Fatiha Nmira

Yuying Liu

Jean-Louis Consalvi
email: jean-louis.consalvi@univ-amu.fr

Frederic Andre

Fengshan Liu

PRESSURE EFFECTS ON RADIATIVE HEAT TRANSFER IN SOOTING TURBULENT DIFFUSION FLAMES

Keywords: methane turbulent jet diffusion flame, pressure effects, radiative loss, turbulenceradiation interaction

This article investigates the effects of increasing pressure up to 4 atm on radiative heat transfer in momentum-driven methane turbulent jet flames by using well-established chemical mechanism, combustion, soot production and radiation models. A transported PDF method is used to close properly the soot-production turbulence interaction and the emission Turbulence Radiation Interaction (TRI). A Narrow-Band CK (NBCK) model is used as the gas radiative property model. The absorption TRI is neglected based on the Optically-Thin Fluctuation Approximation (OTFA). In accordance with a previous study dealing with non-sooting hydrogen flames (Nmira et al., JQSRT 220 (2018) 172-179), the 3-atm and 4-atm flames are designed from the atmospheric flame by using a Froude modeling approach that allows to preserve the flame/flow structure as the pressure is increased and hence to isolate the pressure effects on soot production, radiative heat transfer, and TRI. The effects of increasing pressure on radiant fraction result from two competing mechanisms: i) an increase in soot emission that tends to increase the radiant fraction and ii) a reduction in flame transparency that tends to reduce it. For the present flames, the first mechanism dominates the second, resulting in an increased radiant fraction with increasing the pressure. The TRI effects on flame radiative loss are also governed by competing mechanisms. The enhancement mechanism is due to gas emission TRI and temperature self-correlation effects on soot emission whereas the reduction mechanism is caused by the negative correlation between soot volume fraction and temperature.

The former dominates whereas the latter becomes increasingly important with increasing the pressure. This limits the increase in the global radiative loss due to TRI as the pressure is increased. In addition, numerical simulations show that the TRI effects can reduce the local radiative loss in regions of high soot concentration of the 4 atm flame.

𝜒 𝑅 radiant fraction [-] Δℎ 𝑐 heat of combustion [J kg -1 ]  dissipation rate of the turbulent kinetic energy m 2 •s -3 ]

𝜁

Introduction

Many practical combustion systems, such as gas turbine combustors and piston engines, involve turbulent sooting flames and operate at elevated pressures for reasons of optimal efficiency and small size. Increasing the pressure affects the flow structure and the chemical reaction rates [START_REF] Modest | Radiative Heat Transfer in Turbulent Combustion Systems: Theory and Applications[END_REF].

It modifies also the radiative properties of the medium. Pressure affects gas radiation through enhancements in both gas density and line broadening [START_REF] Modest | Radiative Heat Transfer in Turbulent Combustion Systems: Theory and Applications[END_REF][START_REF] Pal | k-Distribution methods for radiation calculations in high-pressure combustion[END_REF][START_REF] Pearson | Efficient representation of the absorption line blackbody distribution function for H2O, CO2 and CO at variable temperature, mole fraction, and total pressure[END_REF][START_REF] Chu | Effects of total pressure on non-grey gas radiation transfer in oxy-fuel combustion using the LBL, SNB, SNBCK, WSGG, and FSCK methods[END_REF]. Calculations made in 1D parallel plate geometry containing mixtures of CO2 and H2O representative of oxy-combustion showed that the first mechanism is predominant whereas the second, although significant, has a smaller influence on radiative heat transfer (RHT) [START_REF] Chu | Effects of total pressure on non-grey gas radiation transfer in oxy-fuel combustion using the LBL, SNB, SNBCK, WSGG, and FSCK methods[END_REF]. Previous studies also showed that k-distribution methods and the mixing model developed in Ref. [START_REF] Modest | Assembly full spectrum k-distribution from a narrow band database: effects of mixing gases, gases and non-gray absorpting particles and non-gray scatters in non-gray enclosures[END_REF] can be extended to high pressure gas mixtures [START_REF] Pal | k-Distribution methods for radiation calculations in high-pressure combustion[END_REF][START_REF] Pearson | Efficient representation of the absorption line blackbody distribution function for H2O, CO2 and CO at variable temperature, mole fraction, and total pressure[END_REF][START_REF] Chu | Effects of total pressure on non-grey gas radiation transfer in oxy-fuel combustion using the LBL, SNB, SNBCK, WSGG, and FSCK methods[END_REF][START_REF] Chu | Calculations of radiative heat transfer in axisymmetric jet diffusion flame at elevated pressures using different gas radiation models[END_REF]. The radiative property of soot is also influenced by pressure rise primarily through an enhancement in soot production and a change in soot volume fraction distribution.

