Abhijit Sarkar
email: abhijit.sarkar@thomson.net

Laurent Blondé

Patrick Le Callet

Florent Autrusseau

Jürgen Stauder

Patrick Morvan

STUDY OF OBSERVER VARIABILITY IN MODERN DISPLAY COLORIMETRY: AN ANALYSIS OF CIE 2006 MODEL

Keywords: Color Vision, Cone Fundamentals, Color Matching, Observer variability, Display colorimetry

CIE 2006 model presents a convenient framework for calculating the cone fundamentals, and thus the color matching functions, for various ages of an average observer. CIE 2006 model incorporates three major physiological factors affecting observer variability, namely optical densities for the ocular media absorption, macular pigment absorption, and visual pigments in the outer segments of the photoreceptors. However, it does not have a provision for a peak-wavelength shift in the photopigment absorption spectra, a significant but difficult-to-model contributor to observer variability. In the context of color reproduction with modern emissive displays based on different technologies, we performed a theoretical analysis of various aspects of CIE 2006. Out of the above four factors, L-cone peak-wavelength shift was found to be the second most significant contributor to observer variability, after ocular media absorption. Excluding this factor from the CIE 2006 model, while understandable, can undermine the usefulness of age-dependent observer color matching functions.

INTRODUCTION

In 2006, CIE's (Commission Internationale de l'Eclairage) technical committee TC 1-36 published a report [START_REF] Cie | Fundamental Chromaticity Diagram with Physiological Axes -Part I[END_REF] (described hereafter as CIE06) on the choice of a set of Color Matching Functions (CMF) and estimates of cone fundamentals for the color-normal observer. The CIE06 model, largely based on the 1959 Stiles-Burch 10° CMFs [START_REF] Stiles | colour-matching investigation: final report[END_REF][START_REF] Stockman | The spectral sensitivities of the middle-and long-wavelength-sensitive cones derived from measurements in observers of known genotype[END_REF] , defines 2° and 10° reference observers, and provides a convenient framework for calculating average cone fundamentals for any field size between 1° and 10° and for an age between 20 and 80. Each set of CIE06 cone fundamentals can be converted to CMFs through a linear transformation. In order to assess the merits of this new set of age-dependent CMFs, it is pertinent to investigate whether the CIE06 framework incorporates all major sources of observer variability, particularly the ones that are age-independent. In fact, if this is not the case, observer variability within a given age-group can significantly undermine the usefulness of considering an agedependent observer in color imaging applications. The relevance of this issue is obviously quite dependent on the application context. While in case of cross-media color reproduction the significance of observer variability is questionable [START_REF] Oicherman | Effect of observer metamerism on colour matching of display and surface colours[END_REF] , for application contexts relevant to the display and media and entertainment industries (content creation and distribution in particular), this can be rather a critical issue, so a better modeling of observer variability is highly desirable.

In this work, we performed a theoretical analysis of various aspects of CIE06 cone fundamentals from the perspective of display colorimetry. In the first part of our analysis, we altered the CIE06 model in four different ways to explore the contributions of various physiological factors to the overall variations in the cone fundamentals, and ultimately on the predicted observer variability while viewing colors on various displays. Since CIE06 is mainly based on Stiles-Burch 10° CMFs, and since we also use them as the "ground-truth" for evaluation of the CIE06 model, 10° field size has been used throughout the analysis, unless otherwise stated.

CIE06 PHYSIOLOGICAL FACTORS

Three physiological factors have been incorporated in the CIE06 framework, in the form of optical density functions for: a) lens other ocular media absorption, b) macular pigment absorption, and c) visual pigments in the outer segments of photoreceptors. While these physiological factors are important contributors to observer variability, there is another important but more complex source of variability that has not been included in the CIE06 model. A number of studies have suggested that individual differences in the color vision are partly due to the variations in the peak wavelength (λ max ) of the cone photopigment [START_REF] Webster | Reanalysis of λ max variations in the Stiles-Burch 10-degrees color-matching functions[END_REF] . These differences can be due to individual variability, but can also be due to a variation in genetic composition or polymorphism, for example, a single amino-acid substitution (Alanine for Serine) at position 180 of the L-cone photopigment opsin genes [START_REF] Sharpe | Opsin genes, cone photopigments, color vision, and color blindness[END_REF] .

In order to explore the effects of various physiological factors on the cone fundamentals, we considered four hypothetical cases, where there is: i) no ocular media absorption (relevant optical density set to 0), ii) no macular pigment absorption (relevant optical density set to 0), iii) very high absorption in the outer segments of visual pigments, (peak optical density of low-density absorption spectra arbitrarily set to 1), and iv) a wavelength shift of the L-pigment peak absorption toward shorter wavelength, without changing the shape of the curves in wavenumber scale (peak of lowdensity absorption spectra increased by 250 cm -1 ). Case (iii) signifies very high photoreceptor self-screening, resulting in the broadening of low-density absorption spectra 1 , while case (iv) signifies L-cone peak-wavelength shift described earlier.

Here, we used the same amount of peakwavelength shift used by Webster and MacLeod 7 in their analysis of Stiles-Burch data, which translates to about 7.7 nm shift at 560 nm.

In each of these four cases, modified CIE06 cone fundamentals for various field sizes and ages were computed, and were compared to corresponding CIE06 cone fundamentals under normal conditions. The difference between the two sets of functions indicates the contribution of a given physiological factor. Because of space constraint in this paper, we show only a few representative plots. wavelength toward longer wavelengths. In both cases, the effect increases with age. Fig 2 shows the log plots of L-and S-cone fundamentals with and without macular pigment absorption for various field sizes. Because of macular pigment absorption, the slightly reduces for the L-function (similar trend for M-) between 420 and 520 nm, and increases at higher wavelengths for the Sfunction. Clearly, the effect is less significant for 10°, but increases as the field size is reduced. Fig 3 illustrates the effect of visual pigment absorption under conditions described in cases (iii) and (iv) above. The effect of unity peak optical density is to increase the sensitivity both below and above the peak wavelength, while for the peak wavelength-shift the sensitivity reduces above the peak wavelength (since the shift is in the wavenumber scale).

DISPLAYS

In our analysis, we used four displays, two of which are included in this paper. The first is a Sony BVM32 Cathode Ray Tube (CRT) display widely used as a reference studio display (hereafter referred to as Ref-CRT). The display has a peak white luminance of 107.4 cd/m 2 . The second is a Hewlett-Packard DreamColor LP2480zx professional 30-bit Wide-Gamut Liquid Crystal Display with LED backlight (hereafter referred to as WG-LCD), and a peak white luminance varying between 40-250 cd/m 2 .

The spectral power distributions of the two displays are shown in fig 4 . There is a significant difference in the spectral power and characteristics between the two displays, so we can hope to see very different effects of observer variability on them. WG-LCD is representative of modern wide-gamut displays with peaky primaries.

