Daniel Giesecke
email: d.giesecke@ifas.tu-braunschweig.de

Udo Stark

Rubén Harms

Jens Friedrichs

S Y Mpo

O Tatin

Design and Optimization of Compressor Airfoils by Using Class Function / Shape Function Methodology

Keywords:

This paper describes a new design method for high Reynolds number subsonic compressor blade sections for industrial gas turbines and compressors. The focus is on the middle and end stages, where the Reynolds numbers are about 2 to 6 × 10 6 and the Mach numbers between 0.4 and 0.8. The new design method combines i) a parametric geometry definition method, ii) a fast blade-to-blade flow solver, and iii) an optimization tool with a suitable objective function. The development of a new blade section is based on a conventional NACA-65 design, subsequently modified to an optimized CSM profile, where CSM means Class Function / Shape Function Methodology.

The new profile shapes are obtained by superimposing a camber line and a thickness distribution. Both the camber line and the thickness distribution are prescribed as analytical functions to cut down the CPU-time for geometry set up and to guarantee smooth geometries. Numerical calculations are performed by applying the two-dimensional blade-to-blade solver MISES. The optimization method used in this paper is the single-objective genetic algorithm (SOGA) from the DAKOTA library. The objective function consists of 5 components and takes into account the whole loss polar. The corresponding computing time is relatively short -that is 1 to 2 days.

At high Reynolds number, the new profiles show decreased design point losses and increased operating limits compared to corresponding results using conventional NACA-65 profiles. In addition, the present results show close agreement with those produced by so-called high performance profiles of the relevant literature.

INTRODUCTION

High Reynolds numbers and moderate subsonic Mach numbers are typical for the compressor middle and end stages of industrial gas turbines and compressors. The Reynolds numbers of the order of 2 to 6 × 10 6 [START_REF] Schreiber | Effects of reynolds number and freestream turbulence on boundary layer transition in a compressor cascade[END_REF] are much higher than the corresponding aeroengine numbers of about 0.6 to 1.2 × 10 6 [START_REF] Schreiber | Effects of reynolds number and freestream turbulence on boundary layer transition in a compressor cascade[END_REF]. When this was realized some twenty years ago, several new methods were developed taking into account the high Reynolds number and high turbulence effects on boundary layer transition [START_REF] Koeller | Entwicklung einer fortschrittlichen Profilsystematik fuer stationaere Gasturbinenverdichter[END_REF][START_REF] Sieverding | Design of industrial axial compressor blade sections for optimal range and performance[END_REF]. The new blade sections typically show a front loaded pressure distribution, a boundary layer transition (bypass transition) next to the pressure minimum at about 7 to 10% true chord, and a suction side deceleration whose gradient becomes increasingly lower towards the trailing edge. These features usually guarantee a successful design with low design-point losses and low off-design losses and extremely wide loss polars. However, the existing methods have generally long computing times of up to two weeks [START_REF] Sieverding | Design of industrial axial compressor blade sections for optimal range and performance[END_REF] for a complete loss polar. One of the reasons for this is probably the relative large number of geometric parameters (10 in [START_REF] Koeller | Entwicklung einer fortschrittlichen Profilsystematik fuer stationaere Gasturbinenverdichter[END_REF] and 20 in [START_REF] Sieverding | Design of industrial axial compressor blade sections for optimal range and performance[END_REF]) which will all have to be optimized in a time-consuming optimization process. A new method, described in this paper, has only three geometric parameters that have to be optimized. This, together with an analytical formulation of the thickness distribution and camber line, leads to relatively short computing times of two days only. The paper concludes with the introduction and description of the two test cases -one of them a redesign of Profile No. 4 of the Design Case No. 4 in [START_REF] Koeller | Entwicklung einer fortschrittlichen Profilsystematik fuer stationaere Gasturbinenverdichter[END_REF].

METHODS

The new design method combines i) a parametric geometry definition method, ii) a fast blade-to-blade flow solver, and iii) an optimisation tool with a suitable objective function. In the present work, the development of a new blade section is based on a conventional NACA-65 design subsequently modified to an optimized CSM profile, where CSM means Class Function / Shape Function Methodology [START_REF] Kulfan | Fundamental parametric geometry representations for aircraft component shapes[END_REF]. The individual design steps may be summarized as follows: i) design of a conventional NACA-

X f = 0.35 X f = 0.40 X f = 0.45 X f = 0.50 X f = 0.55 X f = 0.60

Blade Section Geometry

The challenge for the geometry method is to produce a fast analytical procedure with a small number of parameters for the design of subsonic, high Reynolds number compressor blade sections. The section shapes are obtained by superimposing a camber line and a thickness distribution. Both the camber line and the thickness distributions are prescribed by analytical functions to cut down the CPU-time for geometry set up and to guarantee smooth geometries.

The camber lines are so-called generalized parabolic arc lines [START_REF] Schlichting | Aerodynamics of the Airplane[END_REF] of the form

Y c (X) = a • X(1 -X) 1 + bX (1)
with a and b representing the following abbreviations

a = 1 X 2 f f c and b = 1 -2X f X 2 f c x t t 2
x y ∆y T E γ with parameters f /c as maximum ordinate (expressed as fraction of chord) and X f as chordwise position of the maximum ordinate, see Fig. 1. As an example, Fig. 2 shows an evaluation of the above formula for f /c = 0.1 and various values of the chordwise position X f of the maximum ordinate. The thickness distributions, as seen in Fig. 3, are derived from the following formulas representing the Class Function / Shape Function Methodololgy [START_REF] Kulfan | Fundamental parametric geometry representations for aircraft component shapes[END_REF] in its simplest form (scaling factor omitted).

Y t (X) = C(X) • S(X) + X • ∆Y T E . (2)
with Class Function

C(X) = √ X • (1 -X) (3)
and Shape Function

S(X) = KR • (1 -X) + 1 KR • X, (4)
where KR is a shape parameter determining the leading edge radius R LE

S (0) = 2R LE , (5)
the trailing edge angle γ S (1) = tanγ + ∆Y T E [START_REF] Drela | A User's Guide to MISES 2[END_REF] and the maximum thickness location X t

Y ′ t (X t ) = (X t ) • (1 -X t ) • S ′ (X t ) (7)
+ -

(X t ) + 1 -X t 2 √ X t • S (X t ) + ∆Y T E (8) Y ′ (X t ) = 0. ( 9
)
As an example, Fig. 4 shows an evaluation of the above formulas for t/c = 0.1, ∆Y T E = 0 and various values of the shape parameter KR.

Altogether, the geometry model has finally only three parameters: two for the camber line and one for the thickness distribution for each blade design.

Blade-to-Blade Flow Solver

The blade-to-blade flow solver MISES 2.63 [START_REF] Drela | A User's Guide to MISES 2[END_REF] has been selected as the flow code for the optimization. This code describes the inviscid flow using the steady Euler equations [START_REF] Giles | Newton solution of steady twodimensional transonic flow[END_REF], while the viscous effects are modeled by the integral boundary layer equations [START_REF] Drela | Two-dimensional transonic aerodynamic design and analysis using the euler equations[END_REF]. The coupled system of nonlinear equations is solved by a Newton technique. The boundary conditions are defined by those of the cascade to be designed and optimized. The design inlet Mach number is kept constant while the inlet angle is varied between positive and negative stall. Boundary layer transition is predicted using the criterion of Abu-Ghannam / Shaw [START_REF] Abu-Ghannam | Natural transition of boundary layers-the effects of turbulence, pressure gradient, and flow history[END_REF] in a slightly modified version [START_REF] Drela | MISES Implementation of Modified Abu-Ghannam / Shaw Transition Criterion[END_REF] to achieve a better modeling of the flow physics at transition. The solver has been sufficiently validated for cascade flows at low and high Reynolds numbers, [START_REF] Schreiber | Advanced highturning compressor airfoils for low reynolds number condition -part II: Experimental and numerical analysis[END_REF][START_REF] Sonoda | Aerodynamic characteristics of supercritical outlet guide vanes at low reynolds number conditions[END_REF] and [START_REF] Koeller | Entwicklung einer fortschrittlichen Profilsystematik fuer stationaere Gasturbinenverdichter[END_REF][START_REF] Benini | Development of high-performance airfoils for axial flow compressors using evolutionary computation[END_REF] respectively, and at various turbulence levels with and without turbulence grids [START_REF] Schreiber | Effects of reynolds number and freestream turbulence on boundary layer transition in a compressor cascade[END_REF]. AVDR effects are taken into account by a hyperbolic tangent variation of the streamtube thickness [START_REF] Stark | The combined effect of axial velocity density ratio and aspect ratio on compressor cascade performance[END_REF][START_REF] Stark | Ebene Verdichtergitter in quasi zweidimensionaler Unterschallströmung[END_REF].

Optimization Method

The optimization method used in this paper is the singleobjective genetic algorithm (SOGA) from the DAKOTA library [START_REF] Adams | Dakota, A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis: Version 6.4 User's Manual[END_REF]. The SOGA starts with a random generation of profiles within the limits of the geometric parameters (50 samples per generation). The generated individuals will be evaluated by means of an objective function and passed or not passed to the next generation (up to 19). The accepted individuals will be promoted, the others will be killed. The latter fail to produce the required exit angle β 2 in specific limits.

The objective function of the present investigation consists of 5 components and takes into account the whole loss characteristic. This function has previously been used by Köller et al. [START_REF] Koeller | Entwicklung einer fortschrittlichen Profilsystematik fuer stationaere Gasturbinenverdichter[END_REF], by Sieverding et al. [START_REF] Sieverding | Design of industrial axial compressor blade sections for optimal range and performance[END_REF]

β 1 0.8 ∆β 1 ∆β 1,stall ω D 2 ω D ω 80 ∆β 1 β 1,D ω Figure 5
. Parameters of the objective function, adapted from [START_REF] Sieverding | Design of industrial axial compressor blade sections for optimal range and performance[END_REF] and is defined as follows:

OBF = C 1 ω D ω D,ref + C 2 ∆β 1,ref ∆β 1 + C 3 ∆β 1,stall,ref ∆β 1,stall + C 4 ω 80 ω 80,ref + C 5 σ 80 σ 80,ref (10)
Each component of the function has been normalized using a corresponding reference value (index ref ) of a suitable reference profile. An illustration of all components is shown in Fig. 5, where ω D is the design loss coefficient at the design inlet angle β 1,D . A loss coefficient of twice the value of ω D defines the operating range ∆β 1 on the loss characteristic. The difference between the (positive) stall point and the design point is generally known as stall margin ∆β 1,stall or safety against stall. For the inner 80 % of the operating range the average value ω 80 and the standard deviation σ 80 are introduced to achieve a loss characteristics as flat as possible. The coefficients C are so-called weighting factors by which the relative importance of a component can be changed. Any change has to be specified by the user and validated by test runs. The results of this paper have been produced with the following weighting factors: C 1 = 1, C 2,4,5 = 0.5 and C 3 = 2.