Experimental investigations in laminar axisymmetric coflow diffusion flames revealed that soot production scales as P n with n varying on the whole between 1 and 3, depending on both the fuel and pressure range [START_REF] Mccrain | Measurements of the soot volume field in laminar diffusion flames at elevated pressures[END_REF][START_REF] Karataş | Soot formation in high pressure laminar diffusion flames[END_REF][START_REF] Gülder | Unified behaviour of maximum soot yields of methane, ethane and propane laminar diffusion flames at high pressures[END_REF].

Most of the studies related to the radiative structure of turbulent diffusion flames including the Turbulent-Radiation Interaction (TRI) effects were performed at atmospheric pressure [START_REF] Modest | Radiative Heat Transfer in Turbulent Combustion Systems: Theory and Applications[END_REF][START_REF] Coelho | Numerical simulation of the interaction between turbulence and radiation in reactive flows[END_REF][START_REF] Coelho | Turbulence-Radiation Interaction: From Theory to Application in Numerical Simulations[END_REF] and very few studies were devoted to the pressure effects on RHT and TRI. Cai et al. [START_REF] Cai | High fidelity radiative heat transfer models for high-pressure laminar hydrogen-air diffusion flames[END_REF] compared a radiative model based on the Full Spectrum Correlated-k (FSCK) model coupled to spherical harmonics methods to a photon Monte Carlo method coupled with a line-by-line spectral model (PMC/LBL) in laminar hydrogen/air diffusion flames at pressures of 1, 5 and 30 atm. They found that the optically-thin approximation is not valid at high pressures and the first order spherical harmonic method (P1) is adequate except at very high pressures. Some other recent studies investigated RHT in high-pressure environments relevant to piston engines. Ferreyro-Fernandez et al. [START_REF] Fernandez | Soot and spectral radiation modeling for high-pressure turbulent spray flames[END_REF] investigated numerically RHT in a high-pressure spray flame fueled with n-dodecane by using a transported PDF and PMC/LBL model. They showed that the redistribution of energy is dominated by CO2 and H2O is mainly responsible for the radiative heat loss. These findings were used in another work to propose a computationally efficient stepwise-gray model that provides predictions within 10% deviations from the results of accurate spectral radiation models [START_REF] Paul | A simplified CFD model for spectral radiative heat transfer in high-pressure hydrocarbon-air combustion systems[END_REF].

Investigating the pressure effects on a given process is a difficult task, since modifying the pressure may affect the flow characteristics and, therefore, lead to a misleading interpretation. It is therefore desirable to isolate the pressure effects on the investigated process. As an example, such kind of methodology was applied to investigate experimentally the pressure effects on soot production in laminar coflow diffusion flames. The experimental conditions were designed to maintain the flow residence time, which is recognized to affect significantly soot production, unchanged as the pressure is increased, allowing to isolate the pressure effects on soot production [START_REF] Karataş | Soot formation in high pressure laminar diffusion flames[END_REF]. In the framework of turbulent diffusion flames, RHT and TRI are mainly controlled by the flow residence time, flame geometry and turbulent intensity. In a previous study, it was shown numerically that the Froude modelling analysis, which maintains the Froude number constant as the pressure varies, preserves these quantities [START_REF] Nmira | Pressure effects on radiative heat transfer in hydrogen/air turbulent diffusion flames[END_REF]. The Froude modelling method was applied to investigate the pressure effects on RHT and TRI in 'non-luminous' hydrogen flames. The numerical results showed that the increase in radiative loss with increasing pressure is limited by an enhancement in the flame optical thickness. It was also shown that the TRI effects on radiative losses are on the whole enhanced by pressure rise.

The objective of this study is to investigate the effects of increasing pressure on RHT and TRI in turbulent sooting diffusion flames. To isolate the pressure effects, the flames at elevated pressures were designed from the atmospheric counterpart by using the Froude modeling strategy reported in Ref. [START_REF] Nmira | Pressure effects on radiative heat transfer in hydrogen/air turbulent diffusion flames[END_REF].

