domain: spi sde

text:
Benchmark of ecodesign LCA tools at neighborhood scale THE 11TH INTERNATIONAL CONFERENCE ON LIFE CYCLE MANAGEMENT

Keywords: LCA softwares, Scientific literature BIM-LCA, CIM-LCA, "Bottom-up" 20 LCA Databases reviewed

Context Purpose Methodology for tool corpus creation • Present a review of LCA tools at neighborhood scale • Identify tradeoffs between accuracy, usability, adaptability and completeness • Suggest levers to improve tool accuracy, usability, adaptability and completeness • Propose research perspectives thanks to identified blind spots in ecodesign LCA tools

Case study presentation

Our case study is based on "La Vallée" neighborhood located in Châtenay-Malabry, France.

Tool corpus (88 items)

• Capacity to perform neighborhood scale analysis or large building stock (multiple buildings scale) • Specialization in building industry and/or neighborhood design • Scientific articles using this software • Still in operation in 2023

Selection criteria

• 5 Softwares • Methods used to address accuracy, usability, adaptability and completeness thanks to a case study • Identify limits of such tools (levers to improve and blind spots)

Benchmark

Results -Tradeoffs between accuracy, usability, adaptability and completeness

Qualitative analysis

Research perspectives:

• LCA & Geospatial Data (BIM, CIM, GIS and related formats) coupling improvements could ease data gathering and reduce uncertainty for LCA analysis. • Tools must be capable to assist municipalities and decision makers before the design stages to build strategy for neighborhood environmental impact reduction.

• During design early design phases, tools must be fast and allow easy modification to let a diversity of development options to be explored. But a risk of improving usability is to lower completeness and adaptability, this needs to be avoided as much as possible.

• We can also see that socio-economic factors and open spaces and in particular green spaces (specifically trees) could be better implemented in those tools.

• Geovisualization of LCA impacts is not currently supported by any of the benchmarked tools.

However, the incorporation of such visualization could provide LCA specialists with valuable insights into the locations where different scenarios exhibit significant impacts.

• Finally, the multidisciplinary nature of a neighborhood needs a coherent and wide set of indicators, avoiding pollution displacement and representing all the protected areas, thus the tool used must be able to take into account a wide variety of thematic. • The benchmark is based on one person using each software to perform the same case study. To complete our usability analysis, standards such as ISO 25010 should be used.

Fig. 1 :

 1 Fig. 1 : Benchmark methodology

Fig. 3 :

 3 Fig. 3 : Tools score synthesis

Levers to improve Examples

Synthesis & Research perspectives


Sarah Christin

Patrice Meunier

Stéphane Le Dizès

Baroclinic critical layer in a viscous stratified boundary layer flow on an undulated tilted surface

The present paper investigates theoretically and experimentally the boundary layer generated by a stably stratified fluid flowing horizontally along a surface tilted in the transverse direction and deformed by sinusoidal undulations with crests perpendicular to the flow direction. In the absence of undulations, a weak transverse velocity, proportional to the normal velocity is created such that the flow remains purely horizontal. In the presence of undulations of amplitude h, a stronger transverse flow is generated that exhibits a singular behavior at the critical altitude where the frequency of the perturbation matches the buoyancy frequency of the fluid. This baroclinic critical layer was previously analysed by Passaggia et al. (JFM 751, 663-684, 2014) for a boundary layer flow with a small sliding velocity on the surface. Here, the no-slip boundary condition of the experimental flow is applied. For this purpose, we solve the viscous sub-layer to obtain a complete theoretical model for the solution in the critical layer without any adjusting parameter. The theoretical predictions for the transverse velocity are compared with experimental measurements, and a good quantitative agreement is demonstrated. Compared to the sliding case, the no-slip boundary condition on the surface reduces the amplitude of the critical layer solution by a factor Re -1/3 . As a consequence, the transverse velocity has a maximum in the critical layer of order h, but it still induces a shear rate of order hRe 1/3 .

Introduction

Boundary flows of stratified fluids are often studied in an oceanographic context. Indeed, they have a strong influence on sediments dispersion [START_REF] Rebesco | Contourites and associated sediments controlled by deep-water circulation processes: State-of-the-art and future considerations[END_REF]) but also on the global ocean dynamics. For instance, they affect the global overturning circulation of oceans [START_REF] Kuhlbrodt | On the driving processes of the atlantic meridional overturning circulation[END_REF]) via the currents appearing on Antarctic coastal slopes [START_REF] Baines | Observations and modeling of antarctic downslope flows: A review[END_REF][START_REF] Thompson | The antarctic slope current in a changing climate[END_REF]. This last phenomenon has motivated numerous gravity current studies, which are also relevant for atmospheric applications such as katabatic winds [START_REF] Baines | Mixing in flows down gentle slopes into stratified environments[END_REF][START_REF] Monti | Observations of flow and turbulence in the nocturnal boundary layer over a slope[END_REF][START_REF] Shapiro | Dense water cascading off the continental shelf[END_REF][START_REF] Baines | Mixing regimes for the flow of dense fluid down slopes into stratified environments[END_REF]. Boundary layers are places of intense dissipation [START_REF] Beckebanze | Damping of quasi-twodimensional internal wave attractors by rigid-wall friction[END_REF][START_REF] Davis | Energy budget in internal wave attractor experiments[END_REF][START_REF] Sen | Global energy dissipation rate of deep-ocean lowfrequency flows by quadratic bottom boundary layer drag: Computations from currentmeter data[END_REF]), but also regions where internal gravity waves [START_REF] Garrett | Internal tide generation in the deep ocean[END_REF] and intense meanflow (Le Dizès 2020) are generated.

For large Reynolds numbers, the Blasius boundary layer flow that develop on a plane surface is known to be unstable [START_REF] Schlichting | Boundary-Layer Theory[END_REF]. This 2D viscous instability does not disappear in the presence of stratification (Wu & Zhang 2008a;[START_REF] Chen | Instability of a boundary layer flow on a vertical wall in a stably stratified fluid[END_REF]). However, another instability, of inviscid nature, appears as soon as the stratification and the shearing directions are no longer aligned [START_REF] Candelier | Inviscid instability of a stably stratified compressible boundary layer on an inclined surface[END_REF][START_REF] Chen | Instability of a boundary layer flow on a vertical wall in a stably stratified fluid[END_REF]. This instability which is characterised by an internal gravity wave emission has been observed in other contexts [START_REF] Lindzen | Instability and wave over-reflection in stably stratified shear flow[END_REF][START_REF] Dizès | Radiative instability in stratified vortices[END_REF]. In the present experimental study, the Reynolds number will be too small for any of these instabilities to be present.

Stratified boundary layers are also particularly sensitive to topography. Wu & Zhang (2008b) showed for example that an obstacle can induce a coupling between internal waves and viscous instability modes of the boundary layer. Besides, the presence of an inclination angle between the stratification direction and the boundary has strong consequences on the flow [START_REF] Garrett | Boundary mixing and arrested Ekman layers: Rotating stratified flow near a sloping boundary[END_REF], influencing transport and mixing, as observed by [START_REF] Phillips | An experiment on boundary mixing: mean circulation and transport rates[END_REF] and also reported by [START_REF] Baines | Observations and modeling of antarctic downslope flows: A review[END_REF]. Recently, [START_REF] Puthan | Energetics and mixing in buoyancy-driven near-bottom stratified flow[END_REF] numerically demonstrated that a density perturbation in a stratified fluid above an inclined surface can lead to the generation of a mean oscillating flow along the slope at the frequency N sin α. Furthermore, in the case of a corrugated surface, inclination enables the flow to avoid surface roughness without vertical displacement (inhibited by stratification) by going around the obstacle horizontally. The two swerve regimes (going around with a pure horizontal motion, or above) have been discussed by [START_REF] Maccready | Stratified flow along a corrugated slope: Separation drag and wave drag[END_REF].

The precise structure of a stratified boundary layer developing above an undulated inclined wall has been adressed by [START_REF] Passaggia | Response of a stratified boundary layer on a tilted wall to surface undulations[END_REF] using numerical simulations. Considering small sinusoïdal undulations of a surface, with crest line perpendicular to the flow direction, and slip boundary conditions, they predicted the generation of a strong transverse flow at a specific location in the boundary layer. They further showed that this flow is associated with a resonance mechanism between the Brunt-Väisälä frequency and the forcing of the undulations at a critical point singularity of the inviscid equations.

Such a baroclinic critical layer has been observed in other contexts. [START_REF] Boulanger | Structure of a tilted stratified vortex[END_REF] showed that it was excited when the axis of a vortex was tilted with respect to the direction of the stratification. They demonstrated that a strong axial flow, localized in the baroclinic critical layer was created and could be destabilized [START_REF] Boulanger | Instability of a tilted vortex in stratified fluid[END_REF]. [START_REF] Wang | Nonlinear dynamics of forced baroclinic critical layers[END_REF], 2021) argued that similar critical layers could be responsible of the complex nonlinear dynamics observed in accretion disks [START_REF] Marcus | Three-dimensional vortices generated by self-replication in stably stratified rotating shear flows[END_REF].

The main objective of the present work is to observe experimentally the baroclinic critical layer in a boundary layer flow and to compare the experimental data with the critical layer predictions. For this purpose, we will also need to extend the theoretical analysis of [START_REF] Passaggia | Response of a stratified boundary layer on a tilted wall to surface undulations[END_REF] to account for the no-slip boundary condition on the undulated surface.

The paper is organised as follows. In section §2, experimental details are given: the set up and vizualisation techniques are first described and the plate design explained. The main parameters of the study and their chosen definition are presented. Section §3 focuses on the derivation of the stratified boundary layer flow on a flat inclined wall, and first comparisons with experimental measurements. This solution constitutes the base flow that is perturbed in section §4 by small undulations of the plate. The extension of [START_REF] Passaggia | Response of a stratified boundary layer on a tilted wall to surface undulations[END_REF] analysis is carried out here. In particular, we solve the viscous sub-layer needed to apply the no-slip boundary condition. A parameterless expression of the transverse velocity in the baroclinic critical layer is obtained and compared to experimental measurements. A good qualitative and quantitative agreement is demonstrated. Section §5 summarizes the main results and briefly discusses an application to a real atmospheric flow.

A stratified boundary layer experiment

A brief description of the experiment is provided in this section. More details on the facility, stratification method and visualisations' subtleties can be found in the PhD thesis of Christin (2021).