EFFECT OF PHYSIOLOGICAL FACTORS ON DISPLAY COLOR PERCEPTION

In order to simulate the effect of various physiological factors when viewing color stimuli on various displays, chromaticities of these stimuli for a given display and a given set of cone fundamentals must be computed. In this analysis, 24-Colorchecker (CC) color patches were used as the stimuli. XYZ tristimulus values of the CC patches were computed using 10° standard observer and D65. 3x3 primary tristimulus matrices of the two displays, computed from the measured XYZ values of the display primaries, were used to convert the XYZ of the CC patches to the linearized RGB values corresponding to two displays. The product of the linearized RGB values and the spectral data of the display primaries gave the spectral power distributions of the CC patches for a given display. Finally, these spectral data were vectorially multiplied by a given set of 10° CMFs, resulting in the XYZ tristimulus values for the CC patches, the given display and a given set of CMFs. The 10° CMFs were computed from various normal and modified CIE06 10° cone fundamentals. Corresponding CIELAB values were computed using these XYZ values and the XYZ of the white, computed using spectral power distribution of the display white and the CMFs under consideration.

Since the 3x3 matrix for converting the CIE06 cone fundamentals to CMFs has not yet been made available by the CIE committee TC 1-36 8 , an approximate 3x3 matrix was computed using the available 1964 10° x-bar, y-bar, z-bar functions and the cone fundamentals for the average 10° Stiles-Burch observer, which was used throughout the analysis. Since all the CIE06 cone fundamentals (and thus the CMFs) are essentially based on 10° Stiles-Burch data (except for the S-function at higher wavelengths), this approximation is not likely to significantly affect the results and the inferences. wavelength-shift in the low-density absorption spectra [case (iv)] greatly affects the perception of cyan. Comparatively, difference in color perception is not as drastic in case of Ref CRT, supporting suggestion of other researchers 9 that modern displays with narrow-band primaries are more susceptible to observer than broadband CRTs. In terms of the maximum and average ∆E* 00 values for the two displays, ocular media absorption has the most significant effect on variations in color perception, followed by the peak-wavelength shift in the L-cone photo-pigment absorption.

CONCLUSION

If two observers of similar ages but having different peak wavelengths of L-cone photo-pigment absorption spectra are asked to evaluate colors on a modern wide-gamut display with narrow-band primaries (and to a lesser extent, in case of a broad-band display), they are likely to disagree on several color perceptions, particularly in the cyan and blue region of the color space. However, CIE06 model will be unable to predict any variation. This discrepancy is likely to propagate to color matches across displays.

Fig 1

 1 Fig 1 shows log plots of L-and S-cone sensitivities with and without ocular media absorption for various ages. For L-(similar in case of M-), the effect of ocular absorption is more significant at lower wavelengths. In case of the Sfunction, the effect is to shift the peak

Fig 1 :Fig 2 :Fig 3 :

 123 Fig 1: L-and S-cone fundamentals with and without ocular absorption, as described in case (i)

Fig 5

 5 Fig 5 shows the color difference (∆E* 00 ) values for CC patches between the chromaticities obtained from normal cone fundamentals and those obtained from modified cone fundamentals corresponding to the four cases described in Section 1. All values correspond to 10° field size and age 32. For the wide-gamut LCD, while ocular media absorption affects the perception of blue the most, peak

Fig 4 :

 4 Fig 4: Spectral Power Distribution of the two displays used in the analysis

Fig 5 :

 5 Fig 5: Simulated ∆E*00 plots for patches shown on two displays and computed for CMFs derived from normal and modified cone fundamentals, as described in cases (i) through (iv) [top to bottom respectively]


Julien Marty
email: julien.marty@onera.fr

Daaa

Lionel Castillon
email: lionel.castillon@onera.fr

Pierric Joseph
email: pierric.joseph@ensam.eu

Numerical investigations on the rotating stall in an axial compressor and its control by flow injection at casing

Keywords: Active flow control, compressor, rotating stall, air injection

The operating range of a compressor is limited by surge or rotating stall line, among others. Numerical simulations must accurately predict these phenomena. This study is based on the experimental compressor CME2, which is a low-subsonic axial compressor. This compressor is tip-critical as the rotor tip is responsible for the rotating stall. This paper shows that the rotating stall onset flow rate is well captured by CFD, compared to experiments. After ten revolutions, all cells are merged and only one cell remains, as in experiments.

Active flow control improves compressor performance and extends the stable operating range. In the present configuration, flow injection is performed at the casing. In the simulation, the insertion of the actuators is carried out through hybrid meshes: structured mesh for blade passages and unstructured mesh for each actuator. For the some stalled operating points of baseline configuration, there is no rotating cells in the controlled configuration. Thus, the rotating stall is delayed at lower flow rate, as expected by the use of active flow control and is in agreement with the experiments.

INTRODUCTION

The stable operating range of compressors used in aeronautical engines is limited by the choke line, at high flow rate, and surge line, at low flow rate. Surge and rotating stall occurring at low flow rate must be avoided for safety reasons as they can lead to the engine failure. Thus, engine manufacturers design compressors using a safety margin, called surge margin, in order to ensure a stable operating conditions at all flight conditions. Marty 2 TURBO-22-1280

Compressor design relies mainly on computational fluid dynamics. Thus, in order to accurately predict the rotating stall or surge limit, this phenomenon must be well understood. Experiments allow addressing this topic. Poesgen and Gallus [START_REF] Poensgen | Rotating Stall in a Single-Stage Axial Flow Compressor[END_REF] investigated the rotating stall of a single stage axial compressor and determined the three-dimensional pattern of a rotating-stall cell and its unsteady behavior. The onset of rotating stall depends on the tip gap size as shown by Inoue et al. [START_REF] Inoue | Short and Long Length-Scale Disturbances Leading to Rotating Stall in an Axial Compressor Stage With Different Stator/Rotor Gaps[END_REF] and by Hewkin-Smith et al. [START_REF] Hewkin-Smith | The Role of Tip Leakage Flow in Spike-Type Rotating Stall Inception[END_REF]. Dodds and

Vahdati [START_REF] Dodds | Rotating Stall Observations in a High Speed Compressor -Part I: Experimental Study[END_REF] performed static and rotating measurements and showed the coexistence of high and low cell counts and the effect of stallrotor interaction leading to acoustic mode generation. Numerical simulations must also predict the rotating stall onset and cell development. Dodds and Vahdati [START_REF] Dodds | Rotating Stall Observations in a High Speed Compressor -Part II: Numerical Study[END_REF] investigated numerically also the rotating stall in a second article. They showed that the high and low cell counts are well predicted by numerical simulations, suggesting that RANS simulations are sufficient to predict some important patterns of part-span rotating stall behavior. Marconcini et al. [START_REF] Marconcini | A Three-Dimensional Time-Accurate Computational Fluid Dynamics Simulation of the Flow Field Inside a Vaneless Diffuser During Rotating Stall Conditions[END_REF] investigated the rotating stall of an industrial centrifugal compressor and showed that steady simulations based on a full-annulus configuration are able to predict the operating curve and the stall onset is in good agreement with experimental value of flow coefficient. The ability of numerical simulations to predict rotating stall was also shown by Choi et al. [START_REF] Choi | Validation of Numerical Simulation for Rotating Stall in a Transonic Fan[END_REF] on a transonic fan.