RESULTS

Test Case 1

In the present work, the development of a new blade section starts with a conventional NACA-65 profile which is subsequently modified to an optimized CSM profile. This development is best documented in table form with four columns for the different stages of the evolution, The new design method in this paper has been tested (Test Case 1) on the design of a high Reynolds number subsonic compressor blade section for industrial gas turbines and compressors. The design specifications belong to the Test Case 4 of reference [START_REF] Koeller | Entwicklung einer fortschrittlichen Profilsystematik fuer stationaere Gasturbinenverdichter[END_REF], and are shown here in Table 1, first column. The Mach number, Reynolds number and turbulence level are M a 1 = 0.44, Re 1 = 2.5 × 10 6 and T u 1 = 3% respectively. For further details see Table 1, columns 2, 3 and 4.

Figure 6 presents all three profiles -the conventional, the reference and the optimized profile. The corresponding blade pressure distributions are shown in Fig. 7 at design air inlet angle (β 1 = 47 • ). An inspection of the optimized pressure distribution led to the conclusion that optimization at high Reynolds numbers and turbulence levels of the present investigation inevitable ends up with a front loaded profile.

As indicated in Fig. 7, boundary layer transition (bypass transition) starts right after the leading edge on the suction side and somewhat downstream on the pressure side for both the conventional NACA-65 and the optimized CSM profile. Deceleration of the turbulent suction side boundary layer begins shortly after the leading edge for the optimized profile and at about 25% chord length for the conventional profile with a gradient becoming increasingly smaller or steeper respectively.

The controlled diffusion of the optimized CSM profile leads to a 0.8% reduction in design point losses. At off-design, the controlled diffusion concept remains essentially valid, specially at the higher inlet angles. There, the conventional profile is the first to reach the operational limit, while the optimized CSM profile generates a significantly increased operating range (+45%) and stall margin (+18%), cf. Fig. 8. The increase in operating range and stall margin is only partly due to decreasing deceleration gradients. The main cause turns out to be the reduced leading edge sharpness, which makes the profile less sensitive to off-design air inlet angles.

Figure 9 shows the optimized CSM profile of this paper together with the corresponding profile 4 of reference [START_REF] Koeller | Entwicklung einer fortschrittlichen Profilsystematik fuer stationaere Gasturbinenverdichter[END_REF]. A comparison revealed noticeable differences, in opt. airfoil -Exp., [START_REF] Koeller | Entwicklung einer fortschrittlichen Profilsystematik fuer stationaere Gasturbinenverdichter[END_REF] Figure 10. Optimized Test Case 1 performance in comparison with Test Case 4 profile of [START_REF] Koeller | Entwicklung einer fortschrittlichen Profilsystematik fuer stationaere Gasturbinenverdichter[END_REF] particular close to the leading edge, where camber and curvature of the optimized CSM profile are relatively low. Numerical performance curves for both profiles are shown in Fig. 10, where they demonstrate a remarkable agreement in spite of considerably different profile shapes. In addition, Fig. 10 presents a comparison between numerical and experimental Test Case 4 results. The agreement, however, is not as good as before for the numerical results. These differences were already observed and described in [START_REF] Koeller | Entwicklung einer fortschrittlichen Profilsystematik fuer stationaere Gasturbinenverdichter[END_REF], but a clear-cut explanation is still missing.

Test Case 2

The new design method of this paper has been developed for high Reynolds number applications. Nevertheless, it has been assumed that the new method might also be useful at lower Reynolds numbers (aeroengine Reynolds numbers or lower). This has been checked in M a 1 = 0.12 X f = 0.5 -→ 0.45 -→ 0.43 Re 1 = 3.5 × 10 5 f /c = 0.1 (C) -→ 0.1 (P) -→ 0.1065 (P) 2 for Test Case 2. The design specifications appear in the first column, specially the Mach number M a 1 = 0.12, the Reynolds number Re 1 = 3.5 × 10 5 and the turbulence level T u 1 = 1%.

β 1 = 30 • X t = 0.4 -→ 0.2281 -→ 0.2128 β 2 = 0 • KR = 2.0 KR = 2.84 t/c = 0.1 λ = 13 • -→ 11
The following three columns 2, 3 and 4 present, as before in Test Case 1, the main geometric parameters (X f , f /c and X f (KR)) for the conventional NACA-65 profile, the reference CSM profile and the optimized CSM profile. All three profiles are shown in Fig. 11, the corresponding design point (β 1 = 30 • ) pressure distribution in Fig. 12 and the complete loss and turning characteristics in Fig. 13, together with experimental results for the NACA-65 profiles.

The most noticeable difference between the conventional NACA-65 and the optimized CSM profile is again the upstream moved location of the maximum thickness for the optimized CSM profile, cf. Fig. 11. Transition of the laminar boundary layer under Test Case 2 condition (low Reynolds number, low turbulence level) happens to occur via laminar separation bubbles downstream the pressure minimum on both sides of the profiles, cf. Fig. 12. In spite of a smaller deceleration gradient and a reduced leading edge sharpness, it is the optimized CSM profile that shows 13% higher design point losses compared to those for the conventional NACA-65 profile with relatively long portions of laminar boundary layers, cf. Fig. 13. However, thanks to the smaller deceleration gradient and the reduced leading edge sharpness, the predicted operating range and stall margin increased by more than 55 and 65% respectively.

CONCLUSION

In summary, a new method has been presented for the design of high Reynolds number, subsonic compressor blade sections for industrial gas turbines and compressors. By carefully selecting the number of design and optimization parameters, the combination with a genetic algorithm led to a competitive design method.

At high Reynolds numbers, the new profiles show decreased design point losses and increased operating limits compared to corresponding results using conventional NACA-65 profiles. In addition, the presented results show close agreement with those produced by so-called high performance profiles of the relevant literature. At low Reynolds numbers, the new profiles show slightly increased design point losses but again considerably in- However, for the chosen design approach further research is required. Applying it to airfoils for aeroengine application will be part of future research. Furthermore, cascade experiments would validate the new method and numerical results.

NOMENCLATURE

  Figure 1. Camber line parameters

Figure 2 .

 2 Figure 2. Camber lines of max. camber f /c = 0.1 at various chordwise positions

Figure 3

 3 Figure 3. Thickness parameters

Figure 4 .

 4 Figure 4. Thickness distributions of max. thickness t/c = 0.1 at various chordwise positions X t (KR)

M a 1 =Figure 6 .

 16 Figure 6. Test Case 1 profiles in comparison

Figure 7 .

 7 Figure 7. Test Case 1 pressure distributions at design inlet angle β 1 = 47 •

Figure 8 .Figure 9 .

 89 Figure 8. Test Case 1 overall performance characteristics

  -CFD,[START_REF] Koeller | Entwicklung einer fortschrittlichen Profilsystematik fuer stationaere Gasturbinenverdichter[END_REF]

Figure 11 .

 11 Figure 11. Test Case 2 profiles in comparison

Figure 12 .

 12 Figure 12. Test Case 2 pressure distributions at design inlet angle β 1 = 30 •

Figure 13 .

 13 Figure 13. Test Case 2 overall performance characteristics

Table 1 .

 1 Test Case 1

	Design Specifications	NACA65	CSM ref.	CSM opt.

Table 2 .

 2 Test Case 2

	Design Specifications	NACA65	CSM ref.	CSM opt.


Silvio Geist
email: silvio.geist@hsu-hh.de

Niklas Neupert

Franz Joos

S Y Mpo

O Tatin

Measuring the Effect of Moist Droplet-laden Flow on Transonic Compressor Cascade Performance

Keywords:

is study provides insight on the influence of air humidity on droplet laden flow as it occurs during inlet fogging and in steam turbines. D-LDA is used to gather air-only data at three different humidities to rate humidity-depended cascade losses such as shock and profile boundary layer losses. Likewise, change in aerodynamics is explained using a combination of isentropic and Rayleigh flow. en, water droplets are added to the flow field and data is collected using D-PDA. Again losses are evaluated. Regarding the droplet-free flow, effects and relations found in literature could be confirmed for the presented cascade flow. However, the results further indicate that for droplet-laden flows favourable humidity effects are decreased and may be neglected since the major loss arises from the increased boundary layer thickness.

INTRODUCTION

Nowadays, power demands are met by an increasing amount of renewable resources. However, peak loads are still covered by conventional approaches such as gas turbines which have to respond quickly on possible supply fluctuations of the renewables. In that regard, high-fogging is a common procedure to enhance gas turbine power output. Small-sized water droplets, sprayed into the engine's inlet, are intended to evaporate, thus absorbing heat from the surrounding flow. As a result the specific work needed for compression is reduced. For instance this increase in flexibility helped to fulfill the stringent requirements of the UK grid code by providing an additional extra output of MW for the Marchwood power plant within a short period of time [ ]. Apparently, high-fogging is particularly useful in hot regions where the cooling potential is high. Regarding, e.g. equatorial areas, operating conditions may not only be hot, but also at a high level of humidity. Furthermore the procedure itself changes the air's humidity. As a result regions of different saturation occur within the flow field, giving rise to condensation. Especially transonic flow fields, as they occur in the tip region of compressor blading, in turbine passages, nozzels and around airfoils may be altered strongly that way. Hence, understanding the interaction of the two opposing processes of evaporation and condensation is essential for realistic design of turbomachinery under atmospheric conditions. Schnerr and Dohrmann [ ],[ ] calculated transonic flow around an airfoil and found that increasing humidity decreases shock strength and shi s the shock position downstream. is is due to local condensation of water vapour. If moist air is accelerated along the airfoil's suction side, static pressure and temperature will decrease. at reduces the air's dew-point and encourages condensation. During condensation, heat is released into the surrounding gas, which decreases the local Mach number and eventually reduces shock intensity. . Unfortunately, no reference measurement data were on hand for the transonic compressor case which underlines the need of the present study. Both measurements and simulation indicate increased total pressure losses in condensation cases, although shock strength and position remained almost unaltered. Concerning compressors, however, they state a rise in passage static pressure which drives the shock downstream and strengthens it. Although moisture effects on transonic flow is studied thoroughly, the interaction of both moist and droplet-laden flow in the vicinity of shocks is not well understood. Hence, it is of great relevance for compressor power enhancement through fogging or wet steam turbine operation.

is study presents measurements of transonic compressor cascade flow velocity utilising a D-LDA/PDA system. Performance quantities deflection, velocity ratio and profile loss are used to evaluate the different flow conditions. Furthermore, to illustrate the underlying physics the throughflow is expressed as a combination of isentropic and Rayleigh flow.

