Victor Chéron

Jorge César Brändle De Motta

Jean-Bernard B Blaisot

Thibaut Ménard

J C Brändle De Motta

Analysis of the effect of the 2D projection on droplet shape parameters

Keywords: Shape parameters, Droplets characterization, Atomization

Introduction 1.Motivations

Atomization of liquid is used in a variety of applications including combustion chambers, medical sprays, fire suppression, and agricultural irrigation. Atomization is usually divided in two fragmentation steps: primary atomization of the main liquid flow into detached liquid structures, and secondary atomization of these structures into spray droplets. Understanding these fragmentation processes is of tremendous interest and requires the analysis and characterization of the different liquid elements encountered during these processes. Important information is obtained by characterizing the shape of the liquid elements, to classify them into ligaments, elongated or spherical droplets and then follow the space-time evolution of these characteristics in a given atomization process.

Shadowgraphy or backlight imaging techniques are, among other methods, particularly good candidates to carry out shape characterization analysis in spray applications [START_REF] Bachalo | Phase/doppler spray analyzer for simultaneous measurements of drop size and velocity distributions[END_REF], van Beeck and Riethmuller, 1996, Schober et al., 2002, Bothell et al., 2020]. A projection on the image plane of the liquid phase elements is obtained by these techniques. The liquid-gas interface is then obtained by analyzing the contour of the liquid elements in this 2D projection along with different parameters such as the projected area, the sphere-equivalent diameter. Several shape parameters are derived from this contour. It should be noted that these parameters, while useful to characterize liquid structures in a spray, are obtained from 2D projections and the shape of liquid elements is intrinsically three-dimensional. This 2D projection results in a loss of information in experimental data.

It is often assumed that elongated liquid elements have a symmetry axis parallel to the image plane and assumed spherical when the apparent shape is circular. However, these assumptions can be misleading, as illustrated in the image sequence of a water jet emanating from a triple-disk nozzle in figure 1. This jet, extensively analyzed by our research group using different shape parameters [START_REF] Dumouchel | Laser diffraction measurement of nonspherical drop sprays[END_REF] and multi-scale analysis [Dumouchel andBlaisot, 2013, Dumouchel et al., 2015], is characterized by a plane liquid sheet that disintegrates asymmetrically in ligaments and droplets. First, the plane geometry of the liquid core, see position I in the figure, is difficult to comprehend only from the image and without additional information.

Second, the droplet at position II has a near circular projection, suggesting a spherical shape. Nevertheless, the sequence shows that the droplet follows oscillations after the break-up of the ancestor ligament and exhibit a Figure 1: Shadowgraphy image sequence (25 000 frames/s) of a liquid jet [START_REF] Dumouchel | Laser diffraction measurement of nonspherical drop sprays[END_REF].

The field of view is 3.5 mm × 6.4 mm. Refer to text for roman letters.

These examples show the deviation due to the projection of a real 3D shape that can be introduced in the analysis of 2D experimental images. To the authors' knowledge, few experiments have been performed to get 3D shape characteristics of liquid-gas interface elements in two-phase flows, and most of them are devoted to bubble dynamics and breakup [START_REF] Ravelet | On the dynamics and breakup of a bubble rising in a turbulent flow[END_REF], Masuk et al., 2019]. The lack of 3D experimental data in sprays is due to the challenging environment of the atomization processes: i.e. high velocity, large number of droplets, optical access limitations, etc., along with the absence of diagnostic techniques that is capable of capturing reliable 3D information in such conditions. The analysis of 3D shapes is more common in granular and porous material communities where each structure can be extracted individually, such as in grain analysis [START_REF] Rorato | Sphericity measures of sand grains[END_REF]. A common way to compute the 3D shape parameters is by analyzing three orthogonal projection of the structure [START_REF] Blott | Particle shape: a review and new methods of characterization and classification[END_REF].

The spray community call on CFD to get 3D information on atomizing flows [START_REF] Dumouchel | Multiscale analysis of atomizing liquid ligaments[END_REF].

This is also what is done here to analyze the systematic error in shape analysis resulting from the 2D projection in experiments. The paper combines an analytic study under the spheroid assumption, numerical simulations for detailed analysis of realistic droplets, and, an application to experimental data.

Shape parameters under consideration for 2D and 3D analysis are detailed in §1.2. In §2, the CFD flow solver and the detection algorithm are briefly described. At the end of this section, the droplet database obtained from the CFD solver used is presented. Two test cases are considered for CFD, yielding to two typical atomization conditions, i.e. (i) isotropic and (ii) anisotropic deformed droplets. In §3, the shape parameters are analytically analyzed under the spheroid assumption and the limits of this assumption is commented using the droplet database. In §4, the analysis of shape parameters is done on the droplet database, first for isotropic droplets and next for anisotropic deformed droplets. The systematic deviation due to the projection step is highlitghted in this section, showing a correlation with the new method introduced in section §3 for analyzing 2D shape parameters based on the spheroid assumption. In §5, this new procedure is applied to a droplet database obtained from an experimental setup. A correlation between the shape parameter probability density function of the distriutions and a log-normal distribution is shown, creating a possibility to reproduce a 3D

shape parameter distribution from 2D information under certain hypotheses.

Summary of morphological parameters

A liquid drop can take on an infinite number of distinct shapes. A significant number of shape parameters have been created in the literature to account for this large variety of shapes. As pointed out by [START_REF] Ghaemi | Assessment of parameters for distinguishing droplet shape in a spray field using image-based techniques[END_REF], the nomenclature diverges among authors. As a result, the definitions of the morphological or shape criteria utilized must be clarified. These definitions are first detailed for 2D analysis, and then for 3D analysis.

The 2D case refers to the experimental works, mainly using image for shape analysis. In such cases, a kind of projection is performed between the real 3D droplet and the 2D image object. The 3D case refers to CFD data that can provide such information not reachable from the experiment. Even if the dimension of the shape parameter space can be considered as infinite, practical morphology analyzes concentrate on a few parameters.

The four parameters under consideration in this study are resumed in table 1 with their mathematical definition in 2D and 3D, and their ranges of variation. The shape parameter values are illustrated in figure 3 for two objects.

2D parameters

Shape parameters are defined from the value of primary parameters measured on the contour of a droplet D such as the contour length (perimeter p D ), the contour area (A D and maximum and minimum lengths a 2D and b 2D measured on two particular points on the contour (see below). Four shape parameters were selected in this study for their relevance to the objectives of this work. The primary parameters used for the 2D definition of these parameters are illustrated in figure 2 on an arbitrary shape droplet D. As the final shape of any droplet in a spray is a sphere, the 2D projected droplet is compared with a circle C of the same projected surface area A D , with its center coinciding with the droplet centroid. This circle is called the equivalent circle within this paper.

Thus the equivalent radius, R EQ,2D , is defined as the radius of the equivalent circle, i.e. R EQ,2D = A D π (see the horizontal dashed circle figure 2).

Aspect ratio α 2D = b 2D a 2D
The aspect ratio [START_REF] Eriksson | Comparison between and evaluation of some methods for the assessment of the sphericity of pellets[END_REF], also called centricity [START_REF] Ghaemi | Measurement of droplet centricity and velocity in the spray field of an effervescent atomizer[END_REF], is the ratio between two lengths, b 2D and a 2D , where b 2D < a 2D , i.e. 0 ≤ α 2D ≤ 1. Different computations of these lengths can be found in the literature. It is for example expressed as the lengths of the smallest rectangle that contains the object [Baert et al., 1992, Malot andBlaisot, 2000]. A similar implementation is the computation of a 2D as the length of the segment between the two most distant points on the object contour and b 2D the maximum distance between two points connected by a segment perpendicular to the first segment [START_REF] Mayor | Microstructural changes during drying of apple slices[END_REF], Kashdan et al., 2007]. In these two configurations, a disc or a square have an aspect ratio of 1. Other authors consider the extremes distances from the centroid of the object to the contour [Adrian, 1991, Adrian, 2005]. In such a case the aspect ratio of a square is

√ 2
2 . This later definition is the one used in the present paper. It can be noted that for the particular case of an ellipse, these distances are measured on two perpendicular lines and the different definitions of a 2D and b 2D lead to the same value of the aspect ratio, α 2D .

Uniformity η 2D = a 2D -b 2D R EQ,2D
The uniformity parameter [START_REF] Blaisot | Droplet size and morphology characterization for dense sprays by image processing: application to the diesel spray[END_REF] is the difference a 2D -b 2D normalized by R EQ,2D . It is a length-based parameter that can be useful to discriminate small surface ripples from elongated droplets. A small value indicates small scale ripples compared to the object size whereas a large value is associated to ligaments.

