] and [START_REF] Berland | Numerical study of screech generation in a planar supersonic jet[END_REF].

Identification of linear dynamical system

As mentioned in 2, equation 12 can be used to identify the coefficients of a linear dynamical system without time-resolved information on the flow. We want to demonstrate a use of the model by applying equation 12 to PIV data from the same flow.

Domain of study and POD analysis

The PIV data is decomposed using POD so that the data can be written as :

u(x, t) = Ns n=1 α (n) (t)Φ (n) (x), ( 25
)
where u is the velocity vector. Since the results of 2D PIV contains vertical and horizontal velocity components, the POD spatial modes are now vectors. The POD is performed on a sub-region of the PIV measurement plane, as presented in figure 16. This region is chosen because this is where the jet flapping amplitude is best captured by the PIV. As can be seen on the same figure, the shock waves are weaker in this area, making it easier for the POD to capture the flapping motion of the jet. This region is slightly different than the area used for the previous analysis, but they overlap in the range 5.0 < x/h j < 7.6. Assuming the convection velocity to be a global scale of the jet flow, which is reasonable for the large scales of interest here, it should not be an issue to perform this analysis on two different, but close, regions of the jet.

The cumulative energy of the computed POD modes Two first POD spatial modes Φ (1) (x) and Φ (2) (x), left and right respectively. For sake of clarity, colors represent the vector length is mainly organized in a circular disposition, indicating that these two modes represent a cyclic phenomenon of a single frequency.

Linear coefficients of dynamical system

The scalar products Φ (i)

,x (x), Φ (j) (x) are presented in figure 19. As predicted by equation 15, we can see that the scalar products are close to zero for i = j meaning that the POD modes are orthogonal to their derivatives. This is again a clear sign that the POD analysis mostly captures the advective motion of the flow. The equation 18 states that, in the case of an advec- Fig. 19 Scalar products values Φ (i) ,x (x), Φ (k) (x) . Note that values out of a grid point are interpolated.

tive motion, the coefficient of a linear dynamical system of POD modes can be obtained using the scalar product of the spatial POD modes and their spatial derivatives. Since these coefficients can be related to the frequency of this simple dynamics (see equation 24), it is possible to compute the expected values of the scalar products Φ (i) ,x (x), Φ (j) (x) . Main flow features oscillate at a known pulsation ω = 2πf s (see figures 8 and 13) and the convective velocity U c is also known (see figure 15), we would expect that Φ 19, the largest values of the scalar product are obtained for Φ

(1) ,x (x), Φ (2) (x) = ω/U c 80m -1 . From figure
(1) ,x (x), Φ (2) (x) and Φ (2)
,x (x), Φ (1) (x) . However, the maximum value is of about 60m -1 which is lower than expected. This may rise for different reasons: either the convective speed of the structures capture by the POD is not equal to the one measured with the schlieren images, or the computation of the spatial derivatives amplified the noise contained in the modes and the POD modes do not perfectly project onto their spatial derivatives. The latter is most probable, the PIV data contains only 200 samples which may not be sufficient to converge the POD analysis to an acceptable uncertainty. The linear dynamical system identified is presented in the figure 20. The initial condition was chosen so that the overall dynamics is representative of the entire data and the model was run using a fourth order Runge-Kutta algorithm. We clearly see that the derived dynamical system well passes in the average circle described by the measurement points. However, it is clear that a simple linear dynamics does not suffice to represent well the more complicated dynamics extracted with the POD modes. This shows that the POD modes extracted do not only represent a convective motion, and thus the POD-advection model can only represent a portion of the original dynamics. Phase portrait of temporal POD modes α (2) (t) as a function of (α (1) (t)). The line is the linear dynamical system identified with the method presented in paragraph 2. The red dot is the initial condition of the dynamical system.

Conclusion

The POD is a widely used technique that was originally designed in order to educe the most energetic structures in the flow. Applying the POD to the advection equation results in a system of equations that links the temporal and spatial POD modes, and their respective derivative to the convection speed of the flow.

We have shown in this paper that these results can be used to estimate the convection velocity of coherent features. For this purpose, the data does not have to be time-resolved nor need to originate from a complicated measurement campaign. Indeed, dual-time information obtained with Schlieren images can suffice to obtain estimate of the convection velocity.

An original feature of the model developed in this paper is that it allows for the construction of a linear dynamical system even if the original data is only made of independent realizations. The only a priori knowledge of a convection speed allows to reproduce and simulate the main dynamics of eligible flow fields, i.e. flows governed by convective motion where a simple dynamics seems to dominate.This can help in order to understand at least part of the dynamics, as was the case here, of complicated flows, especially if time-resolved data cannot be obtained. The present paper focuses on presenting the methodology and how it can used with experimental data. However, since purely experimental were used in this study, any conclusions regarding the uncertainty and bias of the proposed method is a combined effect of both the measurement uncertainties and the method itself. Although the results showed in this paper are encouraging, some more analysis, beyond the scope of this paper, would be required to educe the sensitivity of the method with respect to experimental parameters.

acknowledgments

The authors wish to thank the Region Poitou-Charentes and DGA-Spae for supporting part of this study.

Willert CE, Gharib M (1991) Digital particle image velocimetry. Experiment in FLuids 10: [181][182][183][184][185][186][187][188][189][190][191][192][193]

Fig. 1

 1 Fig. 1 Schematic diagrams of the wind tunnel (top) and detail of the supersonic nozzle (bottom).

Fig. 2

 2 Fig. 2 Schematic diagram of the location of the pressure sensors

Fig. 3

 3 Fig. 3 Gray-scale histograms of 5 different images before histogram specification (a), and after (b)

Fig. 4

 4 Fig. 4 Velocity histogram of a typical PIV snapshot. The bin width represents 4 m/s.

Fig. 5

 5 Fig. 5 Schematic diagram of the PIV acquisition system

Fig. 6

 6 Fig. 6 Comparison of centreline mean (top) and RMS (bottom) velocity profiles obtained with both LDA and PIV.

Fig. 7

 7 Fig. 7 Spark schlieren picture composed of two chosen independent images, taken from the top view.

Fig. 8

 8 Fig. 8 Power spectral density of wall pressure fluctuations inside the wind tunnel (Sensor 1).

Fig. 9

 9 Fig. 9 Definition of the domain of interest on which the POD analysis is performed.

Fig

  Fig. 10 (a) POD eigenspectrum, (b) partial amount of energy contained in the first n modes n.

