Clément Patat
email: clement.patat@coria.fr

Françoise Baillot

Jean-Bernard Blaisot

Eric Domingues

Cl Ément Patat

Franc ¸oise Baillot

Éric Domingues

Responses of lean swirling spray flames to acoustic pressure and transverse velocity perturbations

Keywords: Combustion instabilities, transverse acoustic forcing, spray flames, swirling flows

Introduction

Gas turbines and aircraft engines are mostly operated with swirling spray flames. To ensure the sustainability of these devices, a significant effort must be produced to find technological solutions that reduce their environmental footprint. A promising solution is to operate in lean combustion, which can lead to a decrease in the emission in soot and nitrogen oxides, and a growth in combustion efficiency. Nevertheless, combustors operating in lean conditions are more susceptible to develop thermo-acoustic instabilities. In annular combustors, these instabilities generally couple with azimuthal modes of the cavity, which can be reproduced experimentally in academic configurations, notably through transverse acoustic forcing. The acoustic field produced inside the combustor may be of various types: standing [START_REF] O'connor | Flame and flow dynamics of a self-excited, standing wave circumferential instability in a model annular gas turbine combustor[END_REF][START_REF] Vignat | High amplitude combustion instabilities in an annular combustor inducing pressure field deformation and flame blow off[END_REF], spinning [START_REF] Mazur | Characteristics of self-excited spinning azimuthal modes in an annular combustor with turbulent premixed bluff-body flames[END_REF][START_REF] Bourgouin | Characterization and modeling of a spinning thermoacoustic instability in an annular combustor equipped with multiple matrix injectors[END_REF], or a combination between different modes [START_REF] Bourgouin | A new pattern of instability observed in an annular combustor: The slanted mode[END_REF]. In the case of standing modes, flames may be subjected to acoustic perturbations in pressure or velocity, depending on their position in the acoustic field. The main effect of pure pressure fluctuations is to induce axial flow fluctuations that are symmetric with respect to the injector axis [START_REF] O'connor | Transverse combustion instabilities: Acoustic, fluid mechanic, and flame processes[END_REF]. These axial fluctuations are generally dominant in the gaseousflame response. However, when located between a pressure antinode and a velocity antinode, a premixed V-flame was shown to undergo dissymmetric perturbations that might be linked to the crossed contribution of the acoustic pressure and its gradient. Indeed, considering only the pressure gradient cannot explain the intensity of the dissymmetry [START_REF] Lespinasse | Responses of v-flames placed in an hf transverse acoustic field from a velocity to pressure antinode[END_REF].

Transverse velocity oscillations can also lead to a positive thermo-acoustic coupling via the flame displacement at a velocity antinode [START_REF] Schwing | High-frequency instabilities in cylindrical flame tubes: feedback mechanism and damping[END_REF]. In addition, other types of flame responses can be observed, e.g., high-amplitude instabilities have been shown to induce the extinction of swirling spray flames near the pressure nodal line [START_REF] Vignat | High amplitude combustion instabilities in an annular combustor inducing pressure field deformation and flame blow off[END_REF], showing that transverse velocity fluctuations may have a strong effect on the flames. This is consistent with results obtained for a premixed V-flame [START_REF] Lespinasse | Responses of v-flames placed in an hf transverse acoustic field from a velocity to pressure antinode[END_REF].

Liquid-fuel combustion is also of major interest since most gas turbines are operated with liquid fuels. The presence of a spray introduces supplementary complexity due to the need for the fuel to evaporate and to mix with the air before combustion. Moreover, the presence of two distinct phases implies that their response to an acoustic perturbation is not identical. For an airblast atomizer and in non-reactive conditions, longitudinal acoustic forcing has been shown to generate droplet number density waves [START_REF] Gajan | Investigation of spray behavior downstream of an aeroengine injector with acosutic excitation[END_REF]. The drop size distribution is modified during the acoustic cycle by the higher sensitivity of small droplets to flow velocity fluctuations [START_REF] Giuliani | Influence of pulsed entries on a spray generated by an airblast injection device: an experimental analysis on combustion instability processes in aeroengines[END_REF], but also by the unsteady air momentum. In particular, for an airblast atomizer, a correlation between drop Sauter diameter D 32 and air velocity has been found, indicating that D 32 diminishes as the air velocity increases [START_REF] Eckstein | Experimental study on the role of entropy waves in low-frequency oscillations in a rql combustor[END_REF]. Since evaporation is more efficient for small droplets, the evaporation dynamics is modified with the drop size distribution and the air-fuel mixing can be produced more rapidly with small droplets. Thus, these modulations of the spray generates fluctuations of the local equivalence ratio, which is known to be a mechanism for thermoacoustic instability [START_REF] Huang | Dynamics and stability of lean-premixed swirl-stabilized combustion[END_REF]. In particular, the modification of the evaporation dynamics has been shown to influence self-excited combustion instabilities [START_REF] De La | Investigations on the self-excited oscillations in a kerosene spray flame[END_REF]. Nevertheless, the evaporation rate could be reduced in regions of high droplet number density through droplet clustering, which would therefore act as a damping effect [START_REF] Réveillon | Evaporating droplets in turbulent reacting flows[END_REF]. A more recent numerical work has shown the capability of a liquidfuel film created on the walls of the injection system to couple with the thermo-acoustic oscillation, leading to selfsustained combustion instability [START_REF] Schiavo | Effects of liquid fuel/wall interaction on thermoacoustic instabilities in swirling spray flames[END_REF]. However, few studies examine spray flame response to both pressure and velocity perturbations in a transverse acoustic field. Consequently, it is of great interest to examine the response of swirling spray flames at various locations of a standing transverse acoustic field. This work aims at understanding how flames located at various positions of a standing transverse acoustic field respond to the acoustic perturbation. It proposes to evaluate the spray flame response via a linear array of three swirling spray flames, while the central flame is placed either at a pressure antinode (PAN) or at an intensity antinode (IAN) of the acoustic field. The latter location combines pressure fluctuations, a pressure gradient and transverse velocity fluctuation, all of them presenting significant amplitudes. A brief description of the experimental setup and postprocessing methods is firstly provided. The global flame response is evaluated at both locations, then the spacetime flame response is examined through high-speed camera. Finally, an investigation of the spray response is performed.

Materials

Experimental setup: TACC-Spray cavity

The TACC-Spray setup consists in an acoustic cavity in which a linear array of five swirling injectors of exit diameter D exit = 8 mm is placed. The acoustic cavity of length L c = 880 mm, height h c = 200 mm and width e c = 55 mm is surmounted by a convergent part of height 60 mm and exit width 10 mm, which avoids outer air to go inside the cavity. Air and liquid n-heptane are supplied to the three central injectors while the two side injectors are fed with air only (see Fig. 1). This configuration helps to stabilize the lateral flames far from the side walls, and therefore helps to keep the central flame in an environment similar to that of the annular combustor MICCA-Spray [START_REF] Vignat | High amplitude combustion instabilities in an annular combustor inducing pressure field deformation and flame blow off[END_REF]. Moreover, this setup allows to move the flames inside the cavity, such that it is possible to examine the flame responses at several positions of the forced standing transverse acoustic field corresponding to the 2T1L mode of the cavity (see Fig. 2). Here, the experiments are carried out while the central flame is placed either at a pressure antinode (PAN) or at an intensity antinode (IAN) of the acoustic field. For all measurements, the excitation frequency is tuned to the resonance frequency of the cavity, f r , which is in the range 715-745 Hz here. The operating point, characterized by a flame power of 5.9 kW and an equivalence ratio φ = 0.85, is set such that the flame response at PAN has a significant amplitude (see [START_REF] Baillot | Saturation phenomenon of swirling spray flames at pressure antinodes of a transverse acoustic field[END_REF]). Analyzer (PDPA) system, consisting in a source emitting at λ = 488 nm and λ = 532 nm coupled with a 250mm-lens and a reception probe equipped with a 500mm-lens and oriented with an angle of 30 • with respect to the axis of the source.

Diagnostics

Measurements are performed at various radial locations r = r/(D exit /2) placed on a line passing by the injector axis, which is located at a height ẑ = z/D exit = 0.26 above the injector exit. Each measurement is performed during 4 s (i.e. ∼ 3000 acoustic cycles), allowing to collect enough data to ensure statistical convergence.

Post-processing methods

Flame describing function (FDF)

The flame describing function (FDF) framework is used to quantify the global flame response relatively to the acoustic perturbation. Measurements performed with the PM give the OH*-intensity signal I, which can be expressed by the sum of mean (I) and fluctuating (I ) components. The reduced oscillation amplitude I rms /I, also called flame response, shows the capability of the flame to oscillate around its mean intensity. The local acoustic pressure perturbation P loc is divided by the aerodynamic pressure based on the velocity, ρ air U 2 bulk . The pressure fluctuation and the flame intensity oscillation at the excitation frequency f r , Ȋ and P are calculated in the frequency domain. Therefore, the FDF is written:

F(f r , Π loc ) = Ȋ/I Ploc /ρ air U 2 bulk = G(Π loc )e iφ(Π loc ) (1)
Since the dimensions of the acoustic cavity are kept identical for all experiments, f r is similar and the FDF is evaluated at a fixed frequency. As shown by eq. ( 1), the FDF can also be expressed through a gain G and a phase φ. The gain indicates the efficiency of the coupling between acoustics and combustion and the phase expresses the phase delay between the pressure perturbation and the flame response.