Active flow control is an effective way to delay or even to suppress the rotating stall. D'Andrea et al. [START_REF] D'andrea | Rotating Stall Control of an Axial Flow Compressor Using Pulsed Air Injection[END_REF] investigated pulsed air injection on an axial compressor and showed that the hysteresis loop associated to rotating stall can be removed. Margalida et al. [START_REF] Margalida | Active Flow Control in an Axial Compressor for Stability Improvement: On the Effect of Flow Control on Stall Inception[END_REF] succeeded in delaying rotating stall via flow injection at casing, upstream of the rotor and deeply investigated on the onset mechanism of rotating stall and its modification by active flow control. Weigl et al. [START_REF] Weigl | Best Paper Award-Controls and Diagnostics Committee: Active Stabilization of Rotating Stall and Surge in a Transonic Single-Stage Axial Compressor[END_REF] stabilized rotating stall using active feedback control based on twelve air injectors and showed that, injecting 3.6% of the designed compressor mass flow reduces up to 11% the stalled mass flow.

Bae et al. [START_REF] Bae | Active Control of Tip Clearance Flow in Axial Compressors[END_REF] studied also the active flow control of tip clearance flow in a linear cascade with three kinds of actuators. More recently, Ashrafi et al. [START_REF] Ashrafi | Delay of Rotating Stall in Compressors Using Plasma Actuators[END_REF] delayed the rotating stall inception using an annular plasma actuator based on dielectric barrier discharge.

Most studies rely on experiments. Few research works are based on numerical simulations to investigate the effect of active flow control on rotating stall inception and development. Marty et al. [START_REF] Marty | Numerical and Experimental Investigations of Flow Control in Axial Compressors[END_REF] studied several flow control techniques to improve the stall margin of a compressor, but focused mainly on passive flow control such as axisymmetrical grooves, non-axisymmetrical slots, or even recirculation approach. The work of Halawa et al. [START_REF] Halawa | Optimization of the Efficiency of Stall Control Using Air Injection for Centrifugal Compressors[END_REF] aimed at optimizing the efficiency of stall control in centrifugal compressor using air injection and showed that an injection of 1.5% of inlet mass flow rate at 30 degrees delays the stall onset from 4 to 3.8 kg/s. The investigation of Gmelin et al. [START_REF] Gmelin | Active Flow Control Concepts on a Highly Loaded Subsonic Compressor Cascade: Résumé of Experimental and Numerical Results[END_REF] relies on a linear cascade and the use of steady, pulsed or synthetic jets to control corner separation via slots located at the endwall or at the blade suction side. They found an excellent agreement between experiments and simulations in terms of control effectiveness. The objective of this paper is to study the impact of active flow control on the rotating stall onset of an axial compressor, thanks to steady and unsteady RANS simulations. More precisely, this paper aims at investigating the ability of CFD to capture the one-stall-cell pattern on the axial compressor and the delay of stall onset due to the active flow control. This requires an accurate prediction of the rotating stall onset and evolution of stall cells. The paper is organized as follows: first, the experimental test bench is described and then, the numerical setup is detailed. Numerical results are analyzed in two steps: validation of numerical setup on the baseline configuration (without active flow control) and effect of injected flow on rotating stall onset and evolution.

MATERIALS AND METHODS

Experimental setup

The test case considered in this study is the CME2 axial compressor located at Arts et Métiers (Lille, France). This is a low speed, single stage machine representative to a high-pressure compressor of an aero-engine in terms of stage loading [START_REF] Day | Stall, Surge, and 75 Years of Research[END_REF]. Table 1 summarizes the main geometrical characteristics and some flow properties for the rotational speed considered in this study. A dedicated test bench using a 200 kW electric motor and a speed multiplier allows operating the compressor and setting the requested rotational speed. A throttling vane is located after the compressor. It allows moving its working point along its performance curve up to the stall inception. The entrance of the circuit consists in a tranquilization chamber followed by a converging section, allowing also the measurement of the mass flow rate through the stage. Figure 1 depicts the whole installation. One particularity of this setup is to be able to perform repeated stall tests and to operate in rotating stall regime without any damages thanks to its moderate pressure ratio and to its mechanical robustness.

As an experimental tool, this test bench hosts a complete and modular set of instrumentation. It is relevant here to mention a first differential pressure sensor (First Sensor BTEM50025) allowing measuring the mass flow rate at the converging section and another TURBO-22-1280 one, a First Sensor BTEM50350, devoted to the pressure increase between rotor inlet and stator outlet. One estimates the measurement uncertainties of this setup to ±0.012kg.s -1 and ±1.3Pa after in house calibration [START_REF] Margalida | Active Flow Control in an Axial Compressor for Stability Improvement: On the Effect of Flow Control on Stall Inception[END_REF].

FIGURE 1: CME2 COMPRESSOR AND ASSOCIATED TEST BENCH [9]

To complete the description of the experimental setup, it is worth mentioning that this test bench includes a complete active flow control with pulsed jets [START_REF] Margalida | Active Flow Control in an Axial Compressor for Stability Improvement: On the Effect of Flow Control on Stall Inception[END_REF]. This particular setup used 20 slots machined all around the casing of the compressor, each slot hosting a pair of pulsed jets (i.e. 40 jets in total). Each pulsed jet consists in an actuator (a Matrix 820 magnetic valve, rated to 180 nL/min and 500 Hz maximum), paired with a Coandă shaped nozzle with a 10 × 0.5 mm² slot (see Figure 2). Each injector is located 10 mm upstream the rotor leading edge, and is able to rotate along its axis with 15° steps. Thanks to the valves characteristics, up to 2.5% additional mass flow rate can be added to the main flow with near sonic velocity. Each injection block is spaced by 13°, and in a given block, injectors are spaced by 5° (see Figure 3).