Fig. 11

 11 Fig. 11 Comparison of original image (a) and its 4 modes partial reconstruction (b).

Fig. 12

 12 Fig. 12 Phase portraits of the 3 first modes. The arrows represent the time derivatives of the modes.

Fig. 14

 14 Fig.14Comparison of correlations a (n) a(p) ,t /λ (n) (a) and scalar products Φ(n) ,x (x), Φ (p) (x) (b). Values out of the grid points are interpolated.

Fig. 15

 15 Fig. 15 Estimation of convective velocity of the coherent structures

Fig. 17

 17 Fig. 17 POD spectrum (top), cumulative energy of the POD modes obtained from the PIV data (bottom)

  Fig. 17 POD spectrum (top), cumulative energy of the POD modes obtained from the PIV data (bottom)

  Fig.20Phase portrait of temporal POD modes α (2) (t) as a function of (α (1) (t)). The line is the linear dynamical system identified with the method presented in paragraph 2. The red dot is the initial condition of the dynamical system.

Table 1

 1 Characteristics of the flow.

		Supersonic jet Surrounding flow
	Dimensions (mm)	150 × 30	500 × 500
	Mach number	1.45	0.2
	Fully expanded Mach number 1.52	-
	Velocity (m/s)	380	60
	Total pressure (10 5 Pa)	3.7	1
	Total temperature (K)	260	290

  Mean longitudinal velocity in the z = 0.5h j plane. The sub-region used for the POD analysis is represented by the black rectangle.

							25
							20
							15
							10
							5
							λ n
							n -0.27
		4					(u-u e )/∆u
							1.00
							0.95
		2					0.89 0.79 0.74 0.84	10 0	10 1	n	10 2
	y/h	0					0.68 0.63 0.58
							0.53 0.47	1
							0.42
		-2					0.37 0.32
							0.26
							0.21
							0.16	0.8
		-4					0.11 0.05
		2	4	6	8	10	0.00
	x/h Fig. 16 is presented in figure 17(bottom). As can be seen the 5	Σλ n /Σλ	0.6 0.4 0.2	5 modes	19 % de l'énergie of the energy 19%
	first POD modes represent only 19% of the total energy
	only two distinct modes arise from the POD spectrum of the signal. This low convergence, and the fact that	0	10 0	n 10 1	10 2
	(see figure 17(top)) clearly shows that, contrary to the
	schlieren data set, the PIV data contains more informa-
	tion than the simple flapping motion of the jet. Hence,
	in this case, the application of the POD-Galerkin model
	is most likely to be limited to a small number of POD
	modes, which is acceptable in the sense that only the
	main flapping motion is of interest here.
	The domination of the first POD modes is thus less
	clear than for the schlieren images (figure 10(b)), but
	still the 2 first POD modes have an energy of an order
	of magnitude greater than the higher order modes (see
	figure 17(top)). Additionally, mode 1 and 2 almost have
	the same amplitude, which is attributed to the fact that
	they are representative of a convective motion.
	The spatial POD modes Φ (1) (x) and Φ (2) (x) are pre-
	sented in figure 18. It is obvious from this figure that
	the POD modes mainly contain fluctuations of veloc-
	ity of a single wavelength. It can also be seen in fig-
	ure 18 that Φ (1) (x) and Φ (2) (x) are in phase quadra-
	ture, which is expected if the linear combination of
	Φ (1) (x) and Φ (2) (x) is meant to reproduce a propagat-
	ing pattern. The phase portrait α (2) (t) = f (α (1) (t)) is
	presented in figure 20. As also observed in the anal-
	ysis of Edgington-Mitchell et al. (2014), the data is
	slightly scattered, which is expected giving the slow
	convergence of the POD spectrum (see figure 17), but


Camille Gay
email: camille.gay@ap-hm.fr

Jeanne Perrin

Blandine Courbiere

Florence Bretelle

Aubert Agostini

Impact of salpingectomy for ectopic pregnancy on the ovarian response during IVF stimulation

Keywords:

Introduction: Ovarian reserve is a major prognosic factor for Medical Assisted Procreation. Tubal surgery, realised close to mesosalpinx and ovarian vascularization, could impare ovarian function. However, salpingectomy is currently used to treat ectopic pregnancies or hydrosalpinx before IVF attempt. Disponible studies on this subject are unclear. The aim of this study is to evaluate the impact of salpingectomy for ectopic pregnancy on the ovarian response during IVF attempt. Material and methods: It was a single center comparative study. Included patients were the one receiving oocyte puncture for IVF attempt, with a history of unilateral tubal surgery: salpingectomy for ectopic pregnancy. We conducted a case-control study, comparing the sonographic parameters of the surgery ovary (case) to those of the safe ovary (control) during the first IVF attempt after salpingectomy. The ovarian sonographic response was evaluated according to the follicular antral count on day 3 and the sonographic follicular count on trigger day. Results: 55 patients were included. There was no significant difference in the number of recruited follicles on the operated side versus control side (p = 0.85 for >14 mm follicles, p = 0,46 for 10 to 14 mm, p = 0,52 for total amount of recruited follicles). There was no significant difference for the follicular antral count neither (p = 0.79). Discussion: In our population, there was no significant difference in the sonographic ovarian response to IVF stimulation between the ovary on the operated side and the control ovary among patients treated by unilateral salpingectomy for ectopic pregnancy.