. METHODS is section briefly presents the test rig's structure before mentioning measurement instruments used. at includes pressure and temperature probes to monitor operating point, moisture sensors to gain humidity information as well as a description of the optical D-LDA/PDA system. en, the measurement campaign is outlined. Finally, this study's key performance parameter, loss coefficient, is introduced.

Test Rig To clarify the impact of droplets on moist flows and vice-versa nine blades with profiles similar to modern compressors' outer cross-sections have been mounted within a cascade as shown in Fig. .  at creates a similar flow field, as it can be found in the tip regions of real gas turbine T ot,1

K

Table . Test rig parameters summary compressor stages. Previous studies have shown a nearly undisturbed flow around the centre blades regarding the lower and upper test section walls. Hence, all measurements are conducted around the fourth and the fi h blade counting from the lower wall. e accompanied wind tunnel is situated at the Laboratory of Turbomachinery at the Helmut-Schmidt-University in Hamburg, Germany. Outside air is compressed by a radial compressor which is then led into the test rig's se ling chamber. Excess air may be bypassed to adjust mass flow precisely. e se ling chamber is a boiler-shaped cylinder which is 1.6 m in diameter. Several sieves equalise flow before it is accelerated through a nozzle to enter the test section. Since, the inlet Mach number of Ma 1 = 0.95 is close to unity and increased compared to the design value of Ma design = 0.89 a shock system is always existent on the blades' suction sides as displayed in Fig. . Outlet air is purged in a water-separator before it is discharged through the chimney. Build-in variable vanes back pressure the flow to set a pressure ratio across the measurement section. Notable crosssection parameters are listed in Tab. and illustrated in Fig.

for convenience. Further details of the test rig can be found in [ ]. Note that the velocity triangle in Fig. only underlines the substitution of the relative velocities w 1/2 through the cascades air flow. Hence, the cascade is not rotating or moving.

Instrumentation Overall, three air-only data sets at different specific humidities, namely X L = 3 g/kg, X L = 6 g/kg and X L = 11 g/kg were measured at different blade loadings while the inlet Mach number and total temperature were kept constant. is is monitored using five pressure taps and two hard-wired resistance thermometers on each side wall. e inflow angle was set using a crank mechanism which rotates the cascade's blades around the centre blades leading edge. However, inflow angles used for calculation and evaluation were measured using the more precise D-LDA along traverse in Fig .

To ensure a well defined specific humidity two moisture sensors have been utilised. e first measures the water content of the air drawn from the environment. Based on that measurement demineralised water is sprayed in the air supply's outlet to adjust moisture at the second sensor position in front of the cascade. Due to high duct temperature and residence time within the duct connecting the supply with the se ling chamber the spray is evaporated entirely therein. To maintain a virtually D flow in the test section, the sidewall boundary is re-energised using compressed air which is injected via a slit nozzle system in front of the cascade [ ].

In addition to the air-only measurements each parameter set was measured with a droplet mass fraction of ξ m = 0.01.

at is the ratio of droplet mass to total mass. To do so, water was injected into the se ling chamber via a manifold ofimpingement nozzles (BETE PJ ) generating a spray with a mean diameter D 10 = 7.0 µm and a Sauter mean diameter D 32 = 38.7 µm at the cascade's entrance equaling those utilised in industrial overfogging applications [ ].

A non-invasive D-LDA/PDA technique was used to collect the flow field's velocity components. Sidewalls made of acrylic glass allow visual access. Light was supplied by a 5 W Argon-Ion laser. A D probe with a lens of 160 mm focal length and no beam expansion was used. at resulted in a measuring volume of 0.078 mm x 0.077 mm x 0.65 mm for the 514.5 nm beam and 0.074 mm x 0.073 mm x 0.62 mm for the 488 nm beam. To conduct both, LDA and PDA measurements with only one setup a side sca er configuration was chosen which included a fiberPDA photomultiplier and a Dantec BSA signal processor to process the photomultiplier output. e system is capable of measuring velocities with an error of less than 6 % and droplet diameters with less than 3 %. It was traversed half a chord length (c = 50 mm) in front of the blades leading edges to gather inlet informa-tion. A er traversing through the passage the profiles wake was eventually measured at traverse , situated half a chord length downstream the trailing edge. Based on that data performance parameters, profile loss and isentropic Mach number distribution through the passage were extracted and evaluated.

To account for local temperature change while traversing through the passage the speed of sound is determined based on the measured local velocity and inlet total temperature which is measured within the se ling chamber. Consequently, the general Mach number

Ma = w √ T κR ( )
is altered by substituting local temperature using

T tot = T + w 2 2c p ( )
which eventually yields

Ma = w T tot - w 2 2c p κR ( )
as the local isentropic Mach number discussed herein.

Loss coefficient

is parameter has been used in previous research [ ] and is repeated here for convenience.

ω loss = 2 Θ w t cosβ 2 w 2 w 1 2 ( )
where

Θ w = ∫ Θ 99 1 - w 2 w 2 w 2 w 2 δΘ ( )
is the wake momentum thickness. e loss coefficient is based on velocity measurements and is the only applicable source of information about the losses in the cascade since reliable measurements of total pressure were not possible due to droplet disturbance. Eqn. consists of two main terms, mean deceleration over the cascade and the momentum wake losses accounted in Eqn. . Hence, traverse two velocity distributions with narrower wakes containing a high momentum, generate fewer losses than a comparable fast outflow with a broad wake. e question arises on which distance to measure such a wake. Scholz [ ] states up to % deviation when simply averaging outflow velocity and angle at an arbitrary downstream position.

erefore, he proposes a method to compute the equivalent homogeneous outflow where wake has vanished. Values w 2 and β 2 used during loss computation have been computed according to that method. ese terms will be of great importance later on. Due to the fact that ω loss was derived by the integration of the wake, only the loss of momentum was accounted for. us, the evaluation of losses based on momentum deficit sufficed when droplets are added to the flow. However, only the smallest droplets (< 2 µm) which followed the airflow closely were taken into account during the evaluation. Note that thermodynamic effects between the phases, such as heat transfer, were omi ed when it came to loss analysis. In conclusion, this paper rates the aerodynamic losses due to heat transfer between the phases, but not the heat transfer loss itself. is might be a goal of future publication.

. RESULTS AND DISCUSSION

e aim of this study is to provide information and insights on how compressor cascade's wake and shock losses change for air of different humidity in the vicinity of droplets. us, air-only measurements are first presented and evaluated.

ese results illustrate flow features as they are stated in the literature which verifies the over-all approach. Here, the change in state values is illustrated utilising a combination of isentropic and Rayleigh flow to explain underlying physics.

en, data including droplets are presented and compared to air-only flow to derive coupling effects between the substances. Finally, performance data are compared to rate the influence of humidity on both cases. Please note that these data originate from a greater campaign. For more measurement data, including different cross-sections, sprays and surface wetabilities be referred to [ ].

. Air-only results

First, differences in isentropic Mach number distribution due to humidity at a specific inflow angle are examined which eventually leads to a diabatic treatment of the flow. Second, humidity impact on the cascade performance is evaluated based one the loss coefficient ω loss thus considering wake and profile boundary layer loss. compressor passage shape is very similar to a diffuser. Consequently, when flow is accelerated around the leading edge (x/c = 0) reaching supersonic conditions, as it is the case in that study, the diffuser shaped passage accelerates the flow even further (0.25 > x/c > 0) as in the divergent part of a Laval nozzle. When the shock delays flow into the subsonic region (x/c > 0.25) it is in turn slowed down by the divergent passage shape, forming the well known Mach number distribution. Now, comparing distributions at different humidity levels, it is apparent that an increased humidity lowers the pre shock Mach number and thus reduces shock strength.

Diabatic Flow

is behaviour may be expressed by a flow as proposed by Schneer [ ] who researches stability of compressible fluids with energy supply. It holds the following conditions:

• flow is of converging diverging nozzle type

• one-dimensional flow • variable cross-sectional area • heat exchange.
Schnerr incorporates both Hugoniot equations for shock and Rayleigh equations for heat exchange treatment. at results in formulations describing state variable change through the passage which are extensions of the well known adiabatic shock solutions. e continuous heat addition through droplet condensation during supersonic flow conditions increases static pressure, thus reducing dynamic pressure, and local speed of sound. [ ],[ ] Both effects evoke a reduction of the maximum Mach number and the preceding Mach number slope. However, measurement data do not account for the temperature effect since Mach number is calculated based on inlet total temperature and no information on field temperature is available. Still, theory depicts measurements well, thus giving a meaningful aid when understanding the physical background. Keeping that in mind wake loss is examined over the inflow angle range within the next section. Loss coefficient is highest at specific humidity X L = 3 g/kg over the entire range of measured data, whereas humidity in general reduces the loss coefficient. e reason for this are reduced shock losses due to the aforementioned theory. More precisely, regarding ω loss , wake momentum thickness Θ w is reduced through a decreased wake height as it is shown in Fig. . As for the loss curve, the overall trend is related to the altered blade loading and the, in this particular case more important, curve's offsets are induced by the change in humidity. At higher blade loadings , i.e. β 1 > 149°an increase of specific humidity further decreases loss since wake heights Θ w for X L = 6 g/kg and X L = 11 g/kg diverge with the la er having the smaller slope. Considering small inflow angles there is still a notable wake momentum thickness reduction when compared with dry flow. A further humidity increase has no significant effect. Yet, note the intersection at β 1 = 148.5°. Surprisingly, focusing again on Fig. , X L = 6 g/kg depicts smaller loss for β 1 < 149°and losses re-increase for X L = 11 g/kg . e reason is the alteration of mean deceleration through humidity illustrated in Fig.

.

For high blade loading the passage shock is situated near the leading edge which is why mean deceleration at the back traverse remains constant for all three humidity levels since curvature acceleration is large compared to the humidity effect. at is illustrated clearly for large inflow angles in Fig.