Numerical setup

The numerical simulations presented in this article have been accomplished using ONERA's CFD software elsA [START_REF] Cambier | elsA: an efficient object-oriented solution to CFD complexity[END_REF][18] [START_REF] Cambier | The Onera elsACFD software: input from the research and feedback from industry[END_REF],

which has been developed since 1997 and is co-owned by ONERA and SAFRAN. elsA is a multi-application aerodynamic code based on a cell-centered finite volume method for structured and unstructured meshes. Solving the compressible, three-dimensional RANS equations, elsA allows simulating a wide range of aerospace configurations such as aircrafts, space launchers, missiles, helicopters and turbomachines. Therefore a wide range of numerical tools, turbulence models and boundary conditions are available. In the past few years, an important work has allowed the extension of the multi-block structured solver in elsA to an hybrid grid solver, in which structured (ijk-based) and unstructured blocks may coexist within the same computational domain [START_REF] De La Llave Plata | elsA-Hybrid: an all-in-one structured/unstructured solver for the simulation of internal and external flows[END_REF]. Structured zones may be kept for the sake of efficiency and of accuracy in viscous layers, whereas unstructured zones may enable an easier mesh generation and adaptation process. The latest improvements have been achieved by the work of Soismier [START_REF] Soismier | Improvements in the multiblock hybrid CFD solver elsA-H for turbomachinery flow simulation[END_REF]. In order to compute URANS computations of the CME2 configuration with the injectors at the casing, two different grids are considered. The first grid is a mesh corresponding to 1/10 th of the circumference (36°), as illustrated in Figure 4. Indeed, the blade counts of the compressor (30 rotors, 40 stators) and the number of injectors (20 couples) enable to perform such computational domain reduction in order to reduce the computational cost, especially for the computations of the stable operating points of the compressor map. This first mesh includes 50 blocks and 15 million cells. The second grid is the one used for the computations of operating points in the unstable region of the compressor map. Since the flow is no more periodic in space and in time, it is necessary to use a full annulus grid, obtained by duplicating ten times the previous mesh in order to reach a 360° sector, leading to a grid composed of almost 500 blocks and 150 million cells. The non dimensionalised wall cell (y+) is less than 1, and a grid sensitivity analysis had been previously performed [START_REF] Baretter | Experimental and Numerical Analysis of a Compressor Stage under Flow Distortion[END_REF] on this compressor case, showing that grid independency is obtained and that the current grid is acceptable.

The two types of grid are "hybrid", in the sense that they allow both structured and unstructured parts. As illustrated in the close up view in see Figure 5, the mesh of the channels (upstream part in red, rotor part in green, and stator part in blue) is meshed with structured blocks using a classical "O-nH" topology, while the mesh of the injectors is meshed with unstructured blocks. Indeed, due to the complexity of the shape of the injectors, the use of unstructured elements, such as tetrahedra, pyramids, prism and hexahedra eases the mesh generation. The unstructured blocks of the injector (in black) are connected to the structured mesh of the upstream cylindrical channel (in red) thanks to conformal matching boundaries. As reminded in [START_REF] Castillon | A hybrid structured/non structured grid strategy for the CFD modeling of technological effects on complex Turbomachinery applications[END_REF], this "hybrid" grid approach offers several advantages. First of all, it requires much less effort compared to a full structured multiblock grid generation, which can be time expensive and in some cases impossible. Moreover, compared to a full unstructured grid approach, the hybrid grid approach has the advantage of keeping the "O-nH" type grid topology (well adapted to turbomachinery flows) in most of the part of the blade channel. One last advantage of this approach is that the conformal matching between the structured and non-structured zones avoids any conservation loss issue, which can be encountered with the Chimera technique [START_REF] Castillon | A Technological Effect Modeling on Complex Turbomachinery Applications With an Overset Grid Numerical Method[END_REF], or flow filtering issues, which can appear with mismatched abutting interface techniques.

RANS and URANS numerical simulations are performed on the two computational grids using the upwind scheme of Roe [START_REF] Roe | Approximate Riemann solvers, parameter vectors and difference schemes[END_REF] associated to the MUSCL "minmod" limiter. The Harten coefficient value is set as small as possible (Roe=0.01) in order to be at the same time robust and as little dissipative as possible. The first order accurate Backward-Euler scheme is used for time integration, coupled to an LU implicit phase. For the URANS computations, the physical time step has a value close to 1s corresponding to 18,000 time steps to perform a single rotation. Previous study [START_REF] Baretter | Experimental and Numerical Analysis of a Compressor Stage under Flow Distortion[END_REF] showed that the chosen time step values enables a satisfactory capture of the main unsteady phenomenon, in particular the rotating stall phenomena The one equation Spalart-Allmaras model [START_REF] Spalart | A One-Equation Turbulence Model for Aerodynamic Flows[END_REF] is used for turbulence, previous studies having shown its ability to capture rotating stall phenomenon [START_REF] Gourdain | Numerical Simulation of Rotating Stall in a Subsonic Compressor[END_REF]. Concerning the boundary conditions, their locations are depicted in Figure 7, especially inlet/outlet planes and mixing planes. At the inlet of the computational domain an axial flow direction is imposed with uniform values for the stagnation temperature (288 K) and stagnation pressure (101,325 Pa). At the inlet of the injectors (maximum radius location of the black injector domains on Figure 4), atmospheric stagnation temperature (288 K) and mass-flow are imposed (2g/s for each injector, corresponding to 80 g/s for the whole circumference, which is close to 2% of the stall mass flow rate). For these two inlet boundaries, very low levels of turbulent quantities are imposed (close to 0). Some of the calculations are accomplished without any injection through the injectors, by keeping the domains of the injectors and replacing the injection condition with a solid wall treatment on the upstream frontier of the injectors at the maximum radius location. Adiabatic wall conditions are imposed on all of the walls (blades, casing, hub, injector walls). Finally, downstream of the computational domain, a subsonic outlet condition is used in which the static pressure is imposed with a radial equilibrium law, the downstream pivot static pressure used for the radial equilibrium law integration being defined through a valve law (Ps=Pref+relax point (the mass-flow decreases as the relax parameter increases. The steady computations are accomplished using a mixing plane treatment while it is replaced by a sliding join condition for the URANS computations. The position of the rotor/stator interfaces is located between the three blade rows, as highlighted in Figure 5, with the three different rows represented respectively in red, green and blue.

(Q/Qref) 2 ),

RESULTS AND DISCUSSION

Baseline configuration

The first step of this study consists in evaluating the ability to predict the rotating stall limit of the CME2 compressor by comparison to experimental data. This is performed through the calculation of the static pressure rise at casing, across the stage, between planes located 73 mm upstream and 225 mm downstream of the rotor leading edge, respectively. The static pressure is then averaged in the circumferential direction as simulations are performed in relative frame. Figure 8 shows the evolution of static pressure rise coefficient along with the flow coefficient. The experimental values are compared with different RANS calculations (performed on different computational domains) and with the URANS computation (performed on the 360° sector). Experimentally, the rotating stall of compressor occurs at a flow coefficient of 0.4. The stall onset is relative to sudden drop of the static pressure rise coefficient, as shown by the blue arrow. First steady RANS simulations based on one single blade channel (SBC) are able to predict this operating limit (green line). The computational domain of these simulations is not prepared to receive the active flow control device, as the red part is 2/30periodic instead of 2/20-periodic, and thus, they do not contain the upstream mixing plane between red and green parts in Figure 5 and in Figure 7. These simulations capture well the stall onset. Nonetheless, due to the presence of injectors, which are similar to a fixed row, the upstream mixing plane is required and this influences the static pressure rise by delaying the rotating stall occurrence up to 0.36

and by decreasing the level of static pressure rise over the stable operating range. As a result, the discrepancy in terms of stall onset with experiments comes from the added mixing plane that acts as a boundary condition placed very close to the rotor and not from the solver ability to capture the rotating stall. Similar results are obtained with 1/10 th of the circumference, up to the stall limit. Small discrepancies appear beyond this limit as the difference in terms of periodic conditions (1/10 th vs. single blade channel) leads to different boundary