Facility

The experimental setup, sketched in figure 1(a), consists of a PMMA tank 4 m long, 1 m wide and 1 m deep, filled with salty water up to a height H ∼ 85 cm. A trolley is mounted on rails and moved with a ball screw connected to a motor MAC800 D2 of JVL, allowing a horizontal translation of the plate along the tank. The velocity of the trolley is kept constant during each run and vary in the range 1.79 ⩽ U ⩽ 2.98 cm.s -1 .

As shown in the sketch, the plate is fixed to the trolley thanks to four supporting arms. The study is done on the lower side, so that the arms do not perturb the region of interest. The plate consists of a plane entrance zone where the boundary layer can develop before reaching the undulations. This region is needed to obtain a sufficiently large boundary layer width δ in the undulation zone. The corrugated part consists in 5 undulations, of half crest-to-crest height h and wavelength λ (wavenumber k), and is the zone of interest in this study. It is parametrized as ξ(x) = h sin k(x -L u ), for x > L u , where L u is the distance between the leading edge of the plate and the beginning of the sinusoidal part. The transition between plane and undulated parts is smoothed thanks to a bending radius.

Another important element of the present work is α, the angle made between the outward normal of the plate surface e z and the upward vertical e Z . It generates a transverse inclination of the plate and is necessary for the critical layer to exist.

Figure 1(a) also shows a second angle, γ = 2 • , made between the horizontal and the longitudinal direction of the plate. This very small angle is needed to relaminarize the boundary layer as it will be seen in the subsection 2.4.

Stratification

The tank is filled with salty water with a salt concentration increasing linearly with depth. The Brunt-Väisälä frequency is then defined as:

N = - g ρ 0 ∂ρ lin ∂Z ,
where ρ lin is the fluid density, ρ 0 the density at the mean plate height, g the gravity, and Z the coordinate along the vertical. This stratification is obtained using a technique described in the PhD thesis of [START_REF] Bosco | Etude du sillage stratifié d'un cylindre[END_REF] and improved by [START_REF] Christin | Etudes expérimentales en fluide stratifié : couche limite et interactions fluidestructure[END_REF]. It consists in dividing the tank into two parts, one with pure water and the other one with uniformly mixed strongly salty water, and letting the two fluids mix through holes of diameter 0.5 cm drilled in the separation slab. This technique is preferred to the usual "two tanks method" because it requires only one tank, which is far more convenient considering the huge volume used here.

Once realized, the stratification can hold for months if the experiments performed in the tank do not generate a very violent mixing. The density profile is measured by collecting fluid samples along a tank wall in the middle of its length, every 5 cm. It is done once every day before experiments are carried out, thanks to an Anton Paar densitometer DMA 35. Between two consecutive experimental days, the profile is barely observed to change. The slope estimation induces uncertainties on N on the order of 5 %.

Final experiments which lead to the results of section §4.3 are made in a stratification with constant frequency N = 0.85 ± 0.04 rad.s -1 .

Measurement techniques

Two measurement techniques have been used and recorded with a same camera Sony α7s taking 25 frames per second.

Firstly, the flow was qualitatively observed by a shadowgraph method (see setup in yellow on figure 1(b)) which consists in observing the flow illuminated from the back, through a lens. It is based on the fact that the flow induces inhomogeneities in the density field resulting in variations in the light refraction index which reveals the flow structure. The contrast can be adjusted by moving the camera around the focal point of the lens. On the camera is mounted a FE 2.8/50 MACRO lens, while the external lens used to focus the rays on the camera sensor is a PCX condenser lens from Edmund Optics of focal length 50 cm and diameter 25 cm.

The final results, revealing the critical layer, are obtained thanks to a Particle Image Velocimetry (PIV) system shown in green on figure 1(b). This method requires to seed the flow with particles (polydisperse hollow glass spheres from TSI) that are then enlighted by a laser sheet (Z40M18B-F-532-Ip20 from Z LASER of 532 nm wavelength), and to treat the images thanks to a software (DPIVSoft on MATLAB) to deduce the velocity field. This experimental technique is quantitative but complex in its execution, for its purely experimental part as well as its data post-processing.

The seeding particles have been carefully chosen to be good tracers of the flow, meaning that their density (between 1.05 and 1.15 g.cm -3 ) is close to the one of the ambiant fluid (ρ ∈ [1 ; 1.12] g.cm -3 ) to avoid inertial effects, and their diameter between 8 and 12 µm is lower than the size of the smaller structure of interest in the flow. In order to have a precise flow field, a macro lens of focal length 2.8 mm and diameter 90 mm is mounted on the camera. The laser sheet extends in the crossflow direction, placed such that it cuts the plate on its center part along the e y axis. Furthermore, it makes an angle β = 20.0 ± 0.3 • with the longitudinal axis of the plate e x , shown on the bottom sketch (entitled "side view") of figure 1(b). This peculiar disposition enables a scan of the velocity in the whole boundary layer depth since the velocity field is steady. It however requires a complex recontruction of the flow field in real space (see [START_REF] Christin | Etudes expérimentales en fluide stratifié : couche limite et interactions fluidestructure[END_REF]). As the camera axis is aligned with e z (we neglect the very small angle γ), PIV gives the x and y components of the velocity.

The present study needs PIV to be accurate for small transverse displacements (of few mm.s -1 ) while measuring large longitudinal velocities (up to ∼ 3 cm.s -1 ) close to the plate. In order to do so, for the highest considered velocities, the image post processing is done twice. Firstly with the regular recorded images, and secondly with displaced images to deduce the apparent velocity of the flow close to the plate. The final flow field considered is the concatenation of both processes.

Design of the plate

The first test plate aimed at simply observing how the boundary layer develops. It consists in a 30 cm long flat part, bevelled at 20 • at its entrance tip, followed by undulations with h = 1 cm and λ = 10 cm.

A shadowgraph visualization of the flow above this plate is shown on figure 2(a). The flow is clearly fully turbulent from the very beginning of the plate (called "the leading edge" in the further discussion), with the presence of recirculation bubbles. Besides, the boundary layer is observed to separate from the surface, generating a black stripe (called "the front" hereafter) evidenced by the white arrow in the shadowgraph. This effect is understood as a consequence of lee waves generated by the leading edge, as the front position scales with 2πU ∞ /N . These waves could possibly generate an adverse pressure gradient leading to the boundary layer separation. The undulation amplitude is also too large as it creates a boundary layer separation after each crest.

All these issues have been solved by working on the design and position of the plate. First, the leading edge has been slimmed and shaped with a smooth curve on its downward part. Second, the whole plate has been inclined in its longitudinal direction of a very low angle γ = 2 • with respect to the horizontal. This forces the current lines to stay attached to the surface and strongly inhibits the generation of the so-called front. Finally, the undulation height h has been reduced to 2.5 % of their wavelength λ.

The latter has been fixed to 10 cm so that the perturbation amplitude is not too small, but such that 5 complete undulations can still be shaped on the plate (the length of the plate is limited by the tank dimensions). The plane entrance zone has been fixed at 91 cm so that the boundary layer is of the largest possible extent. This eases visualizations and ensures that the undulations are indeed within the boundary layer extent.

The final plate is machined in black polyoxymethylene (POM). A shadowgraph of the flow is presented on figure 2(b). The boundary layer flow is clearly nicer when compared with the first plate. No more turbulence is observed, the flow is laminar and follows almost all the undulations. Thin lines seem to be emitted from the plate but they do not correspond to a genuine boundary layer separation, as checked by PIV measurements. Note that shadowgraph visualisations do not show any evidence of a critical layer above the plate: this will require PIV.

Parameters of the study

This study presents results for a unique plate designed to optimize the formation of a critical layer, as it has been discussed in the previous subsection. It is 141 cm long (including a plane entrance zone of 91 cm), 60 cm wide and 2 cm thick after the slim leading edge part. The undulations have a wavelength of λ = 10 cm and a height h = 2.17 0.58 ± 0.02 149 ± 6 5.2 ± 0.5 0.52 6 2.34 0.62 ± 0.03 173 ± 8 5.2 ± 0.5 0.5 7 2.5 0.61 ± 0.02 180 ± 6 5.75 ± 0.5 0.52

Table 1. Summary of experimental data. N = 0.85 ± 0.04 rad.s -1 , λ = 10 cm and α = 118.1 • . The critical velocity Uc = N λ sin α/(2π) is equal to 1.19 cm/s.

0.25 cm. The transverse inclination angle α is kept constant equal to 118.1 • . The natural frequency associated with stratification is set to N = 0.85 ± 0.04 rad.s -1 , by taking the mean density ρ 0 ∼ 1.024 g.cm -3 as the reference density. These quantities lead to a constant theoretical critical velocity U c = λN sin α/2π = 1.19 cm.s -1 . By fitting the measured longitudinal velocity at mid-length of the undulations by U ∞ F ′ (z/δ emp ) with F defined in (3.6) (see section §3 for justification), an empirical boundary layer depth δ emp is determined, varying between 0.58 and 0.75 cm (see table 1). It is the reference scale which will be used to non dimensionalize lengths and to calculate the two non dimensional parameters characterizing the flow, namely the classical Reynolds Re and Froude F r numbers:

Re = U ∞ δ emp /ν, (2.1a) F r = U ∞ /N δ emp . (2.1b)
The kinematic viscosity ν is 1.015±0.005.10 -6 m 2 .s -1 , as the salted water temperature is T ∼ 21.7 • C.

A summary of the parameters associated with each experiment is given in table 1.

Base flow

Theoretical developments

A stationary stratified boundary layer flow above a tilted wall is considered. This first section aims at properly calculating the base flow of this physical situation in the case Re ≫ 1.

The wall is inclined in the cross-stream direction, such that its outward normal e z makes an angle α with the upward vertical e Z , which is the stratification direction (see figure 1(a)).

The fluid is assumed to have both constant kinematic viscosity ν and Brunt-Väisälä frequency N , which is associated with a linear stable stratification

ϱ lin = ϱ 0 (1 -N 2 Z/g),
where g is the gravity. The diffusivity of the quantity generating the stratification is supposed very low compared to the viscosity of the fluid, such that its effect can be neglected. It is a valid assumption for brine solutions like the ones experimentally considered in this paper.

In the absence of motion, the balance between pressure and density terms prescribes:

P lin = ϱ 0 gZ(1 - N 2 2g Z). (3.1)
In the following, only pressure and density deviations from this hydrostatic equilibrium, denoted with regular letter without index, are considered.