. e main reason for this is that the passage shock moves towards the trailing edge for small inflow angles which alone strongly influences w 2 . Furthermore the condensation additionally pushes the shock downstream. Post-shock Mach number and thus back traverse velocity is much higher due to the reduced shock strength. at in turn decreases cascade performance. However, for high blade loadings shock position is located near the leading edge and w 2 is nearly constant due to the curvature acc-and deceleration.

. Droplet-laden results is section discusses measurement results when droplets (ξ m = 0.01) are present within the flow. It has to be noted at that point that the vicinity of droplets, i.e. the existence of a fluid phase changes flow properties such as density, compressibility and speed of sound as stated by Brennen [ ]. He encourages to use so called effective values describing an equivalent homogeneous distributed gas. Such effective

loss coefficient ω loss [-] X L =3 g/kg, ξ m =0.01 X L =6 g/kg, ξ m =0.01 X L =11 g/kg, ξ m =0.01 X L =3 g/kg, ξ m =0.00 X L =6 g/kg, ξ m =0.00 X L =11 g/kg, ξ m =0.00
Figure .

Comparison of loss coefficients for air-only and droplet-laden flow at various incidence angles values are basically ideal gas values weighted with particle loading, i.e. ξ m . As a consequence, for a droplet-laden flow with ξ m = 0.01, effective Mach number is about 1.3 % higher compared to isentropic flow conditions. However, for a clear description of the humidity effect, droplet-laden results are still expressed using isentropic inflow Mach number.

According to theory those droplets act as heat sinks when evaporating on the one hand and on the other hand as condensation nuclei. Hence, a change in the net heating could be expected.

In fact, comparing the loss coefficients as displayed in Fig. all wet flow curves not only generate more losses, but cluster as well.

is is in distinct contrast to the aironly results. Especially, when consulting the Mach number through the passage shown in Fig. which basically shows distributions as expected. First, a high specific humidity decreases the pre shock Mach number and shock strength. Second, droplets evaporate and cool the flow. is accelerates the flow resulting in an about 5 % higher peak Mach number of Ma 1 ≈ 1.25 for X L = 3 g/kg compared to Ma 1 ≈ 1.2 at air-only conditions. For humid air this cooling potential is reduced due to saturation limit, resulting in smaller slopes for cases X L = 6 g/kg and X L = 11 g/kg. Consequently, that does not explain similar loss coefficients at different humidities. Regarding deflection a decrease between 5 % and 15 % can be stated which is similar to air-only flow. A quick view on the comparison of deceleration ratios in Fig.

excludes that term as the reason as well. No significant change between air-only and wet cases at constant humidity is recognisable. In fact the constant, high loss is caused by a thickened wake momentum thickness as shown in Fig. .  at loss is dominant compared to the performance gain due to reduced shock loss.

. Conclusions

As thermodynamic effects between the phases are neglected so far, the evaluation of losses based on momentum deficit Comparison of wake momentum thickness Θ w for air-only and droplet-laden flow over inflow angle at different humidities stays valid when droplets are added to the flow. at is why cascade losses do not account for the potential decrease of evaporation due to the simultaneous condensation at the moment. A study examining the combined performance gain or loss remains to be carried out. Yet, the following conclusions can be made.

General

• Change in state variables for both air-only and dropletladen flow of low or high humidity can be described using a combination of isentropic and Rayleigh flow. at reveals a contrary influence of heat sinks through evaporation and heat sources through condensation.

e systems loss is then determined by both decreased shock losses through heat addition, acceleration due to evaporation and momentum transfer from flow to droplets which directly impacts wake shape.

Air-only flow

• Loss coefficient is highest at dry conditions over the entire range of measured data, whereas humidity in general reduces the loss coefficient. e reason for this is a reduction of shock losses resulting in a smaller and less decelerated wake. As a result loss is reduced by 25 % at high and intermediate loading and X L = 11 g/kg and about 30 % at low blade loading and X L = 6 g/kg • e swap in the maximum loss from high to intermediate humidity at low blade loading originates from the downward shi ed shock position. High post shock Mach numbers at high humidity cannot be levelled and impact the mean flow velocity w 2

• Deflection is reduced between 5 % and 15 % whereas deceleration ratio is altered only between 1 % and 4 %.

Both effects are favourable for the overall loss reduction. However a lower deflection decreases the mere cascade performance output as well.

Droplet-laden flow

• e loss is dominated by increased momentum loss due to a thicker boundary on the suction side which is larger than the reduction in shock loss. us, the loss increases and is approximately 30 % higher for inflow angles β 1 > 150°. at increase gradually shrinks with smaller inflow angles to about 10 % at β 1 > 146°.

• e maximum isentropic Mach number is higher compared to air-only flow since droplets evaporate and accelerate the flow. Humid air is less accelerated because the amount of evaporation is limited by the saturation level. However, the effect of condensation is still observable when comparing pre shock Mach numbers at different humidities.

Two different goals have been issued during the research presented. e first adopts a promising description of compressible gas flow, including condensation and reviews it in the vicinity of a discrete fluid phase. e second concerns the impact of condensation on two-phase flow compressor cascade performance. Experiments reveal that for air-only flow ambient humidity has to be considered in the design process, particularly during numerical simulations. e condensation potential strongly alters the shock position and is favourable concerning losses, but reduces compressor stage's key parameters such as deflection. As for droplet-laden flows favourable humidity effects are decreased and may be neglected since the major loss arises from the increased boundary layer thickness with respect to the loss coefficient. However, a direct measurement of pressure based performance was not possible at the time. Last, published experimental data on that topic are rare so that data gathered herein may be of value to fellow researchers.

Figure .

 . Figure . Illustration of the measurement section

Figure .

 . Figure . Schlieren photograph of air-only flow at β 1 = 153°and X L = 3 g/kg; clearly visible is the normal shock located at about one third of the total chord length

Figure .

 . Figure . Illustration of cascade nomenclature

Figure .

 . Figure . Isentropic Mach number over passage traverse at β 1 = 154°for different humidities

Figure .

 . Figure . Top: Scheme of compressor blade passage Bo om: Mach number distribution at the dashed line for a combined isentropic /Rayleigh flow and pure isentropic flow with varying cross-section according to [ ]

  Figure illustrates such an abstract Mach number distribution along the dashed line.

Figure .

 . Figure .Loss coefficient for dry and humid air flow at various incidence angles

Figure .

 . Figure . Isentropic Mach number over passage traverse β 1 = 154°1

[

  ] N. Neupert. Experimentelle Untersuchung einer tropfenbeladenen Strömung in einer ebenen Verdichterkaskade. PhD thesis, Helmut-Schmidt-Universität, Hamburg, . [ ] G. H. Schnerr. Unsteadiness in Condensing Flow: Dynamics of Internal Flows with Phase Transition and Application to Turbomachinery. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, . E. Brennen. Fundamentals of multiphase flow. Cambridge University Press, Cambridge [England], New York, .

  With decreasing inflow angle curves diverge and eventually characteristics of the intermediate and high humidity cross at β 1 = 148.5°. at indicates that mean back traverse

	wake momentum thickness Θ [m] w	0.006 0.008 0.01 0.012 0.014 0.016 0.018	X L = 3 g/kg X L = 6 g/kg X L =11 g/kg
		0.004
			146	148		150	152		154
				inflow angle β 1	[°]
	Figure . Wake momentum thickness Θ w over inflow angle
	at different humidities
		0.9					X L	=3 g/kg
							X L	= 6 g/kg
	[-]	0.85					X L	= 11 g/kg
	/w 1
	deceleration w 2	0.75 0.8
		0.7
		0.65
		147	148	149	150	151	152	153	154
				inflow angle β 1	[°]
	Figure . Deceleration w 2 /w 1 of equivalent, homogeneous flow based on back traverse data over inflow angle at
	different humidities
	velocity is notably greater in the la er case.


Frances M Brodsky

Franck Perez

Thomas E Kreis

LOOKING BACK TO TRAFFIC FORWARD: REPORT ON A SYMPOSIUM TO HONOR THOMAS KREIS (1946-1998) AND HIS INSPIRATION

to honor the fruits of his legacy from the previous twenty years and to consider where studies of membrane traffic might lead in the next twenty. Thomas was a founding editor of Traffic and his enthusiasm was instrumental to its launch [START_REF] Brodsky | Green Light for Traffic[END_REF][START_REF] Brodsky | Life History of the Journal TRAFFIC, Celebrating Ten Years of Publication[END_REF]. Therefore, we chose this 20 th Anniversary issue of Traffic to report on the symposium that celebrated his influence on the community and to share the emerging discussion about cell biology of the future.

Thomas was an outstanding cell biologist who pioneered dynamic analysis of intracellular membrane traffic, contributing seminal research on microtubules and their role in intracellular transport (see for example [START_REF] Kreis | Stress fiber sarcomeres of fibroblasts are contractile[END_REF][START_REF] Allan | A microtubule-binding protein associated with membranes of the Golgi apparatus[END_REF][START_REF] Kreis | Oligomerization is essential for transport of vesicular stomatitis viral glycoprotein to the cell surface[END_REF][START_REF] Kreis | Microtubules containing detyrosinated tubulin are less dynamic[END_REF][START_REF] Duden | b-COP, a 110 kD protein associated with non-clathrin-coated vesicles and the Golgi complex shows homology to b-adaptin[END_REF][START_REF] Pierre | CLIP-170 links endocytic vesicles to microtubules[END_REF][START_REF] Pepperkok | b-COP is essential for biosynthetic membrane transport from the endoplasmic reticulum to the Golgi complex in vivo[END_REF][START_REF] Scales | Visualization of ER-to-Golgi transport in living cells reveals a sequential mode of action for COPII and COPI[END_REF][START_REF] Perez | CLIP-170 highlights growing microtubule ends in vivo[END_REF]. He was a wonderful colleague to members of the cytoskeleton and the membrane traffic communities and was also very active in these communities. At the time of his passing, Thomas was Chair of the Cell Biology Department in Geneva, served on a number of advisory and journal boards and as co-editor of the encyclopedic Guidebook to the Cytoskeletal and Motor Proteins [START_REF] Kreis | Guidebook to the cytoskeletal and motor proteins[END_REF].