Irregularity ι 2D = p C p D
This parameter is defined as the ratio between the perimeter p C of the equivalent circle C and the contour perimeter p D of the object [START_REF] Podczeck | Evaluation of a standardised procedure to assess the shape of pellets using image analysis[END_REF]. As based on the total length of the object contour, this parameter is more likely to capture small shape deformations. Naturally, this parameter is very sensitive to small scale deformation of the contour. This can be a drawback for practical application where the interface cannot be accurately obtained (e.g. false small-scale wrinkling due to pixelized interface contour) [START_REF] Mayor | Microstructural changes during drying of apple slices[END_REF].

Symmetric difference shape parameter (SDS) ψ 2D = A D∪C -A D∩C A D
The SDS parameter introduced by [START_REF] Malot | Droplet size distribution and sphericity measurements of low-density sprays through image analysis[END_REF] is defined by the symmetric difference between D and C (dashed areas on the figure 2), normalized by the object area A D . This parameter is null for a perfect disc and is bound by 2, a value reached when the object and the equivalent circle have no common part.

The definitions and the range of values for all these parameters are given in table 1. An illustration of the particular values taken by the parameters for three directions of projections of two typical shapes; a prolate spheroid and a tube, are given figure 3. It is recalled to the reader that a meticulous description of 2D shape parameters and their application is provided in [START_REF] Ghaemi | Assessment of parameters for distinguishing droplet shape in a spray field using image-based techniques[END_REF]. The influence of the direction of projection is commented in the presentation of the 3D morphological parameters.

3D parameters

Whereas many definitions can be found in the literature about 2D shape parameters, to the authors knowledge, the existing 3D morphological parameters have not been studied in the past in the framework of atomization. This is mainly due to the fact that experimental techniques provide only 2D information. In the present study, the 2D parameters have been extended to the 3D domain reachable from CFD to obtain the 3D shape parameters. The main quantities used to build 3D parameters are: the minimal and maximal distance from the 3D centroid to the 2D interface, b 3D and a 3D , respectively. The total surface and volume of the droplet, S D and V D respectively. These latter quantities are used as equivalence in 3D to the perimeter and projected area in 2D for the shape parameter definitions.

Figure 2: Definition of distances and shape parameters construction for the projection of a peanut-like droplet, vertical dash lines, and its equivalent circle, horizontal dash lines. The union of these two shapes, A D∪C contains these two areas and the intersection of these two shapes, A D∩C , in filled white.

Aspect ratio and uniformity are constructed in their 3D extension with similar lengths (i.e. 1D information).

The 3D equivalent radius is defined from the equivalent sphere that has the same volume than the volume of the droplet V D , i.e. R EQ,3D = 3 3V D 4π . The total surface of the equivalent sphere S S = 4πR 2 EQ,3D , is used to compute the irregularity. Similarly to its 2D version, the accuracy of the irregularity depends on the resolution of the interface. Finally, the equivalent symmetric difference shape parameter (SDS parameter) [START_REF] Malot | Droplet size distribution and sphericity measurements of low-density sprays through image analysis[END_REF]] is defined from the common volume between the droplet and the equivalent sphere.

This parameter is very sensitive to numerical accuracy since every cell has to be checked and compared to the equivalent sphere. In this paper, an additional step is used for each cell containing the interface of the equivalent sphere and the droplet. It consists in a subdiscretization of the Cartesian cell with 15 points per direction, improving the accuracy of the computation of the volume of intersection in each cell. et al., 1997] Uniformity

Parameter 2D Expression 3D Expression Bounds References Aspect ratio α 2D = b 2D a 2D α 3D = b 3D a 3D [0, 1] [Eriksson
η 2D = a 2D -b 2D R EQ,2D η 3D = a 3D -b 3D R EQ,3D
[0, ∞[ [START_REF] Blaisot | Droplet size and morphology characterization for dense sprays by image processing: application to the diesel spray[END_REF] Irregularity et al., 1999] SDS parameter [START_REF] Malot | Droplet size distribution and sphericity measurements of low-density sprays through image analysis[END_REF] Table 1: 2D and 3D morphological parameters analyzed throughout the present communication.

ι 2D = p C p D ι 3D = S S S D [0, 1] [Podczeck
ψ 2D = A D∪C -A D∩C A D ψ 3D = V D∪S -V D∩S V D [0, 2[ [
To illustrate the values that the shape parameter can take, two particular objects are considered, i.e. a prolate spheroid representative of a droplet and a tube representative of a ligament. The values taken by the 2D and 3D parameters for these objects are shown in figure 3. It clearly shows the dependence of these parameters on the object shape. Indeed, the tube can be seen as an elongation of the prolate spheroid but its 3D parameter values are drastically different from those of the prolate spheroid. The figure shows also the sensitivity of the 2D parameter values on the object orientation relative to the projection plane. It can easily be guessed from these examples that a small variation of the main axis orientation can strongly modify the values for all the 2D shape parameters, for the 3D tube as well as for the prolate spheroid. Morphological values for the 3D and each projection are given, respectively the aspect ratio; α, the uniformity; η, the irregularity; ι, and the SDS parameter; ψ.

Numerical procedure

Direct numerical simulation is an ideal tool to study the shape evolution of droplets. The projection of 3D simulated droplets generates experimental shadowgraphy-like images that can be analyzed to understand the deviation between 3D and 2D parameters. Two numerical experiments are treated in this paper. The present section introduces the CFD solver, the projection approach and the two configurations.

Flow solver

The in-house solver Archer has been used to perform the direct numerical simulations analyzed in the present communication. This solver has been developed during the last 15 years and has been successfully applied to the study of atomization process [START_REF] Lebas | Numerical simulation of primary break-up and atomization: Dns and modelling study[END_REF], two-phase turbulence [START_REF] Duret | Dns analysis of turbulent mixing in two-phase flows[END_REF] and microfluidic [START_REF] Charpentier | Capillary phenomena in assemblies of parallel cylindrical fibers: From statics to dynamics[END_REF].

This solver uses a projection algorithm to solve Navier-Stokes equations over a staggered Cartesian mesh.

The sharp interface is treated using a Coupled level set Volume-of-Fluid approach [START_REF] Ménard | Coupling level set/vof/ghost fluid methods: Validation and application to 3d simulation of the primary break-up of a liquid jet[END_REF]. Convective terms are discretized using a fifth order WENO scheme to treat the interface discontinuities. The viscous terms are discretized following the method proposed by [START_REF] Sussman | An improved level set method for incompressible two-phase flows[END_REF]]. The Poisson equation is solved using a conjugate gradient solver [START_REF] Tanguy | Application of a level set method for simulation of droplet collisions[END_REF], and it includes the pressure drop due to surface tension, that is discretized using the ghost fluid method [START_REF] Fedkiw | A non-oscillatory eulerian approach to interfaces in multimaterial flows (the ghost fluid method)[END_REF]. A momentum conservative approach is also implemented to ensure the consistency between the mass and momentum equations in Navier-Stokes, [START_REF] Vaudor | A consistent mass and momentum flux computation method for two phase flows. application to atomization process[END_REF].

The choice of Archer is done since its capabilities to simulate atomization under harsh environment and ensuring an accurate description of the liquid-gas interface have been shown.

Detection, projection and morphological parameter computation

In Archer, a detection algorithm based on the developments of [START_REF] Herrmann | A parallel eulerian interface tracking/lagrangian point particle multi-scale coupling procedure[END_REF] is used to extract each individual droplet in 3D. This algorithm provides for each cell a tag corresponding to the droplet number. To ensure that the shape analysis of the droplet does not consider other structures, each droplet is transported to a numerical box, and, the liquid of any cell containing a different tag is removed. If necessary, the reinitialization algorithm included in Archer is applied to ensure that the level set is accurately representing the interface. Then, the 3D morphological parameters are computed in this domain, following the methodology given by [START_REF] Chéron | From droplets to particles: Transformation criteria[END_REF], Chan et al., 2021].

Once each droplet is isolated, the projection is done to reproduce experimental results from shadowgraphy analysis. The strategy is to consider each cell as a single pixel, and, to use the information from the projection of the 3D field. In this case, the level set is used. A 16 bits gray value for each pixel is given, based on the 256 2 possible gray values. The pixel identified at the interface has the threshold value of 65536/2. It can be identified with the iso-value 0 of the level set field. These experimental-like images are sent to the same algorithm used to analyze experimental images [START_REF] Fdida | Drop-size measurement techniques applied to gasoline sprays[END_REF]. Finally, 2D morphological parameters are calculated for each direction of 2D projection (i.e. 3 per droplet).

Droplet immersed in a Homogeneous Isotropic Turbulent-like flow

The first numerical experiment consists in the study of a single droplet immersed in a Homogeneous Isotropiclike Turbulence, (HIT-like). Interactions between turbulent flows and droplets are studied using such approach by different authors [START_REF] Trontin | Direct numerical simulation of a freely decaying turbulent interfacial flow[END_REF], Perlekar et al., 2012, Duret et al., 2012, Dodd and Ferrante, 2016].