From now, when no scaling is specified, velocities will be non-dimensionalized by U ∞ (the uniform longitudinal velocity outside of the boundary layer), lengths by δ (the boundary layer width), pressures by ϱ 0 U 2 ∞ and density by ϱ 0 U 2 ∞ /gδ. The dimensionless equations of motion, under the Boussinesq approximation, are :

∇.u = 0, (3.2a) (u.∇)u = -∇p + 1 Re ∆u -ρe Z , (3.2b) (u.∇)ρ = u.e Z F r 2 , (3.2c)
where u is the dimensional velocity field, u, v and w being respectively the longitudinal, transverse and normal components in the plate reference frame. Considering a transverse α angle forces the introduction of a transverse velocity v as ρ varies along the vertical Z which includes components in both e z and e y directions. This imposes a three dimensional velocity field but as the wall is supposed infinite in the e y direction, unknown functions can be kept independent of the y variable.

By analogy with the classical Blasius problem on a flat plane (x, z), we introduce the spatial scales

x = Re x, z = z , (3.3)
and the scaling :

u B = ūB , (3.4a) v B = Re -1 vB , (3.4b) w B = Re -1 wB , (3.4c) p B = Re -2 pB , (3.4d) ρ B = Re -2 ρB , (3.4e)
where the "B" index has been added to refer to the Blasius base flow. Using these equalities, the complete system (3.2) can thereby be reduced at leading order to:

∂ ūB ∂ x + ∂ wB ∂ z = 0, (3.5a) ūB ∂ ūB ∂ x + wB ∂ ūB ∂ z = ∂ 2 ūB ∂ z2 , (3.5b) ūB ∂v B ∂ x + wB ∂v B ∂ z = -ρ B sin α + ∂ 2 vB ∂ z2 (3.5c) ūB ∂ wB ∂ x + wB ∂ wB ∂ z = - ∂ pB ∂ z -ρB cos α + ∂ 2 wB ∂ z2 , (3.5d) ūB ∂ ρB ∂ x + wB ∂ ρB ∂ z = Re 2 F r 2 (cos α wB + sin αv B ) . (3.5e)
where it is implicitly assumed that Re/F r and α are O(1).

The two first equations (3.5a,b) correspond to the classical system governing a Blasius boundary layer (e.g. [START_REF] Schlichting | Boundary-Layer Theory[END_REF]). They constitute an independent system for the two velocity components ūB and wB that can be solved in term of a self-similar variable η = z/ √ x using a single function F (η) satisfying

2F ′′′ + F ′′ F = 0, (3.6)
with the boundary conditions: F ′ (0) = F (0) = 0 and F ′ (η → ∞) = 1 (a prime denotes a derivative with respect to η). The relation between F (η) and the velocity components ūB and wB is:

ūB = F ′ , (3.7a) wB = ηF ′ -F 2 √ x . (3.7b)
Observing now the rest of system (3) suggests the origin of the other scalings: the one for v is the same as for w because of (3.5e), which forces the scaling for ρ in (3.5c), itself leading to the scaling for p in (3.5d).

In the present paper, we shall present experimental results obtained for a fixed angle α = 118.1 • and Froude numbers of the order of unity. We shall therefore be in the limit Re ≫ F r for which the last equation (3.5e) reduces to the right-hand side. This means that the transverse and normal velocities will be proportional in order to cancel the vertical velocity. Streamlines will remain horizontal, as it is often the case in stratified fluids. In this large Re/F r limit, the transverse velocity associated with the Blasius flow will therefore be vB

= - cos α sin α wB = - cos α sin α ηF ′ -F 2 √ x . (3.8)
Once vB is calculated, the density field ρB is obtained through (3.5c). The Blasius equation (3.6) is needed to simplify some terms, such that one can write:

ρB = cos α sin 2 α F ′ F -ηF ′2 -2F ′′ 4x 3/2 . (3.9)
Besides, combining (3.5c) and (3.5d) with (3.8) allows us to express the pressure gradient as a function of the density only:

∂ pB ∂ z = - 1 cos α ρB , (3.10) and obtain pB = 1 x sin 2 α 2F ′ - F 2 2 + η sF ′2 (s)ds .
(3.11)

The system (3.5a-e) has now been fully solved in the experimental parameter range, that is: finite sin α, F r of order one and large Re. These solutions are plotted in Fig. 3 as a function of the normal component. Interestingly, the normal and transverse velocity components and the density perturbation do not vanish at infinity but rather converge toward a finite value. The pressure increases linearly at large z.

In the appendix, we provide some informations on another limit, obtained when α → 0 or F r large, that can also be solved explicitly.

Comparison with experimental measurements

As detailed in section §2.3, a PIV technique has been set up to measure both longitudinal and transverse velocity fields (u and v). Without undulations, the transverse velocity v scales as Re -1 and is thus too small to be measurable. Only the longitudinal velocity u has therefore been measured. It has been plotted for experiment 5 in figure 3 where it is also compared to the theoretical Blasius solution.

The experimental measurement has been taken 5 cm before the start of the undulations, after the boundary layer has developed over L =86 cm from the front edge. A good agreement is observed between theoretical and experimental profiles as figure 3 shows. For z < 0.9, the reflexion of the laser on the plate creates a bright area on the images which perturbs the PIV measurements and leads to wrong velocities. Above the laser reflexion zone, for z > 0.9, both profiles match with a maximum uncertainty of the order of 5%, which confirms the base flow theoretical profile. The spatial diffusion of the boundary layer due to viscous effects suggests that the boundary layer width should be δ theo = νL/U ∞ = 0.58 cm, which is larger than the δ fit = 0.49 cm obtained in figure 3 from the best fit. A similar difference between fitting and theoretical prediction of the order of 20 % is observed for all performed experiments. This difference partly comes from the small longitudinal tilt angle γ = 2 • of the plate. Indeed, in the absence of stratification the streamwise velocity above a tilted plate is given by the Falkner-Skan solution [START_REF] Drazin | Hydrodynamic stability[END_REF]

ūB = F ′ F S z x(1-m)/2 m + 1 2 with F ′′′ F S + F F S F ′′ F S + β|1 -F ′2 F S | = 0 (3.12)
where β = 2γ/180 and m = β/(2 -β). As shown on Fig. 3(a), this solution is almost undistinguishable from the Blasius solution if x is increased by 10%. It means that the tilt angle γ = 2 • decreases the thickness by 5% which is thus equal practically to 0.55 cm rather than 0.58 cm. This prediction is closer to the measured thickness δ fit = 0.49 cm although still being 10% larger. The remaining difference probably comes from the rounded leading edge.

For the final plots and the calculation of quantitative predictions presented in section §4.3, the thickness δ emp is measured by fitting the longitudinal profile with Blasius in the middle of the sinusoïdal part of the plate, rather than upstream of the undulations as done here for δ fit . It has been checked that the longitudinal flow is not modified by the undulations and that it still matches the Blasius profile. The empirical values are given in table 1. ). In (a), blue dotted symbols correspond to the experimental longitudinal velocity u at the end of the plane entrance zone (86 cm from the front edge) for experiment number 5 (see table 1) with z dimensionalised by δ fit = 0.49 cm. In (a) the red dashed line corresponds to the Falkner-Skan solution (3.12) for x = 1.1 with a tilt angle γ = 2 • .

Perturbed flow

Experimental transverse velocity field

The tilt angle of the plate breaks the invariance along the y direction and induces a transverse velocity v which is enhanced by the plate undulations. This allows its measurement by PIV. A raw transverse velocity field is displayed in figure 4. The undulations are darkened to ease the visualisation.

First, one can note that the transverse velocity is still very weak. Indeed, its amplitude is of the order of only 5% of the maximum longitudinal velocity U ∞ . This measurement was thus extremely hard to achieve since the noise had to be reduced below 1%. The fine post processing described in section 2.3 was necessary to obtain such an accuracy. Then, one can observe in this field alternate bands of v above the plate upstream of the undulated part. As it was shown in Christin ( 2021), these waves are orographic waves emitted by the front edge. They are of no interest for the present study. Now, focusing on the field above the undulations, inclined lobes of alternatively positive and negative transverse velocity can clearly be identified. They are within the boundary layer and exhibit the wavelength of the undulations. This pattern becomes more pronounced downstream: two undulations seem to be sufficient to settle this oscillating regime forced by the topography.

Although the orographic waves have a smaller amplitude than the inclined lobes, they are not negligible. To better reveal the lobes, data are then filtered with a spatial Fourier filter along x at the wavenumber of the undulations in figure 5, which shows 3 wavelengths to give a more detailed image of the velocity field. The Fourier filtered velocity is obtained as

v c (z) cos(kx) + v s (z) sin(kx)
where for each z the Fourier coefficients are defined as

v c (z) = xmax xmin v(x, z) cos(kx)dx/ xmax xmin cos 2 (kx)dx and v s (z) = xmax xmin v(x, z) sin(kx)dx/ xmax xmin sin 2 (kx)dx
with x min the starting point of the undulations and x max the largest position where experimental data are available. The specific pattern develops slightly above the surface of the plate, around z = 1 and have velocity lobes inclined with a small positive angle with respect to the horizontal. Positive lobes are located above the ridge of the undulation whereas negative lobes are located above the depression.

This measured transverse field corresponds to the perturbed transverse flow induced by the undulations. This pattern localized away from the boundary is reminiscent of a critical layer singularity [START_REF] Passaggia | Response of a stratified boundary layer on a tilted wall to surface undulations[END_REF]). In the next section, we provide a complete theory explaining the presence of this pattern. The theory developed in [START_REF] Passaggia | Response of a stratified boundary layer on a tilted wall to surface undulations[END_REF] is extended to account for a no-slip boundary condition and applied to the Blasius profile configuration for a quantitative comparison.

Theoretical developments

The base flow is perturbed with low-amplitude (i.e. h ≪ 1) undulations of the plate, parametrized with a sine function of wavenumber k = O(1) (the wavelength is supposed to be of the same order as the boundary layer width). It is also assumed that the length over which the boundary layer has developed before arriving on the corrugations is large enough so that the boundary layer flow is settled and its width is approximatively constant on the extent of the undulations. The slow longitudinal variable x of the Blasius boundary layer is now related to a local variable x:

x = Re(x -1) = x -Re.

(4.1)

The location x = 1 (where η = z) corresponds to the position in the middle of the undulations. It is there that the boundary layer width δ emp has been estimated and where the flow is analysed. Perturbations being searched as spatial Fourier modes, physical quantities are written as:

u tot = u B + 1 2 hk ue ikx + c.c. , (4.2a
)

p tot = p lin + p B + 1 2 hk pe ikx + c.c. , (4.2b
)

ρ tot = ρ lin + ρ B + 1 2 hk ρe ikx + c.c. . (4.2c)
where u, ρ and p are functions of the normal spatial variable z only.