His loss was mourned in obituaries that appeared in Nature [START_REF] Simons | 98[END_REF] and Trends in Cell Biology (14). In tribute to Thomas the 2018 symposium in his honor was supported by the journal Traffic, the Institut Curie, University College London, the French Society for Cell Biology (SBCF), the LabEx CelTisPhybio and Donna St George for the Thomas Kreis Legacy. Participants (Figure 1) included colleagues, collaborators and former lab members from the USA, Europe and Israel, who all supported their own travel and accommodation. PhD students from the Institut Curie, as well as local colleagues from Paris attended. We were especially thrilled to have Donna St George (widow) and Eric Kreis (son) join us from the USA and Switzerland for the symposium.

The symposium included talks from cell biologist colleagues with particular reference to how their work was inspired by interactions with Thomas, as well as personal reminiscences. What was also special for everyone present was a focus on where the science discussed was going. Thomas foresaw how important dynamic microscopy imaging would become for understanding intracellular transport, which indeed played out over the past twenty years. The participants strove to identify equivalent new developments and challenges for the next twenty, which produced the following predictions and directions.

Molecular cell biology of the past twenty years has realized the definition of basic membrane traffic pathways and identified not only key molecules, but also the intracellular molecular machines involved. Basics of their structure and regulation by signaling have emerged and their dynamics visualized. Much of this work was enabled by genetic studies in yeast and model organisms and biochemical reconstitution of trafficking steps (recognized by the Nobel Prize of 2013 to Rothman, Schekman and Südhof), as well as developments in imaging and structural studies. It is clear that the next frontier is to integrate this knowledge in multi-scale models to understand how these processes are linked. This challenge includes connecting our understanding of molecular machines with their functions in intracellular pathways, linking twodimensional studies of cell function to three-dimensional interactions in organs and tissues, and linking cellular pathways to physiological processes dependent on organ and tissue interaction (direct and at a distance). To achieve the latter, it will be necessary to consider how intracellular pathways affect whole body metabolism, immune system and nervous system function, and development of complex organisms.

Additionally, the converse must be considered: how physiological and genetic conditions influence cellular processes. For example, how circadian rhythms, gender and aging affect intracellular pathways is only just beginning to be appreciated. The resulting cellular plasticity will of course feed forward to change systemic conditions. Currently the field has linked genetic mutations with cellular and developmental phenotypes in model systems and with diseases in humans. Defects in membrane traffic are associated with cancer, diabetes, cognitive disability, musculoskeletal disease, cystic fibrosis and microbial pathogenesis. However, to truly understand how a single mutation influences disease the problem of multi-scale integration from the molecular to the physiological will need to be addressed. This will also hold true for understanding how pathogens hikack intracellular pathways and, conversely, what we can learn from this about infection and about host function.

To meet the multi-scale integration challenge, there are cutting edge approaches that need to become routine, as well as a need for development of new technologies. The contributions of computational data analysis and modeling, as well as chemistry and physics to understanding how intracellular pathways impact biology are opening up new ways of thinking. The need to understand phenomena like mechano-transduction and the effects of small metabolites underscore the importance of integrating physics and chemistry with cell biology. These cross-disciplinary approaches will also lead to new tools, as physics and computational advances have already expanded imaging capabilities and intracellular chemical modification has enabled local probing of protein interaction. Further developments in chemical biology and ways of measuring physical properties of cells and tissues will be key to multi-scale integration. The community wish list also includes better methods of acute interference in cellular processes to study their influence in real time, as well as live cell imaging at super-resolution. Last, while the use of cell models will continue to dominate the field, in particular thanks to the revolution of genetic approaches like CRISPR/CAS9 and the progress of high resolution imaging, it is clear that cell biology will move to more complex systems that benefit from the same revolutions. Stem cells, organoids, tissues as well as full organisms will be used in dynamic imaging and functional studies aiming again at better integration. The cell biology of membrane traffic will also continue to gain from other fields of biology.

Studying host-pathogen interactions has revealed specific functions of intracellular pathways by identifying pathways that are exploited for infection and thereby uncovered new regulatory processes. The power of phylogenetic comparison and population genetic analysis has already become apparent and further exploitation of these considerations for intracellular pathways will help understand tissue specialization. This evolving appreciation of diversity will build on decades of reductionist analysis to discover how biological complexity functions in health and disease.

A major concern and challenge is connecting "big data" with strategic "small" science. This is already being confronted as we try to utilize proteomics, metabolomics, genetic diversity and other data sets for analysis of intracellular pathways. With these approaches, cell biology has, in some ways, returned to old-fashioned data collection.

But, although the modern Darwin might now use algorithms for analysis, data interpretation still relies on acute observation and search for novel patterns. While the importance of statistics is paramount, need for intuition and thoughtful testing of emerging hypotheses has not disappeared. Here again, gene editing and acute perturbation of cellular functions and pathways will be essential tools in the near future.

So, the imagination that inspired Thomas Kreis to take new approaches and open up new avenues remains a key feature of our future enterprise. And above all, remembering Thomas, we were reminded of the importance of having fun while pursuing it.


Florin Avram
email: florin.avram@orange.fr

Jacky Cresson
email: jacky.cresson@univ-pau.fr

] Keller

BEYOND WENTZELL-FREIDLIN: SEMI-DETERMINISTIC APPROXIMATIONS FOR DIFFUSIONS WITH SMALL NOISE AND A REPULSIVE CRITICAL BOUNDARY POINT

Keywords: dynamical systems, small noise, linearization, semi-deterministic fluid approximation AMS classification: AMS 60J60

We extend below a limit theorem [3] for diffusion models used in population theory.

§1. Introduction

A diffusion with small noise is defined as the solution of a stochastic differential equation (SDE) driven by standard Brownian motion B t (•) (defined on a probability space and progressively measurable with respect to an increasing filtration)

       dX ε t = µ(X ε t )dt + √ εσ(X ε t )dB t , t ≥ 0, X ε 0 = x 0 = ε, X ε t ∈ I := (0, r) (1)
where 0 < r ≤ +∞, ε > 0, µ : I → R, σ : I → R >0 and µ, σ satisfy conditions ensuring that (1) has a strong unique solution (for example, µ is locally Lifshitz and σ satisfies the Yamada-Watanabe conditions [18, (2.13), Ch.5.2.C]). § When ε → 0, (1) is a small perturbation of the dynamical system/ordinary differential equation (ODE):

dx t dt = µ(x t ), t ≥ 0, (2)
which will also be supposed to admit a unique continuous solution x t , t ∈ R + subject to any x 0 ∈ (0, r), and the flow of which will be denoted by φ t (x). A basic result in the field is the "fluid limit", which states that when (1) admits a strong unique solution, the effect of noise is negligible as ε → 0, on any fixed time interval [0, T ]: where x t is the solution of (2) subject to the initial condition x 0 . ¶ Although interesting, this result does not give any understanding of the asymptotic behavior of the diffusion process for times converging to infinity; in particular, it does not tell us how the diffusion travels between equilibrium points (which requires times converging to infinity). Following [START_REF] Barbour | Escape from the boundary in markov population processes[END_REF][START_REF] Baker | Persistence of small noise and random initial conditions in the wright-fisher model[END_REF], we go here beyond Theorem 1, by analyzing the way a diffusion process leaves an unstable equilibrium point. Precisely, we make the following assumptions: Assumption 1. Suppose from now on that l = 0, µ(0) = 0, µ ′ (0) > 0, which makes zero an unstable equilibrium point of (2) and of [START_REF] Alvarez | Optimal sustainable harvesting of populations in random environments[END_REF].

Note that under Assumption 1, the Freidlin-Wentzell theorem 1 implies that the solution of (1) started from a small positive initial condition X ε 0 = ε > 0 converges to zero on any fixed bounded interval sup

t≤T X ε t ¶ ---→ ε→0 0, ∀T ≥ 0.
Assumption 2. Put now a(x) = σ 2 (x), and assume that a(0) = σ(0) = 0, a ′ (0) > 0, which makes 0 a singular point of the diffusion (1)-see for example [START_REF] Cherny | Singular stochastic differential equations[END_REF].

Remark 1. Note that a ′ (0) > 0 rules out important population theory models like the linear Gilpin Ayala diffusion [START_REF] Liu | Sufficient and necessary conditions on the existence of stationary distribution and extinction for stochastic generalized logistic system[END_REF] with

µ(x) = γx 1 -( x x c ) α , σ(x) = √ εx ⇔ a(x) = εx 2 , γ > 0, x c > 0, α > 0, (3)
which includes by setting α = 1 another favorite, the logistic-type Verlhurst-Pearl diffusion [START_REF] Giet | The logistic sde[END_REF][START_REF] Evans | Protected polymorphisms and evolutionary stability of patch-selection strategies in stochastic environments[END_REF][START_REF] Alvarez | Optimal sustainable harvesting of populations in random environments[END_REF]. Recently, a new type of limit theorem [START_REF] Baker | Persistence of small noise and random initial conditions in the wright-fisher model[END_REF] was discovered when T → ∞ under Assumptions 1, 2, when x ε 0 converges to the unstable equilibrium point of (2). Following [START_REF] Baker | Persistence of small noise and random initial conditions in the wright-fisher model[END_REF], let

T ε := 1 µ ′ (0) log 1 ε (4)
denote the solution of the equation φ t,lin (x 0 ) = x 0 e µ ′ (0)t = 1 where φ t,lin (x 0 ) is the flow of the linearized system of (2) in 0, and divide the evolution of the process in three time-intervals:

[0, t c := cT ε ], [t c , t 1 := T ε ], [t 1 , ∞), c ∈ (1/2, 1) (5)
(the restriction c > 1/2 is used in (25)).

It turns out that this partition allows separating the life-time of diffusions with small noise, exiting an unstable point of the fluid limit, into three periods with distinct behaviors: Fisher-Wright diffusion until ^(-1)=100

Figure 1: 6 paths of the Kimura-Fisher-Wright diffusion dX t = γX t (1 -X t )dt + √ εX t (1 -X t )dB t , where x c = 1 is an exit boundary, with ε = .01. On the right, three stages of evolution may be discerned 1. In the first stage, the process leaves the neighborhood of the unstable point. The linearization of the SDE implies that here a Feller branching approximation may be used, and this produces a certain exit law W which will be carried over to the next stage as a (random) initial condition.

2. In the second "semi-deterministic stage" (meaning that paths cross very rarely here), the system moves towards its first stable critical point x c , following the trajectories of its fluid limit (2), again over a time whose length converges to ∞. A further renormalization produces here the main result, the limit exit law [START_REF] Bingham | Continuous branching processes and spectral positivity[END_REF].

3. In the third stage, after the SDE has approaches the stable critical point of the fluid limit, "randomness is regained" -see crossings of paths in figures 1 and 2); (if the process may reach and overshoot the stable critical point, convergence towards a stationary distribution may occur).