Phase-averaging

Acoustically forced signals s(t) present a coherent fluctuation s c (t) at the forcing frequency in addition to the mean s and turbulent fluctuation s (t) components. Therefore, they can be expressed as follows:

s(t) = s + s c (t) + s (t) (2)
Phase-averaging a signal allows to extract its coherent component. However, a sufficient amount of data must be collected in order to cancel the turbulent fluctuation component, whose mean value is assumed to be zero when the quantity of data is high enough. Here, phase-averaging is performed by putting instantaneous data in 20 bins of duration 1/(20f r ) = T /20, giving the phase-averaged signal s (α), where α is the phase angle. s (α) can be expressed as follows for phase angle α i , where i ∈ [START_REF] O'connor | Flame and flow dynamics of a self-excited, standing wave circumferential instability in a model annular gas turbine combustor[END_REF]20] is the bin number: s

(α i ) = 1 N cycles N cycles n=1 1 N i,n data N i,n data j=1 s j (t 0 + nT ) (3)
N cycles is the number of acoustic cycles comprised in the measurement and N i,n data the quantity of data into bin i for acoustic cycle n. Bin i is centered around

α i = 18(i -1) • .
As the bin width is T /20, t 0 + nT is actually considered with a tolerance of ±T /40, which explains why multiple data can be acquired in the same bin during the same acoustic cycle. Therefore, a sufficient amount of data can be collected either by increasing the acquisition duration (thus increasing N cycles ) or by increasing the acquisition frequency (thus increasing N i,n data ). Post-synchronization with the acoustic pressure is performed such that the phase-averaged pressure cycle is written in the form:

P (α) = Asin(ω r t) (4)
where A is the acoustic pressure amplitude and ω r = 2πf r is the acoustic angular frequency. The reconstructed acoustic pressure cycle and its separation in 20 bins is presented in Fig. 3. This pressure cycle is taken as a reference in all this work.

High-speed OH*-emission images

Since flames located at IAN experience both acoustic pressure and acoustic velocity perturbations, they are subjected to modulations in axial and transverse velocities. Axial flow modulation is known to induce a heat release rate oscillation [START_REF] O'connor | Transverse combustion instabilities: Acoustic, fluid mechanic, and flame processes[END_REF], while transverse velocity modulation can displace the flames laterally, modifying the spatial repartition of heat release rate. High-speed imaging of the central flame is performed to evaluate the OH*-intensity I cam OH * (x, z, t) integrated along the y direction (see Fig. 1). Various postprocessings performed on OH* images are described in the following.

Dynamic mode decomposition: Dynamic mode decomposition (DMD) is carried out to extract dynamic modes of the flame OH*-intensity. This algorithm was introduced by Schmid in [START_REF] Schmid | Dynamic mode decomposition of numerical and experimental data[END_REF]. For each measurement, a set of 2500 images is taken to perform DMD, corresponding to about 90 acoustic cycles.

Transverse displacement of the flame: Since the transverse velocity modulation can induce a lateral displacement of the flame, the OH*-intensity distribution in the left and right sides of the injector axis can be periodically modified. To evaluate the flame response to the transverse velocity modulation, the flame is separated in two parts, which are delimited by a vertical line passing by the injector axis. Integration is performed as follows:

I k (t) = N width k i=1 N height k j=1 I cam OH * (x i , z j , t) (5)
where k corresponds to the zone in which integration is performed (left part, right part, or the whole flame), Induced longitudinal OH*-intensity wave: To extract the longitudinal wave, I cam OH * (x, z, t) is firstly integrated along x direction to result in a function of z and t only. This allows to characterize the vertical location of the most reactive regions in the flame and their time evolution. Phase-averaging is then performed by putting the instantaneous OH*-images in 20 bins, each being constructed with around 300 instantaneous images. The resulting OH* signal is then normalized by means of the number of images in a bin, giving the phaseaveraged signal I cam OH * (z, α). This procedure is similar to that described in eq. ( 3). This allows to define a quantity measuring the coupling between acoustics and combustion:

N
R(z, α) = P I cam OH * (z, α) (6

PDPA measurements

Vertical velocity U f z and radial velocity U f r of the fuel droplets are measured with the PDPA system at various radial locations r above the injector exit. Phase-averaging is performed on these data to extract the modulation at f r (see eq. ( 3)). As the PDPA system acquires data only when droplets pass through the measurement volume, uneven sampling is performed and N i,n data (r) depends on both i and n, in addition to the radial location. Phase-averaging N i,n data corresponds to measuring the mean droplet count during a time interval of width T /20 over all the measurement duration. Therefore, the local phase-averaged droplet count N (r, α) in bin i is expressed as follows:

N (r, α

i ) = 1 N cycles N cycles n=1 N i,n data (r) (7)
It gives information about the modulation of the droplet count over the acoustic cycle, but also about the statistical convergence of results. The modulation of the droplet count generates pockets with unequal droplet number density, leading to fluctuations of the local equivalence ratio and droplet clustering. In the following, phase-averaged signals are always indicated between brackets and their dependence on α is omitted for the sake of clarity.

Global flame response to acoustics

The global flame response I rms /I measured with PM is shown in Fig. 4 when the central flame is placed either at PAN or at IAN. Plotting the flame response relatively to the reduced acoustic pressure amplitude at PAN, Π ref , allows to show how flames at various locations in the cavity respond Prepared using sagej.cls in a same transverse acoustic field. Here, I rms /I is higher when the central flame is located at PAN. However, three different regimes are distinguished: a linear growth up to Π ref = 0.35, followed by a transition zone characterized by a weaker growth of the flame response up to Π ref = 0.47, and the beginning of a saturation zone above [START_REF] Baillot | Saturation phenomenon of swirling spray flames at pressure antinodes of a transverse acoustic field[END_REF]. On the contrary, when the central flame is located at IAN, its response increases linearly up to Π ref = 0.47. This result may indicate that the flame mainly respond to the local acoustic pressure, whose values at IAN are about 60% of those at PAN. To verify this assumption, the flame response is examined relatively to the local acoustic pressure, Π loc .

Here, the comparison between the flame responses at PAN and IAN is only possible in the linear zone, due to the lower value of Π loc reached at IAN. For a same Π loc , I rms /I is quite similar at both locations. However, for Π loc > 0.2, the flame response at IAN is slightly lower than that at PAN. This suggests that when Π loc ≤ 0.2, the flame response at IAN is mainly induced by pressure effects, but other effects may interfere above. It is also interesting to use the FDF framework presented above since it relates the flame response to the acoustic perturbation (see eq. ( 1)). Here, the acoustic pressure is taken at the flame location since the FDF should be calculated with perturbations effectively experienced by the flame. Results are presented in Fig. 5 at PAN and IAN. The gain is similar at both locations when Π loc < 0.2, but it slightly higher at PAN than at IAN when Π loc > 0.2, which confirms the observation made above. The phase is sensitive to a change in the flame location and affected by the increase of acoustic amplitude. Here, this indicates a phase delay between the acoustic pressure and the heat release rate about 0.15π longer at IAN than at PAN. However, its closeness to zero for both locations expresses the in-phase oscillation between the pressure and the heat release rate. This indicates that the flame couples positively with the acoustic perturbation at both locations.

Flame response over space and time

To understand the space-time flame response of the flame, it is worth examining its most energetic dynamic mode of oscillation with the DMD method performed on OH*images. In Fig. 6, one can see that the flame response at PAN for Π loc = Π ref = 0.22 is symmetric: the longitudinal OH*intensity wave has a weak amplitude at the flame foot, which increases as the wave propagates downstream, at the same velocity for the left and right sides of the flame. The more intense spots found on the left side of the flame indicates that this side responds a bit more to acoustics. However, a slight dissymmetry in OH*-intensity is also noticed on the unforced flame, indicating that the dissymmetry of the flame response is likely to be induced by effects other than acoustics, e.g., dissymmetry in fuel atomization and therefore in air-fuel mixing. At IAN, the response is estimated for two acoustic amplitudes: i) Π loc = 0.12 ⇔ Π ref = 0.22, which indicates the response of a flame located in the same acoustic field than the flame at PAN (as shown by the same value of Π ref ); ii) Π loc = 0.21 ⇔ Π ref = 0.36, which gives the response of the flame subjected to the same local pressure perturbation amplitude Π loc . But, the transverse velocity modulation becomes significant for this second acoustic amplitude and may displace the flame laterally. The flame response loses its symmetry for the two acoustic amplitudes examined: the OH*-intensity remains stronger in the left side of the flame (towards PAN), but also propagates faster than that in the right side, resulting in a tilted flame response. This is consistent with results obtained by [START_REF] Lespinasse | Responses of v-flames placed in an hf transverse acoustic field from a velocity to pressure antinode[END_REF] for non-swirling laminar premixed flames. However, the flame response remains mostly linked to a longitudinal mode induced at the injector exit.

It is interesting to capture the effects of the transverse velocity modulation on the flame, since, as shown by [START_REF] Schwing | High-frequency instabilities in cylindrical flame tubes: feedback mechanism and damping[END_REF] at a velocity antinode, this displacement can feed the thermo-acoustic coupling. Moreover, the flame lateral displacement can induce blow-out [START_REF] Vignat | High amplitude combustion instabilities in an annular combustor inducing pressure field deformation and flame blow off[END_REF]. To assess these effects, the proportion of OH*-intensity in the right side of the flame divided by the mean value without acoustics, compared to that for a lower Π loc , but the phase of the modulation is identical. It is clearly shown that the gaseous phase experiences a lateral displacement induced by the transverse velocity modulation. Since the flame is displaced towards regions with a high acoustic pressure amplitude when its heat release rate is high, the instantaneous Rayleigh criterion is increased compared to a non-displaced flame; but the flame is also displaced towards regions where the acoustic amplitude is low when its heat release is low: therefore, the instantaneous Rayleigh criterion is decreased compared to a non-displaced flame. Consequently, when integrated over the acoustic cycle, the Rayleigh criterion should be unmodified by the transverse velocity modulation compared to a non-displaced flame, as soon as the mean flame lateral location does not change.

I
Therefore, the space-time thermo-acoustic coupling evaluated through the abovementioned quantity R can be compared between PAN and IAN in Fig. 8. At PAN, a strong coupling is found for 20 mm < z < 40 mm. Indeed, as shown by Fig. 6, an OH*-intensity wave propagates downstream and reaches high OH*-levels in this region in phase with the acoustic pressure. For 40 mm < z < 60 mm, a negative coupling between acoustics and combustion is observed, arising from the further propagation of the OH*-wave. These two regions are the main contributors to the resulting thermo-acoustic coupling but the positive coupling in the lower regions is much more intense, therefore integrating R over the whole flame gives a positive value, indicating the potential instability of this operating point. At IAN, the same behaviour is found for the same local acoustic amplitude (Π loc = 0.21). This indicates that the flame at IAN contributes to sustain the instability through the pressure coupling. However, as shown for Π ref = 0.22, this contribution is much less intense compared to that of the flame at PAN for the same acoustic field. This suggests that flames located at PAN can drive the instability via the strong pressure coupling at this location, whereas flames at IAN may help in sustaining the instability by injecting energy into the acoustic mode, although at a lower level.