In the bulk of the boundary layer (that is for z = O(1)), the perturbations are expected to be described by inviscid equations. This region corresponds to the inviscid outer layer in the sketch of the different regions shown in figure 6. Viscous effects are present in two localized regions. Very close to the surface of the plate, in a viscous sub-layer, they are needed to apply the no-slip boundary condition at the surface of the plate. The width of this layer is obtained by balancing inertial and viscous effects. For a flat plate, this width is O(Re -1/3 ) owing to the presence of a regular critical point at the wall (i.e. U B (0) = ω/k = 0) (see [START_REF] Drazin | Hydrodynamic stability[END_REF]. For an undulated plate, we obtain the same width scaling as long as h ≪ Re -1/3 . The solution of this viscous sub-layer will provide the boundary conditions for the inviscid outer solution. Viscous effects will also be needed in the baroclinic critical layer to smooth the singularity that appears in the inviscid solution.

The viscous sub-layer

In order to describe the perturbation in a viscous sub-layer of O(Re -1/3 ) width, it is natural to introduce the new normal variable z = z -h sin kx Re -1/3 , (4.3) such that z = 0 corresponds to the deformed boundary. As mentioned above, we further assume that h ≪ Re -1/3 such that the problem remains linear at leading order. In the viscous sub-layer, velocity, pressure and density perturbations are expanded as:

u = ũ, (4.4a) v = Re -1/3 ṽ, (4.4b) w = Re -1/3 w, (4.4c) p = Re -1/3 p, (4.4d) ρ = Re -2/3 ρ. (4.4e)
The boundary conditions on the velocity perturbations are obtained by expanding ūB close to the boundary:

ūB ∼ ū′ B0 η = ū′ B0 zRe -1/3 + h sin kx , (4.5)
where ū′ B0 stands for the derivative of ūB with respect to η for η = 0. The boundary condition u tot (z = 0) = 0 then gives

ũ(0) = i ū′ B0 k , ṽ(0) = 0, w(0) = 0. (4.6)
Now introducing (4.4) and (4.5) in the governing equations (3.2) the following non-dimensionalized system is obtained:

ikũ + d w dz = 0, (4.7a) ikũū ′ B0 z + wū ′ B0 = -ik p + d 2 ũ dz 2 , (4.7b) ikṽū ′ B0 z + ρ sin α = d 2 ṽ dz 2 , ( 4
d 3 ũ dz 3 -ikū ′ B0 z dũ dz = 0. (4.8)
Without surprise, we obtain the equation describing perturbations in a viscous critical layer [START_REF] Drazin | Hydrodynamic stability[END_REF]. As already mentioned above, this comes from the fact that the position z = 0 is a regular critical point for the stationary perturbations generated by the wall undulations. Such a critical layer at the wall is also obtained in the asymptotic structure of Tollmien-Schlichting waves in the lower branch of the stability diagram [START_REF] Lin | The theory of hydrodynamic stability[END_REF]. It also corresponds to the lower deck obtained in the triple deck theory describing boundary layer separation [START_REF] Smith | Laminar flow over a small hump on a flat plat[END_REF]Wu & Zhang 2008a;[START_REF] Dong | Receptivity of inviscid modes in supersonic boundary layers due to scattering of free-stream sound by localised wall roughness[END_REF].

Solving this equation allows us to obtain ũ, which gives w thanks to (4.7a), then ṽ through (4.7e), the constant p using (4.7b) and finally ρ using (4.7c).

As explained in the appendix of [START_REF] Drazin | Hydrodynamic stability[END_REF] textbook, recessive solutions of homogeneous Airy equations such as (4.8) can be expressed in terms of generalized Airy functions A k (z, p), where p denotes the integral order (if p > 0) of the Airy function. The longitudinal velocity ũ is then searched under the form:

ũ = C 0 A 1 (βe iπ/6 z, 1) + C 1 ,
with C 0 and C 1 being two complex constants, and

β = (kū ′ B0 ) 1/3 . (4.9)
Applying the boundary conditions (4.6a-c), the system (4.7a-e) can then be fully solved. The solution in the viscous sub-layer is found to be

ũ(z) = iū ′ B0 k A 1 (β 1/3 e iπ/6 z, 1) A 1 (0, 1) , (4.10a) ṽ(z) = -cot αβ -1 ū′ B0 e -iπ/6 A 1 (βe iπ/6 z, 2) -A 1 (0, 2) A 1 (0, 1) , (4.10b) w(z) = β -1 ū′ B0 e -iπ/6 A 1 (βe iπ/6 z, 2) -A 1 (0, 2) A 1 (0, 1) , (4.10c) p(z) = - β 2 ū′ B0 k 2 A 1 (0, 2) A 1 (0, 1) e iπ/3 , (4.10d) ρ(z) = cos αβ ū′ B0 sin 2 α iβe -iπ/6 z A 1 (βe iπ/6 z, 2) -A 1 (0, 2) -e iπ/6 A 1 (βe iπ/6 z) A 1 (0, 1) .(4.10e)
As explained above, from this solution, we can obtain the boundary conditions to apply to the outer solution. The condition of matching implies that the sub-layer solution as z → ∞ should correspond to the inviscid outer solution as z → 0.

Using the property that lim z→∞ A 1 (z, 2) = 0 and relations between A 1 (0, p) and the Γ function (see [START_REF] Drazin | Hydrodynamic stability[END_REF], we in particular obtain the value of the normal velocity of the outer solution at the wall that should be given by

w(0) = lim z→∞ Re -1/3 w = Re -1/3 β -1 ū′ B0 3 1/3 Γ (4/3) e -iπ/6 . (4.11)

The outer layer

The inviscid layer lying above the viscous sub-layer, the so-called outer layer, has been studied by [START_REF] Passaggia | Response of a stratified boundary layer on a tilted wall to surface undulations[END_REF]: it is in this region that the baroclinic critical layer appears. In [START_REF] Passaggia | Response of a stratified boundary layer on a tilted wall to surface undulations[END_REF], the normal velocity that was forcing the solution in the outer layer was O(1) and proportional to a prescribed longitudinal velocity. Here, it is obtained from the condition of matching with the viscous sub-layer. It is therefore weaker and of order Re -1/3 as prescribed by (4.11).

In the outer layer, velocity, pressure and density perturbations vary with respect to the spatial variables x and z, both non-dimensionalized with δ and exhibit a scaling prescribed by the matching with the sub-layer:

(u, ρ, p) = Re -1/3 (û, ρ, p).
(4.12)

Despite the amplitude factor, the analysis of [START_REF] Passaggia | Response of a stratified boundary layer on a tilted wall to surface undulations[END_REF] can still be applied. One just has to consider a Blasius base flow instead of the tanh profile they considered. As shown in that paper, one obtains after manipulating the system (3.2) a single (Rayleigh) equation for the normal velocity ŵ:

d 2 ŵ dz 2 - ū′′ B ūB ŵ -k 2 1 -(kū B F r) 2 sin 2 α -(kū B F r) 2 ŵ = 0. (4.13)
The boundary condition at z = 0 is now rigorously prescribed by the condition ŵ(0) = lim z→∞ w which gives thanks to (4.11):

ŵ(0) = β -1 ū′ B0 3 1/3 Γ (4/3) e -iπ/6 . (4.14)
The other condition at infinity is that the perturbation should either vanish or be an outgoing wave. These conditions at z equals zero and infinity fully determine the function ŵ if one knows how to treat the branch point singularity present at the position zc where

(kū B (z c )F r) 2 = sin 2 α. (4.15)
At this point, the function ŵ is finite but its derivative ŵ′ exhibits a logarithmic singularity. As explained in [START_REF] Passaggia | Response of a stratified boundary layer on a tilted wall to surface undulations[END_REF], this logarithmic function can be defined as in the classical stability theory [START_REF] Lin | The theory of hydrodynamic stability[END_REF]): the branch cut should be fixed in the upper complex plane (because ū′ Bc > 0). This means that the singularity can be avoided by integrating equation (4.13) in the lower complex plane above the singular point.

Once ŵ is obtained, the other components can be deduced from it. For the transverse velocity v and the density ρ, we get:

v = - sin α cos α ŵ sin 2 α -(kū B F r) 2 , (4.16a) ρ = ikū B cos α ŵ sin 2 α -(kū B F r) 2 . (4.16b)
By contrast with ŵ that remains finite at r c , the outer expressions (4.16ba,b) of v and ρ are singular at r c . The singularity is therefore stronger than at a regular critical point. It corresponds to a so-called baroclinic critical point (e. g. [START_REF] Wang | Nonlinear dynamics of forced baroclinic critical layers[END_REF]. It is associated with a local resonance of the inertial frequency ω -kū B of the perturbation (where the forcing frequency ω vanishes in our case since the undulations are stationary) with the local Brunt-Väisälä frequency ±N sin α in the shear plane. As for a regular critical layer, the regularization of the solution is possible by considering viscous effects in a small region of width O(Re -1/3 ) around the baroclinic critical point. This region corresponds to the viscous baroclinic critical layer.

The viscous baroclinic critical layer

This part is formally strictly equivalent to the one presented by [START_REF] Passaggia | Response of a stratified boundary layer on a tilted wall to surface undulations[END_REF] and the reader is advised to refer to this article to have more details on the analysis.

To describe the fields in the vicinity of the critical position zc , viscous effects need to be reintroduced to smooth out the singularity. The scaling becomes:

u = Re -1/3 ǔ(ž), (4.17a)
v = v(ž) + Re -1/3 vs (ž), (4.17b)
w = Re -1/3 ŵc + Re -2/3 w(ž), (4.17c)

p = Re -1/3 p(ž), (4.17d) ρ = ρ(ž) + Re -1/3 ρs (ž), (4.17e)
where ž = (z -zc )Re 1/3 is the critical layer variable. By introducing this scaling in the governing equations (3.2), one can show that the system can be reduced to an inhomogeneous Airy equation for v:

d 2 v dž 2 -2ikū ′ Bc vž = - i ŵc sign(sin α) cos α F r , (4.18)
where ū′ Bc is the derivative of ūB with respect to η, evaluated at the critical position zc . The other velocity components, the pressure and the density are then given by ρ (4.19d) Note that the log(Re) correction terms that were provided in [START_REF] Passaggia | Response of a stratified boundary layer on a tilted wall to surface undulations[END_REF] are here implicitly included in each expression.