The following result was obtained first in [START_REF] Baker | Persistence of small noise and random initial conditions in the wright-fisher model[END_REF], for the "Kimura-Fisher-Wright" diffusion, and extended subsequently to diffusions with bounded volatility.

Theorem 2. Fluid limit with random initial conditions [START_REF] Baker | Persistence of small noise and random initial conditions in the wright-fisher model[END_REF]. Let X ε t satisfy Assumption 1, (1), and X ε 0 = ε > 0. Suppose in addition that the diffusion coefficient σ(•) is continuous and bounded, as well as its first derivative, and that µ(•) satisfies the following drift condition:

µ(y) -µ(x) ≤ µ ′ (0)|y -x|, x, y ∈ R + .
Let Y t denote the solution to the scaled linearized equation

dY t = µ ′ (0)Y t dt + a ′ (0)Y t dB t , Y 0 = 1 =⇒ Y t = 1 + t 0 µ ′ (0)Y s ds + t 0 a ′ (0)Y s dB s , (6)
known as Feller branching diffusion.

Then, it holds that :

(A) X ε T ε ¶ ---→ ε→0 φ(W), (7)
where (i) the random variable W is the a.s. martingale limit

W := lim t→∞ e -µ ′ (0)t Y t = 1 + ∞ 0 e -µ ′ (0) 2 s a ′ (0)Y s dB s (8)
(ii) φ(x) denotes the limit of the deterministic flow pushed first backward in time by the linearized deterministic flow φ t,lin (x) = xe µ ′ (0)t near the unstable critical point 0

φ(x) = lim t→∞ φ t φ -t,lin (x) = lim t→∞ φ t xe -µ ′ (0)t , x ≥ 0. (9)
(B) Also, for any T > 0, sup

t∈[0,T ] X ε T ε +t -x t ¶ ---→ ε→0 0, ( 10
)
where x t is the solution of (2) subject to the initial condition X 0 = φ(W).

Remark 2. Note that W depends only on the local parameters µ ′ (0), a ′ (0) of the diffusion at the critical point. Assume from now on, without loss of generality that a ′ (0) = 1 (recalling however that this is the only part of the stochastic perturbation that survives in the limiting regime), and let γ := µ ′ (0) > 0 [START_REF] Cattiaux | Quasi-stationary distributions and diffusion models in population dynamics[END_REF] denote the Malthusian parameter.

In the one -dimensional case, the Laplace transform of W ∞ is well known [START_REF] Pardoux | Probabilistic models of population evolution[END_REF] and easy to compute. Indeed, letting u t (λ) = -1

x log(E x [e -sY t ] denote the cumulant transform of this branching process, and solving the Riccati-type equation

∂u(t) ∂t = γu(1 - a ′ (0) 2γ u),
yields an explicit expression:

E x e -sY t = exp         - xse γt 1 + sa ′ (0)
2γ (e γt -1)

        , s > 0 (12)
see, e.g., [23, Ch 4.2, Lem. 5, pg. 24].

One may conclude from the explicit ( 12) that

E 1 e -sW ∞ = lim t→∞ u t (se -γt ) = exp - 2sγ/a ′ (0) 2γ/a ′ (0) + s , (13)
and one may check that W ∞ is a Poisson sum with parameter 2γ/a ′ (0) of independent exponential random variables

W ∞ = N 2γ j=1 τ j , τ j ∼ Expo(2γ/a ′ (0)). ( 14
)
Remark 3. Computing the limit ϕ(s) := Ee -sW ∞ = lim t→∞ Ee -se -γt Y t is a famous problem in the theory of supercritical branching processes. Recall that 1. For Galton-Watson processes, ϕ(s) satisfies the Poincaré -Schroeder functional equation ϕ(ms

) = p(ϕ(s)), m = p ′ (1) (15)
where p(s) is the probability generating function of the progeny [2, I.10(5), Thm I.10.2].

2. For continuous time branching processes, letting Ψ(s) = p(s)s denote the branching mechanism, and θ(s) = ϕ(s) -1 denote the functional inverse, it holds that

θ(1) = 0, θ ′ (s) θ(s) = Ψ ′ (1) Ψ(s) , 0 ≤ s ≤ 1 =⇒ θ(s) = (1 -s)e -1 s ( Ψ ′ (1) Ψ(u) + 1 1-u )du , 0 ≤ s ≤ 1 (16)
see [2, III.7(9-10), p.112] and

sϕ ′ (s) = Ψ ′ (1) -1 Ψ(ϕ(s)), ϕ(0) = 1. ( 17
)
For example, for binary splitting with branching mechanism Ψ(s) = s 2s, we find

sϕ ′ (s) = ϕ(s)(ϕ(s) -1) =⇒ ϕ(s) = 1 1 + s , θ(s) = 1 -s s
with W ∞ exponential with parameter 1, and for geometric branching with parameter 1u we find

θ(s) 1-2u = (1 -s) 1 u (1 -u(1 + s)) 1 1-u .
The example of k-ary fission is also explicit-see [8, p. 218] and [19, p. 119].

Problem 1. Extend the results of [START_REF] Barbour | On the emergence of random initial conditions in fluid limits[END_REF] from birth-death to Markov discrete space with finite number of transitions upwards and downwards. Solve numerically the Schroeder equation.

3. For the continuous state case, letting -κ(s) = ln E[e -sW ∞ ] denote the logarithm of the Laplace transform and Ψ(s) denote the branching mechanism, it holds that

s'(s) = Ψ ′ (0) -1 Ψ(κ(s)) (18)
see [START_REF] Bingham | Continuous branching processes and spectral positivity[END_REF]Cor. 4.3] and also the Appendix, for the multi-type case. Also [START_REF] Bingham | Continuous branching processes and spectral positivity[END_REF]Thm 4.2], it holds that the functional inverse θ(s) = κ(s) -1 satisfies

θ ′ (s) θ(s) = Ψ ′ (0) Ψ(s) , 0 ≤ s ≤ Ψ ′ (0) =⇒ θ(s) = se s 0 ( Ψ ′ (0) Ψ(u) -1 u )du , 0 ≤ s ≤ Ψ ′ (0). ( 19
)
For example, for the Feller branching diffusion with branching mechanism Ψ(s) = γs -1 2 s 2 , we find

θ(s) = 2γs 2γ -s , κ(s) = 2γs 2γ + s .
Remark 4. The main part of Theorem 2 is the equation ( 7) which identifies the limit after the second stage

X ε T ε = Φ ε T ε (ε) ¶ ---→ ε→0 lim t→∞ φ t φ -t,lin (W) = φ(W), (20)
Φ ε t (x) denotes the flow generated by the SDE (1).

Note that φ depends only on the dynamical system µ.

By [3, Prop. 4.1], it is a nontrivial solution of the ODE µ ′ (0)x φ ′ (x) = µ( φ(x)), φ(0) = 0 (21)
which is equivalent to the Poincaré functional equation

φ(xe γt ) = φ t ( φ(x)) ⇔ µ( φ(x)) = φ(γx) (22)
arising in Poincaré conjugacy relations for dynamical systems. Interestingly, this is the same type of equation as [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF], minus the restriction that v(•) be a Bernstein function.

The inverse w(x) = φ(x) -1 when φ(x) satisfies ( 21) is given by

w(x) = xe x 0 γ µ(u) -1 u du , 0 ≤ x ≤ γ. (23)
(21), [START_REF] Liu | Sufficient and necessary conditions on the existence of stationary distribution and extinction for stochastic generalized logistic system[END_REF] suggest possible generalizations to multidimensional diffusions (and possibly to jump-diffusions (where a CBI might replace the Feller diffusion in the limit).

Remark 5. Part 2. of Theorem 2 follows immediately by a simple change of time: letting

X ε t = X ε T ε +t , and B t = B T ε +t -B T ε one obtains from (1) X ε t = X ε 0 + t 0 f ( X ε s )ds + t 0 εσ( X ε s )d B s ,
and the result follows from (7) by the fluid convergence Theorem 1. This part may be viewed as describing "short transitions" (invisible on a long time scale) between the second and third stages.

Remark 6. The limit [START_REF] Bingham | Continuous branching processes and spectral positivity[END_REF] describing the position after the second stage has been established in [START_REF] Baker | Persistence of small noise and random initial conditions in the wright-fisher model[END_REF] for one dimensional distributions with bounded σ(x). This assumption seems however restrictive, since for typical diffusions whose fluid limit φ t (x) admits a stable critical point x c , the probability of leaving the neighborhood of the stable point x c is very small as ε → 0. This intuition is confirmed by simulations -see Figure 2.

The remark 6 suggests the relation of our problem to that of studying the maximum of X t . More precisely, we would like to establish and exploit the plausible fact that ∀θ > 1

lim ε→0 P[T θx c < T ε |X 0 = ε] = lim ε→0 P[ sup 0≤t≤T ε X ε t > θx c |X 0 = ε] = 0, (24)
where x c is the closest critical point towards which the diffusion is attracted, and T θx c is the hitting time of θx c ; clearly, [START_REF] Skorokhod | Asymptotic methods in the theory of stochastic differential equations[END_REF] renders unnecessary the assumption that the diffusion coefficient σ(•) be bounded.

A weaker statement than ( 24), but still sufficient for a slight extension, is provided in the elementary Lemma (3) below.