Numerical model

The turbulent jet diffusion flames are modelled by using the computational model described in detail in Refs. [START_REF] Consalvi | Simulations of sooting turbulent jet flames using a hybrid flamelet/stochastic Eulerian field method[END_REF][START_REF] Consalvi | Effects of soot absorption coefficient-Planck function correlation on radiative heat transfer in oxygen-enriched propane turbulent diffusion flame[END_REF][START_REF] Consalvi | Transported scalar PDF modeling of oxygen-enriched turbulent jet diffusion flames: Soot production and radiative heat transfer[END_REF]. A brief description is given below.

The model is based on a hybrid flamelet/transported composition PDF approach. The local thermochemical properties of a reactive flow is determined by a set of scalar variables, namely the mixture fraction, , the enthalpy defect, 𝑋 𝑅 = ℎ -ℎ 𝑎𝑑 , the scalar dissipation rate, , the soot mass fraction, YS, and the soot number density, NS. The scalar dissipation rate  is assumed to be statistically independent of the other scalars and its PDF is modeled by a Dirac- function.

The one-point, one-time joint composition PDF of  𝑋 𝑅 , YS and NS is then transported. The interaction by exchange with the mean (IEM) model is used to model the molecular mixing for mixture fraction and enthalpy defect. On the other hand, the effective Schmidt number for soot is very high and soot is not effectively transported by molecular species. As a consequence the soot scalars are not mixed. The flamelet libraries are generated using the full chemical kinetic scheme of Qin et al. [START_REF] Qin | Combustion chemistry of propane: a case study of detailed reaction mechanism optimization[END_REF]. Soot production is modelled by a semi-empirical acetylene/benzenebased soot model [START_REF] Lindstedt | Simplified soot nucleation and surface growth steps for non-premixed flames[END_REF]. Turbulence soot interaction (TSI) is modelled 'exactly' by the joint composition PDF.

A time marching technique is used until a statistically stationary state is reached. The conservation equations for momentum, turbulent kinetic energy and the dissipation rate of turbulent kinetic energy are solved in cylindrical coordinates using a finite-volume method on a staggered grid. For the convective terms, the third-order upwind QUICK combined with ULTRASHARP limiter [START_REF] Leonard | Why you should not use hybrid, power-law or related exponential schemes for convective modelling. There are much better alternatives[END_REF]. A second-order central difference scheme is used for diffusion terms. Pressure-velocity coupling is solved using the Iterative PISO algorithm [START_REF] Chow | Selection of differencing scheme on simulating the sprinkler hot-air layer problem[END_REF]. The joint composition PDF transport equation was solved by using the Stochastic Eulerian Field (SEF) method developed by Valiño [START_REF] Valiño | A field Monte Carlo formulation for calculating the probability density function of a single scalar in a turbulent flow[END_REF].

Absorption TRI is neglected based on the OTFA. Previous studies showed the validity of OTFA over a broad range of configurations including very large highly-sooting jet fires [START_REF] Mehta | Radiation characteristics and turbulence-radiation interactions in sooting turbulent jet flames[END_REF][START_REF] Consalvi | Absorption turbulence-radiation interactions in sooting turbulent jet flames[END_REF] and non-luminous turbulent jet diffusion flames with pressures up to 30 atm [START_REF] Nmira | Pressure effects on radiative heat transfer in hydrogen/air turbulent diffusion flames[END_REF]. Emission TRI is modelled 'exactly' by using the joint composition PDF. The spectral gas and soot radiation is modelled using a NBCK. The NB soot absorption coefficient is computed from the Rayleigh theory coupled to the correlations of Ref. [START_REF] Chang | Determination of the wavelength dependence of refractive indices of flame soot[END_REF] for the refractive index. The gas radiation model considers a database of mixed NB k-g distributions generated from the Line-By-Line database of Pearson et al. [START_REF] Pearson | Updated correlation of the absorption line blackbody distribution function for H2O based on the HITEMP2010 database[END_REF]. The mixing scheme of Modest and Riazzi [START_REF] Modest | Assembly full spectrum k-distribution from a narrow band database: effects of mixing gases, gases and non-gray absorpting particles and non-gray scatters in non-gray enclosures[END_REF] is used to deal with the mixtures of CO2 and H2O. For each pressure of interest, the mixed NB k-g distributions are stored in the database for the 10-Gauss quadrature points used to solve the NB Radiative