= -i v F r , ( 4
To solve equation (4.18), it is convenient to introduce the generalized Airy functions B k (z, 0) (see [START_REF] Drazin | Hydrodynamic stability[END_REF], that satisfy the inhomogeneous Airy equation:

d 2 dz 2 -z B k (z, 0) = 1, (4.20)
and behave as 1/z for large |z| (for particular values of arg(z)) to be able to match with the solution in the outer region. An adequate choice of the cubic root of 2ikū ′ Bc allows us to use the function B 1 (z, 0) and write the solution as v = i ŵc sign(sin α) cos α where vceq is defined in (4.22).

(2kū ′ Bc ) 2/3 F r B 1 (-i (2kū ′ Bc ) 1/3 ž, 0). ( 4
Resulting theoretical fields are presented in figure 7, on the left side, to directly be compared with the corresponding experimental ones obtained thanks to a spatial Fourier filtre applied on PIV data as introduced in §4.1.

It is important to emphasize that the theory has no adjustable parameter except the boundary layer width δ which is slightly modified to better fit with the base flow profile.

Even in the absence of direct fitting, a good qualitative agreement is observed in figure 7 between the experimentally measured fields and the ones predicted by the theory. Indeed, the inclined lobes of alternatively positive and negative transverse velocity generated slightly away from the plate surface are captured. Whereas the theoretical critical layer is thinner than the one effectively observed, both fields show a constant inclination angle Ω (around 10 • ) of the lobes, which are located at a position that gets closer to the plate surface when the towed velocity increases (from experiment 2 to 6). The order of magnitude also seems to be recovered.

Quantitative results

These qualitative agreements can be quantitatively checked.

In order to do so, filtered and theoretical transverse velocity profiles along the normal axis z of a typical experiment (the number 4 in table 1) are plotted in figure 8. The first two graphs, shown in (a) and (b), correspond to the profile at two specific longitudinal positions (x = λ/4δ and 0 respectively) where the imaginary part and the real part of the transverse velocity should be obtained according to the theory. The third graph, shown in (c), presents the norm of v. Note that the theoretical position of the maximum of |v| is slightly smaller than zc . This is due to the moderate value of the Reynolds number and the decreasing behavior of the outer solution |v|. These plots show some previously noted deviations between data such as the larger extent of the experimental critical layer compared to the predicted one, especially visible in (b) and (c). Also, an amplitude difference of the order of 50% is also observed in (c): the amplitude of the velocity in the critical layer is larger than expected. However, the distance from the plate at which the critical layer starts is remarkably well predicted, as well as its first-half profile (up to z ∼ 1.5), which is particularly striking in the second plot. This better agreement for lower z position could be explained by the larger amount of experimental data over which the Fourier filter is applied at these heights. Indeed, as it can be seen in figure 4, the visual range is more and more cropped away from the plate as x increases, so that the complete velocity field over the whole critical layer is only obtained on the first undulations (especially for lower towing velocity experiments). Furthermore, it is on these early undulations (especially the two first ones as it can be seen in figure 4) that the critical layer settles: then the filtered profile resulting from this first part is expected to deviate from the theoretical prediction.

To have a more systematic comparison for each experiment, four characteristic quantities are defined in figure 9 and plotted in figure 10. The first one is the position of the critical layer zc experimentally defined as the z position of the norm maximum value, itself corresponding to the second checked quantity. The third one is the angle Ω made by the velocity lobe with respect to the plate surface, and the last one is the phase φ max indicating the relative position of the lobes compared to the undulations. Again, theoretical and experimental results show a good agreement, especially considering the completeness of the theory that does not require any direct fitting between data.

The upper left plot confirms the lowering position of the critical layer with the increase of kF r (and then in U ∞ as it is the only varying parameter). It goes from +∞ when kF r → sin α (i.e U ∞ tends to the critical velocity), to 0 when kF r increases. This can be easily understood as the critical velocity (kept constant in these experiments as k, N and α are fixed) is reached for smaller z when the boundary layer profile has to match a higher velocity at "infinity". Experimental values all lie above the theoretical line. It could be explained by the fact that the Blasius boundary layer is not a perfect model above the undulations which tend to widen the boundary layer thickness by generating a larger normal velocity than expected above a plane wall. Still, experimental values are close to the theoretical line and seem to validate the model.

Concerning the critical layer amplitude (upper right graph), the theoretical predicted value is globally smaller than the one observed, but the model seems more and more accurate with increasing kF r. The stronger deviation for lower towing velocity experiments could be explained by the same phenomenon as discussed above, about the cropped visual range. However, considering the range of potential theoretical values suggested by the dotted lines indicating the extreme k values experimentally studied, the model seems to catch the essence of the critical layer amplitude behaviour.

The lower graphs show the phase φ max on the left and the inclination angle Ω of the lobes on the right which are theoretically expected to be constant for kF r > 1.4, that is in global agreement with experimental measurements.

Conclusion

In this paper, a coupled theoretical and experimental approach has been presented to study a stratified boundary layer above an inclined sinusoidal wall.

The experimental study of a stratified boundary layer generated by a towing plate has been found to be particularly challenging. The shape of the leading edge of the plate, its overall longitudinal inclination and the height/wavelength ratio of the undulations had to be empirically studied and adjusted to limit boundary layer separation, and lee waves generation. These issues have been solved using a thin rounded leading edge, a longitudinal angle of the plate of 2 • and undulations with h = 0.25 cm and λ = 10 cm. Shadowgraph was performed to check the qualitative laminar aspect of the boundary layer. The flow was then measured by PIV, which is a quantitative visualisation technique set up so that the longitudinal and transverse velocity fields could be obtained on the whole width of the boundary layer. The wide range of velocities involved and the specific position of the laser sheet used to scan the boundary layer have led to a complex data post processing.

The longitudinal velocity profile has been convincingly fitted with the Blasius profile predicted by the theory. The base flow obtained by analysis was then perturbed with small undulations of the plate, and the perturbations searched in the form of monochromatic Fourier modes. A viscous sub-layer has been added in the perturbation analysis to take into account the no-slip boundary condition and obtain rigorous boundary conditions for the inviscid outer problem that was already considered in [START_REF] Passaggia | Response of a stratified boundary layer on a tilted wall to surface undulations[END_REF]. This sub-layer analysis was necessary to be able to compare quantitatively the experimental data with the theory. It leads to a complete theory with no adjustable parameter except the boundary layer width δ obtained by fitting the Blasius profile on the measured longitudinal base flow.

Theoretical transverse velocity fields have been compared to raw experimental data and experimental data filtered at the wavelength of the undulations. A good qualitative agreement has been demonstrated. Both show localized lobes of alternatively positive and negative transverse velocity, detached from the surface of the plate and with a barely constant low inclination angle with respect to the plate surface.

Besides, the critical layer can be characterized by four parameters which are: its posi-tion above the plate z c , v max (the maximum transverse velocity in a lobe), the inclination angle of the lobe Ω and their relative position compared to the plate undulation defined by the phase φ max . Experimental and theoretical values of these quantities have been compared and, again, a convincing agreement has been shown. It is worth emphasizing that the viscous sub-layer introduces a reducing factor of order Re -1/3 in the amplitude of the transverse velocity in the critical layer compared to the one found by [START_REF] Passaggia | Response of a stratified boundary layer on a tilted wall to surface undulations[END_REF]. This significantly reduces the expected amplitude of the phenomenon: the transverse velocity is now of order h whereas it was predicted as O(hRe 1/3 ) in the previous paper. This explains why only weak transverse velocities have been observed. The generated shear rate is larger than the velocity since it is here of order hRe 1/3 . However, it must remain smaller than O(1) for the linear analysis done in this paper to be valid. For larger amplitudes h, it is necessary to do a non-linear analysis of the viscous sub-layer. Intuitively, we may expect a forcing of the normal velocity outside of the sub-layer independent of the Reynolds number. This would lead to an amplitude of the velocity inside the critical layer proportional to Re 1/3 , as for a slip boundary condition. This would justify the use of slip boundary conditions, which is commonly used for the study of lee waves [START_REF] Legg | Mixing by oceanic lee waves[END_REF][START_REF] Bühler | Waves and mean flows[END_REF]). However, it will also generate harmonics and modify the mean velocity profile. The problem is thus far more complex and we leave this for future work. Furthermore, it will be necessary to look at the critical layer in a turbulent boundary layer in order to describe real geophysical flows for which Reynolds numbers can be of order 10 8 . Declaration of Interests. The authors report no conflict of interest.

  Figure 1. (a) Schematic of the experimental setup indicating the main axes. The horizontal/vertical coordinates (X, Y, Z) are rotated with an angle α around the X-axis to give the new coordinates (x, y, z). The plate is in the (x, y) plane such that its outgoing normal ez is tilted with an angle α with respect to eZ (i.e. with an angle π -α with respect to g). The plate is tilted around the y axis by a small angle γ = 2 • to prevent boundary layer separation. The undulations have a wavelength λ and an amplitude h. (b) Schematic of the visualization setup including both shadowgraph (in yellow) and PIV (in green).

Figure 2 .

 2 Figure 2. Compilation of shadowgraph pictures of the flow above the entrance zone of the plate for two different leading edge profiles at towing speed U∞ = 2.65 cm.s -1 and N = 1.06 ± 0.06 rad.s -1 . (a) α = 180 • , γ = 0 • . (b) α = 168.7 • , γ = 2 • . The transverse angle α has been observed to have no influence on the boundary layer quality.

Figure 3 .

 3 Figure3. Normal profiles of the longitudinal velocity (a), the normal and transverse velocity (b), the density (c) and the pressure (d). Solid black lines correspond to the stratified Blasius solution (3.7-3.11) for x = 1 with F solution of (3.6). In (a), blue dotted symbols correspond to the experimental longitudinal velocity u at the end of the plane entrance zone (86 cm from the front edge) for experiment number 5 (see table1) with z dimensionalised by δ fit = 0.49 cm. In (a) the red dashed line corresponds to the Falkner-Skan solution (3.12) for x = 1.1 with a tilt angle γ = 2 • .

Figure 4 .

 4 Figure 4. Raw data of the transverse velocity field above the plate for experiment number 4.

Figure 5 .

 5 Figure 5. Transverse velocity field for experiment number 4, after applying a Fourier filter at the wavelength of the undulations along x.

Figure 6 .

 6 Figure 6. Scheme of the different layers studied.

  (4.7b) with respect to z yields a homogeneous Airy equation on dũ/dz:

Figure 7 .