The methodology is to use a tri-periodic domain to develop a turbulent flow with two immiscible phases. The main difference in the present configuration is that only one droplet is considered. Thus, its evolution can be studied without dealing with other phenomena such as breakup or coalescence. In addition, the high resolution of the droplet can be guaranteed, and, thousands of droplets can be studied with low numerical cost. The main drawback of this configuration is that the turbulence is not fully developed. Since the authors are not analyzing here the turbulent properties, but only the shape evolution of a droplet interacting with turbulent-like flow, this drawback is reasonable.

The initial turbulent carrier flow is generated through single phase simulation with a linear forcing scheme [Rosales andMeneveau, 2005, Duret et al., 2012]. This initial forcing is held during 5 eddy turnover, in order to obtain a statistically steady state. Then, a fully-resolved solid particle is added on the domain. This particle ensures a no slip/no penetration condition on a spherical region. After 1 eddy turnover time, the solid constraint is relaxed and the particle is transformed into a droplet. This initialization process avoids any nonphysical shear flow at the interface at start. The droplets are thus initially spherical. The statistics are recorded after an eddy turnover time, T e , and snapshots are obtained at every T e /10. In this series of simulation, half of the droplets are studied maintaining the turbulence forcing, and, the other half are studied in a decaying turbulence. This method has been used in two previous studies [START_REF] Chéron | From droplets to particles: Transformation criteria[END_REF], Chen et al., 2019]. In this configuration, approximately 4000 droplets are studied.

In order to generate this wide database, the mesh resolution is kept relatively low, 64 3 . To have an accurate As it will be seen later, the droplets in this database are mostly spherical. Thus, these droplets can characterize the deformation due to turbulence in the secondary atomization area.

Droplets generated from airblast atomization

In order to obtain droplets close to those observed near the injector, a second numerical experiment is done.

The experiment chosen is a planar airblast atomizer that produces several droplets shapes through different breakup mechanisms. By nature, this atomization process is anisotropic. For this reason, one can expect that, statistically, the projection of the droplets in each direction does not provide the same results. Here, three directions are considered for projection: the streamwise direction Z, and, the spanwise directions X -Y . The influence of the anisotropy on the shape parameters is studied later in §4.2.

The airblast considered here is based on the already studied numerical configuration [START_REF] Cordesse | Validation strategy of reduced-order two-fluid flow models based on a hierarchy of direct numerical simulations[END_REF].

The injected liquid has a density of ρ l = 12.25 [kg.m -3 ], and, a dynamic viscosity of µ l = 1.11 × 10 -4 [P a.s].

The density ratio and the viscous ratio, with respect to the carrier phase, are ρ * = 10 µ * = 6.10. The ratio of injection height against the surface length is 1.25%, momentum ratio is

ρ l V inj 2 ρgV carrier 2 = 6.65 [-], Weber number is W e inj = ρ l l * V inj 2 σ = 20904 [-].
The simulation has been performed on 1024 processors. The mesh size is set to ∆x = 103, 13 [µm]. The droplet collection is done after 0.2 seconds of atomization once the statistically steady state is reached. The simulation covers 0.13 seconds with a snapshot taken every 2.2 × 10 -3 seconds.

The droplets are individually studied using the algorithm presented in §2.2. More details can be found in [START_REF] Belhadef | Pressure-swirl atomization: Modeling and experimental approaches[END_REF], Cordesse et al., 2020].

A snapshot of the airblast is given in figure 4 (left). This figure is obtained applying the projection of the liquid phase over the X -Z plane, perpendicular to the stream-wise direction. As observed, the atomization process generates a wide poly disperse spray. Then, a filter depending on the numerical resolution of the droplets is used to avoid analyzing inaccurate droplets since the computation of morphological parameters is subject to large errors for underresolved droplets [START_REF] Chéron | From droplets to particles: Transformation criteria[END_REF]. A droplet with an equivalent radius smaller than 2 mesh size, i.e. R EQ,3D < 2∆x, is then not included in the database. In addition, a filter based on the aspect ratio α 3D is applied. Only the droplets with an aspect ratio larger than α 3D > 0.2 are studied. That allows to remove the large ligaments and also the liquid core. Both filters are illustrated in the morphologicalsize map shown in figure 4 (right). Through these filters, the analysis covers 40% of the total structures (green kept, red filtered, figure 4 (right). That corresponds to 1200 droplets. In this figure, different snapshots of the liquid core can be observed, they are located at R EQ,3D ∼ 45.

In this database the limit α 3D = 1 is not reached, implying that the population of droplets is far from spherical droplets. This database is then a perfect complement to isotropic database presented in §2.3. ∆x .

On the spheroid assumption

As mentioned, a droplet shape can take different aspect that should need as many shape parameters to be thoroughly described. From a practical point of view, a reduced set of shape families are considered. The first one, in terms of relevance to spray analysis, is the spheroid shape, also called ellipsoid of revolution. A droplet close to the spherical shape can be considered as a spheroid. Spheroids can be classified in two distinct families;

named prolate (cigar-like) and oblate (pancake-like). Each family represents a specific atomization process, i.e. the prolate case is observed when a sphere is flattened under the action of aerodynamic forces whereas the oblate case is observed when a droplet undergoes a large drift velocity, relatively to the surrounding gas [Green, 1975, Clift et al., 1978]. Spheroid shapes can also be associated to the surrounding gas turbulence properties, as mentioned by [START_REF] Mukherjee | Droplet-turbulence interactions and quasi-equilibrium dynamics in turbulent emulsions[END_REF] : "Axial strain tends to stretch droplets into prolate ellipsoids (cigar-like objects), while bi-axial strain would shape them into oblate ellipsoids (flat pancake-like objects).". In the absence of other effects than the surface tension and the kinetic forces, these two shapes are observed since following the Lamb's theory, a spheroidal-like droplet oscillates between oblate and prolate shape [Lamb, 1881]. In the latter theory, the shapes are not described in terms of spheroid but in terms of spherical harmonics. Based on these observations, the effect of the projection on the shape parameters of the droplets under the spheroidal shape assumptions is explored in the present section.

Morphological parameters under spheroid assumption

To assess the assumption of considering droplets as spheroids, called hereafter the spheroid assumption, relations between shape parameters are derived for the oblate and prolate spheroids. For 2D parameters, the following analytical relations (Eqs 1-3) were derived for an ellipse by [START_REF] Blaisot | Droplet size and morphology characterization for dense sprays by image processing: application to the diesel spray[END_REF] for uniformity, irregularity and SDS parameters as function of the aspect ratio where the approximation of the ellipse circumference given by [Peano, 1887] is used for the irregularity parameter:

η E 2D (α 2D ) = 1 -α 2D √ α 2D , (1)
ι E 2D (α 2D ) = 3(1 + α 2D ) 4 √ α 2D - 1 2 -1 , (2)
ψ E 2D (α 2D ) = 4 π arcsin 1 1 + α 2D -arcsin α 2D 1 + α 2D . (3)
A criterion based on these relations, in the form of a given tolerance, defines a domain in the shape parameter space that they have called the "elliptic shape family". This is used to estimate the deviation of a droplet shape from an ellipse, since a droplet whose shape parameters are outside this domain is regarded as not ellipsoidal.

Since the projection of a 3D spheroid is a 2D ellipse (see details later in §3.3), the aforementioned relations are later extended to a 3D definition. These relations can be used to estimate the deviation of the droplet from a spheroid using a combination of 3D shape parameters. In this context, it is necessary to distinguish the prolate and the oblate families, since it modifies the analytical relationship between the parameters. Indeed, the prolate shape is defined by a 3D = λ 1 > λ 2 = b 3D and the oblate shape by a 3D = λ 2 > λ 1 = b 3D where λ 1 is the semi-axis along the symmetry axis and λ 2 is the perpendicular semi-axis. Thus, the volume of a spheroid, V = 4 3 πλ 2 2 λ 1 , or the equivalent radius, R EQ,3D = (λ 2 2 λ 1 ) 1/3 , have a different expression in terms of a 3D and b 3D , depending on the spheroid family. With this in consideration, the analytic relation for uniformity, irregularity and SDS 3D parameters are derived as functions of the aspect ratio, for prolate and oblate spheroid families:

η P 3D (α 3D ) = 1 -α 3D α 2/3 3D , (4)
η O 3D (α 3D ) = 1 -α 3D α 1/3 3D , (5)
ι P 3D (α 3D ) = 2α -2/3 3D 1 + arcsin √ 1-α 2 3D α 3D √ 1-α 2 3D , (6)
ι O 3D (α 3D ) = 2α 2/3 3D 1 + α 2 3D √ 1-α 2 3D arctanh 1 -α 2 3D , (7)
ψ 3D (α 3D ) = 2 1 -α 2/3 3D 3 1 -α 2 3D . (8)
It can be noticed that SDS parameter, based on volume information, does not depend on the spheroid family. Following the work of [START_REF] Blaisot | Droplet size and morphology characterization for dense sprays by image processing: application to the diesel spray[END_REF], the relations given by Eqs. 5-8 are used to assess the ellipse/spheroid assumption.