Transfer Equation (RTE), 9 mole fractions of CO2 and 9 mole fractions of H2O, xCO2 and xH2O, ranging from 0 to 1 (0, 0.05, 0.1, 0.2, 0.3, 0.4, 0.6, 0.8, 1), 28 temperatures, T, ranging from 300 to 3000 K (with a uniform step of 100 K), and 115 NBs with a non-uniform spectral resolution optimized to minimize the variation of the blackbody intensity over a NB. At each computational node and for each NB and quadrature point, the gas absorption coefficient is extracted from the database by using a linear interpolation with respect to xCO2 and xH2O and a spline interpolation on T. For each NB band, the NB soot absorption coefficient is then added to the interpolated gas absorption coefficient. The RTE is solved by the Finite Volume Method (FVM) using the mapping technique developed by Chui et al. [START_REF] Chui | Prediction of radiative transfer in cylindrical enclosures with the finite volume method[END_REF] for axisymmetric configurations. Computations are carried out using the first-order UPWIND spatial discretization scheme and an angular mesh with 12×16 control angles.

The capability of the numerical model described above to predict the flame structures, soot production, and RHT was extensively validated in Refs. [START_REF] Consalvi | Simulations of sooting turbulent jet flames using a hybrid flamelet/stochastic Eulerian field method[END_REF][START_REF] Consalvi | Effects of soot absorption coefficient-Planck function correlation on radiative heat transfer in oxygen-enriched propane turbulent diffusion flame[END_REF][START_REF] Consalvi | Transported scalar PDF modeling of oxygen-enriched turbulent jet diffusion flames: Soot production and radiative heat transfer[END_REF] by considering axisymmetric turbulent jet diffusion flames covering wide ranges of turbulence level, residence time, fuel sooting propensity, oxygen index and pressure. In particular, this validation exercise has considered the methane turbulent jet diffusion flames at 1 and 3 atm investigated experimentally by Brookes and Moss [START_REF] Brookes | Measurements of soot production and thermal radiation from confined turbulent jet diffusion flames of methane[END_REF] and the predictions were found to be in satisfactory agreement with the experimental data [START_REF] Consalvi | Simulations of sooting turbulent jet flames using a hybrid flamelet/stochastic Eulerian field method[END_REF], showing the capability of the numerical method to predict accurately the effects of moderate increase in pressure on the flame structure and soot production. As a consequence, the present study is limited to methane and to a pressure range of 1-4 atm. Another reason to limit the range in pressure at 4 atm is that the Froude modeling may not preserve fully the flame structure if the pressure would be further increased (see section 3.2).

Results and discussions

Radiant fraction

The radiant fraction, 𝜒 𝑅 , which represents the fraction of the heat release rate (HRR) that is lost from the flame by thermal radiation, is a relevant metric to quantify the radiative loss from diffusion flames. It is defined as:

𝜒 𝑅 = 𝑄 ̇𝑒𝑚 -𝑄 ̇𝑎𝑏𝑠 𝑄 ̇𝑐ℎ (1)
where

𝑄 ̇𝑒𝑚 = ∫ 4𝜋〈𝜅 𝑝 𝐼 𝑏 〉 𝑉 𝑑𝑉, 𝑄 ̇𝑎𝑏𝑠 = ∫ ∫ 𝜅 𝜂 𝜂 𝑉
𝐺 𝜂 𝑑𝜂𝑑𝑉, and 𝑄 ̇𝑐ℎ = 𝜌 𝐹 𝑢 𝐹 𝜋 𝑑 𝐹 2 4 ⁄ Δℎ 𝑐 represent the total emission, the total absorption and the chemical HRR due to combustion, respectively.

Eq. 1 can be rearranged to yield [START_REF] Consalvi | Modeling of large-scale under-expanded hydrogen jet fires[END_REF]:

𝜒 𝑅 = 4𝑉 𝑓𝑙 𝜋𝑢 𝐹 𝑑 𝐹 2 ⏟ 𝐼 • 𝑄 ̇𝑒𝑚 ′′′ 𝜌 𝐹 ∆ℎ 𝑐 ⏟ 𝐼𝐼 • (1 - 𝑄 ̇𝑎𝑏𝑠 𝑄 ̇𝑒𝑚 ) ⏟ 𝐼𝐼𝐼 (2)
where 𝑉 𝑓𝑙 is the flame volume and 𝑄 ̇𝑒𝑚 ′′′ = 𝑄 ̇𝑒𝑚 𝑉 𝑓𝑙 ⁄ is the emission per unit flame volume. Term I represents a global residence time [START_REF] Turns | Oxides of nitrogen emissions from turbulent jet flames: Part I-Fuel effects and flame radiation[END_REF], Term II is the inverse of a characteristic emission time scale, and Term III characterizes the flame transparency with 𝑄 ̇𝑎𝑏𝑠 𝑄 ̇𝑒𝑚 ⁄ being a measure of the flame optical thickness.