Introduction

Increasingly more patients resort to medical assistance for procreation Medical Assisted Procreation (MAP). Ovarian reserve, and thus ovarian integrity, is one of the principal prognostic factors. Many of our patients have a history of tubal surgery, some of them having been performed for ectopic pregnancies, for whom salpingectomy is the benchmark treatment when the contralateral fallopian tube is healthy [START_REF] Clausen | Conservaive versus radical surgery for tubal pregnancy. A review[END_REF][START_REF] Yao | Current status of surgical and nonsurgical management of ectopic pregnancy[END_REF]. In addition, we know that the presence of a hydrosalpinx reduces pregnancy rates in MAP [START_REF] Katz | Deleterious effect of the presence of hydrosalpinx on implantation and pregnancy rates with in vitro fertilization[END_REF], and salpingectomy is recommended in this situation to improve implantation rates [START_REF] Kontoravdis | Proximal tubal occlusion and salpingectomy result in similar improvement in in vitro fertilization outcome in patients with hydrosalpinx[END_REF][START_REF] Zeyneloglu | Hydrosalpinx and assisted reproduction: options and rationale for treatment[END_REF][START_REF] Johnson | Laparoscopic salpingectomy for women with hydrosalpinges enhances the success of IVF: a cochrane review[END_REF]. Surgical impact on ovarian reserve and function must be known to determine indications, benefits and risks.

Ovarian vascularization is provided by the ovarian and uterine arteries and is close to the fallopian tube and mesosalpinx. Thus, fallopian tube surgery may impair ovarian vascularization and function. This outcome has been confirmed with animal studies [START_REF] Beyth | Ferility following excision of the mesotubarium superius in the rabbit[END_REF], but the result from human studies remains unclear.

Available studies on the impact of salpingectomy on the ovarian response during IVF are contradictory [START_REF] Fan | Effect of salpingectomy on ovarian response to hypersimulaion during in vitro fertilization: a meta-analysis[END_REF][START_REF] Kotlyar | The Effect of Salpingectomy on Ovarian Function[END_REF]. These contradictions are partly because the impact might be different according to the indication of the surgery [START_REF] Kotlyar | The Effect of Salpingectomy on Ovarian Function[END_REF]. The access to uniform studies, the evaluation of the impact on the ovary according to the indication of the salpingectomy and the use of direct parameters for evaluation of the ovarian response are necessary.

To study the direct impact on the ovary, it would be interesting to study sonographic follicular modifications during IVF stimulation. Only 4 studies have analysed this parameter; Tal and Xi's studies [START_REF] Tal | Ovarian response to gonadotropin simulation in repeated IVF cycles after unilateral salpingectomy[END_REF][START_REF] Xi | Ovarian response to gonadotropins after laparoscopic salpingectomy for ectopic pregnancy[END_REF], which did not find a reduction in the homolateral ovarian response among respectively 26 and 76 operated patients, and Gelbaya and Orvieto's studies, which highlighted a decrease in ovarian response among 40 and 15 patients per study12,13. However, the included patients had various salpingectomy indications, and that is why it would be interesting to evaluate the sonographic follicular modifications among patients treated by salpingectomy for the same indication. The principle aim of our study is to evaluate the sonographic impact on the homolateral ovarian response after salpingectomy for ectopic pregnancy (EP) during IVF stimulation.

Material and methods

This was an observational single centre comparative study that was conducted in the clinicobiological MAP centre of the university hospital of Marseille, France, between 1 January 2008 and 31 March 2018. Eligible patients were those treated for IVF +/-ICSI in the centre who received oocyte puncture during this period, whether or not embryo transfer was performed. Included patients were those who had a history of unilateral tubal surgery: salpingectomy for EP before IVF attempt. The excluded patients were those who received an other type of tubal surgery, bilateral tubal surgery or ovarian surgery.

We conducted a case-control study, comparing the sonographic parameters of the surgery ovary (case) to those of the safe ovary (control) during the first IVF attempt after salpingectomy for EP. The ovarian sonographic response was evaluated according to the follicular antral count on day 3 and the sonographic follicular count (divided in more than 15 mm and 10 to 14 mm) on trigger day. Sonography were realised with the same ultrasound scanner, in our center.

Clinico-biological and stimulation-related data were extracted from the medical file: age, smoking status, BMI, IVF indication, and ovarian reserve parameters (AMH, follicular antral count FAC, and FSH on day 3). Salpingectomy information was collected: indication, side, time between surgery and IVF attempt, and laparoscopy or laparotomy. Finally, IVF attempt information was collected: number and size of recruited follicles visible with sonography on each ovary, day of the ovulation trigger, plasmatic oestradiol on the day of trigger, total gonadotrophin dose, protocol (short, long, antagonist), number of oocytes collected and number of mature oocytes.

The chosen threshold for follicular size (between 10 and 14 mm and more than 14 mm) is the usual threshold used in published studies [START_REF] Tal | Ovarian response to gonadotropin simulation in repeated IVF cycles after unilateral salpingectomy[END_REF][START_REF] Gelbaya | Ovarian response to gonadotropins after laparoscopic salpingectomy or the division of fallopian tubes for hydrosalpinges[END_REF][START_REF] Orvieto | Does salpingectomy affect the ipsilateral ovarian response to gonadotropin during in vitro fertilization-embryo transfer cycles?[END_REF]. Our study received favourable opinion from the Aix Marseille University ethics committee (2018-27-09-002). Qualitative values are compared with test or Fishers exact test. The live birth rate and secondary outcome measures were compared between the operated side and the safe control side. Qualitative values were formulated as headcount and percentage, and continuous variables as averages with standard deviation (or median and min max). The results were given as the odds ratio with a confidence interval of 95%. Analyses were performed with Microsoft Excel 2010. The significance level was established at 5%.

Results

During this period, 55 patients were included. Patients were aged an average of 34 years (AE 5).

Ovarian reserve parameters on day 3 were FSH 7.5 UI/L (AE 2.3), AMH 3.5 ng/ml (AE 2.7), and FAC 16.2 (AE 10.5). A total of 41 patients (75%) had a tubal indication for IVF. Data concerning IVF follicular response were available for all patients. Data according to follicular antral count were available for 46/55 patients (84%), and data concerning the surgery approach was available for 19/55 patients (35%). Patient characteristics are summarized in Table 1. The delay between surgery and IVF attempt was 3.3 AE 3.8 years. There was no significant difference in the number of more than 14 mm recruited follicles: 4.2 AE 2.5 on the operated side vs 4.1 AE 2.4 control side (p = 0.85). There was no significant difference for 10 to 14 mm recruited follicles between the groups (3.1 AE 3.6 vs 2.6 AE 2.9, p = 0.46), for the total amount of recruited follicles (7.2 AE 4.9 vs 6.7 AE 4.1; p = 0.52), or for the follicular antral count (8.3 AE 4.7 vs 7.9 AE 5.3, p = 0.79) (Table 2).