Analysis of the numerical droplet database

The 3D shape parameter distribution are compared to the oblate and prolate family curves (Eqs. 5-8) in figure 5, for the droplets generated from isotropic and anisotropic numerical experiments. It is recalled that only the droplets verifying α 3D > 0.2 are considered.

Values for uniformity parameter η 3D (figure 5a) mostly belong to the range [η O 3D , η P 3D ] (upper script are used to represent the oblate spheroid and prolate spheroid shapes, respectively. In this article, the letter Q is used to represent both prolate and oblate spheroids when the distinction is not done). As η 3D mainly depends on the primary lengths a 3D and b 3D , i.e. on two particular points in the contour, this shape parameter is not strongly related to the true shape of a droplet, i.e. many droplets of different shapes and similar size can share the same values of a 3D and b 3D and the same uniformity parameter value. Thus, one can always find a spheroid S P for given a 3D and b 3D values. Out of range values, i.e η 3D < η O 3D or η 3D > η P 3D , are due to R EQ,3D that can be significantly different from the one of the spheroid S P for any droplet. The large range of values spanned by uniformity between oblate and prolate families makes it a good candidate for classifying spheroids. However, for the reason mentioned just above, the value pair (α 3D , η 3D ) is not sufficient to identify a droplet as a spheroid as these two parameters depends manly on the sole primary parameters a 3D and b 3D .

Irregularity parameter ι 3D depends on area primary parameters, and is thus more strongly related to the shape of a droplet than η 3D . The values are broadly scattered around the prolate curve ι P 3D (α 3D ) (see figure 5b), suggesting that droplets in the numerical database are mainly of the prolate family. Nevertheless, to establish a rigorous shape classification only one shape parameter is not sufficient but a combination of parameters must be used in order to define an effective criterion.

The SDS parameter ψ 3D is defined from volume-based primary parameters and as such it is strongly related to the true shape of a droplet, as is ι 3D . It can be observed on figure 5c that the values covered by ψ 3D span largely over the spheroid curve, particularly for low aspect ratios, for the anisotropic set. Thus it can be guessed that this numerical set is not composed only of spheroid droplets. In contrast, the isotropic set clearly shows spheroid-like characteristics for SDS parameter. Moreover, irregularity and uniformity parameter values present also spheroid-like characteristics for this set.

It can be concluded from this first simple analysis of the shape parameter values that the anisotropic set presents a wider variation of shapes than the isotropic set. The isotropic set seems mainly composed of spheroid-like droplets whereas the anisotropic set includes also non-spheroid droplet. As an example of the different cases, the droplets of appendix B are highligted with four different symbols in figure 5. Droplets A and B are the less distorted droplets, i.e. for all morphological parameters the values are closed to the analytical spheroid relations. Droplets C follows the prolate spheroid relation for the uniformity, η 3D , and irregularity ι 3D . However, this droplet is far from the unic spheroid relation in the case of the SDS parameter ψ 3D . The most distorted one, droplet D, also follows the prolate spheroid relation for the first two morphological parameters. Also, the analysis of the SDS parameter against the aspect ratio, figure 5c, shows a good agreement with the spheroid relation. This is not expected when considering the rendering proposed in appendix B, showing the necessity to consider the shape analysis with several morphological parameters.

The 2D parameters obtained from the projection along the three main axes of the computation domain are now considered. These parameters are compared to the analytic relations for the elliptic shape given by Eqs. 1-3. The result of these comparisons is given in figure 6 for both isotropic and anisotropic databases. Isotropic projections are represented in black and the anisotropic projections are colored with respect to the direction of projection, X (red), Y (blue) and Z (green).

The same conclusion can be drawn from the observation of the three figures, i.e. the droplets of the isotropic set remain closer to the spheroid family than those of the anisotropic set. This is verified for projections of distorted droplet from the isotropic set, see the projections of droplet D from appendix B on figure 6. It is also noticeable the different behavior of the projection along the Z axis compared to the two other direction for the anisotropic set.

From the 3D and 2D analyses, it can be conclude that the population of droplets in the two numeric sets is not composed of spheroid-like droplets only. In the next subsection the effect of the projection on the calculation of 2D morphological parameters is explored for spheroids. The uniformity parameter is selected since it does not permit to classify the morphological shape of an object alone. This parameter is therefore investigated in detail in this study because the conclusions drawn for this parameter are likely to be extrapolated to other morphological parameters more representative of the droplet shape, for instance the irregularity or the SDS parameters. Then, the extension of the uniformity analysis to other parameters is considered as a perspective.

Projection of spheroids

The main goal of this section is to estimate the systematic bias in shape analysis performed on 2D images, introduced by the projection effect. This is done considering 3D spheroids of known properties. A spheroid is defined by its symmetry axis, e z , and two lengths; λ 1 the semi-axis length along e z and λ 2 the semi-axis length perpendicular to e z . As mentioned in §3.1, λ 1 = a 3D and λ 2 = b 3D for a prolate spheroid and λ 1 = b 3D and λ 2 = a 3D for an oblate spheroid, a 3D and b 3D being the maximum and minimum distances from the centroid to the object surface, respectively. The projection of a spheroid is an ellipse, and, due to the axisymmetry, the ellipse properties depend only on the angle θ between e z and the projection direction (normal to the projection plane) (see figure 7).

The ellipse is defined by two lengths, λ 1,2D and λ 2,2D , that corresponds to the two axis. These lengths can be computed from the projection of the ellipse:

λ 1,2D = λ 1 λ 2 λ 2 2 sin 2 (θ) + λ 2 1 cos 2 (θ) , (9)
λ 2,2D = λ 2 . ( 10
)
These two lengths provide the maximum and minimum distances from the centroid to the contour of the 2D projected ellipse,

a 2D = λ 1,2D , (11)
b 2D = λ 2,2D = λ 2 = b 3D , (12)
for the prolate spheroids, and

a 2D = λ 2,2D = λ 2 = a 3D , ( 13
) b 2D = λ 1,2D , (14)
for the oblate spheroids.

Expressions for a 2D (θ) and b 2D (θ) depends on the spheroid family (prolate or oblate). These equations obviously shows that a 2D = a 3D and b 2D = b 3D for any projection angle but θ = π/2. For θ = 0, corresponding to a projection axis parallel to e z , the projection is a disk leading to

a 2D = b 2D = λ 2 .
The variation of a 2D (θ) and b 2D (θ) is illustrated in figure 8 for two spheroids, one prolate and one oblate, of same major semi-axis a 3D = 1 and of same uniformity η 3D = 0.25. The 2D equivalent radius R EQ,2D = √ a 2D b 2D is also plotted and compared to R EQ,3D = (λ 2 2 λ 1 ) 1/3 (see §3.1) in this figure. It is recalled that the projection area of a spheroid is an ellipse. Two additional equivalent radii are proposed here: R EQ,2D,O = (b 2D a 2 2D ) 1/3 for a presumed oblate droplet and R EQ,2D,P = (a 2D b 2 2D ) 1/3 for a presumed prolate droplet. Whatever the spheroid family, a 2D and b 2D vary between a 3D and b 3D . One can also remark that for all angles:

b 2D = λ 2 = b 3D for prolate spheroids and a 2D = λ 2 = a 3D for oblate spheroids. It can be observed in figure 8 a bell curve for the 2D equivalent radii for a prolate, and, the inverse bell curve for an oblate object. For both spheroid cases, the equivalent radius R EQ,2D , evolves below and above R EQ,3D . For the prolate spheroid, the additional radius for a presumed prolate droplet R P EQ,2D,P ≤ R EQ,3D , the equality being reached for θ = π/2.

For the oblate spheroid the radius for a presumed oblate droplet R O EQ,2D,O ≥ R EQ,3D and here again equality is reached for θ = π/2. For the prolate spheroid R O EQ,2D,O is always greater than R EQ,2D . It can be noticed that whatever the spheroid family R P EQ,2D,P ≤ R EQ,2D ≤ R O EQ,2D,O . This property is used below.

The error on the estimation of the primary lengths, a 2D , b 2D and R EQ,2D with respect to the 3D values, described in this section, has an influence on the shape parameters obtained from the projection step (2D from 3D).

Figure 8: Evolution of a 2D , b 2D ,R EQ,2D and R P EQ,2D or R O EQ,2D against π for a spheroid with an initial uniformity parameter of η 3D = 0.25, left: prolate spheroid, right: oblate spheroid. Limits of the 3D spheroid are given: a 3D , b 3D , and R EQ,3D .