Spray response to acoustics

Earlier work on the same experimental setup showed that at PAN, acoustic pressure modulation induced longitudinal modulations of the count, vertical velocity and size of the droplets resulting in the generation of a droplet number density wave simultaneously to the vertical velocity modulation [START_REF] Baillot | Saturation phenomenon of swirling spray flames at pressure antinodes of a transverse acoustic field[END_REF].

Droplet velocity modulation

At a PAN of a transverse acoustic field, vertical velocity modulation is known as an effect of the pressure oscillation.

Since the behaviour of the Rayleigh source term is similar at PAN and at IAN, it is interesting to examine the fuel droplet vertical velocity modulation at both locations for a same local acoustic pressure amplitude. Here, the phase-averaged vertical and radial velocities, U f z and U f r respectively, are evaluated for a same Π loc when the central flame is placed at PAN or at IAN. Velocity modulations can be observed more easily by looking at fuel droplets which have a low Stokes number St = ρ f d 2 U f z 18µ air Dexit , characterizing the ability of droplets to follow the air flow. Here, ρ f is the fuel density, d the droplet diameter and µ air the air dynamic viscosity, which is calculated at T = 320 K for |r| ≤ 1 and T = 750 K for |r| > 1. Profiles of the Stokes number calculated for small droplets are shown in Fig. 9. Two profiles are presented: one for droplets of size 3 µm, whose vertical velocity U f z is assumed to be the mean velocity of droplets of diameter in the range 1-5 µm; another for droplets of size 7.5 µm, whose vertical velocity is assumed to be the mean velocity of droplets of diameter in the range 5-10 µm. At all radial locations, the Stokes number of the 3 µm droplets is lower than 0.1 while that of the 7.5 µm droplets does not exceed 0.55. Therefore, these droplets should follow air flow fluctuations relatively well. The phaseaveraged velocities are thus calculated by means of velocities of droplets satisfying 1 µm ≤ d ≤ 10 µm, which allows to obtain enough data for statistical convergence. Moreover, they constitute about 65% of the droplet population (see Fig. 10). Fig. 11 shows the modulation amplitude of vertical The modulation amplitude progressively decreases towards |r| 1, then remains weak up to |r| 1.5. So, the spray is characterized by an internal region which experiences a strong vertical velocity oscillation, and an external region in which the vertical velocity oscillation is weak. The limit between these two regions can be approximately located at |r| = 1, which is the vertical projection of the nozzle exit edge. The similarity of the profiles at PAN and IAN indicates that the modulation amplitude of the droplet vertical velocity is insensitive to effects of the pressure gradient noticed in laminar non-swirling premixed flames [START_REF] Lespinasse | Responses of v-flames placed in an hf transverse acoustic field from a velocity to pressure antinode[END_REF]. As shown in Fig. 12, the phase between the pressure perturbation at the flame location and the vertical velocity modulation is also similar at PAN and IAN. In the internal region of the spray, U f z oscillates in phase opposition with the pressure. In the external region, it deviates from phase opposition, but the weak modulation amplitude may increase uncertainty about the results. The modulation amplitude of U f r (see Fig.

Generation of droplet spatial number density wave

The phase-averaged droplet count N is evaluated at various radial locations via the method described in eq. ( 7). Indeed, the oscillation in droplet population may generate a modulation of the equivalence ratio, leading to a heat release rate oscillation. A high droplet count may also generate clustering effects, thus reducing evaporation. Fig. 13 shows results for N at PAN and IAN for Π loc = 0.2. For both locations in the acoustic field, the response to the acoustic perturbation is similar at the nozzle exit, supporting the assumption that the spray response mostly depends on the pressure perturbation. A densely populated region is found for |r| > 1 when the acoustic pressure is positive. This high number of droplets is injected with a low mean vertical velocity of about 15-20 m/s. It is followed, when the acoustic pressure is negative, by a decrease of N (r, α), which is approximately divided by a factor 2. For 0.5 < |r| < 1, a high number of droplets is injected when the acoustic pressure is negative, but their mean vertical velocity is in the range 35-43 m/s. Simultaneously, droplets are measured for |r| < 0.5, which is an unpopulated region without acoustics. But, when the acoustic pressure is positive, this region empties and a weak spatial density is even found up to |r| = 0.6. At the same time, droplets are less rapid. Therefore, at the nozzle exit, the internal region (r ≤ 1) is characterized by the injection of many fast droplets, followed by less slower droplets, while the external region (r > 1) is characterized by an oscillating droplet count, but droplets remain relatively slow during all the acoustic cycle. This complex injection pattern results in a spacetime modification of the droplet number density, leading to fluctuations of the local equivalence ratio after evaporation, which may however be diminished if dense droplet clusters are present [START_REF] Réveillon | Evaporating droplets in turbulent reacting flows[END_REF]. The strongest flame response found for 20 mm < z < 40 mm indicates that this region is particularly sensitive to the modulations induced by acoustics. Thus, understanding the interaction between the droplet number density waves generated in the internal and external regions and the flame response requires to examine the resulting equivalence ratio fluctuations and their transport to the flame front.

Conclusion

A linear array of three swirling spray flames is submitted to transverse acoustic perturbations. The central flame is placed either at a pressure antinode (PAN) of the standing acoustic field, or at an intensity antinode (IAN), which combines perturbations in pressure, transverse velocity and pressure gradient. The Flame Describing Function, calculated with respect to the local pressure, has a similar gain but a slightly different phase at both locations. At PAN, the flame response is symmetric while at IAN, a dissymmetry is observed, which may be linked to the periodic lateral displacement of the flame induced by the transverse velocity modulation. Space-time flame dynamics shows that at both locations, the flame response is dominated by the middle part of the flame, where a periodic OH*-intensity wave fluctuates in phase with the pressure oscillation, thus feeding the thermo-acoustic coupling. It is shown that the flame at PAN is the main contributor to thermo-acoustic coupling, but the flame at IAN also injects energy in the acoustic mode. Analysis of the spray shows the weak sensitivity of fuel droplets to the transverse velocity modulation at the injector exit. However, a complex injection pattern induced by the pressure perturbation generates droplet number density waves at the injector exit, inducing local equivalence ratio fluctuations at higher locations. Further understanding requires to examine the resulting equivalence ratio fluctuations and their transport to the flame zone.

Figure 1 .

 1 Figure 1. Top view of the acoustic cavity: compression drivers (CD), injectors: red (fuel + air), blue (air)

Figure 2 .

 2 Figure 2. Comparison between the experimental and the simulated acoustic fields inside the cavity, and the two locations of the central flame (injector B) in the present study. PAN/IAN: pressure/intensity antinode.

Figure 3 .

 3 Figure 3. Reconstructed acoustic pressure cycle and its separation in 20 bins for phase-averaging. Bin width: T /20.

  width k and N height k are respectively the width and height in pixels of zone k. Phase-averaging is then performed on I lef t , I right and I f lame as indicated by eq. (3), giving the resulting phase-averaged signals I lef t , I right and I f lame respectively. If the flame is displaced laterally, the OH*-intensity integrated in one lateral part will be modified due to the displacement induced to acoustics. However, it can also be modified during the acoustic cycle due to effects linked to the axial velocity modulation. These effects can be suppressed by evaluating the reduced signals I lef t = I lef t / I f lame and I right = I right / I f lame . These values are finally divided by the mean values without acoustics, I lef t ,0 = I lef t 0 /I f lame 0 and I right ,0 = I right 0 /I f lame 0 .

  ) which is linked to the Rayleigh criterion. Negative values of R indicate stable combustion. Positive values of R indicate that combustion injects energy into the acoustic mode, which is necessary to create an instability. Nevertheless, this is not a sufficient condition, since R must excess the sum of damping and flux divergence to trigger an instability.

Figure 4 .

 4 Figure 4. Flame response Irms/Imean for the central flame placed at PAN (blue squares) or IAN (orange triangles). (a) plot vs. acoustic amplitude at PAN Π ref ; (b) plot vs. local acoustic amplitude Π loc .

Figure 5 .

 5 Figure 5. Flame describing function for the central flame placed at PAN (blue squares) or IAN (orange triangles). (a) gain G(Π loc ); (b) phase φ loc .

  right /I right ,0 , is compared between PAN and IAN. At IAN, the right side is located towards a velocity antinode. Data are shown in Fig. 7(b) for the same Π ref , i.e. for the same acoustic field. The associated reconstructed pressure cycle is shown in Fig. 7(a). I right /I right ,0 is modulated with an amplitude slightly higher at IAN than at PAN, but in phase opposition. When the pressure is growing, the flame at IAN is displaced towards PAN, which results in a decrease of I right /I right ,0 . The contrary is observed when the pressure decreases. When the flame is subjected to a same pressure perturbation, i.e. a same Π loc (see Fig. 7(c)), the amplitude of I right /I right ,0 at IAN is multiplied by 1.4

Figure 6 .

 6 Figure 6. Dynamic mode decomposition of OH*-emission images of the central flame. The location of the flame and the acoustic amplitude are changed accordingly to the indications. The range of the color scale is different for each series.

Figure 7 .

 7 Figure 7. Evolution of the proportion of OH*-intensity in the right side of the injector axis reduced by the mean value without acoustics, I right /I right ,0 over a reconstructed acoustic pressure cycle, reproduced twice for the sake of clarity. Left: reconstructed acoustic pressure cycle; middle: I right /I right ,0 at PAN (blue squares) and at IAN (orange triangles) for similar values of Π ref = 0.22; right: I right /I right ,0 at PAN (blue squares) and at IAN (green triangles) for similar values of Π loc 0.22.

Figure 8 .