 7 Figure 7. Theoretical (vp) and Fourier-filtered experimental transverse velocity fields for two wavelengths of experiments number 2, 4 and 6 (from the top to the bottom). For details see table 1. The four other fields are presented in Christin (2021).

Figure 8 .

 8 Figure 8. Filtered (dots) and theoretical (full lines) transverse velocity profiles for experiment number 4 for two x positions and the norm.

2Figure 9 .Figure 10 .

 910 Figure 9. Scheme defining the quantities experimentally measured and theoretically computed for comparison in figure 10.

† Email address for correspondence: meunier@irphe.univ-mrs.fr

Appendix A. Inclined boundary layer flow in a stratified fluid

In section §3.1, we have given the equations that govern the stationary boundary layer flow on an inclined surface in a stratified fluid. We have provided the general scalings, obtained when Re ≫ 1, Re/F r = O(1) and α = O(1), for which the Blasius solution is recovered for the longitudinal and normal velocities. We have further shown that when Re/F r is large, the system of equations can be fully solved in terms of the self-similar function involved in the Blasius equation. This limit, called limit 1, corresponds to the experiments we have analysed. There exists another limit, called limit 2, obtained for small angles, or large Froude numbers that can also be completely solved and that we want to present now.

For this purpose, it is useful to introduce the new parameter R α defined by

The square of this parameter measures the ratio of the magnitudes of the v term with respect of the ρ terms in equation (3.5e). The limit 1 was obtained when the balance in that equation was between w and v terms, which means R α ≫ 1. This limit is obtained for small α if the condition Re sin ≫ 1 is also satisfied. The other limit of interest is obtained when R α ≪ 1 and F r ≫ 1. In this limit, the balance in equation (3.5e) is between w and ρ terms. The adequate scalings in this limit are then

)

which give the parameterless system:

(A 3f)

As for limit 1, (A 3a) and (A 3b) constitute the Blasius equations for u * B and w * B which are then known quantities. Self-similar forms can also be obtained for the other quantities. p * B1 and ρ * B can be deduced from the Blasius profile using (A 3d) and (A 3f) respectively. Then, v * B and p * B2 are obtained from (A 3c) and (A 3e). The different quantities can be analytically expressed as functions of the Blasius solution F and its derivatives:

Both regions where limits 1 and 2 are obtained have been indicated in figure 11 for values of α of order 1 (left), small (center) and null (right). Note that when α = 0, only the limit 2 is obtained when F r ≫ 1. In that case, v naturally vanishes, but a pressure correction (p * B2 ) associated with the density correction is still present.


Élise Maigné

Céline Noirot

Julien Henry

Yaa Adu Kesewaah

Ludovic Badin

Sébastien Déjean

Camille Guilmineau

Arielle Krebs

Fanny Mathevet

Audrey Segalini

Laurent Thomassin

David Colongo

Christine Gaspin

Laurence Liaubet

Nathalie Vialaneix
email: nathalie.vialaneix@inrae.fr

ASTERICS: A Simple Tool for the ExploRation and Integration of omiCS data

Keywords: Web user interface, Statistical analyses, Data integration, Omics

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Background

The rapid development of omics acquisition techniques has induced the production of a large volume of heterogeneous and multi-level omics datasets measured on the same individuals. Information of biological interest is obtained using predictive and exploratory statistical methods performed on one omic level or, more widely from the so-called integration methods. The latter methods, which have been increasingly developed in the past few years, aim at providing an understandable representation of the combined information provided by several omic levels or at extracting information about interlevel crosstalk. Some of these methods are already available under the form of packaged software or libraries, and especially as R packages (like STATegRa https:// doi. org/ doi: † Élise Maigné and Céline Noirot have been contributed equally to this work. [START_REF] Ge | iDEP: an integrated web application for differential expression and pathway analysis of RNA-Seq data[END_REF]. 18129/ B9. bioc. STATe gRa, mixOmics [START_REF] Rohart | mixOmics: an R package for 'omics feature selection and multiple data integration[END_REF], MOFA [START_REF] Argelaguet | Multi-omics factor analysis-a framework for unsupervised integration of multi-omics data sets[END_REF][START_REF] Argelaguet | MOFA+: a statistical framework for comprehensive integration of multi-modal single-cell data[END_REF], mixKernel [START_REF] Mariette | Unsupervised multiple kernel learning for heterogeneous data integration[END_REF][START_REF] Brouard | Feature selection for kernel methods in systems biology[END_REF], among others; see https:// github. com/ mikel ove/ aweso me-multi-omics for a more complete list or [START_REF] Ritchie | Methods of integrating data to uncover genotype-phenotype interactions[END_REF][START_REF] Bersanelli | Methods for the integration of multiomics data: mathematical aspects[END_REF][START_REF] Cantini | Benchmarking joint multi-omics dimensionality reduction approaches for the study of cancer[END_REF]). However, the use of these packages still requires to learn a programming language and to have access to sufficient statistical knowledge to choose method parameters and interpret outputs. Hence, several web based applications have been deployed to perform (sometimes advanced) analyses on omics data, like SHAMAN https:// shaman. paste ur. fr/ [START_REF] Volant | SHAMAN: a user-friendly website for metataxonomic analysis from raw reads to statistical analysis[END_REF], dedicated to differential analysis of metagenomic data, or iDEP http:// gelab. org/ idep/ [START_REF] Ge | iDEP: an integrated web application for differential expression and pathway analysis of RNA-Seq data[END_REF], GENAVi https:// junkd nalab. shiny apps. io/ GENAVi/ [START_REF] Reyes | GENAVi: a shiny web application for gene expression normalization, analysis and visualization[END_REF], IDEAmex http:// www. uusmb. unam. mx/ ideam ex/ [START_REF] Jiménez-Jacinto | Integrative differential expression analysis for multiple experiments (IDEAMEX): a web server tool for integrated RNA-seq data analysis[END_REF], DIANE https:// diane. bpmp. inrae. fr [START_REF] Cassan | Inferring and analyzing gene regulatory networks from multi-factorial expression data: a complete and interactive suite[END_REF], all providing complete analysis pipelines for RNAseq or other count data, mostly oriented toward differential analysis.

We present ASTERICS, an intuitive and interactive web application that aims at making exploratory and integration analysis workflows easily available to biologists and nonstatisticians. Contrary to existing tools, ASTERICS

• is not dedicated to a specific type of omics but can perform specialized normalization or differential analyses for different types of omics (including omics obtained as compositional data such as proteomics or metabolomics, bulk transcriptomics as microarray or RNA-seq data, metagenomics,...); • is not dedicated to a specific type of analysis (e.g., differential analysis) but can handle exploratory analysis (PCA, clustering, etc), differential analysis as well as different types of data edition (including correction of missing values, log-transformation, scaling or sample selection); • allows the integration of multiple omics, i.e., it includes exploratory analysis able to explain the typology of individuals described by omics and/or characters simultaneously obtained at different levels of the living organisms; • does not only include a single and straight analysis workflow but allows the user to perform complex analyses workflows with various editions, selections based on statistical analysis results, data normalizations,... while keeping a comprehensive track of the history and dependency of performed operations and analyses.

Implementation

Code organization

ASTERICS frontend is developed in Vue.js https:// vuejs. org/ and uses the CSS framework Bulma https:// bulma. io/. This allows a flexible and modular organization of the different screens and analyses, as well as an easy management of persistant sessions and multiple user sessions. The backend is controlled with flask https:// flask. palle tspro jects. com/, pyRserve https:// pypi. org/ proje ct/ pyRse rve/ and Rserve https:// www. rforge. net/ Rserve/ to launch analyses performed with R scripts. R package versions are controlled using renv https:// CRAN.R-proje ct. org/ packa ge= renv to ensure stability and reproducibility of the analyses. An R wrapper runs the function corresponding to the analysis chosen by the user, converts the interactive output plots and the output tables in JSON to serve the interface and maintain a database of datasets and analyses.

The main R packages used for the analyses are: missMDA [START_REF] Josse | missMDA: a package for handling missing values in multivariate data analysis[END_REF], FactoMineR [START_REF] Lê | FactoMineR: an R package for multivariate analysis[END_REF], fastcluster [START_REF] Müllner | fastcluster: fast hierarchical, agglomerative clustering routines for R and Python[END_REF], SOMbrero [START_REF] Villa-Vialaneix | Stochastic self-organizing map variants with the R package SOMbrero[END_REF], mixOmics, VIM [START_REF] Kowarik | Imputation with the R package VIM[END_REF], edgeR [START_REF] Robinson | edgeR: a Bioconductor package for differential expression analysis of digital gene expression data[END_REF][START_REF] Mccarthy | Differential expression analysis of multifactor RNA-Seq experiments with respect to biological variation[END_REF], and sva [START_REF] Leek | The sva package for removing batch effects and other unwanted variation in high-throughput experiments[END_REF]. ggplot2 [START_REF] Wickham | ggplot2: Elegant Graphics for Data Analysis[END_REF] and plotly [START_REF] Sievert | Interactive Web-Based Data Visualization with R, plotly, and shiny[END_REF] are also used to produce high quality and interactive graphics.

Availability and deployment

ASTERICS is distributed under GNU General Public License Version 3 and its source code are available at https:// forge mia. inra. fr/ aster ics/ aster ics, including recipes for building self-contained docker images. In addition, an open issue repository is available for users wanting to report bugs or suggest improvements https:// forge mia. inra. fr/ aster ics/ aster ics-issues.

ASTERICS is available online at https:// aster ics. miat. inrae. fr and does not require any user account creation. It can also be deployed for local usage or on servers using the 3 docker images provided with the project repository https:// forge mia. inra. fr/ aster ics/ aster ics/ conta iner_ regis try 1 , following the installation instructions provided in the README file of the project repository. In particular, the section "Run docker locally" of this page provides a 5-minute local installation procedure for users having docker installed on their computer.

Features

Figure 1 gives a complete overview of the application: users connect to ASTERICS simply providing an email on which they receive an information about their project ID (and URL), which can be shared with collaborators or re-used later to continue their analysis. The email is also used to warn the user about his/her workspace Fig. 1 Overview of ASTERICS web interface. The figure illustrates the different features of ASTERICS, including data upload and edition, analyses with interactive plots, comprehensive display of the workspace, exportation of data, plots and reports, and help pages providing advice for choice of analyses and options and tips for interpretation 1 A fourth docker image is provided to be used as a base image for R and packages but is not useful for deployment. automatic deletion a few days before it is performed. Workspaces, including data, user email and performed analyses, are deleted after 30 days of inactivity.