Discussion

In our study, there was no significant difference in the ovarian sonographic response during IVF protocol among patients treated by unilateral salpingectomy for EP. Some limitations exist. The study does not allow the evaluation of the number of oocytes of each ovary, and thus, the evaluation of the final result of the IVF attempt. Data concerning follicular antral count for each ovary were available only for 46 patients (86%). All the other ovarian reserve parameters were available for all patients. Another limitation concerns surgery data. Indeed, some details of the surgery were not accessible, including the level of expertise of the surgeon, tools used, tubal condition at surgery time, and the safeguarding of the mesosalpinx. Surgical approach (laparoscopy or laparotomy) was available for 19 of 55 patients (35%). however, the strict inclusion of only patients treated by salpingectomy for EP generates a uniform group and a close surgical technique.

However, our study provides a contribution to the existing literature. The aim was to evaluate sonographic follicular response. Few studies have used this evaluation criterion [START_REF] Tal | Ovarian response to gonadotropin simulation in repeated IVF cycles after unilateral salpingectomy[END_REF][START_REF] Xi | Ovarian response to gonadotropins after laparoscopic salpingectomy for ectopic pregnancy[END_REF][START_REF] Gelbaya | Ovarian response to gonadotropins after laparoscopic salpingectomy or the division of fallopian tubes for hydrosalpinges[END_REF][START_REF] Orvieto | Does salpingectomy affect the ipsilateral ovarian response to gonadotropin during in vitro fertilization-embryo transfer cycles?[END_REF]. Sonographic data were not used in the latest meta-analyses [START_REF] Fan | Effect of salpingectomy on ovarian response to hypersimulaion during in vitro fertilization: a meta-analysis[END_REF][START_REF] Yoon | Does salpingectomy have a deleterious impact on ovarian response in in vitro fertilization cycles?[END_REF], and this limitation has been highlighted. This was a case-control study comparing the ovarian response within one patient (self-control study) and avoiding confounding factors that might exist between cases and controls (environmental impact, age, genetic, BMI, toxins, etc.).

Most of the accessible studies analyse heterogeneous populations in terms of surgical technique (salpingectomy, salpingotomy, and tubal exclusion, etc.) and its indication (EP, hydrosalpinx, endometriosis, and sterilisation, etc.), including Yoon's meta-analysis (patients treated by salpingectomy for EP of hydrosalpinx) and studies by Fan, Noventa and Kotlyar (salpingectomy for any indication, comparing uni-or bilateral surgery to a control population or self-control study) [START_REF] Fan | Effect of salpingectomy on ovarian response to hypersimulaion during in vitro fertilization: a meta-analysis[END_REF][START_REF] Kotlyar | The Effect of Salpingectomy on Ovarian Function[END_REF][START_REF] Yoon | Does salpingectomy have a deleterious impact on ovarian response in in vitro fertilization cycles?[END_REF][START_REF] Noventa | Salpingectomy before assisted reproductive technologies: a systematic literature review[END_REF]. Most of the studies evaluated small samples, with multiple evaluation criteria (mature follicle, retrieved oocytes, obtained embryos, pregnancy rates, and ovarian reserve parameters). The use of pregnancy rates or obtained embryos is debatable criteria because they are dependent on many other ovarian reserve parameters (age, toxins, endometrial parameters, and father's parameters).

The punctured oocyte number only evaluates the oocyte puncture results. This number depends on the ovarian response but also on the puncture gesture, which could be impaired in the case of salpingectomy (post-surgery adhesions, modification of the position of the ovary and accessibility, etc.). Thus the ovarian evaluation, thanks to the study of sonographicrecruited follicles during stimulation, could be a good representation of global ovarian function and reserve. In 2012, Mutlu found that the most reliable ovarian reserve parameter was the antral follicle count and, thus, a sonographic parameter [START_REF] Mutlu | Antral follicle count determines poor ovarian response better than aniMüllerian hormone but age is the only predictor for live birth in in vitro fertilization cycles[END_REF]. The sonographic monitoring of ovarian response could be a good evaluation of ovarian function and could allow a different approach from published studies.

In our study, the inclusion of patients who have all been treated by salpingectomy for EP permitted us to obtain a uniform population and a close surgical technique. Salpingectomy is probably the most invasive technique, in an emergency and tubal suffering context, which would have the most important impact on the ovarian function, as Kotlyan's study suggests it does [START_REF] Kotlyar | The Effect of Salpingectomy on Ovarian Function[END_REF]. Our population was quite important (55 patients) and is the second population with comparable patients available in the literature to this day, after Xi's study (76 cases) [START_REF] Xi | Ovarian response to gonadotropins after laparoscopic salpingectomy for ectopic pregnancy[END_REF]. The delay between surgery and IVF attempt was 3.3 AE 3.8 years, quite a long period that permits the avoidance of confounding factors involved in early post-operative and post-ectopic pregnancy (inflammation, healing process, and ovarian collateral vascularization development, etc.). Only the first IVF attempt after surgery was studied to avoid bias induced by an overrepresentation of women having had several cycles, who present with poorer prognosis, statistically, and an overrepresentation of women having or not having a different ovarian response on the case or control side.

Our results are concordant with those found in studies by Dar, Tal and Xi [START_REF] Tal | Ovarian response to gonadotropin simulation in repeated IVF cycles after unilateral salpingectomy[END_REF][START_REF] Xi | Ovarian response to gonadotropins after laparoscopic salpingectomy for ectopic pregnancy[END_REF][START_REF] Dar | Ovarian function before and after salpingectomy in artificial reproductive technology patients[END_REF]. In 2002, Tal did not find a significant difference in the ovarian response in 26 patients treated by salpingectomy for EP when compared to a control group of patients treated for male infertility (number of follicles, obtained oocytes and embryos). In 2012, Xi found an increase in the necessary gonadotrophin dose after salpingectomy but no difference in the number of mature follicles or obtained oocytes in a population of 76 operated patients (before and after salpingectomy) and 80 control patients. In 2000, in a population of 26 patients treated by unilateral salpingectomy for EP, Dar found no ovarian response reduction during IVF when comparing before vs. after surgery or case vs. control side (length of stimulation, gonadotrophin dose or obtained oocytes).