 8 Figure 8. Evolution of the quantity R measuring the coupling between acoustics and combustion over height during the acoustic cycle reproduced twice. The location of the flame and the acoustic amplitude are changed accordingly to the indications.

Figure 9 .

 9 Figure 9. Profiles of Stokes number measured at ẑ = 0.26 calculated for droplets of diameter 3 µm and 7.5 µm.

Figure 10 .

 10 Figure 10. Normalized fuel drop size distribution of the whole spray at ẑ = 0.26.

Figure 11 .

 11 Figure 11. Profiles of the modulation amplitude at ẑ = 0.26 for Π loc = 0.2 of: (a) phase-averaged droplet vertical velocity, U f z ;

Figure 12 .

 12 Figure 12. Profile of the phase relationship between P at the central flame location and U f z at various radial locations r. U f z is calculated on droplets of diameter comprised between 1 and 10 µm.

  11 (b)) at IAN is as weak as at PAN, and the low levels may be attributed to noise. These results indicate that the liquid phase only responds to the pressure perturbation at the injector exit, similarly at PAN and at IAN; neither effects of the transverse velocity nor of the pressure gradient are measured at IAN in the vicinity of the nozzle exit.

Figure 13 .

 13 Figure 13. Evolution of the droplet count N (r, α) over a reconstructed acoustic pressure cycle, reproduced twice for the sake of clarity. Left: Reconstructed acoustic pressure cycle; middle: N (r, α) at PAN for Π loc = 0.2, fr = 715 Hz; right: N (r, α) at IAN for Π loc = 0.2; fr = 729 Hz. White zones indicate a droplet count too low for validation.

Acoustic pressure fluctuations are measured with Brüel & Kjaer Type 4182 microphones at different locations inside the acoustic cavity. In particular, it is measured at the central flame position, which gives the local acoustic pressure P loc , and at a reference position which is the central PAN, giving P ref . Thus, when the central flame is located at

  PAN, P loc = P ref . The rms acoustic pressure is calculated and reduced by the bulk aerodynamic pressure, which gives a non-dimensional parameter Π loc = P rms loc /ρ air U 2

	bulk
	showing the balance between acoustics and aerodynamics,
	where ρ air is the air density and U bulk the bulk velocity at
	the injector exit, both calculated at 20 • C. In the following,
	the non-dimensional parameter Π loc is taken as indicator
	of the acoustic forcing level. Its highest attainable value
	is determined by the acoustic system. Higher values are
	reached at PAN than at IAN, as shown in Fig. 2. Global
	flame dynamics is measured with a photomultiplier (PM)
	Hamamatsu H6779 equipped with an OH*-filter and coupled
	with a UV-visible lens to focus the flame image on the sensor.
	High-speed OH*-imaging is performed to investigate the
	space-time flame dynamics with a Phantom V2012 camera
	equipped with a Lambert HiCATT 25 intensitifier, a UV-
	100 mm lens and an UG11 filter. Each series consists of
	6,000 images acquired at 20,000 fps. Spray dynamics is
	examined with a two component-Phase Doppler Particle

CORIA-UMR-6614, Normandie Universit é, CNRS, INSA and Universit é de Rouen, 76801 Saint-Étienne-du-Rouvray, France

Prepared using sagej.cls [Version: 2017/01/17 v1.20]

Prepared using sagej.cls

Acknowledgements

This work was supported by the French National Research Agency (ANR) through the project FASMIC ANR-16-CE22-0013, Région Normandie and French Ministry of Higher Education and Research. The authors wish to thank RENADIAG for the technical support provided to this research.


Faouzi Moussa

Christophe Kolski

Patrick Millot

Artificial intelligence approach for the creation

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

I -INTRODUCTION

The technological evolution in the continuous industrial processes is characterized by the installations increasing complexity, the control and supervision process centralization and the computer more and more frequent use in order to control this process. Thus, the current trend, in the control rooms of automated processes, is the information presentation on graphical display screens. For this purpose, many tools of graphical edition have appeared to cater for industrial synoptic designers needs. Then, lots of ergonomic problems inherent in the use of graphical displays in control rooms appeared. These problems can result from gaps in the way of presenting the graphical data on the screen. This is often due to the fact that the designer has -in most of the cases -little or no ergonomic knowledge. Therefore, an ergonomic corrective method has to be implemented so as to improve the data presentation. The expert system SYNOP developed by [START_REF] Kolski | Contribution à l'ergonomie de conception des interfaces graphiques homme-machine dans les procédés industriels : application au système expert SYNOP[END_REF] -helping the ergonomic evaluation and improvement of the data presentation -helped studying the corrective ergonomic notion thanks to formal techniques, and stressing a few limits of such a corrective ergonomic process. But, these ergonomic problems may also arise from gaps in the analysis of the operator informational needs during his different tasks. Moreover, the data on the screen, meeting the needs, can also be misdistributed on the different displays which form the graphical imagery. After a dynamic evaluation of these displays on the site or in simulation, it's important then to verify the synoptic structure itself and to come back to an ergonomic design stage. The insufficiency of formal processes in this field underlines the interest of present researches concerning the study of ergonomic design method based on the processing formalization of ergonomic knowledge.

To begin with, this paper briefly presents the expert system SYNOP. This system is integrated in a global ergonomic methodology of graphical displays design, and is based on a formal approach of corrective ergonomy point. Then, the paper describes our first research works concerning the study of a new ergonomic design methodology based on the model of an expert system of ergonomic design.

II -FORMAL METHOD OF CORRECTIVE ERGONOMICS: APPLICATION TO THE EXPERT SYSTEM SYNOP

In order to take into account the human factors while designing the graphical interfaces, the helping methods often consist in the use of ergonomic guidelines such as [START_REF] Cakir | Les terminaux à écran : agencement, ergonomie, organisation[END_REF][START_REF] Mc | Human factors in engineering and design[END_REF] or [START_REF] Scapin | Guide ergonomique de conception des interfaces homme-machine[END_REF], or in the best case, in the consultation of an expert in ergonomics (see [START_REF] Lucongsang R | Démarche ergonomique intégrée à une méthode d'informatisation[END_REF]. At present, tools which integrate formalized ergonomic knowledge, such as SYNOP, begin to be studied in the research laboratories.

SYNOP is an expert system, presently, under the form of a laboratory model. Its purpose is to evaluate and to improve automatically the displays and to give the designer advices, according to the ergonomic rules of data graphical display on the screen. The ergonomic rules are centralized in its knowledge bases and formalized in production rules [START_REF] Kolski | Towards an intelligent editor of industrial control views, using rules for ergonomic design[END_REF]. We distinguish two types of ergonomic rules: (i) General rules dealing with basic ergonomic concepts, such as those which are related to the characters size, the use of colors and contrasts, the data repartition on the screen ... (ii) Ergonomic rules which are specific to the treated applications such as the process control, the displays embarked in the cars (Moussa & Kolski, 1989), and so on.

Developed in the LISP language, SYNOP uses several techniques of representation and use of knowledge: a first order logic inference engine [START_REF] Grzesiak | Représentation des connaissances et techniques d'inférence pour le maniement d'objets graphiques : application au système expert SYNOP[END_REF], frames [START_REF] Minsky | A framework for representing knowledge[END_REF] and semantic networks [START_REF] Bonnet | L'intelligence artificielle : promesses et réalités[END_REF]. It is at present interfaced with a tool of creation and animation of graphical displays using the GKS norm. Figure 1 represents the SYNOP structure.

Figure 1: Stucture of the expert system SYNOP Therefore, SYNOP includes a "static" evaluation module of graphical displays which purpose is to evaluate and correct some displays outlines. This is done according to ergonomic rules arranged in its knowledge bases. Thereby, SYNOP, as a tool of evaluation in "static", integrates in a general methodology of graphical Man-Machine interface design. This methodology is described on the figure 2 [START_REF] Kolski | Contribution à l'ergonomie de conception des interfaces graphiques homme-machine dans les procédés industriels : application au système expert SYNOP[END_REF][START_REF] Millot | Les approches intelligence artificielle dans les systèmes de supervision[END_REF][START_REF] Taborin | Cooperation entre Opérateur et Système d'Aide à la Décision pour la Conduite de Procédés Continus : Application à l'Interface Opérateur[END_REF]. The aim of the first stage is the selection of the set of data which are necessary to process control, from among all the available data. This set constitutes the operator's informational needs. It results from an analysis of human tasks in the different operational contexts of the process [START_REF] Hollnagel | Principles for Dialogue design in Man-machine systems[END_REF][START_REF] Woods | Aiding Human performance: From Cognitive Analysis to support systems[END_REF], or from an analysis of the probable future activity [START_REF] Daniellou | Ergonomie et démarche de conception dans les industries de processus continus, quelques étapes clés[END_REF] if the process is only in the stages of design. The second stage consists in defining the decomposition framework of the pictures while taking constraints imposed by the software and hardware limits of the graphic support selected for the application into account. Then the third stage allows us to obtain the synopsis schedule. This is constituted with a framework of displays deduced from the structural and functional decompositions of the operator's informational needs [START_REF] Taborin | Cooperation Between Man and Decision Aid System in Supervisory Loop of Continuous Processes, 8[END_REF]. For instance, one of the displays provides the operator with a concrete representation of a cooking system; another displays the values of a temperature and two pressures. The fourth stage defines the mode of graphic information presentation for every display of the synopsis. Thus the animation functions are selected, placed on the screen, associated with a static graphic environment, color coding, etc. The fifth stage consists of the graphic creation of the displays for instance by means of an editor of animated graphic control displays.