FIGURE 9: STATIC PRESSURE CONTOURS AND STREAMLINES IN RELATIVE FRAME AROUND THE ROTOR LEADING EDGE

In order to explain discrepancies between RANS simulations with a single blade channel, without and with the upstream mixing plane, RANS-SBC_w/o_upxmpl and RANS-SBC configurations, respectively, Figure 9 depicts the static pressure field and streamlines computed in the relative frame, at 97% of the relative height in spanwise direction (100% is relative to the casing wall), for the flow coefficient equal to 0.44. Two blue dashed lines are used as lines of reference to make the comparison easier. The comparison of both simulations shows that, as the upstream mixing plane is located in the potential effect of the rotor, this affects the angle of attack viewed by the rotor. The purely axial flow allowed by the presence of the upstream mixing plane is observed closer to the rotor and is not influenced by the static pressure field around the rotor. This leads to a significant effect on the angle of attack. With the upstream mixing plane, this angle is smaller. Thus, this explains why the stall onset occurs at a lower flow coefficient. TURBO-22-1280

Before analyzing the influence of the active flow control, it is important to highlight the occurrence of the rotating stall in the compressor. Experimentally, the rotating stall occurs at the rotor tip and only one cell is observed on the static pressure measurements [START_REF] Margalida | Active Flow Control in an Axial Compressor for Stability Improvement: On the Effect of Flow Control on Stall Inception[END_REF]. Figure 10 depicts the time evolution of the entropy variation across the compressor, from an initial field which is the steady state obtained by RANS simulations. The contours are shown at 90% spanwise, at four different instants (after 1, 3, 5 and 10 revolutions after the start of the unsteady simulation), for the time-averaged flow coefficient of 0.36. Figure 11 depicts, with the same scale, the entropy variation in the rotor/stator axial interface plane. The evolution of entropy rise highlights the alignment of tip leakage vortex to the rotor leading edge plane and the appearance of rotating stall cells. After one revolution, the periodicity of the flow is lost and high loss areas are observed near the tip, close to the trailing edge. These are the footprints of separation in each rotor channel. After three revolutions, ten cells are observed and they are not periodic, some rotor channels being blocked, and others not. The blockage due to stall cell creates an incidence rise on the adjacent blade, leading to stall. As each cell covers a different azimuthal extent and their rotating velocities differ allowing an interaction between rotating stall cells. The low velocity zones created by stall cells rotate in the circumferential direction and gradually tend to merge as they rotate with the rotor blades, leading to two main cells after five revolutions, and a single cell after ten revolutions. Thus, five additional revolutions are necessary to switch from two to one cell. It should be noticed, that, at the flow coefficient 0.4, there is no rotating cell, in the full-annulus simulation (URANS) as shown in Figure 12. After stabilization of the simulation, the flow is still periodic and no separation occurs at the tip. For the other operating points, the higher the throttle is, the faster separations occur and rotating cells merge.

Impact of active flow control

As shown before, injectors are added with hybrid meshes. RANS simulations are carried out on the 1/10 th configuration for all orientations of the injector (Figure 13). The injected mass flow rate, in standard condition, is 2g/s for each injector, that is to say 80g/s for the full-annulus (less than 2% of the mass flow at the rotating stall onset). The experimental data with active flow control based on an axial injection (inj=0°) is included in the graph. The agreement between numerical results and experimental data is good in the stable operating range. However, the static pressure drop occurs too early in the simulations. Figure 14 depicts the static pressure rise coefficient for the rotor, the stator and the stage. This shows that the first decrease of static pressure rise is not due to the stall on the rotor but on a static pressure drop across the stator. Figure 15 depicts the wall streamlines over rotor and stator suction sides and shows that a corner separation over stator suction side occurs before the stall onset over the rotor blade. This too spread separation over the stator suction side at the casing junction with the casing is responsible for the first decrease of static pressure rise coefficient.

In the stable operating range (flow coefficient higher than =0.4), the active flow control does not significantly influence the static pressure rise coefficient across the compressor stage, the only exception being the 30° angle. The effect of injected flow is clearly visible once the rotating stall occurs in the baseline configuration. The reduction of static pressure rise coefficient is delayed to lower flow coefficient as the maximum value reaches =0.47 at =0.39 instead of =0.46 at =0.44. For lower flow coefficient, the static pressure Marty 14

TURBO-22-1280 rise coefficient remains higher than 0.44 up to =0.32 before a sudden drop. By comparison to the baseline configuration, the drop is delayed from =0.36 to =0.32, which is similar to the delay observed on the maximum static pressure rise coefficient. This variation (0.04) is close to the experimental shift of the stall onset (0.05). The effect of the injector orientation is small by comparison to the adding of active flow control in the computational domain. This is due to the use of the mixing plane between injectors (similar to a fixed row) and rotors, which averages the flow in the azimuthal direction. Far from injectors in circumferential direction, injected massflow does not influence the flow orientation. Thus, the flow remains axial in the absolute frame. As the injected flow covers partially the azimuthal extent, this zero angle reduces the area-averaged flow angle due to the mixing plane and finally viewed by the rotors. The mixing plane tends to minimize the effect of flow angle. Thus, unsteady simulations are required to compute more accurately the effect of injector angle. Nevertheless, as this study aims at investigating the effect of flow control on rotating stall, these unsteady simulations are only carried out with the axial injection angle (inj=0°). Unsteady simulations are only performed on the latter one. Numerical results of baseline configuration are also recalled (single blade channel, 1/10 th and full-annulus). The static pressure drop is delayed and reduced by adding the active flow control. This effect is seen quite consistently on the three different types of computations: steady calculations on the 1/10 th mesh, steady and unsteady computations on the full-annulus mesh. Yet the full annulus unsteady computations tend to predict a lower pressure drop than the steady computations.

Thus, RANS simulation are slightly pessimistic. It should be noticed that length of time signal is insufficient for the time average computation. Thus, these operating points are given for information only.

Figure 18 depicts the entropy variation field at 90% span, after five revolutions for baseline (top) and controlled (bottom) configurations. The initial field is still a steady RANS simulation based on mixing plane at the rotor-stator interface. The four operating points are the same as in Figure 12:=0.4, 0.36, 0.32 and 0.28, from left to right, respectively. At stable operating point, the flow is periodic in the baseline configuration while the active flow controls leads to a slight loss of periodicity without any impact on stall limit.