However, it is concerning that these publications contrast with some others that found a decrease in ovarian response. Indeed, Lass in 1998 showed a decrease in the number of follicles and oocytes ipsilateral to the surgery ovary in 29 patients treated by unilateral salpingectomy for EP [START_REF] Lass | Effect of salpingectomy on ovarian response to superovulaion in an in vitro ferilization-embryo transfer program[END_REF]. There was no significant difference in the total number of follicles and oocytes obtained during the stimulation in comparison to a control group of 73 patients.

Although the results of our study and some of the other available studies in the literature are reassuring, the complete safety of this procedure is unsure, especially concerning the longterm consequences on ovarian function. Furthermore, a possible decrease in ovarian response to IVF stimulation would not necessarily mean a decrease in spontaneous fertility; just like the decrease in AMH is not linked to spontaneous time to pregnancy [START_REF] Streuli | AMH concentration is not related to effective time to pregnancy in women who conceive naturally[END_REF]. Nevertheless, caution concerning indications and the quality of tubal surgery is important, all the more concerning infertile women. Surgeon sensitivity to fertility problems is necessary [START_REF] Kalra | Ovarian reserve may be compromised after adnexal surgery: Are we sufficiently fertility-focused in our surgical training? Facts Views[END_REF]. Surgical technique standardization would allow stronger studies and the optimization of practices. Finally, patient information about risks and benefits of techniques is important to make an informed choice.

Conclusion

In our population, there was no significant difference in the sonographic ovarian response to IVF stimulation between the ovary on the operated side and the control ovary among patients treated by unilateral salpingectomy for ectopic pregnancy. More comparative studies on larger samples with reliable methodology and a standardized population are necessary to ensure the safety of this common surgery.

Table 1

 1 Clinico-biological Characteristics.

	Variables : (N = 55 patients)
	Age (years)	34 AE 5
	Smoking	Active 11 (20%)
		Former 7 (13%)
		Never 37 (67%)
	BMI (kg/m 2 )	23 AE 4.3
	Right Salpingectomy	35 (64%)
	Left Salpingectomy	20 (36%)
	Delay between IVF	3.3 AE 3.8
	attempt and
	salpingectomy (years)
	Surgical Technic (19	Laparoscopy 14 (74%)
	patients)	Laparotomy 5 (26%)
	IVF Indication	Tubal cause 41 (75%)
		Ovulatory cause 1 (2%)
		Man cause 1 (2%)
		Poor Ovarian Response 6 (10%)
		Idiopathic Cause 2 (4%)
		Endométriosis 4 (7%)
	Poor Ovarian Response	18 (33%)
	(AMH < 1 ng/ml et/ou
	AFA < 7 )
	Follicular Antral Account	16.2 AE 10.5
	(46 patients)
	FSH on Day 3 (UI/L)	7.5 AE 2.3
	AMH (ng/ml)	3.5 AE 2.7
	IVF or ICSI	FIV 50 (91%)
		ICSI 5 (9%)
	Stimulation Protocol	Long 25 (45.5%)
		Short 14 (25.5%)
		Antagonist 16 (29%)
	Gonadotrophin (UI/L)	2077 AE 760
	Plasmatic Estradiol on	2427 AE 1246
	Trigger Day (pg/ml)
	Trigger Day	10.8 AE 2.1
	Collected Oocytes	9.2 AE 5.6
	Mature Oocytes	7.5 AE 5.0
	Bologna criteria, ESHRE 2011


domain: spi phys

text:
Benjamin Deveaux

Camille Fournis

Vincent Brion

Julien Marty

Antoine Dazin

EXPERIMENTAL STUDY AND MODELLING OF THE TIP-LEAKAGE FLOW FOR AN ISOLATED FIXED BLADE

INTRODUCTION

The space between the blade tip and the casing wall of an axial flow compressor is often referred as tip-gap or tipclearance. The aerodynamic loading of the blade generates a leakage flow through that gap. This flow not only limits the compressors pressure rise and efficiency, but is also associated with the arising of compressor rotating stall and fan broadband noise. Downstream of the blade the total pressure losses come primarily from the tip-leakage vortex [START_REF] Doukelis | The effect of tip clearance gap size and wall rotation on the performance of a high-speed annular compressor cascade[END_REF][START_REF] Brion | Stabilité des paires de tourbillons contra-rotatifs: application au tourbillon de jeu dans les turbomachines[END_REF]. Moreover this vortex exhibits a large streamwise velocity deficit for small gaps in its core that drives the production of turbulent kinetic energy [START_REF] Muthanna | Wake of a compressor cascade with tip gap, part 1: Mean flow and turbulence structure[END_REF].

Many studies show that a larger gap height increases the tip-leakage flow losses [START_REF] Doukelis | The effect of tip clearance gap size and wall rotation on the performance of a high-speed annular compressor cascade[END_REF][START_REF] Flachsbart | Spaltverluste an Tragflügeln[END_REF][START_REF] Storer | An approximate analysis and prediction method for tip clearance loss in axial compressors[END_REF]. A bigger gap height leads to a smaller pressure rise, efficiency and operating range of the compressor [START_REF] Kameier | Experimental study of tip clearance losses and noise in axial turbomachines and their reduction[END_REF].

The effect of the casing boundary layer thickness on the tip-leakage flow has been rarely studied. Storer and Cumpsty [START_REF] Storer | An approximate analysis and prediction method for tip clearance loss in axial compressors[END_REF] have shown that a thicker boundary layer on the casing increased the total pressure loss coefficient without changing its evolution with the gap height.

Several models have been tried to describe the tipleakage flow losses in the past such as the lifting-line theory and the model of a jet.