Then an ergonomic evaluation aims to insure that the created displays answer the informational requirements of the operator. This evaluation can be divided into two stages: the "static" and "dynamic" ones. The "static" stage's objective is to verify that the displays respect ergonomic rules of information presentation on a screen. It is necessary to apply a set of ergonomic criteria related to the human information acquisition capacities. These criteria concern especially: the accuracy of the representation modes chosen for each piece of information; the shape of the different symbols used to represent the components of the process; for instance: specific colours for specific types of information, contrasts between these colors [START_REF] Murch | Physiological Principles for the Effective Use of Color[END_REF], the localization of the different items composing each display, and the arrangement of the windows on the screen. The ergonomic criteria applied are generally expressed in terms of rules in design guidelines. However if numerous rules are general ones, some are specific to a particular industrial area. This is the case of the nuclear area where specific design norms must be applied [START_REF] Taborin | Cooperation entre Opérateur et Système d'Aide à la Décision pour la Conduite de Procédés Continus : Application à l'Interface Opérateur[END_REF]. Presently, research works tend to define methodologies or create tools with a view to fostering a better consideration of these criteria for designing the synopsis [START_REF] Fassote | Conception des Synoptiques sur C.R.T. destinés au Contrôle de Processus[END_REF][START_REF] Elzer | Knowledge-supported generation of control room pictures[END_REF][START_REF] Kolski | An "intelligent" interface approach[END_REF]). The expert system SYNOP intervenes in this stage. The "dynamic" stage is carried out on the industrial site and/or in simulation. Its principal aim is to verify that the structure and the content of each display responds to the operator's informational needs during the execution of his tasks and according to the different contexts of the process. There is currently no particular method to carry out this evaluation, although some general ones can be used: subjective methods like questionnaires, or more analytic ones like cognitive task analysis methods [START_REF] Woods | An evaluation of nuclear power plant safety parameter display systems[END_REF][START_REF] De Keyser | Temporal decision making in complex environments[END_REF] which consists in dividing supervisory tasks into objectives and sub-objectives and in verifying that the content of the displays allows the operator to reach them.

These two ergonomic stages lead either to a validated interface or to being sent back to the previous stages for improving the final system, figure 2. Although the SYNOP intervention in the static evaluation stage helps improving the data display according to some ergonomic criteria (criteria which are formalized in production rules) there are some cases for which the ergonomic rules to apply are very general, subjective, or difficult to implement. SYNOP is then unfit for automatic modifications and deals only with providing the designers with advices. Moreover, knowing that the graphical representation methods of the process different variables are chosen "à priori" and may eventually not correspond to the operational context of the application, SYNOP can not intervene to change them. This is due to the fact that it only corrects the syntactic aspect of the display, and not the semantic aspect. This restriction requires to go further and to take advantage of the acquired experience during these works of ergonomic knowledge formalization in the corrective ergonomic field, in order to move towards a formal process aiming at automating some aspects of the ergonomic design. The study of the problem in this field forms the subject of the following paragraph, which is illustratedregardless of exhaustivity -by some recent works.

III -THE ERGONOMIC DESIGN PROBLEM

According to [START_REF] Scapin | Vers des outils formels de description des tâches orientés conception d'interfaces[END_REF], the Man-Machine interaction is not always clearly dealt with. Indeed, these authors explain that the models which are now proposed are too general, too informal and lack -for the greatest part -organization to be directly usable by the designers. Thus, this deficiency characterizing the ergonomic models causes difficulties to computer scientists and even to ergonoms for the ergonomic recommendation utilization. Effectively, according to [START_REF] Scapin | Vers des outils formels de description des tâches orientés conception d'interfaces[END_REF], these difficulties are of several types, summing up numerous studies in this field. For example: (i) The interface's designer is not always in a position to allow the necessary time -which is often very long -to read completely a manual of recommendations. (ii) Recommendations are often very general and pose problems of access which are too intricate for a beginner. (iii) There are some interactions between the ergonomic criteria, leading to compromises which are difficult to estimate for a beginner. (iv) As in every fields using the exploitation of an expert knowledge, the ergonomic data do not always converge as their form and their contents change according to the design criteria (use performance, learning performance, etc.).

It is therefore necessary to collect the design general principles used here in the ergonomic field of the graphical Man-Machine interfaces. The aim of this collection is to understand better the design problem and to try to bring about formal solutions which are applied to the study and the creation of a computer tool helping the interface's design. [START_REF] Brown | An Approach to Expert Systems for Mechanical Design[END_REF] consider that, generally speaking, there are three categories of design: (i) Category 1: the inventive or creative design. (ii) Category 2: the special case in relation to a certain routine; if there is a potential market, the firm will generally invest in order to change the exception into a routine case. (iii) Category 3: the routine design which, given a well known problem solved several times, tries to find out an optimal function in answer to certain specifications.

III.1 -THE DESIGN GENERAL PRINCIPLES:

Therefore, the difficulty of the design task depends strongly on the category. So, the complexity of the design task would decrease from the first category to third as the rules that are used would get less heuristic and more explicit. Whatever the category, the design, generally speaking, goes through three phases which are independent and especially not sequential: (i) Problem analysis, (ii) originating of answers, (iii) critic and evaluation of answers.

However, this approach is still not very efficient as a problem is never totally dealt with and described, and as some data of the problem appear during the design. For this purpose, Darke (quoted in [START_REF] Lawson | How designers think[END_REF] propounds a methodology which meets the prototypage method used in the software engineering: (i) Classing the problems in importance order, (ii) building a model which answers the most important problems, (iii) observing the reactions and deducing new problems, (iv) going to the first step, or stopping.

Nevertheless, the designer has still to decide on the iteration stopping so that the application does not reach too important sizes.

As for Mantalban (1987), he defines the design as such: "the design is a process which creates and solves the problems dynamically". He specifies that a problem cannot be statically stated, but that it must be considered as "in constant tension with its answer". According to Lawson's most general point of view, the design consists in saying that a designer essentially operates constraints which have three principal characteristics: -Their generator: Lawson distinguishes four types of constraints classified here in: (i) generated by the design, (ii) generated by the customer, (iii) generated by the user of the final product, (iv) imposed by the legislation (the most rigid).

-Their field: It helps creating two categories of constraints: (i) internal: only the relations between the artefact objects are taken into account, (ii) external: relations between the artifact and its environment are taken into account.

-Their function: according to [START_REF] Montalban | Prise en compte de spécifications en ingénierie, application aux systèmes expert de conception[END_REF], the function of a constraint depends on everyone's point of view and, is then dealt with in several proposals. However, Lawson distinguishes four of them: (i) practical: concerns the production reality, the manufacture, etc, (ii) radical: describes the object usefulness, its functionalities, (iii) formal: describes the object aspect, its form, its colors, etc, (iv) symbolic: in order to treat the object as a symbol.

In the field of the design applied to the ergonomy of the graphical Man-Machine interfaces, [START_REF] Scapin | Vers des outils formels de description des tâches orientés conception d'interfaces[END_REF] suggest two different global stages for the development of a system: the design and the implementation. Moreover, the ergonomic design stage has to be ended far before the beginning of the implementation stage. More exactly, by relying upon some of [START_REF] Bailey R | Human performance engineering: a guide for system designers[END_REF]), Hendrick's & al. (1982), [START_REF] Shneiderman | Software psychologie[END_REF][START_REF] Shneiderman | Designing the user interface: strategies for effective human-computer interaction[END_REF]) and [START_REF] Scapin | Guide ergonomique de conception des interfaces homme-machine[END_REF] suggestions concerning the interfaces design, these authors identify some fifteen stages for the design process, going from "the identification of the needs" to "the preparation of the system future evolution". Basing themselves on this cutting out, they insist on the necessity of a modular approach according to a hierarchical structure coming near to the designer steps; which, in their opinion will allow to implement methods of design based on formal models at a medium and long term.

It is therefore interesting to insist on the fact that an ergonomic application is more than an ergonomic dialogue, and that it is effectively true that it is impossible to conceive without thinking conjointly and in interaction the user and the task, which represent the two major parts of the design process [START_REF] Valentin | L'ergonomie des logiciels. ANACT, Collection Outils et Méthodes[END_REF].

"The first major part" of the design process, the user, being the future "potential customer" must be considered with particular care from the starting phase of the design task [START_REF] Shneiderman | Designing the user interface: strategies for effective human-computer interaction[END_REF]. Effectively, a user will be much disposed to accept a system if he has been associated to the specification step and if he notices that a certain number of his proposals has been implemented. Thus, calling for well established techniques (interviews for instance), the most experienced users can take part in the success of the final product by making sure that their requirements have been taken into account [START_REF] Gould | Designing for usability: Key principles and what designers think[END_REF].

As far as the task is concerned -"the second major part" of the design process -it is particularly interesting to notice an increasing research activity in the field of the tasks description formalization. The following paragraph briefly presents several current studies going in this direction.

III.2 -CURRENT STUDIES CONTRIBUTING TO THE FORMALISATION OF THE TASKS DESCRIPTION:

Regardless of exhaustivity, our aim is here to describe briefly four different approaches contributing to the formalization of the tasks description. It will be interesting to note the cutting out of formalism in several levels for each of these methods. Montalban's work (1987) is placed in the frame of a design task. He has developed a processing architecture called A.B.S. (Architecture Based on Specifications), applied to the thermic design of lodgings. His formalism brings in a functional level and operational level: (i) the functional level is represented in a hierarchical way by an arborescence of functions of high level. Preconditions recapitulate the data which are necessary before the function launching. Post-conditions represent the data calculated by a function. (ii) The operational level is made of "models" or "prototypes" of tasks composed of a list of arranged production rules, a group of indicators, a task of exception which is called in case of blocking and its nature. A task is then considered as an instancy of this model completed by: an environment of execution and a task author.

III.2.1 -THE A.B.S. SYSTEM:

The attribute task ensures here the link between the functional level and the operational level. Therefore, the task represents the action or the procedure to execute in answer to a functional need. The most important result, according to Montalban -he has implemented the A.B.S. system with the help of the generator of expert systems SMECI -is the considerable gain brought by the system in terms of flexibility of knowledge use. Moreover, taking into account the functional aspect of the objects ensures the expert system of a better quality of the Man-Machine dialogue, dialogue which is carried out in terms of functionalities. Finally, at the level of the knowledge acquisition, thanks to the A.B.S. system ability to extract a dynamic control structure from a static functional description, the expert system constructor doesn't think about the train of tasks. Pierret-Golbreich's works (1988) in the field of help to the tasks of scientific computation present a common principle in a task formalization. Indeed, the task is formalized under the form of an object describing in a declarative way its functional and operational levels: (i) The functional level refers to the execution conditions of the task (example: input/output type ...), as well as to its effects (parameters calculated by the task). (ii) The operational level refers to the action or to the tasks to be executed in answer to a given functional context.