The flow is certainly different from one passage to another, but there is no sign of blockage in each passage. Thus, the compressor TURBO-22-1280 operates at a stable regime. By throttling the compressor (decrease of mass flow rate), discrepancies between baseline and controlled configurations are more and more observable. At the flow coefficient 0.36, rotating cells are visible on the baseline configuration. The default of periodicity is still present in the controlled configuration and is accentuated by areas of high and low entropy variation alternately. Separations and blockage occur over one or two passages and the one or two next passages are not subjected to these phenomena thanks to the flow injected at casing. Thus, after five revolutions, there is no rotating cells at this mass flow rate while the baseline configuration is stalled. At flow coefficients, 0.32 and 0.28, two rotating cells are visible on the entropy variation field but their development is slowed down by the injected flow. The active flow control delays the rotating stall onset and pushes back the stable operating limit. It should be noticed that the drop of static pressure rise is directly linked to the appearance of the rotating stall cells. In order to highlight the delay of stall onset, time signal analysis is performed for all full-annulus computations. Figure 19 shows the axial momentum signal recorded on a gauge located in the absolute frame of reference on the rotor/stator interface at 90% of span height. All signals are plotted both in time and frequency domain thanks to the use of Fast Fourier Transform. The red and blue signals correspond respectively to the case without and with injection. The frequency domain allows the identification of particular frequencies, such as the rotor passing frequency as the gauge is then located downstream of the rotor: 30 events per revolution are directly linked to the rotor blade number (30).

For the first shown operating point (=0.4), the flow is stable for both configurations (without and with active flow control) and the existing frequencies correspond as expected to the rotor blade passing frequencies and its harmonics (30 blades). For the second operating point (=0.36), rotating stall occurs in the baseline configuration, which explains the appearance of a large spectrum of low frequencies: the flow periodicity is broken and the time evolution is no more driven by the blade passing frequencies. On the contrary, although some low frequencies are visible on the spectrum due to the periodicity loss, the configuration with active flow control remains stable and has a Fourier spectrum close to the one of a stable operating point. This is no longer the case for the third operating point (=0.32) for the simulation with flow injection at casing since the injection does not delay anymore the onset of rotating stall cells. TURBO-22-1280

However the time signals plotted in, Figure 19-c show that the appearance of instabilities is delayed with the injection. Rotating stall occurs after 10 ms without active flow control and after 60 ms with injectors. Thus, in a real compressor, even if active flow control cannot remove indefinitely the stall onset, this increase the available time between a possible precursor detection and stall onset in order to avoid the latter. Finally, for the last operating point, a low frequency corresponding to two events per revolution is clearly and corresponds to the two rotating cells observed after five revolutions. The time signal analysis is thus coherent with the previous analysis of the entropy variation field.

CONCLUSION

The present numerical study focuses on the rotating stall in the axial compressor CME2 and its control. The active flow control corresponds to flow injection at casing thanks to twenty pairs of injectors and based on the Coandă effect in order that the injected flow remains close to the endwall. The injected mass flow rate is smaller than 2% of the mass flow rate at stall onset.

The comparison to experimental data, from single blade channel to 1/10 th to full-annulus configurations, validates the numerical setup. The rotating stall is well captured by steady RANS CFD despite the delay due to the presence of a mixing plane between injectors and rotors.

The analysis of the baseline configuration shows that the rotor is tip-critical as separations occur at blade tip. Moreover, there is an alignment of tip leakage vortex with the rotor leading edge, leading to the compressor stall. For the present compressor, ten revolutions are required to capture the stall pattern composed of one rotating stall cell. Five revolutions are necessary to switch from two to one cell.

Active flow control does not influence significantly the static pressure rise across the compressor. Nevertheless, as expected, it delays the drop of static pressure rise, which is a sign of occurrence of rotating stall and emergence of stall cells. Both steady and unsteady RANS simulations predict this drop, even if steady computations are slightly pessimistic. Even if this statement can be configuration-dependent, the drop is well captured by steady RANS computations over a reduced computational domain (1/10 th in the present case thanks to the spatial periodicity).

In the present study, the rotating stall onset is delayed at least from the flow coefficient 0.4 to 0.36, as rotating stall cells are visible at =0.32 for the controlled configuration. This observation performed on entropy variation field at 90% span is corroborated by the analysis of axial momentum signals recorded by a gauge at the same channel height. Moreover, active flow control slows down the development of rotating stall cells. This statement will be verified experimentally. If confirmed, thanks to tip injection, the slow-down of rotating stall would increase available response time to avoid the stall onset once precursors are detected. TURBO-22-1280

As the effect of injector angle is only investigated on the 1/10 th configuration with steady RANS simulations, which rely on the use of a mixing plane at the rotor-stator interface, this effect will be studied with unsteady simulations on the full-annulus configuration, as further works. This is mandatory to fully validate the capacity of the numerical setup to predict the effect of flow injection at casing.

A future publication will present the power balance of the flow control system. Preliminary analyses of the on-going tests on the CME2 compressor show that flow injection saves energy for certain operating points.

FIGURE 2 :

 2 FIGURE 2: AXIAL POSITONNING AND COMPOSITON OF PULSED JETS

FIGURE 3 :

 3 FIGURE 3: ANGULAR ARRANGEMENT OF THE PULSED JETS AROUND THE CASING

FIGURE 4 :

 4 FIGURE 4: MESH OF THE CME2 COMPRESSOR (1/10TH OF THE CIRCUMFERENCE). GLOBAL VIEW OF THE 3 ROWS

Figure 6

 6 represents close up views of the 5 meshes of the injectors corresponding to the 5 different injections angles (inj) which are investigated : -30°,-15°, 0°,+15° and 30°. In the schematic view, the green rectangle represents the injection angle equal to zero. The flow is injected from the left to the right, that is to say, the flow direction is aligned with the rotational axis in the blade-to-blade view.

FIGURE 5 :

 5 FIGURE 5: MESH OF THE CME2 COMPRESSOR (1/10 th OF THE CIRCUMFERENCE). CLOSE UP VIEW AT THE CASING ZONE

FIGURE 6 :FIGURE 7 :

 67 FIGURE 6: CLOSE UP VIEW OF THE INJECTOR GRID FOR THE DIFFERENT ANGLES OF INJECTION

  where Q is the massflow at the outlet boundary and ()ref values are reference values for static pressure and massflow. This means that for each operating point, the parameter relax is modified in order to change the target mass-flow and the associated operating Marty 9 TURBO-22-1280

  conditions and the converged steady state differs. Four operating points are simulated over the full-annulus configuration (steady RANS and unsteady RANS). Results are similar to previous numerical data, in the stable operating range. Once the rotating stall occurs, discrepancies are observed and can be explained by an insufficient time length used to compute time averaged flow.