In the former case, Lakshminarayana and Horlock [START_REF] Lakshminarayana | Tip-clearance flow and losses for an isolated compressor blade[END_REF] adopted the lifting-line theory to describe the flow through a slotted wing. They observed that, for small gaps, the theory overestimated the induced-drag as it had been noticed earlier by Flachsbart [START_REF] Flachsbart | Spaltverluste an Tragflügeln[END_REF].

In the latter case, for small gaps, Denton [START_REF] Denton | Loss mechanisms in turbomachines[END_REF] and then Storer and Cumpsty [START_REF] Storer | An approximate analysis and prediction method for tip clearance loss in axial compressors[END_REF] adopted an alternative approach which consists in modelling the tip-leakage flow as a jet in crossflow in order to investigate the total pressure losses.

The first objective of this paper is to analyze the effects of gap height and casing boundary layer thickness on the tip-leakage flow. To do so an experimental parametric study is realized in a low-speed wind tunnel.

The second objective of this paper is to model the tipleakage flow as a jet in crossflow to have an estimation of the tip-leakage vortex circulation. The domain of validity of such an approach is evaluated based on the parametric study. In addition to that, the experimental data provides informations on the influence of the tip-leakage circulation on the evolution of the total pressure losses.

DESCRIPTION OF THE EXPERIMENT 2.1 SETUP

Experiments were conducted in the low-speed wind tunnel S2L located at the ONERA center of Meudon. The setup is placed in a wooden test section represented in figure 1. This test section has a length of 1.75 m.

As displayed in figure 2, a NACA0012 straight blade with a chord c = 200 mm is placed perpendicular to a flat plate which models the casing. The distance between the casing and the upper wall of the wind tunnel remains fixed at 560 mm. The blade leading edge is located 660 mm downstream the test section inlet.

This setup can reproduce values of h/c encountered in real compressors and, as shown in table 2, larger gaps are possible. The gap is adjusted using gauge blocks placed at the leading edge of the blade. At the trailing edge, the gap is slightly bigger (+0.1% of chord).

The boundary layer thickness upstream of the blade can be adjusted with the distance L between the leading edges of the casing and the blade. This is done by adding or subtracting flat plate sections in front of the blade. In this study we consider L/c = 1.8 and L/c = 3.3.

The angle of attack α can be adjusted from -15 • to 15 • . Geometrical constraints imposed α < 0 to have measurements in the tip-leakage vortex. Thus the angle of attack α is fixed at -10 • , which provides a high aerodynamic loading.

COORDINATE SYSTEM

As illustrated in figure 2, the x axis is oriented in the streamwise direction and the z axis is perpendicular to the casing pointing upwards. The origin (x, y) = (0,0) is defined as the leading edge of the blade at zero angle of attack (α = 0 • ) and z = 0 corresponds to the casing wall.

5-HOLE PRESSURE PROBE MEASURE-MENTS

As shown in figure 3, velocity and total pressure measurements were realized behind the blade trailing edge with a 5-hole probe. This probe has a diameter of 1.6 mm and the uncertainties associated with the calibration method are ±0.001 for the Mach number and ±0.15 • for pitch and yaw 2, this corresponds to a Mach number of 0.1, therefore the flow can be regarded as incompressible. The free stream turbulence intensity T U∞ , which is measured with a hot wire 1.5c upstream the blade, is about 0.07% .

CASING BOUNDARY LAYER

A two-component laser doppler velocimetry is used to measure the evolution of the boundary layer on the casing. To mitigate the potential effect of the blade on the boundary layer, the gap height is adjusted to its maximum value (h = 26 mm or h/c = 13%) and the angle of attack is set to 0 • . The properties of the boundary layer on the casing are synthetized in table 4.

BLADE PRESSURE DISTRIBUTION

The steady flow around the two-dimensional NACA0012 airfoil is simulated with the finite volume code elsA. The convection flux is discretized with a second order AUSM+(P) scheme [START_REF] Liou | Ausm schemes and extensions for low mach and multiphase flows[END_REF]. The fully turbulent k -ω -SST model of Menter [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF] is used in this computation.

The pressure distribution around the airfoil obtained in the simulation is compared to the measurements in figure 4. There is a good agreement between the simulation and the measurements.

TRANSITION

On the blade suction side the transition from laminar to turbulent flow arises from a laminar separation bubble located between x/c = 2% and x/c = 4%.

On the pressure side the transition is triggered with a 125 µm thick turbulator tape placed at 20% of chord from the leading edge.

On the casing wall, the same type of turbulator tape is placed 7.5% of chord downstream its leading edge.

IDENTIFICATION OF THE TIP-LEAKAGE FLOW REGIMES

The distance between the vortex center and the casing is represented in figure 5. The boundary layer thickness does not have a major effect on the vortex vertical position except for h/c = 0.5%.

If the blade tip is far enough from the casing, the tip-leakage vortex should be similar to a wing-tip vortex. Therefore the position of the vortex relative to the blade In other words the tip-leakage flow seems to have three different regimes depending on the gap height: a small gap regime for h/c < 4.5 %, a large gap regime for h/c > 8 % and an intermediate regime in-between.

In the small gap regime at x/c = 1.05, as shown in figure 6, the tip-leakage vortex remains around y/c = -0.11. For larger gaps the vortex gets closer to the blade when the gap height is increased.

Increasing the casing boundary layer thickness moves the tip-leakage vortex away from the blade. This trend can be explained by doing an analogy with a jet in crossflow.

In fact increasing the boundary layer thickness reduces the average velocity U 0 of the main stream at the gap exit. Moreover based on Rains approach [START_REF] Rains | Tip clearance flow in axial compressors and pumps[END_REF], the gap exit velocity U j depends primarily on the pressure difference between the pressure side and suction side of the blade. Thus the casing boundary layer thickness should have little impact on U j . As a consequence increasing the boundary layer thickness increases the ratio R = U 2 j /U 2 0 which, as explained by Fearn and Weston [START_REF] Fearn | Vorticity associated with a jet in a cross flow[END_REF], extends the penetration of the jet in the crossflow, hence a larger distance between the tip-leakage vortex and the blade.