III.2.2 -HELP TO THE TASKS OF SCIENTIFIC COMPUTATION:

Then, Pierret-Golbreich defines a task as a generic object described by its initial state, its final state and an operator. The operator refers to the operational level described by its inputs, its outputs, its use conditions and its "frame" (the "frame" is here a primitive action, a task of high level, or a list of subtasks).

III.2.3 -THE HIERARCHICAL PLANNING AS A METHOD OF THE TASK ANALYSIS:

Sebilotte (1987) used the hierarchical planning paradigm in artificial intelligence (Sacerdoti, 1987) where each individual stage in a plan is considered as a task with its own inputs, conditions, goals, and so on. Therefore, in the frame of an office task analysis, Sebilotte considers that any given task may be decomposed according to models having four principal levels: (i) the most abstract level representing the general formulation of the task or the goal to be reached, (ii) the level "expert" which describes the procedures or the specific sub-tasks of a precise field, (iii) the highest common level of the general procedures level: -procedures which are independent on the field and used by the subjects in order to execute the specific procedures, and procedures which are very like more general knowledges; (iv) the lowest level or the elementary actions level, which belong to the field of automated activities.

The author underlines three advantages of this approach consisting in formalising a task under the form of a hierarchy of representation: (i) the model of the activities resulting from this, account for the way the expert subjects see the planning of their activity, (ii) the different levels of representation used authorize more flexibility. Indeed, depending on the situations he will be confronted with, the operator will be able to adapt his procedures by choosing the goals he wants to execute at the level of detail that he wants, and to choose the execution order of these goals and sub-goals.

A task model at different levels of details also helps proposing some guides or aids to the user as well as to the beginner, and to control the task progress.

III.2.4 -THE ANALYTIC METHOD OF TASKS DESCRIPTION (M.A.D.):

The work of [START_REF] Scapin | MAD : une Méthode Analytique de Description des Tâches. Acte du colloque sur l'ingénierie des interfaces Homme-Machine[END_REF] takes place in the field of the help to the office tasks. In this aim, the leading concepts introduced in their formalism are those of object task, of action and of structure. The task concept is represented by a generic object called object task, and composed of an initial state, a final state, a goal, pre-conditions and post-conditions. Thus, the object task is the root of two subclasses: The "elementary task" class and the "composed task" class: (i) The elementary task is an irresolvable task, which operational level is characterized by an object method of simple type, that is to say an action. (ii) The composed task is a task which operational level is defined by a structure describing the task body. The structure object is represented by a generic object characterized by the constructor describing the arrangement of the various tasks involved and the constructor arguments. Then several constructors have been defined, such as: SEC: sequential task, PAR: parallel task, ALT: alternative task, ITER: repeated task and FAC: facultative task.

As a conclusion, Scapin and Pierret-Golbreich underline that the formalism will, only with use, demonstrate its possible utility, on the one hand during the evaluative study, on the other hand in the future works which will consist to define some ergonomic design tools, from the tasks description. They wish that this formalism will help underlining in a better way the difference between functional logic and logic of use in the interface design, taking into account the operator features and task, instead of essentially taking into account the functional logic of the application and of the constraints which are imposed by the computer.

III.2.5 -CONCLUSION ON THESE FORMALISATION APPROACHES:

Such works give interesting plans in the field of the Man-Machine engineering, in so far as the present research work aims at basing the design process on the processing modelization of the tasks to perform. Therefore, this work puts in evidence the necessity to check off and to organize in several levels the data related to the tasks, the stage then permitting the creation of an ergonomic data base, which goal is to easy the Man-Machine interface design.

Our present works are going this way and consist in studying a global ergonomic methodology of graphical displays design integrating an ergonomic expert tool in the field of continuous industrial processes.

VI -STUDYING A GLOBAL ERGONOMIC METHODOLOGY OF GRAPHICAL DISPLAYS DESIGN INTEGRATING AN INTERACTIVE EXPERT TOOL

The aim of this formal tool, based on an oriented object expert system architecture, is to assist the graphical displays designer in order to avoid some ergonomic errors with regard to the operator real needs in the displays structuralization, the choice of some methods of graphical representation (curve, bar graph, ...) and of some chosen graphical attributes (color, characters size, ...). So as to do that, the system must integrate a description of the operational various contexts of the process and of the tasks to perform. Therefore, this system in essentially made of 3 modules, figure 3 [START_REF] Moussa | Etude de l'intégration d'un outil expert interactif ergonomique dans la démarche globale de conception d'interface graphique homme-machine[END_REF]): (i) a module of classification of the operator's informational needs, (ii) a module which must define the display description at two complementary levels: a first level of the structural description of the display and a second level of the functional description, (iii) a module of graphical definition and creation of the display. These modules are subsequently explained in the following part.

IV.1 -THE MODULE OF DESCRIPTION OF THE OPERATOR'S INFORMATIONAL NEEDS:

From a description of the operator's informational needs in the various operational contexts of the process, it is then easier to deduce the skeleton of the synopsis in creation. Therefore, the first interactive module must help strictly identifying, -before the graphical creation -the various contexts, the sets of variables, the system users, etc. A processing solution to the problem of knowledge representation can be a data formalization in a first representation under the form of structured objects in relation, by a semantic network supplying the other modules of the channel. Thus, regardless of exhaustivity in our description, it The general coherence of the semantic network can be ensured, for one part, thanks to rules checking for example the total distribution of all occurences of an object class in the other object classes forming the network. For instance: Knowing that a task is created thanks to a whole of activities, we have the following rule: "The union of the activities appearing in the group of tasks must be equal to the group of occurrences of the activities constituting the class of the object "activities".

IV.2 -THE MODULE FOR FUNCTIONAL AND STRUCTURAL DESCRIPTION:

Each task is decomposed in a group of displays to create. This second interactive module must then help the ergonomist to ensure the description of each of the displays according to complementary levels:

-A first level of the structural description of the display: it aims at associating to every kind of display the adequate structure. The ergonomic recommendations proposed by the expert and centralized in knowledge bases help deciding on the right structure to use -For example: spare a zone on the left hand side of the screen for the alarms.

-A second level of the functional description: it aims at defining the set of the tasks which the operator in charge of control and supervision will have to perform across the graphical display. Thus, we determine the functional aspects of the display associated to different tasks; we determine also the description of the procedures which will be implemented to meet the functional level.

For example: the display must help the graphic representation of variables T2 and T4' trend.

At the end of the step, the semantic aspect of each display is completely described and stored into a database -this display is used by the following third module which performs the graphical definition and creation of the display.

IV.3 -THE MODULE FOR GRAPHICAL DEFINITION AND CREATION OF THE DISPLAYS:

The last module of the chain permits to the designer to define and create the display thanks to a graphical editor. A knowledge base involving ergonomic rules related to the ergonomic presentation of information aims at guiding the designer while he chooses the definition of the presentation information: * Type3: A third knowledge base proposes -from conclusions of rules type1-predifined modes of representation included in data base -from conclusions of rules type2-and prints them on the screen. Example: IF mode of representation of type 'curve' is chosen for variable X THEN print in a window (for eventual selection by the designer) the predifined curve with the parameter 'color' selected for variable X.

This stage of definition and graphical creation of the different displays leads to a version of the synopsis, ready for a "dynamic" evaluation on the site or in task simulation.

V -CONCLUSION

In this paper we have presented our current research works concerning the use of formalized ergonomic knowledge. The aim is the graphical Man-Machine interface design in the field of continuous complex processes. The first part has given a idea of our work concerning the use of ergonomic knowledge by the expert system SYNOP, for the help to the static ergonomic evaluation of the graphical information presentation. This expert system currently under the form of a laboratory model, contains a hundred rules related to ergonomic general or specific concepts.

The second part of this paper dealt with an ergonomic design problem after examining the design notion in general. This part was illustrated by the brief presentation of several current research works of the tasks description formalization. These works contribute to the creation -at medium and long run -of formal tools for the Man-Machine graphical interface design.

Finally, the last part concerned the study of a design global ergonomic methodology, based on the use of an expert tool constituted of three major modules: a module for the description of the operator's informational needs, a module for the functional and structural description and a module for the graphical definition and the creation of the displays. The first model of this expert tool is in progress thanks to the expert tool SMECI on MICROVAX.

Figure 2 :

 2 Figure 2: Ergonomic methodology for man-machine interface design

Figure 3 :

 3 Figure 3: A model of an interactive expert system of ergonomic design integrated into a global design methodology

Figure 4 :

 4 Figure 4: Simplified network of objects

*

  Type1: A first knowledge base created to help the designer while he chooses the representation modes. Example: IF the operator has to visualize a variable tendency THEN propose a 'curve' type representation. * Type2: A second knowledge base assists the designer while he chooses graphical attributes. Example: IF the variable X is in a normal mode THEN use green color.


Samuel Bernard
email: s.bernard@biologie.hu-berlin.deh.herzel@biologie.hu-berlin.de

Hanspeter Herzel

Why Do Cells Cycle with a 24 Hour Period?

Keywords: cell cycle, circadian clock, programmed cell death, resonance, tumor suppression, delay dierential equations

A typical proliferating human cell divides on average every 24 h. This division timing allows cells to synchronize with other physiological processes and with the environment. The circadian clock, which orchestrates daily rhythms, directly regulates the cell division cycle and is a major synchronizing factor. There is, however, no evidence that the circadian clock is able to entrain the cell cycle to a 24 h period. We show here, using a computational model for the cell cycle, that cells under circadian control that have an interdivision time close to multiples of 24 h proliferate faster. Moreover, growth of cell populations with a markedly dierent cell cycle time is impaired. We propose that this resonance eect in cell proliferation has a role to play in ecient normal cell proliferation and suppression of tumor growth.