Deviation on uniformity computation due to projection under the spheroid assumption

In this section, the potential deviation of 2D uniformity parameter η 2D from the real 3D value η 3D is evaluated.

It is recalled that uniformity is defined by λ 1 , λ 2 and the equivalent radius. Oblate and prolate spheroid families are considered separately and the evolution of η 2D is analyzed as a function of the direction of projection, θ.

To illustrate the main trends, the curves for η 2D (θ) is shown in figure 9 for the two spheroids considered in §3.3. Two extra uniformity parameters are also introduced: η Q 2D,P is the value obtained for a presumed prolate droplet and η Q 2D,O is the one for a presumed oblate droplet for a prolate, or oblate spheroid, respectively O and P upper script (it is recalled that to simplify the notation, the upper script letter Q represents both oblate and prolate families). In the definition of these extra uniformity, R EQ,2D is replaced by the corresponding R Q EQ,2D,Q .

The main points to retain are: (i) η Q 2D and η Q 2D,Q curves have a bell shape whatever the spheroid family; (ii) minimum value for η 2D equal zero whatever η 3D for both spheroid families (corresponding to θ = 0 and θ = π, i.e. projection equal to a disk); (iii) maximum value for η Q 2D,Q equal η 3D (for θ = π/2) when the right spheroid family is presumed (in this case, 2D and 3D values for a, b and R Q EQ,2D,Q are equal, see figure 8).

From the previous remarks it can be concluded that the most probable situation is that the 2D uniformity obtained from a projection underestimates the 3D uniformity. To analyze this deviation from a statistical point of view, the 2D uniformity is averaged over all observation orientations, i.e. over solid angle 4π sr. Each observation direction is assigned a probability sin θ 4π dθdϕ where ϕ is the azimuthal angle. Averaged expression are thus given by:

η2D = π 0 a 2D (θ) -b 2D (θ) R EQ,2D (θ) sin(θ) 2 dθ, (15)
and

η2D,Q = π 0 a 2D (θ) -b 2D (θ) R EQ,2D,Q (θ) sin(θ) 2 dθ. (16)
The analytical derivation of these integrals are developed in Appendix A. To illustrate the relation between 2D

and 3D values of the uniformity parameter, a plot of η 3D vs η 2D is shown in figure 10 for a random set of oblate spheroids and a random set of prolate spheroids (each set consists of 4000 elements). For each set, the points

(η P 2D,O , η O 2D,O , η 3D
) and (η P 2D,O , η P 2D,P , η 3D ) are also drawn. The curves for the mean values ηP 2D and ηO 2D are also plotted in figure 10 for both prolate and oblate spheroids as function of η 3D . The following information can be deduced:

• averaged 2D uniformity values (η Q 2D ) generally underestimates the 3D value, except for ηO 2D > η 3D with η 3D > 1.2 for the oblate family. That is even the case when considering the alternative equivalent radius η O 2D,O and η P 2D,P .

• The minimum value for η Q 2D is always 0 (a circular shape can be observed even for very deformed spheroids, as in the case of the projection of deformed prolates, see figure 3). Neverheless, for larger values of η 3D the probability to observe a circle decreases.

• The maximum value of the shape-assumed 2D uniformity η Q 2D,max,Q is equal to η 3D for the well presumed shape.

• The maximum 2D uniformity η Q 2D,max underestimates (overestimates) η 3D for the prolate (oblate) spheroids.

The estimation of the 2D uniformity parameter with the assumption of the wrong family is also added to figure 10. These values are obtained applying the oblate assumption (i.e. using R P EQ,O in the definition of η P 2D ) for the prolate spheroids, and vice versa. The miss-assumption of the spheroid family drastically modifies the deviation between 2D and 3D uniformity parameters, compared to the right assumption. For the prolate spheroids, η P 2D,O is shifted toward lower values of 2D uniformity, which increases the deviation with 3D values. For the oblate case, the results are also worsened since the population is shifted toward higher values of uniformity, resulting in a stronger overestimation of the 3D value. This figure also shows that regardless of the initial spheroid family, the oblate assumption never overestimates the 3D value for the uniformity parameter.

This will be helpful for the analysis of the isotropic and anisotropic databases (see §4.1 and 4.2 respectively)

where the representation of η 3D vs η 2D is also used to analyze both droplet databases. 4 Shape parameter deviation analysis for numerical databases

The deviation in shape parameter estimation induced by 2D projection is analyzed here for two numerical databases, i.e. the isotropic database presented in §4.1 and the anisotropic one presented in §4.2. It is reminded that the isotropic database corresponds to droplets deformed by isotropic turbulence (e.g. secondary atomization area), and the anisotropic database to droplets with a large drift velocity (e.g. primary atomization area). For each database, the length-type primary parameters are first considered before the uniformity shape parameter.

Isotropic database

The isotropic database has been introduced in §2.3. It results from a simulation of an isolated droplet of constant size, i.e. monodisperse droplet population, in a developed Homogenenous Isotropic Turbulence-like flow. For this reason, the projections are independent of the direction of projection.

Length-type primary parameters

The uniformity shape parameter is based on three length-type primary parameters: the equivalent radius R EQ and the maximal and minimal distances to the droplet centroid (a and b). The deviation resulting from the projection on a 2D plane of the 3D droplet contour is first estimated for these three quantities.

First, 2D and 3D equivalent radii are compared. In the isotropic database, all the droplets have the same initial volume (and thus, the same R EQ,3D ). Equivalent radius without assumption R EQ,2D , with prolate spheroid assumption R EQ,2D,P , and with oblate spheroid assumption R EQ,2D,O are considered. These 2D equivalent radii are scaled by R EQ,3D for simplification purpose. The probability distribution functions (PDFs) for the three normalized 2D equivalent radius are plotted in figure 11. The radii obtained from the projection in the three orthogonal directions of the Cartesian mesh of the numerical domain are considered for each droplet. The distribution is constructed with 35 bins. The distributions are Gaussian-like with the following average values:

R EQ,2D R EQ,3D = 1.011, R EQ,2D,P R EQ,3D = 0.953, R EQ,2D,O R EQ,3D = 1.043, and standard deviations: σ R EQ,2D R EQ,3D = 0.066, σ R EQ,2D,P R EQ,3D = 0.071, σ R EQ,2D,O R EQ,3D = 0.080.

showing that the best estimate of R EQ,3D , that would give a distribution centered on R EQ,2D /R EQ,3D = 1 with low dispersion, is given by R EQ,2D on average. The prolate assumption underestimates the radius while the oblate assumption overestimates it.

Figure 11: Probality density functions for R EQ,2D in black, R EQ,2D,P in red (prolate spheroid assumption) and R EQ,2D,O in green (oblate spheroid assumption).

It can be noticed that as the three directions of projection are considered. For a given droplet, any underestimation or overestimation of R EQ,3D in one direction is offset by the projection in the other two directions.

This basic observation has an important impact on the calculation of the maximal and minimal distances that is now analyzed.

The comparison between 2D and 3D values for a and b, is presented in figure 12. These length-type parameters are also scaled by R EQ,3D to simplify the presentation in a way to obtain normalized values lower than 1, i.e. b/R EQ,3D and R EQ,3D /a. A colormap is used to classify the points according to the value of R EQ,2D /R EQ,3D . As explained before, for droplets with R EQ,2D /R EQ,3D 1, an accurate estimation of a and b lengths is expected.

The lengths a 2D and b 2D should be in the range [b 3D , a 3D ], implying points below the first bisector line in figure 12. This is almost verified, the points above the first bisector are due to computational inaccuracies.

The points near to the first bisector mainly correspond to R EQ,2D /R EQ,3D 1. That concerns spherical droplets (e.g. droplet (A) in appendix B), or elongated droplets projected in a direction that minimizes the error.

A correlation between the overestimation of R EQ,3D and the overestimation of b 3D is clearly seen from the colored points in figure 12 left. In other words, if the minimal distance is not correctly captured by the projection, the equivalent radius is overestimated. This situation can be illustrated by droplet C) of appendix B. A correlation between the underestimation of R EQ,3D and the underestimation of a 3D is also seen from the colored points in figure 12 right, i.e. if a 3D is underestimated, then, the equivalent radius is also underestimated (dark blue markers in the figure).

Figure 12: Distribution of b 3D against b 2D scaled by the equivalent 3D radius R EQ,3D (left). Distribution of a 3D against a 2D scaled by the equivalent 3D radius R EQ,3D (right). The color map is associated to R EQ,2D /R EQ,3D values, see figure 11.