High-pressure flame design

The baseline case is a turbulent coflow methane/air diffusion flame at atmospheric pressure with the operating conditions given in Table 1. The nozzle diameter and the fuel injection velocity are 2.5 mm and 133 m/s, respectively. The injection Reynolds number is 18790. The Froude number is defined as [START_REF] Delichatsios | Transition from momentum to buoyancy-controlled turbulent jet diffusion flames and flame height relationships[END_REF]:

Fr = u F Z st 1.5 [( ρ F ρ ∞ ) 0.25 ( T ad -T ∞ T ∞ gd F ) 0.5 ] ⁄ (3)
The baseline flame is in the momentum-dominated regime, since its Froude number is higher than 5 [START_REF] Delichatsios | Transition from momentum to buoyancy-controlled turbulent jet diffusion flames and flame height relationships[END_REF] (see Table 1). In this regime, the flame length is directly proportional to the injection nozzle diameter and can be expressed as [START_REF] Delichatsios | Transition from momentum to buoyancy-controlled turbulent jet diffusion flames and flame height relationships[END_REF]:

𝐿 𝑓𝑙 = 𝐿 * 𝑑 𝐹 (𝜌 𝐹 𝜌 ∞ ⁄ ) 1/2 𝑍 𝑠𝑡 with 𝐿 * = 23 (4)
Eq. 3 shows that maintaining the fuel injection velocity and nozzle diameter constant at an elevated P allows to preserve the Froude number since the adiabatic temperature is very weakly dependent on pressure. Eq. 4 shows that the flame length is then also preserved. A dimensional analysis supports that the flame structure is also preserved in terms of flame geometry, flow residence time, turbulent kinetic energy, its dissipation, and turbulent mixing time and turbulent integral length scales (see Appendix A). This scaling is referred to as Froude modeling and is applied to design the high-pressure flames (see Table 1) based on the atmospheric pressure flame. It should be pointed out that the Froude modeling does not preserve the heat release rate (see Table 1), the soot production rate, and the Kolmogorov time scale (see Appendix A).

Calculations were run for the three flames by considering the parameters given in Table 1. The computational domain considers a nozzle centered in a 600-mm-i.d. blackened stainless steel chamber. The numerical calculations were performed in a computational domain of 0.3 m (r) ×1.2 m (z) with 84 (r) × 93 (z) control volumes. The walls were assumed to behave as blackbody emitting at the computed surface temperature. The peak soot volume fraction increases from 0.02 ppm at 1 atm to 0.8 ppm at 3 atm (see Fig. 1c). Figures 1 (d) to (f) show that these contours are also rather well preserved as the pressure is increased from 1 atm to 4 atm. However, differences can be clearly observed. The radiative losses in the 4-atm CH4 flame are significantly higher than those in the 1-atm flame due to a strong increase in soot volume fraction from about 0.02 ppm at 1 atm to about 4.3 ppm at 4 atm. This induces modification in the temperature distribution (see Fig. 1e) which, explains the observed differences. The differences are more pronounced in temperature than in mixture fraction contours (see Figs. 1 d ande).

Table 2 quantifies the pressure effects on the flame geometry and flow residence time. The flame contour is defined by the isotherm of 1400 K. This value is expected to correspond to a threshold below which soot ceases to be oxidized [START_REF] Markstein | Radiant emission and absorption by laminar ethylene and propylene diffusion flames[END_REF][START_REF] Kent | A quantitative relationship between soot yield and smoke point measurements[END_REF], being then an indicator of the "luminous" flame contour. The mean flame height, flame volume, and flow residence time (Term I in Eq. 1) are 0.688 m ± 0.003 m (±0.5%), 3.2×10 -4 m 3 ± 1.4×10 -5 m 3 (±4.6%), and 490 ms ± 22 ms (±4.5%), respectively. These results indicate that the Froude modelling preserves the flame geometry and the flow residence time within 5% as the pressure is increased from 1 to 4 atm.