Introduction

A fast proliferating human cell divides on average every 24 h. Given that many physiological processes such as activity, psychophysical performance, blood pressure, and liver metabolism show daily variation, it is not surprising that cells cycle in a circadian fashion. The circadian clock, which orchestrates these daily rhythms, consists of a set of genes showing self-sustained rhythmicity in expression with a period of approximately 24 h [START_REF] Schibler | Cellular oscillators: Rhythmic gene expression and metabolism[END_REF]. On a cell population level, the circadian clock regulates DNA synthesis and mitotic activity [START_REF] Bjarnason | Circadian variation in the expression of cell-cycle proteins in human oral epithelium[END_REF][START_REF] Potten | Cell kinetic studies in the murine ventral tongue epithelium: thymidine metabolism studies and circadian rhythm determination[END_REF][START_REF] Smaaland | DNA synthesis in human bone marrow is circadian stage dependent[END_REF], and on a genetic level, tumor suppression [START_REF] Fu | The circadian gene Period2 plays an important role in tumor suppression and DNA damage response in vivo[END_REF], liver regeneration [START_REF] Matsuo | Control mechanism of the circadian clock for timing of cell division in vivo[END_REF], and DNA damage control [START_REF] Gery | circadian gene Per1 plays an important role in cell growth and DNA damage control in human cancer cells[END_REF]. The kinetics of the cell cycle has been studied using cell population-based models [START_REF] Bernard | Analysis of cell kinetics using a cell division marker: Mathematical modeling of experimental data[END_REF][START_REF] Mackey | Unied hypothesis of the origin of aplastic anemia and periodic hematopoiesis[END_REF], and molecular-based models [START_REF] Csikasz-Nagy | Analysis of a Generic Model of Eukaryotic Cell-Cycle Regulation[END_REF]. In this paper, we look at how an external periodic force, the circadian clock, inuences the cell cycle kinetics. We show, using a computational model, that cells with an division time close to multiples of 24 h proliferate faster. Circadian variation of the cell cycle kinetics interacts with the cell division clock and the cell death program to create a resonance eect that promotes certain cell division times. This phenomenon contributes to the emergence of cells with specic cell cycle durations. The selectivity of cell cycle times could play a role in the suppression of tumor development by slowing down the growth of cells that have a perturbed cell cycle.

Model

We use a population-based model for the human cell cycle [START_REF] Bernard | Analysis of cell kinetics using a cell division marker: Mathematical modeling of experimental data[END_REF][START_REF] Burns | On the existence of a G phase in the cell cycle[END_REF][START_REF] Mackey | Unied hypothesis of the origin of aplastic anemia and periodic hematopoiesis[END_REF][START_REF] Smith | Do cells cycle?[END_REF] (Figure 1). To mimic the eect of the circadian clock, we allow the kinetic parameters to vary with a circadian period. We rst describe the model as a stochastic system, where cell fate|death, arrest or division|are random events. We then formulate the model as a deterministic system, valid when the cell population is large.

Cells go successively through four phases: G1, S, G2, and M, at the end of which cells divide and reenter G1 phase. Most of the variability in duration of the cell cycle comes from dierences in length of G1 phase. Durations of S phase (8{12 h), G2 (2{4 h), and M (1 h) are relatively constant under normal conditions. The G1 phase has a minimal duration, during which cells are in G1d phase, with an average duration of 9{11 h. The G1d, S, G2, and M phases have a xed total duration , between 12 and 25 h in human cells [START_REF] Alberts | Molecular Biology of the Cell[END_REF], and constitute the proliferative phase of the cell cycle. The variable part of the G1 phase (called G1s) has a duration exponentially distributed with parameter + . G1s phase cells go in proliferative phase with probability =( + ), or leave the cell cycle by programmed cell death or cell cycle arrest with probability =( + ). Cell progressing in the proliferative phase have constant probability 1 exp( ) to undergo programmed cell death, or apoptosis. The time a cells spend in proliferative phase is exponentially distributed with parameter when a < and otherwise. At time after entering G1s phase, surviving cells divide and go back in the G1s cell pool.

When the cell population is large, stochastic uctuations can be neglected and only the average population needs to be considered. This leads to the deterministic version of the model [START_REF] Bernard | Analysis of cell kinetics using a cell division marker: Mathematical modeling of experimental data[END_REF][START_REF] Mackey | Unied hypothesis of the origin of aplastic anemia and periodic hematopoiesis[END_REF]. The population in G1s phase at time t, N(t), is sucient to describe the dynamics of cell proliferation. Cells in G1s phase leave the cell cycle at a rate and enter the proliferative phase at a rate . Cells in proliferative phase die at a rate , and surviving cells divide a time after entering G1d phase. The G1s population is described by a delay dierential equation with periodic coecients, dN(t

) dt = [ + (t)]N(t) + 2(t)(t )N(t ): (1)
The fraction of cells surviving the active phase is

(t) = exp Z t t (u)du ; 0 (t) 1: (2)
Two cell cycle kinetic parameters, the G1s!G1d transition rate and the apoptosis rate , are subject to circadian uctuation. A variation in the G1s!G1d transition rate is motivated by the periodic expression of proliferation promoting genes such as c-Myc and cyclin D1 [START_REF] Fu | The circadian gene Period2 plays an important role in tumor suppression and DNA damage response in vivo[END_REF]. The variation of the apoptosis rate is relevant in the context of periodic, or chronomodulated cancer therapy [START_REF] Mormont | Cancer chronotherapy: Principles, applications, and perspectives[END_REF]. Detailed molecular models of the mammalian circadian clock [START_REF] Leloup | Toward a detailed computational model for the mammalian circadian clock[END_REF] and the mammalian cell cycle [START_REF] Csikasz-Nagy | Analysis of a Generic Model of Eukaryotic Cell-Cycle Regulation[END_REF] exist, but a link between them has not been made. We therefore consider the circadian clock input as a sine wave. The G1s!G1d transition rate has the form (t) = (1 + A sin(2t=T )); 0 A 1:

(3) For the apoptosis in the proliferative phase, we allow either the apoptosis rate or the survival fraction to vary sinusoidally, (A) Exponential growth of populations N(t). For A = 0 (dashed lines), cells are rapidly desynchronized and the population grows exponentially. For A = 1 (solid lines), cells are synchronized and the population oscillates with a 24 h period in addition to the exponential growth. The initial population N(t) = 1 for t 0. (B)

Average population growth rate as a function of the proliferative phase duration . For A = 0 (dashed line), the growth rate decreases monotonically with respect to . For A = 1 (solid line), the growth rate peaks approximately every 12 h. For C = 0:2 (dotted line), the resonance is also present, albeit in a weaker form. The growth rate is not aected by a periodic (B > 0, same as A = 0). In every case, the circadian variation of parameters does not aect the growth rate for = nT . Parameters are = 0:1 h , = 0:2 h , = 0:8. The coupling coecients A, B, and C are zero unless specied. (C) Growth rate for = 0 and A = 1. or (t) = (1 + C sin(2t=T )); 0 C 1= 1:

(5)

The circadian coupling coecients, A, B, and C, control the strength of the circadian clock on the cell cycle parameters. The parameter C is limited by the constraint that the survival fraction (t) must be between 0 and 1. The period T is xed to 24 h. We simulated the stochastic model with a Gillespie algorithm adapted for delay equations and the deterministic model with the delay equation solver dde23 (Matlab).

Results

Because equation 1 is linear, one would expect an exponential growth for the G1s population, N = K(t) exp( t). With constant coecients, N(t) indeed grows exponentially, with K(t) rapidly converging to a constant (Figure 2 (A), A = 0, dashed lines). The growth rate is slightly larger for = 15 h than for = 22 h (thin and thick dashed lines, respectively). For periodic coecients, the average growth is also exponential, but the population oscillates periodically, with K(t) converging to a T -periodic function. In contrast to the constant case, the growth rate increases as varies from 15 to 22 h when the G1s!G1d transition rate is periodic (Figure 2 (A), A = 1, solid lines). For = 15 (solid thin line), cell deaths outnumber cell births and the population decays. When is increased to 22 h (solid thick line), the population grows faster than any other, despite the longer cell cycle time. This suggests that a rhythmic interacts with the successive waves of cell divisions. The proliferation phase duration is the key parameter that determines the population growth under circadian control.

The population dynamics with respect to behave dierently depending on the values taken by the circadian coupling strengths A, B, and C (Figure 2 (B)). For A = 0 (dashed line), the growth rate decreases monotonically with respect to . For A = 1 (solid line), the growth rate peaks every 12 h approximately, with main peaks at 22.1 and 46.2 h, and secondary peaks at 33.6 and 57.6 h. The maximal growth rate, at = 22 h, is twice as large as for A = 0. When the fraction of cells surviving the proliferative phase is periodic, the growth rate peaks every 24 h, without showing secondary peaks (C = 0:2, dotted line). The situation is dierent for periodic apoptosis. A periodic parameter (B > 0) does not aect the growth rate [START_REF] Clairambault | A mathematical model of the cell cycle and its circadian control[END_REF]. That is, the growth rate has the same values as in the constant case. This is because a periodic rate pushes up the average survival fraction, h per i const , in a -dependent manner, preventing direct analogy between the eects of B > 0 and C > 0. When = nT , the growth rate is independent of A, B, and C (see the intersections of the three lines at = 24 and 48 h in Figure 2 (B)). If is negligible, the resonance eect becomes weaker (Figure 2 (C)). All these results show that cell populations under periodic conditions grow faster for specic proliferation phase durations. The resonance eect relies on the cell death or arrest rate .

For arbitrary ratio between and T , the growth rate is best determined by numerical methods. Some analytical treatment, however, is possible for particular cases. The population growth rate of N is the largest (real) root determined by the eigenvalue problem

dK(t) dt = [ + + (t)]K(t) + 2(t)(t )e K(t ); (6)
where K(t) must be a T -periodic solution. When all the coecients are constant, the eigenproblem reduces to the equation + + = 2e . For periodic coecients, the eigenproblem can be approached by spectral methods [START_REF] Just | On the eigenvalue spectrum for time-delayed Floquet problems[END_REF] or by trying a reduction to a system of ordinary dierential equations. If the proliferative phase duration is an integer multiple of the period T , the latter method leads to the case of constant coecients.