Uniformity shape parameter projection analysis

The plot of η 3D vs η 2D for the isotropic set is shown in figure 13. The analytic maximum and averaged curves for η 2D with and without spheroid family assumption, are included for the oblate and prolate families, respectively in green and red, and the markers are colord by R EQ,2D /R EQ,3D values but no spheroid shape assumption is made to estimate η 2D . It is recalled that a 3D droplet creates three 2D information, based on the three Cartesian direction of projection.

This figure is to compare to the figure 10 (in particular black circles that corresponds to projection without assumption) keeping in mind that the range of values for η is reduced here. As markers only fill the space over the first bisector, it is guessed that the spray is mainly populated by prolate-like spheroids. Also, this does not consider the entire distribution since few points are observed on the right of η P 2D,max curve, which can only be crossed by oblate-like droplets. Generally, the points are centered on the prolate spheroid analytical mean value; ηP 2D , and, limited by the first bisector. One can assimilate the majority of these droplets to prolate-like spheroids. Finally, in contrast to the theoretical framework, here no perfect circular projections are observed for large deformed 3D shapes. For instance, the most circular projection for large deformed droplets is droplet D in appendix 10 that is strongly deformed (η 3D = 0.9) and have a projection in Y direction close to a circle η 2D,Y = 0.13.

The systematic underestimation of the uniformity due to projection can also be understood in terms of primary lengths projections. A correlation was shown in figure 12 between an overestimation of the minor axis b and an overestimation of the equivalent radius R EQ . The combination of these two overestimations leads to underestimate η as can be observed for the orange colored markers in figure 13. A correlation between an underestimation of the major axis a and an underestimation of R EQ was also found. These two underestimations are likely to be compensated (see the definition of η parameter in table 1). The dark blue colored markers in figure 13 are indeed a little closer to the first bisector than orange ones.

The results presented here show the interest of the analogy with the random projection of spheroids. However, only isotropic droplets with a low deformation have been treated here. The next section deals with a more realistic case with anisotropic droplet deformation.

Anisotropic database

The anisotropic database has been introduced in §2.4 and is constituted of polydisperse droplets. As an indicator, the relative span factor is; RSF = D 0,9 -D 0,1 D 0,5 = 0.690. It results from a simulation of airblast atomization with the stream-wise direction along Z axis. For this reason, the projections for each direction are examined individually.

Length-type primary parameters

The PDF for the three 2D equivalent radii (i.e. R EQ,2D , R EQ,2D,P and R EQ,2D,O ) normalized by R EQ,3D are plotted in figure14, for X, Y and Z directions, from left to right, respectively. In contrast to the isotropic database, the volume of the droplets varies.

Here also Gaussian-like distributions are observed. The average and standard deviation of these distributions are given in table 2. As observed for the isotropic case, the best estimate of R EQ,3D is given by R EQ,2D ;

the prolate assumption globally underestimates the equivalent radius whereas the oblate estimate globally overestimates it.

It can be clearly seen that the X and Y projections produce similar statistics, meanwhile a different behavior appears on the projection along the streamwise Z direction. Moreover, results for X and Y directions are similar to the ones for the isotropic case. For the Z direction, mean values are ordered the same way than for the other two directions but the best estimate is given here by the oblate assumption but with the wider standard deviation.

To explain this, let's consider that the droplets are spheroids. Now, in opposition to the theoretical developments on §3.3-3.4, these spheroids have a preferential alignment, the Z direction. Thus, the R EQ,2D

underestimates or overestimates the R EQ,3D (depending on the prolate or oblate spheroid probability due to the oscillatory regime of these droplets), while the X and Y direction produces more accurate estimation since they capture the lengths along the axisymmetry axis. This ideal case explains the larger dispersion of

R EQ,2D R EQ,3D
on Z projection. To go further on this assumption based on an ideal droplet shape, since the main value of

R EQ,2D
R EQ,3D = 0.938 < 1, more prolate spheroids than oblates are present in this database. Indeed, a value higher than one would have yield to a majority of oblate spheroids in the population, as detailed in figure 8. This idealized instance may appear to be a long way from the complexity of the physics behind an airblast atomizer.

However, since the droplets are the result of ligament breaking, and the ligaments in this type of atomizer are aligned along the Z axis, this interpretation sounds appropriate. The initial droplets are thus prolate-like spheroids aligned with the Z axis, which follows the classical oblate/prolate oscillations predicted by linear theory and observed in the literature [González andGarcía, 2009, Moallemi et al., 2016].

[-] X Y Z R EQ,2D R EQ,3D

1.021 1.010 0.938 R EQ,2D,P R EQ,3D

0.898 0.892 0.889

R EQ,2D,O R EQ,3D 1.142 1.128 1.025 σ (-) X Y Z R EQ,2D R EQ,3D
0.071 0.0756 0.1244 R EQ,2D,P R EQ,3D

0.0847 0.0811 0.1105

R EQ,2D,O R EQ,3D
0.1166 0.122 0.1566

Table 2: Mean values and standard deviation of

R EQ,2D
R EQ,3D for different computations of R EQ,2D , left and right, respectively.

(a) (b) (c)

Figure 14: From left to right: histogram of the deviation of the projected equivalent radius: R EQ,2D in black, and their correction based on the prolate spheroid assumption: R EQ,2D,P , in red, and for the oblate spheroid assumption: R EQ,2D,O , in green, for direction of projections; X (figure 14a), Y (figure 14b), and Z (figure 14c).

The minimal and maximal lengths, i.e. a and b, are now considered. The deviation of the 2D to the 3D versions of these primary parameters, scaled by R EQ,3D , is shown in figure 15 for X, Y and Z projection directions, respectively from left to right, with a parameter in bottom row and b parameter in top row. A color map based on R EQ,2D /R EQ,3D values is used to color the marker points, as in the isotropic section and the first bisector is also drawn to clearly identify a b cases.

A correlation between the overestimation of R EQ,3D and the overestimation of b 3D as well as between the underestimation of R EQ,3D and the underestimation of a 3D is observed, as for the isotropic case (see the color distribution in figure 15 which is particularly clear for Z direction (right column)). It is recalled that the PDF for R EQ,2D /R EQ,3D along Z direction shows a larger dispersion (see figure 14 andtable 2). Also, the results observed for the X and Y directions are in the range of the isotropic set. Few points recover a 2D = a 3D , and are located on the first bisector. The length b 2D retrieves the 3D value for several points in the set but the probability that the projection allows to recover the 3D value is not correlated to the ratio R EQ,2D /R EQ,3D , in opposition to the isotropic set. Another divergence with this database is the observation of minimal lengths overcoming the 3D value; b 2D > b 3D . These droplets are consistent since b 2D < R EQ,2D < a 2D . Also, they are far from a spheroidal shape, and the centroid of their projection is shifted which explains this overestimation of the minimal length with respect to the 3D droplet.

Uniformity shape parameter projection analysis

The deviation analysis of the uniformity parameter, η 3D vs η 2D , is shown in figure 16 with X, Y and Z projection directions from left to right, respectively. The same colormap as the one in figure 15 is used to classify the points according to the value of R EQ,2D /R EQ,3D . The range of values for η is almost the double than for the isotropic database. For projections along X and Y the points are mainly located in the 'prolate' assumption zone, i.e. η 2D is evenly distributed in the range [η P 2D /2, η P 2D,max ]. This shows a systematic underestimation of the uniformity value, increasing for higher η 3D . For Z direction, the points are shifted towards lower values of η 2D with values approaching the limit η 2D = 0. From the observations of section §4.2.1 and the uniformity analysis, it is confirmed that the majority of droplets are prolate-like aligned along the Z direction. In other words, an observer looking at the spray in the streamwise direction will see many circular shapes. As discussed before, for these very cases the equivalent radius (here dark blue markers) is underestimated.

It can be shown that the analytic average value ηP 2D for a random set of prolate spheroids (see §3.3) is overestimated for X and Y directions and underestimated for Z directions. This confirms that the orientation of observation has a great influence on shape parameter estimation from 2D projection, which is almost always the case in experiments.

Figure 16: Distribution of the 3D uniformity, η 3D , against η 2D , for the three direction of projections, left to right: X -Y -Z. Lines: 2D averaged uniformity for each kind of spheroid family, ηA 2D , maximum values, ηQ 2D,max , and family correction, ηQ 2D,Q and ηQ 2D,max,Q (it is recalled that Q represents both prolate and oblate spheroid shapes). The color map scales equivalent 2D and 3D radius, see figure 14.

5 Application of the correction on 2D droplets from experimental database

Experimental setup

The experimental database is obtained from a set of more than one hundred images recorded on a water/air spray produced by a shear-coaxial injector [START_REF] Ficuciello | Investigation of airassisted sprays submitted to high frequency transverse acoustic fields: Droplet clustering[END_REF]. The injection condition is characterized by

W e g = ρgU 2 g D l σ
= 400, and Re l = ρ l U l D l µ l = 6600, where ρ g and U g are the density and the bulk velocity of the gas, ρ l , U l , and µ l are the density, the bulk velocity and the viscosity of the liquid, and σ the surface tension between liquid and gas.