Upon connection, users can also automatically load an example dataset extracted from The Cancer Genome Atlas (TCGA, https:// www. cancer. gov/ tcga) and related to breast cancer as already studied in [START_REF] Singh | DIABLO: an integrative approach for identifying key molecular drivers from multi-omics assays[END_REF]. In addition, other data derived from this initial example are available at https:// doi. org/ 10. 15454/ YNMQUY, including information on cancer type and datasets with missing values obtained from the complete dataset with different missing value mechanisms. The complete description of the dataset extractions is also provided in this repository.

Analyses available in ASTERICS are organized into three main menus:

1. Edit contains three submenus to (a) perform simple dataset manipulations (e.g., selecting rows or columns based on manual selection or on a condition, changing a variable type, etc.); (b) explore, clean or impute missing values. Missing values can be either imputed (single PCA imputation as implemented in missMDA or k-NN imputation as implemented in VIM) or rows/columns containing more missing values than a user chosen threshold can be removed from the dataset; (c) perform different types of normalization (from a basic scaling to unit variance to more sophisticated normalizations like the TMM and TMMwsp normalizations of edgeR for RNA-seq data or the correction of an explicit batch effect as implemented in the "ComBat" method of sva [START_REF] Johnson | Adjusting batch effects in microarray expression data using empirical Bayes methods[END_REF]);

2. Explore contains five submenus to perform simple exploratory analyses of the datasets, including:

(a) basic descriptive statistics and tests (numerical and graphical summaries for one, two or all the variables, complex plots including up to five variables or correlation plot between all variables); (b) PCA (with plots for individuals that can be interactively combined with additional metadata variables and plots for variables that can be filtered based on a given correlation threshold) and its extension to mixed data [START_REF] Pagès | Analyse factorielle de donnés mixtes[END_REF]; (c) heatmap or dotplot (for which rows and/or columns can be organized by a hierarchical clustering and that can also be combined with metadata describing individuals); (d) clustering (based on a fast version of hierarchical clustering [START_REF] Müllner | fastcluster: fast hierarchical, agglomerative clustering routines for R and Python[END_REF] or on k-means). Outputs of this analysis include a heatmap and a PCA individual plot based on the initial data where the clusters are displayed with colors; the PCA plot can also be interactively combined with external information to compare the clustering with known groups of individuals; (e) self-organizing maps [START_REF] Kohonen | Self-Organizing Maps[END_REF], a method combining clustering and dimension reduction, that offers various graphics usable to obtain a summarized representation of the typology of the individuals in a dataset;

3. Integrate contains all analyses that are based on data coming from at least two different datasets, in a generic sense (these datasets can be omics datasets or other types of data, including metadata). This includes both analyses designed to explore relationships between at least two datasets containing, each, multiple variables (PLS and MFA) or analyses designed to compare all numeric variables in a given dataset with respect to the levels of a categorical variable from another dataset (PLS-DA and differential analysis). Integrate contains 4 submenus to perform data integration with (a) PLS [START_REF] Abdi | Partial least squares regression and projection on latent structure regression (PLS regression)[END_REF], a method unraveling relationships between two numeric datasets acquired on the same matched individuals. Similarly to PCA, it creates composite variables (e.g., linear combinations of the original variables) in each dataset, so as to maximize the covariance between these composite variables; (b) PLS-DA [START_REF] Ståhle | Partial least squares analysis with cross-validation for the two-class problem: a Monte Carlo study[END_REF], a method, based on PLS, integrating a numeric dataset with a categorical variable. The categorical variable is used as an information on classes (or conditions) on individuals and the method creates composite variables from the numeric dataset so that they separate at best the individuals according to their classes. As for differential analysis, this helps highlight the variables the most involved in the difference between conditions; (c) MFA [START_REF] Escofier | Multiple factor analysis (AFMULT package)[END_REF], a generalization of the PCA for cases where individuals are described by several groups of variables, including categorical ones. The method weights each group of variables in a data-driven fashion so as to balance the contribution of each group in the global PCA; (d) differential analysis of all the numeric variables in a given dataset with respect to the level of an external categorical variable. In this analysis, statistical tests are performed on all the variables of a numeric dataset to compare their means with respect to the levels of a categorical variable (thus describing conditions on individuals). The differential analysis is automatically adjusted to the type of data provided by the user: for generic or continuous data, a normality test can be performed to choose between 1-way ANOVA or Kruskal-Wallis test. For count data, the test defaults to negative binomial GLM tests as implemented in edgeR [START_REF] Mccarthy | Differential expression analysis of multifactor RNA-Seq experiments with respect to biological variation[END_REF]. Also, posthoc tests, comparing all pairs of the factor levels can be performed when the global test is found positive.

Note that PLS-DA and differential analysis have the same purpose but using two very different types of methods: PLS-DA is an exploratory approach using all the numeric variables in a multivariate fashion while differential analysis performs a test for every numeric variable independently and provides statistical guarantee (i.e., provides a p-value to control the probability of false rejection) on the result. The first three (factorial) analyses provides outputs similar to PCA (individual and variable plots, all interactive and that can be customized with external information), in addition to some other plots specific of each analysis (for instance, the individual plot of the MFA can be complemented with an information about the contribution of each type of data on the individual positions).

In addition, a menu My workspace summarizes the current state of the datasets, analyses and dependencies between them by an interactive directed acyclic graph (DAG). It also includes:

• a list of all datasets currently available, which includes the possibility to remove some, along with all the analyses that have been performed on them, to export a selection (in CSV or RDA), or to add new ones. During importation, the user can specify if the dataset corresponds to a specific type of omic (e.g., "Sequencing data (RNA-seq, miRNA,...) counts" or "metagenomics composition (compositional data)") and if these data have been previously normalized or log-transformed. These choices automatically drive sound options in the edition, normalization, or differential analysis menus, with understandable warnings or errors if the user does not perform the best analysis corresponding to what he/she has declared during importation (e.g., for RNA-seq data, users are advised to perform a TMM normalization before performing differential analyses and differential analysis is automatically performed with edgeR GLM model unless the counts have been log-transformed), • a list of all already performed analyses, together with their parameters. Analysis can also be exported in RDA format or as HTML reports. The latter include the workflow DAG, all numeric and graphical results obtained in the analysis and an information on the version of ASTERICS that has produced these results.

ASTERICS has been designed to help non-statisticians and beginners to choose the analyses and options the most appropriate to his/her question. By clicking on one of the three main menus, all analyses available in this menu are described in a simple way with plots and short texts to help the user understand the purpose of the analysis. In the different screens, a question mark button points to a documentation explaining how to set options and how to interpret results of the different screens, with short texts, understandable plots or short videos. This documentation is also gathered in a single document available at https:// aster ics. pages. mia. inra. fr/ user_ docum entat ion/ that provides descriptions of the functions and methods used in the backend to perform the analyses, along with the way the options have been set for these functions. The user documentation includes two case studies, one using the example datasets available in the interface 2 and the other using an in-house dataset obtained to study piglet survival [START_REF] Voillet | Handling missing rows in multi-omics data integration: multiple imputation in multiple factor analysis framework[END_REF][START_REF] Voillet | Integrated analysis of proteomic and transcriptomic data highlights late fetal muscle maturation process[END_REF][START_REF] Lefort | The maturity in fetal pigs using a multi-fluid metabolomic approach[END_REF]. 3

Results

This section presents a case study derived from the case study "Piglet" included in ASTERICS user documentation. This study is focused on fetal development in late gestation in pig species and addresses the question of delayed development in mammalian that may impact survival ability at birth. Two ages at late gestation were studied, 90 and 110 days (birth is expected at 114 days). Two genetic lines were included, LW for Large White, with higher risk of delayed development, and MS for Meishan, more robust. Studied piglets were of four genotypes: LWLW, MSMS, for pure breeds (mother 2 https:// aster ics. pages. mia. inra. fr/ user_ docum entat ion/ case-study-breast-cancer. html.

3 https:// aster ics. pages. mia. inra. fr/ user_ docum entat ion/ case-study-piglet. html.

and father are of the same genotypes), and the crossed piglets LWMS and MSLW (father then mother genotype). Here, we perform an analysis of three metabolomics datasets, respectively obtained on plasma, urine, and amniotic fluid samples of 444 piglets, which is an extension of the study of [START_REF] Lefort | The maturity in fetal pigs using a multi-fluid metabolomic approach[END_REF].

The workspace of the case study is made available under UID piglet_usecase on ASTERICS. 4 Datasets are described in [START_REF] Lefort | The maturity in fetal pigs using a multi-fluid metabolomic approach[END_REF] and correspond to a metabolomic analysis performed with 1 H NMR technique. Raw data ( 1 H NMR spectra) are available in the metabolights database [START_REF] Haug | MetaboLights-an open-access generalpurpose repository for metabolomics studies and associated meta-data[END_REF] under accession number MTBLS1541. Spectra were processed with the ASICS package to identify and quantify metabolites as described in [START_REF] Lefort | The maturity in fetal pigs using a multi-fluid metabolomic approach[END_REF] and metabolite quantifications together with a metadata dataset containing information on the design of the experiment have been made available at https:// doi. org/ 10. 57745/ TCKSTD.

Dataset importation and correction

From My workspace, the four datasets (three metabolite quantifications and information on the experimental design) are imported from CSV files using the button "Add" in the panel "All datasets". For metabolite quantifications, the original datasets were transposed during the upload to stick to ASTERICS requirements that the individuals are in rows. In the experimental design file, one of the variable (N_mother, reporting the mother identifier) was encoded as numerical in the CSV file. The menu "Edit/Dataset edition" allows to correct that problem, using the option "Change variable (column) types". These operations resulted in the workflow DAG of Fig. 2, as obtained from the menu My workspace. The four imported datasets are displayed in blue in rectangular nodes, whereas the corrected design of the experiment is displayed in white, also in a rectangular node. The edition process is also displayed in white, with a round node, as for all the analyses. This simple representation allows to easily understand the relationships between datasets and analyses.

Fig. 2 Workflow from the My workspace menu, as obtained after the importation of the four CSV files and the change of a variable type in the last dataset "Metadata" (that contains the uploaded data describing the design of the experiment) 4 Directly accessible at https:// aster ics. miat. inrae. fr/ works pace/ piglet_ useca se.

Exploratory analysis of the design of the experiment

First, simple exploratory analyses were first performed to understand the structure of the metadata file.