FIGURE 8 :

 8 FIGURE 8: STATIC PRESSURE RISE ACROSS THE COMPRESSOR STAGE FOR THE BASELINE CONFIGURATION

FIGURE 10 :FIGURE 11 :FULL

 1011 FIGURE 10: ENTROPY VARIATION FIELD AT 90% SPAN AT THE FLOW COEFFICIENT =0.36, FOR THE FULL ANNULUS CONFIGURATION

FIGURE 13 :

 13 FIGURE 13: STATIC PRESSURE RISE ACROSS THE COMPRESSOR STAGE FOR THE CONTROLED CONFIGURATIONS WITH DIFFERENT INJECTOR ORIENTATIONS

FIGURE 14 :

 14 FIGURE 14: STATIC PRESSURE RISE ACROSS THE ROTOR, THE STATOR AND THE STAGE FOR THE 1/10 TH SECTOR CONFIGURATION, WITHOUT AND WITH ACTIVE FLOW CONTROL

FIGURE 15 :

 15 FIGURE 15: WALL STREAMLINES OVER THE ROTOR (LEFT) AND STATOR (RIGHT) SUCTION SIDES AT THREE OPERATING POINTS, FOR THE CONFIGURATION WITHOUT ACTIVE FLOW CONTROL.

Figure 16 and

 16 Figure 16 and Figure 17 show the static pressure rise coefficient of the baseline configuration and of the controlled configuration with the zero angle orientation. Computations on controlled configuration are carried out for the 1/10 th sector and for full-annulus.

FIGURE 16 :

 16 FIGURE 16: STATIC PRESSURE RISE ACROSS THE COMPRESSOR STAGE FOR THE BASELINE CONFIGURATION AND CONTROLED CONFIGURATION WITH ZERO INJECTOR ANGLE

FIGURE 17 :FIGURE 18 :

 1718 FIGURE 17: STATIC PRESSURE RISE ACROSS THE COMPRESSOR IN THE UNSTABLE REGIME (FLOW COEFFICIENT LOWER THAN 0.4)

FIGURE 19 :

 19 FIGURE 19: COMPARISON OF AXIAL MOMENTUM SIGNALS (TIME/FREQUENCY DOMAINS, H/H=90%)

TABLE 1 :

 1 CME2 COMPRESSOR MAIN CHARACTERISTICS

	Rotational speed (rpm)	3200
	Design mass flow rate (kg/s)	5.3
	Design axial velocity, LE (m/s)	43
	Rotor blade number	30
	Stator blade number	40
	Casing diameter (mm)	550
	Hub-tip ratio, LE	0.75
	Rotor tip chord (mm)	84
	Rotor tip stagger angle (°)	54
	Rotor tip gap (mm)	0.5

ACKNOWLEDGEMENTS

All simulations have been performed in the framework of the elsA agreement between SAFRAN and ONERA, which are co-owners of this software. This paper is funded by European Union's Horizon 2020 research and innovation program under grant agreement No 886352, project ACONIT.

NOMENCLATURE

RANS


Abhijit Sarkar
email: abhijit.sarkar@thomson.net

Laurent Blondé

Patrick Le Callet

Florent Autrusseau

Jürgen Stauder

Patrick Morvan

Study of Observer Variability in Modern Display Colorimetry: Comparison of CIE 2006 Model and 10°S tandard Observer

Keywords: Color Vision, Cone Fundamental, Color Matching, Observer variability, Display Colorimetry

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

INTRODUCTION

In 2006, CIE's (Commission Internationale de l'Eclairage) technical committee TC 1-36 published a report [START_REF] Cie | Fundamental Chromaticity Diagram with Physiological Axes -Part I[END_REF] (described hereafter as CIE06) on the choice of a set of Color Matching Functions (CMF) and estimates of cone fundamentals for the normal observer for any field size between 1° and 10° and for an age between 20 and 80. The CIE06 model is largely based on the 1959 Stiles-Burch (S&B) 10° CMFs [START_REF] Stiles | colour-matching investigation: final report[END_REF] .

A theoretical analysis of various aspects of CIE06 cone fundamentals from the perspective of display colorimetry was performed. In the first part of the analysis [START_REF] Sarkar | Study of observer variability on modern display colorimetry: an analysis of CIE 2006 model[END_REF] , we showed that the peak-wavelength shift in the long wave-sensitive photopigment absorption spectra, excluded from CIE 2006 model but present in normal observers, is a significant contributor to observer variability in modern display colorimetry. This paper presents the second part of the analysis, addressing two key questions: i) Does CIE06 age parameter correspond to the real ages of 47 S&B observers?, and ii) If we consider a specific age group, which set of CMFs best represents the intra-group average of S&B observer data, those derived from CIE06 or the 1964 10° standard observer?

CIE06 AGE PARAMETER AND REAL OBSERVER AGES

We explored the possibility that predicted CIE06 functions that best match the real observer data may not always be obtained using real observer ages. This could possibly happen because of random observer variability, and/or because of the exclusion of one or more age-independent physiological factors in the CIE06 model [START_REF] Sarkar | Study of observer variability on modern display colorimetry: an analysis of CIE 2006 model[END_REF] . We determined the CIE06 ages that resulted in the best predictions of individual S&B observer data. In this computation, we converted all S&B observer CMFs into corresponding cone fundamentals, using a 3x3 transformation matrix computed from the available average S&B observer CMFs and average S&B observer cone fundamentals [START_REF] Cie | Fundamental Chromaticity Diagram with Physiological Axes -Part I[END_REF] .

Two different methods were used, in which CIE06 age parameter values were computed separately for each observer's three cone fundamentals. In the first method, correlation coefficients between the normalized cone fundamentals for each S&B observer and those corresponding to all possible CIE06 age parameter values between 20 and 80 (a total of 61) were computed. For each S&B observer, the corresponding CIE age was the one yielding the highest correlation coefficient for a given cone fundamental. This process was repeated for all three cone fundamentals and for all 47 S&B observers.

In the second method, the corresponding CIE06 age for each S&B observer was determined by minimizing the RMS errors between various CIE06 functions and real observer data.

In fig 1, the actual ages of 47 S&B observers have been plotted against the CIE06 predicted ages obtained using the correlation coefficient (CORR) method. The second method (RMSE) produced very similar results. No direct correspondence exists between the real and predicted ages that can be explained through a mathematical function. However, with the correlation coefficient method, CIE06 predicted ages for 38 observers were within ±5 years of real observer age for at least one of the three CIE06 cone fundamentals. With the RMSE method, this was true for 34 observers.