Contents. The paper is organized as follows. In Section 2 we offer, based on Lemma 3, a slight extension of Theorem 2 of [START_REF] Baker | Persistence of small noise and random initial conditions in the wright-fisher model[END_REF]. A conjecture (see Problem 2) is presented here as well. We illustrate our new result with the example of the logistic Feller diffusion in Section 3. We include for convenience in Section 4 an outline of the remarkable paper [START_REF] Baker | Persistence of small noise and random initial conditions in the wright-fisher model[END_REF]. §2. An extension of Theorem 2 [START_REF] Baker | Persistence of small noise and random initial conditions in the wright-fisher model[END_REF] Recall now from [START_REF] Baker | Persistence of small noise and random initial conditions in the wright-fisher model[END_REF]

that the restrictive condition σ ∞ < ∞ is used for proving that ‡ σ ∞ < ∞, c ∈ (1/2, 1) =⇒ Φ t c ,t 1 (X ε t c ) -φ t c ,t 1 (X ε t c ) L 2 ---→ ε→0 0, (26)
where t c = cT ε . We will show now that it is possible to remove the condition σ ∞ < ∞ in (26), if only convergence in probability is needed, by assuming rather weak and natural conditions on the scale function s(•). Recall that the scale function s is defined (up to two integration constants) as an arbitrary increasing solution of the equation Ls(x) = 0, where L is the generator operator of the diffusion, and that this function is continuous -see [START_REF] Karlin | Linear birth and death processes with killing[END_REF]Ch. 15,(3.5), (3.6)] (noting that [START_REF] Karlin | Linear birth and death processes with killing[END_REF] denote the scale function by S (•)). Lemma 3. Assume that 0 is an attracting boundary and that r is an unattracting boundary, i.e. that , s(0

+ ) > -∞, s(r-) = ∞. Put X ε = sup 0≤t<∞ X ε t , ( 27
)
where X ε is defined in [START_REF] Alvarez | Optimal sustainable harvesting of populations in random environments[END_REF]. Then: ‡ Let us recall the proof of this important piece of the puzzle. Let Φ s,t (x), φ s,t (x) denote the stochastic and deterministic flows generated respectively by the SDE (1) and ODE (2), put Φ ε t := Φ tc,tc+t (X ε tc ), φ t := φ tc,tc+t (X ε tc ) for brevity, and define δ ε t = Φ ε tφ t . Subtracting equations ( 1) and ( 2) and applying the Itô formula:

E δ ε t 2 = E t 0 2δ s µ(Φ ε s ) -µ(φ s ) ds + t 0 εEσ(Φ ε s )ds ≤ t 0 2γE(δ s ) 2 ds + εt σ ∞ , t ∈ R +
where assumption (2) was used. By Grönwall's inequality

E Φ tc,t 1 (X ε tc ) -φ tc,t 1 (X ε tc ) 2 = E δ ε t 1 -tc 2 ≤ C 1 εt 1 e 2γ(t 1 -tc) ≤ C 2 ε 2c-1 log 1 ε ---→ ε→0 0 ( 25
)
where the convergence holds since c ∈ ( 1 2 , 1).

(A) ∀ε, lim

M→r P ε [X ε > M] = lim M→r s(ε) -s(0) s(M) -s(0) = (s(ε) -s(0)) lim M→r 1 s(M) -s(0) = 0, (28) and (B) c ∈ (1/2, 1) =⇒ Φ t c ,t 1 (X ε t c ) -φ t c ,t 1 (X ε t c ) P ---→ ε→0 0. ( 29
)
Proof. (28) is straightforward. Indeed, recall that the boundary 0 is attracting. Then,

P ε [X ε > M] = P ε [T M < T 0 ] = s(ε) -s(0) s(M) -s(0) ( 30
)
where T 0 , T M are the hitting times of X ǫ t at 0 and M -see (29) follows by a similar argument. Indeed, denote the deterministic and stochastic flows generated by the ODE (2) and SDE (1) (i.e. the solutions of these equations at time t that start at x at time s) by φ s,t (x) and Φ s,t (x), respectively, and put Φ ε := Φ t c ,t 1 (X ε t c ) and φ ε := φ t c ,t 1 (X ε t c ) for brevity and define δ ε = Φ εφ ε . For fixed ε and M, it holds that

∀δ > 0, P ε [|δ ε | > δ] ≤ P ε [X ε T ε ≤ M]P ε [|δ ε | > δ|X ε T ε ≤ M] + P ε [X ε T ε > M] ≤ P ε [X ε T ε ≤ M]P ε [|δ ε | > δ|X ε T ε ≤ M] + P ε [X ε > M].
Letting now ε to 0 makes the first term go to 0 by (26), yielding

∀M < r, ∀δ > 0, lim sup ε→0 P ε [|δ ε | > δ] ≤ lim ε→0 s(ε) -s(0) s(M) -s(0) = 0
where we have used again the continuity of the scale function.

Theorem 4. The conclusions of Theorem 2 still hold under the assumptions of Lemma 3.

Proof. Theorem 2 of [START_REF] Baker | Persistence of small noise and random initial conditions in the wright-fisher model[END_REF] only uses the assumption σ ∞ < ∞ in establishing the unnecessarily strong result (26). Providing weaker conditions for the weaker but still sufficient result (29) establishes therefore our claim.

Problem 2. Note that essential use of s(0) > -∞ was made in (28). We conjecture however that a finer analysis will reveal that the result of Theorem 4 still holds whenever r is "repelling/unattracting", more precisely when it is natural unattracting or entrance, cf. Feller's classification of boundary points [20, Ch. XV]. §3. Examples with lim t→∞ X t /x t = 0: The logistic Feller and Gilpin-Ayala diffusions

We recall now some famous examples for which the conditions of our Lemma 3 hold. The logistic Feller diffusion is defined by

dX t = γX t 1 - X t x c dt + εX t dB t , X t ∈ (0, ∞).
The limit point x c of x t is a regular point for the diffusion; w.l.o.g. we will take it equal to 1. The scale density s

′ (x) = e -2γ ε (x-x 2
2 ) is integrable at 0, but not at ∞, and the speed density [START_REF] Karlin | Linear birth and death processes with killing[END_REF]

m ′ (x) = e 2γ ε (x-x 2
2 ) εx is integrable at ∞, but not at 0, so that the conditions of Lemma 3 hold. § Therefore, fluid convergence with random initial point before T ε [START_REF] Baker | Persistence of small noise and random initial conditions in the wright-fisher model[END_REF] still holds, with the same deterministic flow and random initial condition as for the Kimura-Fisher Wright diffusion studied in [START_REF] Baker | Persistence of small noise and random initial conditions in the wright-fisher model[END_REF]

φ t (x) = xe γt 1 -x + xe γt , φ(x) = x 1 + x , X 0 = W W + 1 (since µ(.), a ′ (0) did not change)-see Figure 2.
In fact, the paths of the logistic Feller and Kimura-Fisher-Wright diffusions are almost indistinguishable up to T ε of each other -see Figure 3. After reaching the neighborhood of x c however, the paths split, reflecting the different natures (regular and exit) of x c for these two stochastic processes.

Some other examples of interest in population theory are the diffusion processes defined by the SDEs

dX t = γX t 1 -( X t x c θ dt + σ X t dB t , σ >, θ > 0, dX t = γX t 1 - X t x c -β X n-1 t 1 + X n t dt + σ X t dB t , β ≥ 0, n ≥ 1,
which are stochastic extensions with square root volatility of deterministic population models introduced by Gilpin and Ayala and Holling respectively. It is easy to check that adding the exponents θ and n does not affect integrability of the scale and speed densities of these diffusions, so that our extension applies. Furthermore, the rescaled flow φ may be computed numerically by [START_REF] Baker | Persistence of small noise and random initial conditions in the wright-fisher model[END_REF]Prop. 4.1] (and even symbolically for small integer values of θ, n).

Moving away from the square root volatility case, an interesting, still open question is to investigate whether analogues of the [START_REF] Baker | Persistence of small noise and random initial conditions in the wright-fisher model[END_REF] result are available for the processes satisfying

dX t = γX t 1 -( X t
x c ) θ dt + √ ε(X t ) α dB t , α > 0. § § Furthermore, conform Feller's boundary classification [START_REF] Karlin | Linear birth and death processes with killing[END_REF], 0 is an exit boundary since s ′ (x)m[x, 1] is integrable at 0, and absorbtion in 0 occurs with probability 1, and ∞ is an entrance (nonattracting) boundary, since m ′ (x)s [1, x] is integrable at ∞-see also [START_REF] Cattiaux | Quasi-stationary distributions and diffusion models in population dynamics[END_REF][START_REF] Bansaye | Diffusions from infinity[END_REF] and [START_REF] Foucart | Continuous-state branching processes with competition: Duality and reflection at infinity[END_REF] for the generalization to continuous-state branching processes with competition.

§ The particular case α = θ = 1 is the famous Verlhurst-Pearl diffusion (VP)-see for example [START_REF] Liu | Sufficient and necessary conditions on the existence of stationary distribution and extinction for stochastic generalized logistic system[END_REF]. Recall that t c = ct 1 with c ∈ (1/2, 1), arbitrary, and note that

X ε T ε = Φ t c ,t 1 (X ε t c ) = Φ t c ,t 1 (Φ t c (ε)
). The idea of the proof is to approximate this random variableby

X ε T ε ≈ φ t c ,t 1 (Φ t c (ε)) ε→0 ---→ φ(W), (31)
with the random variable W from [START_REF] Bingham | On the limit of a supercritical branching process[END_REF]. The proof of [START_REF] Baker | Persistence of small noise and random initial conditions in the wright-fisher model[END_REF] involves several steps 1. The first idea for establishing the approximation φ(W) of X ε T ε is to blow-up the process near the boundary 0

X ε t := ε -1 X ε t ,
which fixes the initial condition to 1 and changes the SDE to

d X ε t = ε -1 µ(ε X ε t )dt + a(ε X ε t ) ε dB t , t ≥ 0, (32)
it is easy to check that a subsequent linearization of the SDE yields where Y t is a Feller branching diffusion started from 1, defined by

X ε t ≈ Y t
Y t = 1 + t 0 µ ′ (0)Y s ds + t 0 √ Y s dB s , t ≥ 0. ( 33
)
One may take advantage then of the well-known nonnegative martingale convergence theorem for the "scaled final position" of the branching process Y t W := lim t→∞ e -µ ′ (0)t Y t .

(34)

Remark 7. Let us note that the linearization for processes satisfying a(x) = O(x 2 ) and failing Assumption 2, like the linear Gilpin-Ayala (3), leads to geometric Brownian motion. In this case, (34) holds with W = 0, and a different approach seems necessary.

2. After "blowing up" the beginning of the path, the second idea is to "look from far away". We want to break the trajectory at a suitably chosen time point

t c < t 1 = T ε = 1 γ log 1 ε (35)
such that before t c , the original process is close to Feller's branching diffusion (33), and convergence to the limit W of the Feller diffusion occurs, i.e.

X ε t c = ε X ε t c = e -γt 1 X ε t c ≈ e -γt 1 Y t c = e -γ(t 1 -τ c ) e -γt c Y t c ≈ e -γ(t 1 -τ c ) W.

The first approximation e -γt c X ε

t c L 1 ---→ ε→0
Y t c follows from the following lemma [START_REF] Baker | Persistence of small noise and random initial conditions in the wright-fisher model[END_REF] showing that the solution of (1) converges, under appropriate scaling, to the Feller branching diffusion (33). Lemma 5. Let X ε t := ε -1 X ε t , where X ε t is the solution of (1) subject to X ε 0 = ε. Then

X ε t L 1 ---→ ε→0 Y t , ∀ t ≥ 0,
where Y t is the solution of (33).