In summary, in the small gap regime the tip-leakage flow has similarities with a jet in crossflow. In one hand the tip-leakage vortex is far from the blade in that regime. In the other hand the ratio between the velocity of the main flow and the jet exit velocity influences the lateral position of the tip-leakage vortex. In the large gap regime the effect of the casing wall vanishes when the gap is increased and the tip-leakage vortex becomes similar to a wing-tip vortex.

GLOBAL ANALYSIS OF THE TIP-LEAKAGE FLOW LOSSES

The global losses in a flow are represented by the mass-flow average of the total pressure loss coefficient C m Pt . Indeed for an incompressible flow around a fixed setup,

C m Pt = S m C Pt ρu x dS ṁ = ρT t∞ 1 2 U 2 ∞ Σ u.∇s dV ṁ (1)
The total pressure loss C Pt is decomposed in two terms in relation [START_REF] Brion | Stabilité des paires de tourbillons contra-rotatifs: application au tourbillon de jeu dans les turbomachines[END_REF]. For an axisymmetric vortex, it can be shown that the first term C vor t e x is proportional to the square of its circulation. Besides, as shown in figure 7, surface averaged value of C vor t e x seems to increase with Γ 2 . For gap heights smaller than 6.0% of chord, which corresponds to Γ 2 /(cU ∞ ) 2 < 0.11, the difference between the surface and mass-flow averages is small. For larger gaps, figure 9 shows that the vortex core has a significant streamwise velocity surplus, hence the deviation of the mass-flow average from the surface average. The second term C wak e is a deficit of u 2

x . Thus C wak e can be interpreted as the contribution of wakes to the total pressure losses.

C Pt =       P s∞ -P s 1 2 ρU 2 ∞ - (u 2 y + u 2 z ) U 2 ∞       C v or t e x +       1 - u x U ∞ 2      C w ak e (2)
The evolution of the mass-flow averaged losses with the gap height is shown in figure 8. The total pressure loss coefficient C m Pt increases linearly for h/c < 4 %, then increases more slowly for bigger gaps and even decreases for h/c = 13 %.

This rise of C m Pt comes mainly from the vortex loss coefficient C m vor t e x which seems to increase linearly with h for h/c ≤ 8 %.

The slower augmentation of C m Pt for h/c > 4 % corresponds to a decrease of the wake loss coefficient C m wak e . Besides, based on figure 9, this reduction of C m wak e is caused by the arising of a streamwise over-velocity in the vortex core which compensates the streamwise velocity deficit present in the other regions of the flow.

For h/c = 13 % the coefficient C m vor t e x seems to stagnate whereas C m wak e continue to decrease, hence the reduction of total pressure loss coefficient for this gap size.

Increasing the casing boundary layer thickness translates C m wak e and C m vor t e x of approximately +0.027 and -0.016 respectively. As a consequence C m Pt is increased of about 0.011, which corresponds to 8 % of the smallest value of C m Pt . In other words the casing boundary layer thickness has no major influence on the evolution of the wake and vortex losses with the gap height. Only the relative contributions of the vortex losses and wake losses to the total pressure losses are impacted by a thickening of the casing boundary layer.

In summary, the increase of the total pressure coefficient C m Pt with the gap height comes from the augmentation of the tip-leakage vortex circulation. Indeed it seems that the C m vor t e x increases linearly with Γ 2 . The contribution of velocity deficit C m wak e to the total pressure losses is significant for small gaps and becomes weaker for larger gaps. Changing the casing boundary layer thickness does not have a significant influence on the evolution of the loss coefficients with the gap height.

THEORETICAL PREDICTION OF THE TIP-LEAKAGE VORTEX CIRCULATION

It has previously been shown that the rise in total pressure losses is mainly attributed to the vortex contribution which seems to increase as the square of the circulation. That is why a theoretical study was performed in order to model the evolution of the tip-leakage vortex circulation Γ with respect to gap height h.

Moreover, the evolution of the losses are not significantly influenced by the casing boundary layer thickness. Therefore, the casing is considered as an inviscid wall in this theoretical study. The casing wall then constitutes a symmetry plane of a slotted wing configuration (see figure 10) where the slot corresponds to twice the gap.

MODELLING THE FLOW AS A JET IN CROSSFLOW

In the small gap regime, the flow is modelled as a jet in crossflow. Jacquin [START_REF] Jacquin | Phenomenological description and simplified modelling of the vortex wake issuing from a jet in a cross flow[END_REF] and Karagozian [START_REF] Karagozian | The jet in crossflow[END_REF] reported the production of a symmetrical counter-rotating vortex pair aligned with the direction of the crossflow: the symmetry plane corresponds to the casing wall. Jacquin [START_REF] Jacquin | Phenomenological description and simplified modelling of the vortex wake issuing from a jet in a cross flow[END_REF] considered an analogy with a wing wake in order to assess the jet contribution to the circulation of this vortex pair. The jet thrust is assimilated to the lift generated by a wing which span is 2z v , the distance between the two counter-rotating vortex cores (see figure 10): where A j = 2hc is the slot area and U 2 j is the mean value of the jet velocity squared. Γ can finally be expressed in terms of U 2 j :

ρU 2 j A j = ρU ∞ Γdz = 2ρU ∞ Γz v (3)
Γ cU ∞ = h z v U 2 j U 2 ∞ ( 4
)
The flow entering the gap gives birth to a separation bubble and can be considered as two-dimensional [START_REF] Rains | Tip clearance flow in axial compressors and pumps[END_REF]. Rains assumed the fluid at rest on the pressure side: P s+ (ξ) and P s-(ξ) are the pressure respectively on the pressure and suction side at chord position ξ (see figure 10). Moore and Tilton [START_REF] Moore | Tip leakage flow in a linear turbine cascade[END_REF] defined an overall discharge coefficient χ D which accounts for the separation bubble and links the jet velocity U j (ξ) to the pressure difference ∆K P (ξ) across the blade:

U j (ξ) U ∞ = χ D P s+ (ξ) -P s-(ξ) 1 2 ρU 2 ∞ = χ D ∆K P (ξ) (5)
Equation ( 5) corresponds to a pressure loss formula in which χ D is a discharge coefficient accounting for the losses inside the gap. Moreover P s+ (ξ) and P s-(ξ) correspond to the pressure respectively on the pressure and suction side of a two-dimensional airfoil. The mean value of the jet velocity squared U 2 j is linked to the two-dimensional lift coefficient C L :