Fig 2 shows chromaticities of matches of equal-energy white computed with CMFs derived from CIE06 cone fundamentals for each S&B observer. CIE06 cone fundamentals were obtained by using corresponding ages from both methods (CORR and RMSE) as well as by using actual observer ages. Matches obtained with real observer cone fundamentals are also plotted. While CIE06 with age correspondence (with either method) yields greater observer variability than CIE06 with actual observer ages, it fails to explain all the variability in the real observer data, particularly in the ydirection. The standard deviation plot of the CIE06 cone fundamentals (averaged over all observers) with respect to the real observer data are also shown in fig 2 .  Mean (central mark), as well as the 25 th and 75 th percentiles (dotted bars) of standard deviations are higher when age correspondence is ignored. The error is higher for L-than for M-or S-. In the rest of the analysis, we use the correlation coefficient method for age correspondence.

STILES-BURCH OBSERVER GROUPS

Out of 47 S&B observers, three age-groups were identified. Six observers with ages between 22 and 23 formed Group-1, ten observers with ages between 27 and 29 consisted Group-2 and another six observers with ages between 49 and 50 were placed in Group-3. In the rest of the analysis, these three observer groups were used. For each group, age correspondence for the average data were established by using the average S&B cone fundamentals for the group and CIE06 cone fundamentals for all possible ages.

COMPARING CIE06 MODEL PREDICTION AND 10° STANDARD OBSERVER WITH INTRA-GROUP AVERAGE OBSERVER DATA

Due to the space constraint in this paper, we only show a few representative plots. Significant variations in x-CMF were observed between CIE06 model predictions and the 10° standard observer in case of Group 1 and 3. While CIE06 predictions worked well for the z-CMF, 10° standard observer showed significant error in all three groups. In general, for the y-CMF, the variations between the CMFs were not statistically significant for any of the three groups. Fig 3 shows the log of wavelength-wise absolute difference of CIE06 predictions and 10° standard observer from the average S&B observer x-CMF for Group-1 (weighted by the same average). Difference of the 75 th percentile of the group observer data from the average indicates the intra-group observer variability. For Group-1, the age correspondence method improves CIE06 prediction of x-CMF and is better than the 10° standard observer, which is also evident from fig 4. In general, CIE06 with no age correspondence produces more or similar errors as the 10° standard observer for the x-CMF for Group-1 and 3. Cone fundamental plots show that in both cases, the average observer prediction error is more significant for the L-than for the M-.

DISPLAYS

In our analysis, we used four displays, two of which are included in this paper, and also described in the companion paper [START_REF] Sarkar | Study of observer variability on modern display colorimetry: an analysis of CIE 2006 model[END_REF] . The first is a Sony BVM32 Cathode Ray Tube (CRT) reference studio display (hereafter referred to as Ref-CRT). The second is a Hewlett-Packard DreamColor Wide-Gamut Liquid Crystal Display with LED backlight (hereafter referred to as WG-LCD). Significant difference in the spectral power distributions of the two displays (fig 5) is likely to have different effects on observer variability. WG-LCD is representative of modern wide-gamut displays with peaky primaries.

DISPLAY COLORIMETRY: WHICH SET OF CMFS IS BETTER?

In order to compare the results obtained by using various CMFs with respect to real observer data, chromaticities of color stimuli for a given display and a given set of CMFs must be computed. Note that display nonlinearity is not relevant for this simulation. As before 3 , 24-Colorchecker (CC) patches were used as the stimuli. XYZ tristimulus values of the CC patches were computed using 10° standard observer and D65. 3x3 primary tristimulus matrices of the two displays, computed from the measured XYZ values of the display primaries, were used to convert the XYZ of the CC patches to the linearized RGB values corresponding to two displays. The product of the linearized RGB values and the spectral data of the display primaries gave the spectral power distributions of the CC patches for a given display. Finally, these spectral data were vectorially multiplied by a given set of 10° CMFs to obtain the XYZ tristimulus values for the CC patches, the given display and a given set of CMFs. CIELAB values were computed using these XYZ values and the XYZ of the white, computed using spectral power distribution of the display white and the CMFs under consideration. In case of CIE06, the xyz CMFs were computed from various normal and modified 10° cone fundamentals. For this, as explained in the companion paper [START_REF] Sarkar | Study of observer variability on modern display colorimetry: an analysis of CIE 2006 model[END_REF] , a 3x3 transformation matrix was computed using the 1964 10° z y x functions and average S&B cone fundamentals, which was used throughout the analysis. ∆E* 00 values for the CC patches were computed separately for each display, between any two sets of CMFs.

Fig 6 shows the color difference (∆E* 00 ) values for CC patches simulated for WG-LCD, and computed with CIE06 CMFs and 10° standard observer, with the intra-group average observer data as the reference. In case of Group-1, 10° standard observer performs better than the CIE06 model. The improvement with age correspondence is related to the x-function prediction error discussed earlier.

For Group-1 and -3, while 10° standard observer produced relatively large errors in cyan/blue, arguably due to the poor match with the average observer in z-CMF, CIE06 model resulted in larger ∆E* 00 values in the longer wavelengths of the color spectrum, mainly due to the x-CMF prediction error. For Group-3, age correspondence method had no significant impact on ∆E* 00 values for CIE06. This indicates that in this case the x-CMF prediction error cannot be rectified by simply choosing different CIE06 age parameter values as model inputs. For Ref CRT, the difference between the CIE06 CMFs and the 10° standard observer was less apparent.

CONCLUSION

The long-wave sensitive (x-) color matching function derived from the CIE06 model did not accurately predict average Stiles-Burch observer data for two subgroups in the age ranges of 22-23 and 49-50. This could be due to random observer variability, but might also reflect the effect of at least one age-independent physiological factor not modeled in CIE06 [START_REF] Sarkar | Study of observer variability on modern display colorimetry: an analysis of CIE 2006 model[END_REF] . The prediction error is reflected in the longer wavelengths of the color spectrum. On the other hand, the shortwave sensitive (z-) function of the 10° standard observer produced significant deviation with respect to the intra-group average observer for all three groups. Subject to further evaluation, the CIE06 model can provide the basis for the refinement of 10° standard observer.

Fig 1 :

 1 Fig 1: Age correspondence between CIE06 and 47 Stiles-Burch observers, actual observer age vs. CIE06 predicted age

Fig 2 :Fig 3 :

 23 Fig 2: Chromaticities of matches of equal-energy white, computed using cone fundamentals from the 47 Stiles-Burch observer data and CIE06 predictions (top), and mean standard deviation of CIE06 cone fundamentals from the 47 Stiles-Burch observer data, with and without age correspondence (bottom)

Fig

  Fig 4: x-CMF for the average Stiles-Burch Observer, CIE06 model predictions and 10° standard observer for Group 1 [top] and Group 3 [bottom]

ACKNOWLEDGMENT

We gratefully acknowledge Prof. Andrew Stockman, Dr. Boris Oicherman and Prof. Alexander Logvinenko for their assistance with the tabulation of Stiles-Burch dataset, and Prof. Stockman and Prof. Françoise Viénot for the valuable discussions on this topic.