Putting these together yields φ t c ,t 1 (X ε t c ) ¶ ---→ ε→0 φ(W).

3. The hardest part is proving that in the second portion [t c , t 1 ], the influence of the stochasticity is negligible, for example that Φ t c ,t 1 (X ε t c )φ t c ,t 1 (X ε t c )

L 2
---→ ε→0 0, as proved in [START_REF] Baker | Persistence of small noise and random initial conditions in the wright-fisher model[END_REF] under the restrictive assumption σ ∞ < ∞.

Putting it all together in one line, one must prove that

X ε t 1 = Φ t c ,t 1 (X ε t c ) ≈ Φ t c ,t 1 (We -γ(t 1 -τ c ) ) ≈ φ t c ,t 1 (We -γ(t 1 -τ c ) ) ¶ ---→ ε→0 φ(W). (37)
To extend [START_REF] Baker | Persistence of small noise and random initial conditions in the wright-fisher model[END_REF], it is sufficient to improve the third approximation step above.

Theorem 1 .

 1 [Freidlin and Wentzell] [15, Thm 1.2, Ch. 2.1] Let X ε t satisfy (1), assume µ, σ satisfy the Lifshitz condition, and that X ε

  [START_REF] Karlin | Linear birth and death processes with killing[END_REF] Ch. 15, (3.1), (3.10)]. Using now the continuity of the scale function s(•) [20, Ch. 15, (3.5), (3.6)] (note that [20] denote the scale function by S (•)) yields lim M→r s(M) = s(r -) = ∞ and the result.

Figure 2 :

 2 Figure 2: 6 paths of the logistic Feller diffusion (x c = 1 is regular) with ε = .01, until T ε and after

Figure 3 :

 3 Figure 3: 6 paths of the logistic Feller and Kimura-Fisher-Wright diffusions with ε = 1/20, before and after T ε

§ For reviews discussing the existence of strong and weak solutions, see for example[START_REF] Breiman | Probability, volume 7 of classics in applied mathematics[END_REF][START_REF] Helland | One-dimensional diffusion processes and their boundaries[END_REF][START_REF] Cherny | Singular stochastic differential equations[END_REF].

Acknowledgement: We thank J.L. Perez for useful remarks and the referee for the help in improving the exposition.


Benjamin Lee
email: benjamin.lee1@monash.edu

Arnaud Prouzeau
email: arnaud.prouzeau@monash.edu

Maxime Cordeil
email: max.cordeil@monash.edu

Tim Dwyer
email: tim.dwyer@monash.edu

FIESTA: A Free Roaming Collaborative Immersive Analytics System

Keywords: Immersive analytics, data visualisation, collaboration, collaborative immersive analytics, virtual reality CCS Concepts, Human-centered computing → Visualization systems and tools

We present FIESTA, a prototype system for collaborative immersive analytics (CIA). In contrast to many existing CIA prototypes, FIESTA allows users to collaboratively work together wherever and however they wish-untethered from mandatory physical display devices. Users can freely move around in a shared room-sized environment, author and generate immersive data visualisations, position them in the space around them, and share and communicate their insights to one another. Certain visualisation tasks are also supported to facilitate this process, such as details on demand and brushing and linking.

Introduction

Immersive analytics explores how augmented and virtual reality (AR/VR) technologies can be used to support data visualisation, analytical reasoning, and decision making [START_REF] Chandler | Immersive Analytics[END_REF]. Collaborative immersive analytics (CIA), by extension,

Interactive Demonstration

ISS'19, November 10-13, 2019, Deajon, Republic of Korea explores how multiple users collaborate in co-located and remote settings [START_REF] Billinghurst | Collaborative Immersive Analytics[END_REF].

Current research has mainly demonstrated prototypes which combine AR head-mounted displays with physical display panels. Augmented Reality over the Tabletop [START_REF] Butscher | Clusters, Trends, and Outliers: How Immersive Technologies Can Facilitate the Collaborative Analysis of Multidimensional Data[END_REF] combines a multi-touch tabletop with an AR 3D parallel coordinates visualisation positioned orthogonally from the table. Multiple co-located users can stand around the tabletop and configure and adjust the visualisation using touch controls. DataSpace [START_REF] Cavallo | Dataspace: A Reconfigurable Hybrid Reality Environment for Collaborative Information Analysis[END_REF] includes an array of 15 adjustable display panels attached to robotic arms. These are integrated with AR, VR, and other mobile devices to visualise and analyse large and complex datasets in a room-sized environment.

CIA has the ability to transform the way users work collaboratively to find insights. What is promising is the concept of a workspace where users can freely choose where and how to work unrestricted from physical limitations. For example, a group may decide not to sit around a stationary conference table, but instead work more actively and take advantage of the space around them using AR/VR technologies. In this sense, the only restriction is the actual size of the room.

We take this concept and develop FIESTA: the Free-range Immersive Environment to Support Team-based Analysis. FIESTA allows an arbitrary number of users to synchronously work together to visualise and find insights in a multivariate data set. These users are placed in a roomsized environment where they are untethered and can freely move in the space around them. In this, they can author and generate immersive data visualisations, position them in the space around them, and share and communicate their insights to one another (Figure 1).

Figure 2:

The visualisation authoring panel used to generate visualisations. In this case it is configured to facet-by "YearBuilt", creating filtered small multiples between fixed yearly intervals.

System Design

Implementation FIESTA was developed as a VR prototype in Unity3D using HTC Vive Pro headsets, controllers, and wearable backpack PCs to achieve tetherless VR. We chose to use VR as current AR technologies are limited in aspects such as field of view and performance. Because of this, users instead see each other as simplified Oculus Avatar1 models with each user a unique colour. FIESTA is intended to be used in a co-located environment, with positional tracking correctly synchronised to minimise collisions, but it can very easily be adapted to be used in remote collaborative settings. While it can support an arbitrary number of simultaneous users, we found that three works comfortably in terms of performance, usability, and practicality.

Authoring Visualisations

A key component of FIESTA is the ability for users to author and generate visualisations to then place in the space around them. To minimise learning time, we use conventional UI designs-buttons, dropdowns, sliders, etc.-which the user interacts with to change visualisation parameters.

We organise this interface using a panel metaphor, as seen on the left side of Figure 2. The right side has the resultant visualisation that is updated to reflect the chosen parameters. The panel itself is a kinematic 3D object which the user can grab and move around the environment and position wherever they see fit, analogous to them moving their workspace. Once a visualisation is deemed to be finished, the user can create a copy of a panel's visualisation by simply grabbing hold of it and pulling away from it, invoking a "tear-out" metaphor (Figure 3). This copy can now be re-positioned in the space around them in a manner similar to the panel. If a user wants to keep working on an already duplicated visualisation, they can "load" a visualisation onto a panel by holding and releasing a visualisation on top of it, in which the panel will reconfigure its buttons such that it matches the loaded visualisation. If they want to destroy a visualisation, they can simply "throw" it at the ground.

Visualisations are generated using the Immersive Analytics Toolkit (IATK) [START_REF] Cordeil | IATK: An Immersive Analytics Toolkit[END_REF]. As multiple users can each simultaneously create multiple copies of visualisations, it is crucial for them to be as optimised as possible. While IATK can create visualisations with three-dimensional axes, we purposefully choose to restrict them to two-dimensions in order to minimise learning and confusion. Numerous parameters are available to customise these visualisations on the panel, such as size-by, colour-by, facet-by (small multiples), and mapping discrete colour palettes to a dimension.

Visualisation Tools

FIESTA also supports a selection of common visualisation tools. For example, users can resize visualisations, rescale maximum and minimum ranges along axes, inspect specific points for details on demand using a laser pointer, and brush points to make linked selections. Inspection is performed by holding down the touchpad on the dominant hand which enables a laser pointer, then pointing it nearby any data point (Figure 4). Brushing is accessed using a spin menu [START_REF] Gerber | The spin menu: a menu system for virtual environments[END_REF] on the non-dominant hand, with two brushing modes: private and shared. This invokes a toolboxlike metaphor, as the user then chooses which mode they wish to use by using the dominant hand. Private selections are only visible to the user (shown in yellow) while shared selections are visible to everyone (shown in the owner's avatar's colour). Selections and deselections made with either mode are automatically linked to all other visualisations in the environment.

Shared Environment

Users are placed in a virtual room that is slightly smaller than the actual size of the room they are in (Figure 5). In general, this room contains four virtual walls, which both acts as a visible boundary to prevent collisions, and allows users to "pin-up" visualisations on the wall. To do this, users can either hold and release or throw visualisations directly onto any wall, after which it will snap neatly into position. These surfaces can aid users in organising their visualisations should they so desire (Figure 6).

We consciously made everything in the environment visible and interactable by everyone-there is no concept of privacy nor are there locking mechanisms in FIESTA. This means users can freely walk over and use other panels or even interrupt one another while working.

Interactive Demonstration ISS'19, November 10-13, 2019, Deajon, Republic of Korea

Conclusion and Future Work

This paper introduces FIESTA: the Free-range Immersive Environment to Support Team-based Analysis. It is a CIA prototype which allows multiple users to freely move around in a shared room-sized environment and collaboratively gather insights in data using immersive visualisations.

Figure 1 :

 1 Figure 1: Multiple users can simultaneously work together to visualise data and find insights in FIESTA.

Figure 3 :

 3 Figure 3: FIESTA uses a "tear-out" metaphor to duplicate visualisations from panels, and these can be freely positioned either in space or on the surfaces of the virtual room.

Figure 4 :

 4 Figure 4: Users can easily get details on demand for individual data points using a laser pointer.

Figure 6 :

 6 Figure 6: The walls in FIESTA can facilitate group discussion and presentation.

Figure 5 :

 5 Figure 5: The real-life perspective of FIESTA being used by three users at once. The virtual room is set to be slightly smaller than the real room.

https://developer.oculus.com/documentation/avatarsdk/latest/concepts/

We plan to publish the results of an in-depth exploratory study using FIESTA with groups of users solving data visualisation tasks in a future paper. There are also many possible improvements we can make to the system. One option is to allow users to create and destroy their own surfaces at will, such as flat tabletops or moveable "whiteboards", which may open up whole new collaboration styles. We are also looking into supporting 3-dimensional visualisations, however we expect this would necessitate major design changes to both the panel-like interface and the laser pointer interaction methods.