U 2 j U 2 ∞ = χ 2 D c 0 P s+ (ξ) -P s-(ξ) 1 2 ρU 2 ∞ c dξ ≈ χ 2 D C L (6)
and inserted into equation ( 4) such that:

Γ cU ∞ = h z v U 2 j U 2 ∞ ≈ h z v χ 2 D C L (7)
As shown in figure 5, the vortex height z v , which varies with respect to h, is fitted based on experimental measurements of the position of the tip-leakage vortex:

z v c = A h c a = 0.35 h c 0.372 (8)
where A and a are empirical coefficients. The discharge coefficient χ D is adjusted to 0.9 in order to fit the experimental values of the circulation. This value of χ D is close to the value of 0.84 obtained by Rains [START_REF] Rains | Tip clearance flow in axial compressors and pumps[END_REF] and supported by Moore and Tilton [START_REF] Moore | Tip leakage flow in a linear turbine cascade[END_REF].

RESULTS AND DISCUSSION

The evolution of the circulation of the tip-leakage vortex versus gap height is given in figure 11. For gap heights lower than 4.5 % of the chord, the model is in good agreement with experimental data. For bigger gap heights, the experimental curve seems to tend towards a constant asymptotic value represented by the horizontal dashed line. This asymptotic value corresponds to the circulation of the tip vortex of an isolated wing computed with Prandtl's lifting-line theory [START_REF] Prandtl | Tragflügeltheorie. II. Mitteilung[END_REF].

Those results also suggest that the tip-leakage flow undergoes several regimes depending on the size of the gap: a small-gap regime which entails the generation of a jet in the main flow, a big-gap regime which corresponds to the formation of a wing-tip vortex and an intermediate regime.

Fearn and Weston [START_REF] Fearn | Vorticity associated with a jet in a cross flow[END_REF] reported that the counterrotating vortices move away from one another as the flow marches downstream and as the ratio U 2 j /U 2 ∞ increases. It suggests that parameters A and a introduced into the model for z v depend on the integration plane downstream of the blade and on C L according to equation (6).

CONCLUSIONS

The influence of gap height and casing boundary layer thickness on the tip-leakage flow was investigated. Velocity and total pressure were measured downstream an isolated fixed blade using a 5-hole pressure probe. This study reveals the existence of a small gap regime for h/c < 4.5 %, a large gap regime for h/c > 8 % and an intermediate regime in-between. In the small gap regime the tip-leakage flow presents similarities with jets in crossflow. In the large gap regime it is comparable to a wingtip vortex. Besides a model of a jet in crossflow gives a good estimation of the tip-leakage vortex circulation for h/c < 4.0%.

The total pressure loss was decomposed in a vortex loss and a wake loss. The vortex loss seems to increase with the square of the tip-leakage vortex circulation and drives the rise of total pressure losses. The wake loss is significant at small gaps and decreases for h/c > 4.5 %. This reduction of wake loss, which comes from the apparition of a streamwise velocity surplus in the vortex core, leads to a slower increase of the total pressure losses with the gap height.

The casing boundary layer thickness does not influence the evolution of the losses with the gap height. On the contrary a thicker boundary layer moves the tip-leakage vortex away from the blade. This effect comes from the augmentation of the ratio between the gap exit velocity and the streamwise velocity, which is associated to the penetration of a jet in a crossflow.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Dimensions of the test section in millimeters

Fig. 4 :

 4 Fig. 4: Pressure distribution around the NACA0012 airfoil for α = -10 • (h = 0). Pressure taps located in z 1.7c

Fig. 5 :Fig. 6 :

 56 Fig. 5: Vortex center vertical position at x/c = 1.05

Fig. 7 :Fig. 8 :

 78 Fig. 7: Correlation between vortex losses and circulation. The value of C vor t e x at h = 0 is extrapolated from the experimental data

Fig. 9 :Fig. 10 :

 910 Fig. 9: Streamwise velocity in the vortex center

Fig. 11 :

 11 Fig. 11: Prediction of the circulation

Table 2 :

 2 Experimental parameters

	U ∞				40 m/s
	Re				550 000
	M				0.1
	T U∞			0.07%
	Airfoil			NACA0012
	Blade rotation axis	x = 0.25c , y = 0
	c				200 mm
	α				-10 •
	h/c			0.5% to 13%
	L/c			1.8 or 3.3
	angles. Moreover the uncertainty on the total pressure loss
	coefficient C Pt is ±0.01.
		0.2
	y/c	-0.2 0.0				5-hole probe → measurement plane
		-0.4
		-0.6	0.0	0.2	0.4	0.6	0.8	1.0
					x/c
	Fig. 3: Measurement plane at x/c = 1.05 (mesh
	resolution: ∆y ≈ 2 mm and ∆z ≈ 1mm)
	3 FLOW CONDITIONS
	3.1 FREE STREAM
	For this experiment, the Reynolds number is fixed at
	550 000. As exposed in table

Table 4 :

 4 Boundary layer characteristics at x = 0

	L/c		1.8	3.3
	δ 0.99 (δ 0.99 /c)	7.02 mm (3.50%) 11.44 mm(5.70%)
	δ *	(δ * /c)	1.09 mm (0.55%) 1.82 mm (0.90%)
	θ *	(θ * /c)	0.82 mm (0.40%) 1.35 mm (0.70%)
	H *		1.33	1.35
	Re θ *		2200	3600
	tip should not vary anymore. As a consequence, for large
	gaps z v should vary like h/c, which seems to be the case
	for h/c > 8 %. For gap heights smaller than 4.5 %, the
	tip-leakage vortex vertical position seems to increase like
	(h/c) 0.372 .

ACKNOWLEDGEMENTS

The present is work supported by the ANR project NumERICCS (ANR-15-CE06-0009). The simulations have been performed in the framework of the elsA threeparty agreement between AIRBUS, SAFRAN, and ON-ERA which are co-owners of this software.