The sensitivity of the growth rate on for periodic coecients is reminiscent of resonance eects in physical systems. In a periodically forced damped oscillator, the amplitude is maximal when the intrinsic and the forcing frequencies coincide. For constant coecients, the fraction of G1s cells, X = N=(N + P ) (where P represents the proliferative population) is a nonlinear damped oscillator. Its intrinsic frequency can be related to the average interdivision time, with the variance in G1s duration slowly desynchronizing the cells and damping out X. If the interdivision time were such that cells minimized their residence times in G1s, a smaller fraction would die or arrest due to the term . Consequently, a larger fraction would reenter the proliferative phase. This is enough to compensate the longer needed to achieve this eect. Together, the probability of a G1s cell to enter the proliferative phase eventually, the rate of entry, and the length of the proliferative phase determine the growth rate. The stochastic formulation of the cell cycle model is best suited to determine the G1 residence times and the probability to enter the proliferative phase.

With the stochastic version of the cell cycle model, cells can be tracked individually. Time courses of the model for = 15 and 22 h have a dynamics similar to the deterministic simulations (Figure 3 (A), (B)). The average population growth rate shows a resonance eect as in the deterministic version of the model (Figure 3 (C)). To understand this, we looked at the average time cells spent in G1 phase before entering S phase (Figure 3 (D)). The mean G1s residence time is 5.0 h for = 15 h and 2.4 h for = 22 h. Shorter residence time for = 22 h means that fewer cells go to apoptosis. The probability of G1s cells to transit to proliferative phase is 0.54 for = 15 h and 0.74 for = 22 h. The dierence is due to the presence of a \shoulder" in the residence time distribution for = 15 h. G1s cell death or arrest () plays an important role and is the major factor shaping of the growth rate resonance. For vanishing , the resonance eect is still present but becomes much weaker (Figure 2 (C)).

Discussion

Molecular evidence points to an important role of the circadian rhythm for ecient cell proliferation and reduction of tumor growth. This can be demonstrated using knock out mice without essential clock genes such as Cry or Per2. Free-running rhythmicity is abolished in Cry -decient mice. After partial hepatectomy, liver regeneration in these mice is slower compared to wild type mice [START_REF] Matsuo | Control mechanism of the circadian clock for timing of cell division in vivo[END_REF]. Cells showed a normal entry to S phase but a marked reduction in mitotic fraction compared to wild type. This implies that a functional circadian clock is necessary for ecient cell proliferation.

Following -radiation, mPer2 null mice developed spontaneous salivary gland hyperplasia and teratomas faster than the wild type mice [START_REF] Fu | The circadian gene Period2 plays an important role in tumor suppression and DNA damage response in vivo[END_REF]. Moreover, epidemiological studies support the idea that circadian rhythms contribute to slowing down tumor growth. Patients with colorectal cancers exhibiting normal daily rhythms had longer survival times than those with disrupted circadian rhythms [START_REF] Mormont | Cancer chronotherapy: Principles, applications, and perspectives[END_REF]. Nevertheless, a direct link between circadian rhythm and development of cancer has not been established. Arrhythmic mice do not systematically display increased sensitivity to cancer, as shown in a Cry1 = Cry2 = mouse model [START_REF] Gauger | Cryptochrome, circadian cycle, cell cycle checkpoints, and cancer[END_REF]. Individual components of the circadian clock might have roles to play that are independent of the circadian rhythm. We suggest here that the presence of a circadian rhythm in cell cycle helps cells that have a normal cell cycle length, in addition to possible eects of individual clock genes.

Circadian regulation of the cell cycle orchestrates normal tissue regeneration and protects cells from oncogenic mutations. We propose that this is done by slowing down the cell cycle. Without circadian regulation, oncogenic mutations would be doubly damaging, because of the proliferative advantage and the possibly reduced anti-tumor mechanisms a short cell cycle length would allow. By shifting the proliferative advantage to slowly dividing cells, the circadian clock hence acts as a tumor suppressor. Slower cells have more time to detect and to repair genetic damages; mutations shortening the cell cycle are thus rendered unfavorable.

Our simulations show that only cells with a proliferative phase slightly shorter than 24 h (20 to 24 h) proliferate faster in presence of a circadian clock. This also holds for cells with between 44 and 48 h, to a lesser degree. The stochastic simulations allow the calculation of the average G1 residence time. For between 20 and 24 h, the G1s phase average residence time ranges from 3.3 to 2.3 h, for a total average cell cycle time between 23.3 and 26.3 h.

In two transplantable mice tumor models, Glasgow osteosarcoma and Pancreatic adenocarcinoma, disruption of circadian rhythmicity accelerated malignant growth [START_REF] Filipski | Disruption of circadian coordination and malignant growth[END_REF]. Ablating the suprachiasmatic nuclei, center of the circadian pacemaker, or subjecting the mice to chronic jet lag resulted in the suppression of sleep-wake rhythms and altered body temperature and lymphocyte count rhythms. Our model provides a simple explanation for the accelerated tumor growth: if tumor cells have a deregulated cycle length (i.e. 20 h), the mere loss of circadian rhythmicity is sucient to increase the tumor growth rate, every other quantities being equal (see Figure 2 (B), for = 12 h, A = 1 and A = 0, for instance).

Other computational models have shown similar resonance eects. Clairambault et al. [START_REF] Clairambault | Circadian rhythm and tumour growth[END_REF] recently studied tumor growth in an age-structured model with periodic parameters, and also found that the growth rate is signicantly inuenced by the ratio between the G1 and the proliferative phase durations. Battogtokh and Tyson [START_REF] Battogtokh | Periodic forcing of a mathematical model of the eukaryotic cell cycle[END_REF] used a simplied molecular model of the eukaryotic cell cycle to show that periodic change of parameters is able to synchronize a population initially randomly distributed over the cell cycle. Using their formalism and adding random cell death, we showed a similar resonance eect for dierent lengths of the cell cycle (Figure 4 in Appendix).

The molecular role of the circadian clock in cell cycle regulation has been claried in recent studies [START_REF] Fu | The circadian gene Period2 plays an important role in tumor suppression and DNA damage response in vivo[END_REF][START_REF] Gauger | Cryptochrome, circadian cycle, cell cycle checkpoints, and cancer[END_REF][START_REF] Gery | circadian gene Per1 plays an important role in cell growth and DNA damage control in human cancer cells[END_REF][START_REF] Matsuo | Control mechanism of the circadian clock for timing of cell division in vivo[END_REF]. Its impact on normal and tumor cell growth, however, can only be assessed when other regulatory mechanisms are taken into account. One of those, programmed cell death, is dicult to quantify because cells undergoing apoptosis are rapidly cleared from the organism, thus the apoptosis rate is generally underestimated [START_REF] Danial | Cell death: Critical control points[END_REF][START_REF] Jacobson | Programmed cell death in animal development[END_REF]. In addition to its role in tissue development and maintenance, it might act together with the circadian clock to coordinate cell proliferation. Programmed cell death, as it is, might nally promote normal cell growth and helps this way to suppress tumor development. A cell division occurs when X crosses downwards the threshold X thresh = 0:05, and at this point the cell mass m is divided equally between the two daughter cells (see [START_REF] Battogtokh | Periodic forcing of a mathematical model of the eukaryotic cell cycle[END_REF] for more details). We introduced random cell death or cell cycle arrest in the G1 phase by calculating the net increase b in proliferative cell number after each cell division (0 b 2). When cells are in G1 phase, cells exit the cell cycle at a rate d, so that the survival fraction after division is b = 2 exp( d t G ), where t G is the time cells spent in G1 (dened as cells having a mass less than 1.2). The external period, T = 88:88 min, was set to be equal to the unperturbed cell cycle length, and we varied the cell cycle length from 40 to 200 min. To change the cell cycle length, the time was rescaled between t ! T =200t and T =40t. The growth rates with or without external forcing were similar except for cell cycle lengths T C = k=2T , k = 1; :::; 4 (Figure 4), where the growth rate under periodic forcing showed a resonance similar to what is seen in the population-based model.

Figure 1 :

 1 Figure 1: Model for the human cell cycle.

Figure 2 :

 2 Figure 2: Resonance eect for periodic G1sG1d transition rate (t) and proliferative phase apoptosis (t).(A) Exponential growth of populations N(t). For A = 0 (dashed lines), cells are rapidly desynchronized and the population grows exponentially. For A = 1 (solid lines), cells are synchronized and the population oscillates

Figure 3 :

 3 Figure 3: Stochastic simulations for periodic G1sG1d transition rate . (A{B) Ten stochastic time series for = 15 and 22 h are shown with their respective deterministic simulations (thick black line). Parameters are = 0:1h , = 0:2h , = 0:01h , initial number of G1s cells for each run n = 100, duration of G1d phase 6 h, A = 1, B = C = 0. (C) Resonance eect in the stochastic model. The mean growth rate was estimated from 20 stochastic runs for each , for A = 0 (thin line) or A = 1 (thick line), the error bars show the standard deviation for dierent runs. (D) Average residence time in the stochastic part of G1 (G1s) of cells entering the S-phase for A = 1.

Figure 4 :

 4 Figure 4: Resonance eect in a molecular-based cell cycle model. Average growth rate of the cell population.Death, or cell cycle arrest, was introduced at the beginning of the cell cycle, corresponding to the G1 phase (d = 0:04). In absence of periodic forcing, the growth rate is inversely proportional to the cell cycle length (dashed line). In presence of periodic forcing, the growth rate shows signicant resonance eect for cell cycle lengths T + = k=2T , k = 1; :::; 4 (solid line). The default cell cycle parameters are A = 0:52, B = 0:52, = 0:005776, k = 0:002, k = 0:0795, k ! = 0:01, k " = 2, k # = 0:05, k $ = 0:04, k % = 1:5, k & = 0:19, k ' = 0:64, k = 0:0025, k = 0:07, k = 0:08, P = 0:15, J = 0:05, f = 2=T , and T = 88:88.

Acknowledgments

We thank B.

Cavajec for carefully reading and commenting the manuscript. This work was supported by the European Commission (BioSim Network, contract No. LSHB-CT-2004.005137).