Using the menu "Explore/Explore variables in a dataset/All variables in a dataset", the content of the design of the experiment dataset (named Metadata_Clean) was summarized. From My workspace, the results can be obtained by simply clicking of the node corresponding to the analysis "UnivariateDataset_1" and further on the button "More" in the panel that opens on this action. For numerical variables, statistical summaries (min, quartiles, mean, maximum, standard deviation, number of missing values, number of unique values) are displayed in a table and the distributions are represented by boxplots (that can be scaled or not, to account for possible differences in scales between the numeric variables). Normality tests (Shapiro-Wilk) indicated that all variables significantly deviate from the normal distribution. For categorical variables (factors), statistical summaries (number of missing values, number of unique values, percentage and number of individuals in the mode level) are displayed in a table and the distributions are visualized with colored barplots. This simple analysis showed that the distributions of age, sex and genotype of the mother were balanced in the experiment, whereas the distribution of the father genotype was unbalanced (383 LW versus 61 MS). This issue is due to the fact that sows were inseminated with mixed LW+MS semen but LW semen shows a higher efficiency for reproduction. On the contrary, the mother genotype was fully controlled by the experimental design.

Relation between 3-fluid metabolome and stage of gestation

The last part of the analysis consisted in the analysis of the stage of gestation effect on the metabolome. Contrary to what was performed in [START_REF] Lefort | The maturity in fetal pigs using a multi-fluid metabolomic approach[END_REF] to address this question (PLS-DA analyses performed independently on each of the fluid), we performed an integrated analysis combining the three fluids using MFA. The effect of the stage of gestation ("Age") was further investigated on plasma metabolome using a differential analysis.

The menu "Integrate/Integrate dataset with MFA" allowed us to perform a MFA for the three metabolomics datasets. The individual plot (first two axes of the MFA with variables "Age" and "TG_F" superimposed as colors and shapes respectively, Fig. 3, top) showed a good discrimination of the stage of gestation on the first axis but not of the fetus genotype "TG_F" (even interactively removing the mixed genotypes from the plot as shown on the right). However, combining the day of gestation (for color) with the mother genotype ("TG_M") allowed us to show a separation between the two mother genotypes at 110 days of gestation on the first axis: Meishan sows (MS) were positionned further on the right of this axis, confirming its relation with piglet maturity (The Meishan breed is expected to be robust and more mature at birth).

The individual plot was put in relation with the variable plot (first two axes of the MFA, correlation threshold of 80%, Fig. 3, bottom), which allows to easily identify metabolites up-concentrated at 90 or 110 days for the different fluids. In particular, glycose-6-phosphate and lactose are up-concentrated in the three fluids at 110 days of gestation whereas fructose and guanidinoacetate (guanidinoacetic acid) are downconcentrated in late (110 days of ) gestation. This confirms previous finding of [START_REF] Lefort | The maturity in fetal pigs using a multi-fluid metabolomic approach[END_REF], which also found these metabolites evolving in the reported directions using independent mixed models for the three fluids. Interestingly, lactose was not found differentially expressed by mixed model in urine in [START_REF] Lefort | The maturity in fetal pigs using a multi-fluid metabolomic approach[END_REF], whereas, the MFA shows a clear (even if weaker) relation of this metabolite with the first axis. Also notable, valine and D-glucose showed opposite evolution in amniotic fluid and plasma (they are less concentrated in amniotic fluid and more concentrated in plasma at 110 days of gestation). In addition, D-glucose is also up-concentrated at 110 days in urine (but to a lesser extent than plasma). This, combined with the expected transfer between the three fluids (from plasma to urine for excretion and to amniotic fluid as a source of nutrients for the fetus), seems to indicate that these metabolites tend to be used by the fetus in late gestation: glucose is used for energy storage (under the form of glycogen [START_REF] Young | Claude Bernard and the discovery of glycogen; a century of retrospect[END_REF]) and essential amino acids, like valine, have already been reported to decrease in amniotic fluid during late gestation in Humans [START_REF] Orczyk-Pawilowicz | Metabolomics of Human amniotic fluid and maternal plasma during normal pregnancy[END_REF].

The menu "Explore/Explore variable in a dataset/2 variables" can be used to further visualize the impact of the stage of gestation on metabolites found important to drive the first axis of the MFA. For instance, used on valine in plasma and amniotic fluid, this menu generates the graphics in Fig. 4, which confirms the findings of the MFA and shows, in addition, that valine is much more concentrated in plasma than in amniotic fluid. Since valine was also not detected in urine, these results tend to indicate that valine, which is not excreted, is used by the fetus in late gestation. Note that both plots can be saved using the corresponding icon for further inclusion in exported report (see next section).

Finally, using the menu "Integrate/Differential analysis", a differential analysis of the plasma metabolome with respect to the day of gestation ("Age") is performed with nonparametric tests (correction for multiple test performed with the Benjamini-Hochberg procedure [START_REF] Benjamini | Controlling the false discovery rate: a practical and powerful approach to multiple testing[END_REF]). The panel "Extract dataset" in this workflow allowed us to export downand over-concentrated metabolites found differential for this analysis (FDR < 5%). These results confirm that (as already discussed before) valine is significantly more concentrated in plasma at 110 days of gestation (adjusted p-value = 1.674e-52).

Detailed results can be exported as CSV files from My workspace and are made available for the reader in the data repository https:// doi. org/ 10. 57745/ TCKSTD. Using these results in MetaboAnalyst 5.0 https:// www. metab oanal yst. ca/ [START_REF] Chong | MetaboAnalyst 4.0: towards more transparent and integrative metabolomics analysis[END_REF], we also confirmed the Galactose metabolism pathway as the most significant pathway for metabolites downconcentrated at 110 days of gestation. This confirms previous findings by [START_REF] Lefort | The maturity in fetal pigs using a multi-fluid metabolomic approach[END_REF]), which also reported that most of the metabolites in this pathway were also found differential in the other two fluids.

Saving results

Finally, in My workspace, the analyses "UnivariateDataset_1", "bivariateAnalysis" and "MFAobj_1" were selected in the panel "All analyses" and a report was obtained by clicking on the button "Export report". These reports are also made available in the data repository https:// doi. org/ 10. 57745/ TCKSTD.

Discussion and conclusions

ASTERICS makes the analysis and combined analyses of datasets easily accessible to people not expert in statistics, data science and programming. Other tools like Orange [START_REF] Demšar | Orange: data mining toolbox in Python[END_REF] might provide a more extensive panel of statistical and machine learning methods but they do not offer the same level of interactivity in plots, do not include data processing methods and analyses specific to certain omics, and require a local installation. Our use case study showed that ASTERICS is able to perform analyses in omics integration settings and to reproduce findings in just a few clicks. Results and plots can be Currently, the version of ASTERICS deployed online is designed to handle small-tomedium datasets (typically of the size of standard RNAseq data) due to limitations in RAM of the deployment server and to the uploading of data over https in uncompressed form. Local deployment does not have this limitation (it is only limited by the computating capacity of the local deployment machine).

Currently, ASTERICS is designed to perform the most common statistical analyses on the most common omics (batch transcriptomics from microarray or RNA-seq experiments, metagenomics, metabolomics or proteomics, in particular as compositional datasets, etc). However, the tool has been programmed in a modular way (both in the frontend and backend) with a full developer documentation so as to allow the extension of ASTERICS to other analyses. Future developments are expected to include more advanced workflows (MOFA, DIABLO, kernel methods), analyses helpful for interpreting results such as gene enrichment analysis and network inference, as well as more recent omics such as single-cell RNA-seq.

Fig. 3

 3 Fig. 3 Individual (top) and variable (bottom) plots obtained from the MFA of the three metabolomics datasets. The right individual plot is interactively obtained from the left one by clicking on the mixed genotype icons of the legend to remove the corresponding data from the plot and increase readability

Fig. 4

 4 Fig. 4 Concentration of valine in plasma (left) and amniotic fluid (right) at the two ages of gestation

Acknowledgements

The authors are very grateful to Jérôme Mariette for his substantial contribution to the first version of the tool. They are also grateful to Valentine Rossi for her contribution to one of the use cases.

Funding

This work has been funded by Région Occitanie https:// www. lareg ion. fr/ (Project "ASTERICS", N° 20008788). The funder had no role in the design of the study and collection, analysis, and interpretation of data and neither in the writing the manuscript.

Availability and requirements

Project name: ASTERICS Project home page: https:// forge mia. inra. fr/ aster ics/ aster ics. (also archived on software heritage under SWHID https:// archi ve. softw arehe ritage. org/ swh:1: dir: 7fbdd c80f2 87596 a6472 d2980 0681e f74b6 82dfe).

Operating systems: platform independent.

Programming languages: R, Python, JavaScript, HTML.

Other requirements: docker for local deployment and none for web usage.

License: GPL 3.

Any restrictions to use by non-academics: none.

Availability of data and materials

The source code of ASTERICS is released under GPL-3 license at https:// forge mia. inra. fr/ aster ics/ aster ics and docker images for its local deployment are also available in this repository. 1 H NMR spectra on which the case study is based are available in the metabolights database [START_REF] Haug | MetaboLights-an open-access generalpurpose repository for metabolomics studies and associated meta-data[END_REF] under accession number MTBLS1541. Processed data (metabolite quantifications), experimental design and output reports of ASTERICS are available at https:// doi. org/ 10. 57745/ TCKSTD. The case study is fully accessible at https:// aster ics. miat. inrae. fr/ works pace/ piglet_ useca se.

Author Contributions

EM, CN, JH, LB, LT, NV contributed to the development of the web part of ASTERICS and to its deployment. EM, JH, YAK, CGu, FM, NV contributed to the development of the R backend of ASTERICS. EM, SD, CGu, AK, LL, NV contributed to the user documentation and use cases. AK designed the layout, page organization and illustrations of ASTERICS. SD, CGu, AS, LT, CGa, LL performed independent tests. DC, CGa, LL, NV designed and supervised the project. NV wrote the manuscript. All authors read and approved the manuscript.

Declaration

Ethics approval and consent to participate Not applicable.

Consent for publication Not applicable.

Competing interests LB, AS, LT, and DC are all employees of the company Hyphen-Stat. The other authors have no conflict of interest.

• fast, convenient online submission • thorough peer review by experienced researchers in your field

• rapid publication on acceptance

• support for research data, including large and complex data types • gold Open Access which fosters wider collaboration and increased citations maximum visibility for your research: over 100M website views per year

•

At BMC, research is always in progress.

Learn more biomedcentral.com/submissions

Ready to submit your research Ready to submit your research ? Choose BMC and benefit from:

? Choose BMC and benefit from:

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.