An example of image of the experiment is shown in figure 17 where a large view of the flow is also displayed. Objects not totally included in the image (i.e. touching an image side) are rejected as well as very big objects (with a projected area greater than the one of a circle of about 2 mm in diameter). A filter based on the contrast of the image is applied (C > 0.1), removing objects with a low SNR (signal-to-noise ratio).

Another filter based on the estimation of the point spread function is also applied to select droplets in a given range of out-of-focus, whatever their size (see [START_REF] Blaisot | Droplet size and morphology characterization for dense sprays by image processing: application to the diesel spray[END_REF] or [START_REF] Fdida | Drop-size measurement techniques applied to gasoline sprays[END_REF]). The database contain around 30000 droplets corresponding to about 300 droplets per image. In figure 18 the size probability density functions based on the 2D equivalent radius are plotted for the raw data and after filter application.

Undoubtedly, this filter concerns mainly the small droplets. Filtering these droplets modifies the relative span factor from 3.755 to 2.458.

Elliptic assumption

Two-dimensional uniformity (η 2D ), irregularity(ι 2D ), SDS (ψ 2D ) and aspect ratio (α 2D ) parameters are extracted from the experimental database. The distribution of the three first parameters (η 2D , ι 2D , ψ 2D ) as functions of the aspect ratio (α 2D ) are compared to the elliptic shape relations (Eqs. 1-3) in figure 19. Markers are colored using a three classes color map based on the equivalent radius R EQ,2D .

The smallest droplets (blue markers) are almost close to the elliptic shape curves (red lines). This is not surprising since these droplets are the most likely to keep a near spherical shape. It can be observed that a large number of big droplets (orange and red markers) are far under the elliptic curve for ι parameter (see figure 19b).

Indeed, due to their size, these droplets are prone to large deformation. Only a few big droplets are away from the elliptic curve for uniformity parameter (see the region near (α 2D , η 2D ) = (0.2, 1.3) in figure 19a). Thus, this parameter is neither able to identify spheroid-like droplet in its 2D version nor in its 3D one. Also, a certain amount of large droplets are away from the elliptic curve for ψ parameter (figure19c) but a more restraining filter that combines these three morphological parameters must be considered to recover object with ellipticlike projections only [Blaisot andYon, 2005, Fdida andBlaisot, 2008]. These distributions are quite similar to the ones shown in figure 6 for the numerical databases. In particular, the X and Y projections for the anisotropic numerical database (Airblast simulation) seems close to the experiment database distributions.

The main differences come from the larger deformation observed in the experiment. This is because the experiment indicates a larger range of droplet sizes. Also, the smallest structures are not recovered in the numerical simulations due to low resolution imposed by the numerical restriction (see [Herrmann, 2010a] for a discussion of this point).

Analysis of the uniformity deviation

In the experiments, the 3D parameters are not available. It was shown in §3 that the range and the mean values for η 2D can be estimated for a population of spheroids randomly oriented with a given value for η 3D . The relation between η 3D and η2D given in Eq. 20 (developed in appendix A and showed as a red solid line in figure 10) can be used to recover 3D parameter values from 2D measurements. Applying this specific equation for Figure 19: Morphological parameters as a function of the 2D aspect ratio, respectively from left to right the uniformity η 2D (figure 19a), irregularity ι 2D (figure 19b), and SDS parameter ψ 2D (figure 19c). Analytical relations are showed in solid red line. The color map is based on the equivalent 2D radius expressed in logarithmic scale the reconstruction implies three assumptions: (i) the droplets are prolate spheroids; (ii) the main axis of the spheroids is randomly oriented; (iii) the distribution of η 3D is a Dirac function. The two first assumptions have been discussed extensively in §3 and §4. The last one is treated here. For a given distribution of η 3D and under assumptions (i) and (ii), there is only one distribution of η 2D but the inverse is not true. From the numerical database, η 3D was estimated from the value of η 2D and the relation given by Eq. 20. This provides a mean value of 2.13 for the estimated η 3D whereas the real 3D mean value is given by η3D = 1.15. Furthermore, the estimation of η 3D from η 2D parameter values of the experimental database leads to outlier about 190. To assess more realistic values for η 3D from η 2D measurements, an alternative approach is proposed, based on equivalent radii and the probability density function for uniformity parameter.

The uniformity parameter is determined from the primary parameters a 2D , b 2D and R EQ,2D (see §1.2). For each droplet, the prolate and oblate assumptions are made to determine the corresponding values for η 20a for the experimental and numerical databases (isotropic and anisotropic). For the three databases, the sets of points are evenly distributed around the mean ηQ 2D , obtained from a set of randomly oriented spheroids (see §4 The ultimate interrogation of this work is to determine the possibility to estimate a 3D parameter from its 2D measurements. The comparison of distributions of η 2D and η 3D for the anisotropic database clearly shows an underestimation of the 3D parameter by the 2D parameter, described beforehand. The present approach consists in analyzing the statistical properties of the parameter distributions. As commented for the considered databases, the uniformity PDF follows a log-normal distribution:

Q 2D (see §3.3). A plot of η Q 2D vs η 2D is shown in figure
P (η, σ, χ) = χ ση √ 2π exp -log 2 (x/χ) 2σ 2
. For instance, the best fit obtained for the numerical anisotropic 3D database, η 3D , gives the values of σ = 0.31 and χ = 1.42, and, is represented in figure 21. This shows an excellent agreement with the pdf of 3D anisotropic uniformity. Under the assumption of a family of spheroid (prolate or oblate), the distribution of the 2D uniformity can be estimated from the distribution of 3D uniformity and the distribution of angles θ. As seen in §4, the prolate assumption is more appropriate to account for atomizing droplets. From the log-normal fit of η 3D distribution and the assumption of prolate droplets, several distributions for η 2D were guessed, one for a random distribution of orientation angles θ (corresponding to the distribution sin θ 2 dθ), and for a fixed angle set to θ = [π/6, π/3, 4π/10, π/2] (corresponding to a Dirac function distribution).

As seen in figure 21, the distributions with a fixed angle provides a log-normal like distribution. The obtained distribution is narrower than the measured one (in other words, σ is smaller). The averaged 2D uniformity decreases when increasing the fixed θ angle. The case θ = π/2 can be considered as the limit where all the droplets are aligned with the streamwise axis. That is why this angle Dirac distribution over-estimates the measured 2D uniformity. The random distribution shows a completely different behavior. The distribution is no longer a log-normal.

This is linked to the fact that in this case the probability to see a perfect circle after projection is not null, neither the probability of a null value for η. In practice, this can occur only for a projection along streamwise Z direction, which is never the case in experiments. This random distribution under-estimated the measured 2D uniformity obtained for X -Y projection (i.e. perpendicular to the streamwise direction). As conclusion, the random and the fixed angle at θ = π/2 distributions surround the true distribution for η 2D as observed in figure 21. A more accurate knowledge of the θ distribution should give a better prediction of the reconstruction of the distribution for η 2D . Here, this distribution has not been computed. The authors consider that it would be important to include this distribution in future measurements as it will allow a better understanding of the relationship between the 2D and 3D parameters.

Conclusion

An accurate characterization of the droplets shape is of main interest to understand the atomization process.

This characterization is often done on 2D projections obtained from experimental setups. As seen in the literature [START_REF] Ghaemi | Assessment of parameters for distinguishing droplet shape in a spray field using image-based techniques[END_REF], different shape parameters can be used to describe these droplets' projections.

Among the several parameters introduced in the present paper, the uniformity parameter is considered to analyze the deviation of the 2D value from the 3D value. Two approaches are pursued: i) an analytical approach based on the assumption that the droplets are spheroids; ii) a numerical approach were numerical atomized droplets, and their projections, are analyzed.

According to these two approaches, the following results are retained:

• Droplets can be considered as spheroids in order to analyze their shape parameters.

• For a given 3D shape, the 2D shape parameters can vary depending on the angle of projection. For the uniformity shape parameter, and under the assumption of spheroidal shape, analytical relations between the 2D and 3D parameters are given in §3.4 and appendix A).

• The kind of spheroid considered (oblate or prolate) modifies strongly the relations between 2D and 3D uniformity parameter. In the present databases, the droplets projections follows the statistical behavior of prolate spheroid. In particular, the 2D uniformity underestimates the 3D uniformity.

• The projection axis should be considered when analyzing a database. The primary lengths and the shape parameters can suffer from a systematic deviation due to this preferential projection angle. In the anisotropic numerical case studied here, the sketch of prolate-like droplets that are aligned preferentially along the injection axis (axis Z) is drawn. As a result, this projection axis exacerbates both the underestimation of the equivalent radius (affecting the size probability distribution function) and the underestimation of the uniformity shape parameter.

In experiments, the projection axis is often perpendicular to the injection axis, which improves the estimation of 3D parameters. However, certain complex atomization configurations do not permit an optimal optical configuration, which can drastically influence the deviation from the 3D results. In order to show the effect of the projection on the shape parameter analysis, the comparison between an experimental database and the generated numerical databases is done in §5. This comparison shows that the uniformity distributions fit a lognormal distribution. Furthermore, the study of the uniformity distributions provides a tool that should allow us B Influence of the correction on singular droplets from the isotropic database.

In this appendix, four droplets are extracted from the isotropic database, illustrating the influence of the pro-

Figure 3 :

 3 Figure 3: Projection of two specific 3D shapes: a prolate spheroid and a tube, respectively left and right.Morphological values for the 3D and each projection are given, respectively the aspect ratio; α, the uniformity; η, the irregularity; ι, and the SDS parameter; ψ.

  representation of morphological evolution, the number of cells across a droplet diameter is set to 32, d d = 32∆x. It gives a cubic box of L = 1.5 × 10 -4 [m] and a droplet diameter of d d = 7.5 × 10 -5 [m]. The physical configuration is set to ρ l /ρ g = 1 and µ l /µ g = 1, giving a Reynolds number of Re = √ kL/ν = 15.73, with k, the turbulent kinetic energy equal to k = 3.6 [m 2 .s -2 ]. The surface tension is the only physical parameter that varies in this database. The motivation is to have several Weber numbers generating different droplet shapes. The surface tension variation goes from σ = [6.75 × 10 -5 , ..., 6.75 × 10 -1 ] [kg.s -2 ], giving a range of Weber number from W e = ρkL/σ = [10 -2 , ..., 10 2 ]. The droplet morphological evolution goes from small surface variation toward highly elongated droplets. To illustrate the typical situations, the visualization of 4 droplets is given in the appendix B.

Figure 4 :

 4 Figure 4: Left: X -Z projection of the liquid volume fraction at a given time step in the airblast atomizer numerical experiment. Right: droplets database from the airblast atomizer based on their aspect ratio, α 3D , and numerical resolution, R EQ,3D

Figure 5 :

 5 Figure 5: 3D morphological parameters as a function of the aspect ratio, respectively from left to right: the uniformity η 3D (figure 5a), irregularity ι 3D (figure 5b), and SDS parameter ψ 3D (figure 5c). Results of the isotropic and anisotropic flows are represented with black triangles and blue circles, respectively. Analytical relations are showed in dashed red line for prolate spheroids, and continuous red line for oblate spheroids. The four droplets extracted from appendix B are represented with square, circle, diamond and down triangle symbols, from A to D, respectively.

Figure 6 :

 6 Figure 6: 2D morphological parameters as a function of the aspect ratio, respectively from left to right; the uniformity η 2D (figure 6a), irregularity ι 2D (figure 6b), and SDS parameter ψ 2D (figure 6c). Results for the anisotropic flow are plotted with color markers based on the direction of projection and black marker are used for isotropic flow. Analytical relations for the ellipse are shown in solid red line. The X-Y-Z directions of projection of droplet D from appendix B are represented with square, circle and diamond symbols, respectively.

Figure 7 :

 7 Figure 7: Prolate and oblate spheroids (left and right). θ is the polar or projection angle and ϕ the azimuthal angle. The lengths λ 1 and λ 2 are the length along the axisymmetry axis and its perpendicular.

Figure 9 :

 9 Figure 9: Evolution of the 2D uniformity, η P 2D or η O 2D , and the corrected 2D uniformity, η Q 2D,P and η Q 2D,O with Q depending on the initial family, for a range of direction of projection θ = [0, π]. The 3D uniformity value is fixed, η 3D = 0.25, left: prolate spheroid, right: oblate spheroid.

Figure 10 :

 10 Figure 10: Plots of η 3D vs η 2D . Lines: 2D averaged uniformity for each kind of spheroid family (prolate: ηP 2D , oblate ηO 2D ), and maximum values (lower script max). Markers: random spheroids sets without (grey circles η P 2D or η O 2D ) and with spheroid family correction (blue cross for prolate η P 2D,P and η P 2D,O , and orange star for oblate η O 2D,P and η O 2D,O ). Left figure: prolate spheroid set. Right: oblate spheroid set.

Figure 13 :

 13 Figure 13: Distribution of the 3D uniformity, η 3D , against η 2D . The color map is associated to R EQ,2D /R EQ,3D values, see figure 11. Lines: 2D averaged uniformity for each kind of spheroid family, ηQ 2D , maximum values, ηQ 2D,max , and family correction, ηQ 2D,Q and ηQ 2D,max,Q .

Figure 15 :

 15 Figure 15: Distribution of b 3D against b 2D scaled by the equivalent 3D radius R EQ,3D (top). Distribution of a 3D against a 2D scaled by the equivalent 3D radius R EQ,3D (bottom). The color map is based on the scaling of the equivalent 2D and 3D radius, see figure 14. An individual scattering of each direction of projection is given for both quantities; from left to right: X-Y-Z.

Figure 17 :

 17 Figure 17: Images of the spray: Air-assist jet on the left and an image of the experimental database on the right. The red rectangle shows the region covered by images of the database.

Figure 18 :

 18 Figure 18: Droplet size distribution of the experimental database. Data removed by applying the out-of-focus filters/morphological filter correspond to the hashed zone.

  Figure 20: a) η Q 2D vs η 2D for prolate (circles) and oblate (crosses) assumptions in the case of isotropic (green), anisotropic (blue) and experimental (red) databases. Solid lines are for ηQ 2D determined on a set of randomly oriented spheroids. b) Probability density function of uniformity (direct and with prolate or oblate assumptions) for each database.

Figure 21 :

 21 Figure 21: Probability density function of the uniformity of the numerical anisotropic experiment.Reconstruction of 2D are given from 3D.

  jection. The lenghts and morphological parameters are also given for each droplet. These droplets (A, B

Figure 22 :

 22 Figure 22: Almost spherical droplet extracted from the HIT-like flow database.

Table 3 :

 3 Morphological parameters of the extracted droplet, see figure22, of the HIT-like flow database.

			2D		3D
		X	Y	Z	-
	R EQ,3D /a	0.975 0.988 0.991 0.972
	b/R EQ,3D	0.973 0.999 0.987 0.971
	R EQ /R EQ,3D	0.999 1.01 0.996	-
	α	0.949 0.988 0.978 0.944
	η	0.053 0.013 0.0225 0.0573
	ι	0.999	1.0	1.0	0.998
	ψ	0.031 0.007 0.012 0.0407
	R EQ,P /R EQ,3D 1.008 1.008 1.002	-
	η P	0.0522 0.012 0.023	-
	R EQ,O /R EQ,3D 0.990 1.004 0.994	-
	η O	0.053 0.013 0.023	-

Acknowledgments

This work was granted access to the HPC resources of IDRIS, TGCC and CINES under the allocation A0092B1010 made by GENCI (Grand Equipement National de Calcul Intensif) and the CRIANN (Centre Régional Informatique et d'Applications Numériques de Normandie) under the scientific project N. 201 704. The authors would thank Alexandre Poux for the fruitful exchanges. We also thank the Normandy region for their funding of this research.

communication.

The authors are convinced that the true understanding of atomization, and in particular of the transition between primary and secondary atomization, requires a detailed characterization of the 3D droplet shape. This will require in the future more experiments using 2D projections, but also numerical simulations, and, the application of optical measurement methods allowing the 3D reconstruction of the droplet shape.

A Analytical integration of uniformity averages

In this appendix we provide the methodology to obtain the integrals given in equations 15 and 16. For space reasons, here we develop only the η2D for prolate spherodids, ηP 2D . At the end of this appendix, the solution for each case is provided.

This equation can be reduced introducing the parameter A = λ 2 λ 1 . First, we consider a prolate spheroid where this parameter is equal to the aspect ratio A = α 3D < 1. In this case the previous equation becomes:

We can reduce the integrals to the interval 0 to π/2 and multiply by two since sin(θ) is symmetric at π/2.

Thus, making an integral by substitution u = sin 2 (θ) we obtain:

These integrals can be computed analytically using hypergeometric function 2 F 1 . Thus the final solution is

given by:

Similar development can be done for the oblate case where A = 1 α 3D > 1. From Eq. 17, the following relation is obtained:

One can notice that the solution is similar to the prolate since the two integrals are switched.

Similarly, the development is done for ηP 2D,P and ηO 2D,O . The solution is given by:

B.2 Droplet B) 1.07 0.587 0.603 0.401 R EQ /R EQ,3D