Effects of pressure on radiant fraction

Table 3 shows that the contribution of soot to total emission increases significantly as the pressure increases. Soot emission is negligible at 1 atm and contributes to about 15 % and 33% of total emission at 3 atm and 4 atm, respectively. Nevertheless, it should be pointed out that soot emission dominates locally in regions of high soot volume fractions in the 4-atm flame, where soot volume fraction reaches about 4 ppm.

On the other hand, the radiant fraction increases moderately with pressure, being enhanced by only 2% as the pressure rises from 1 atm to 3 atm and by about 20% as the pressure is further increased to 4 atm.

This section aims to understand the pressure effects on the radiant fraction. As discussed previously, Term I in Eq. 2 is preserved by the Froude modelling and the effects of pressure on the radiant fraction are due to modifications to Terms II and III.

𝑄 ̇𝑒𝑚,𝐺 ′′′ is expected to increase linearly with P since, on one hand, the Planck-mean absorption coefficient of radiating gases scales with P [START_REF] Modest | Radiative Heat Transfer[END_REF] and, on the other hand, the temperature, species mole fractions, and turbulent intensity are on the whole preserved by the Froude modelling.

Moreover, 𝜌 𝐹 ∝ 𝑃 and consequently, the term 𝑄 ̇𝑒𝑚,𝐺 ′′′ 𝜌 𝐹 ∆ℎ 𝑐 ⁄ is expected to be independent of P. This behavior is illustrated in the second column of Table 4 and in Fig. 3a where

𝑄 ̇𝑒𝑚,𝐺 ′′′ 𝜌 𝐹 ∆ℎ 𝑐 ⁄
is observed to remain constant as P increases from 1 to 3 atm and to increase by about 8% as the pressure increases to 4 atm. The flame structure is not fully-preserved by the Froude modeling as the pressure increases to 4 atm, explaining these small variations.

In an opposite way, Fig. 3a

Effects of pressure on flame transparency

The ratio of total absorption, 𝑄 ̇𝑎𝑏𝑠 , to total emission, 𝑄 ̇𝑒𝑚 , is a measure of the flame selfabsorption. The second column of Table 5 indicates that, as expected, 𝑄 ̇𝑎𝑏𝑠 𝑄 ̇𝑒𝑚 ⁄ is enhanced as the pressure raises, indicating that the rate of increase in 𝑄 ̇𝑎𝑏𝑠 with increasing P is higher than that of 𝑄 ̇𝑒𝑚 . As a consequence, the flame transparency (Term III in Eq. 1) is reduced by pressure rise (see the third column of Table 5). The fourth and fifth columns in Table 5 quantify the contributions of soot and radiating gases to the flame self-absorption, respectively. It appears clearly that both contributions tend to enhance the flame self-absorption as the pressure increases and the rate of increase in the soot contribution is significantly higher owing to the strong increase in soot production. The last three columns of Table 5 show the contribution of radiating gases and soot to the total absorption and total emission. It can be observed that soot is comparatively more transparent than the radiating gases. This conclusion is consistent with that reported in Refs. [START_REF] Mehta | Radiation characteristics and turbulence-radiation interactions in sooting turbulent jet flames[END_REF][START_REF] Rodrigues | Analysis of radiative transfer in a turbulent sooting jet flame using a Monte Carlo method coupled to large eddy simulation[END_REF].

Effects of pressure on TRI

TRI affects the radiative loss mainly through the emission TRI. The gas and soot emission TRI terms can be developed as [START_REF] Coelho | Numerical simulation of the interaction between turbulence and radiation in reactive flows[END_REF]: [START_REF] Mccrain | Measurements of the soot volume field in laminar diffusion flames at elevated pressures[END_REF] TRI contributes to emission through the last three terms in Eq. ( 7). Terms A, B and C represent the effects of the Planck-mean absorption coefficient self-correlation, the effects of the temperature self-correlation, and the effects of the cross-correlation between the Planck-mean absorption coefficient and temperature, respectively.

〈𝜅 𝑝 𝐼 𝑏 〉 ≈
In the case of radiating gases, the three contributions tend to enhance emission [START_REF] Li | Importance of turbulence-radiation interactions in turbulent diffusion je flames[END_REF]. This is illustrated in Table 6 where considering TRI enhances gas emission. Table 6 shows also that this enhancement increases with increasing the pressure which is in accordance with the conclusions drawn previously in Ref. [START_REF] Nmira | Pressure effects on radiative heat transfer in hydrogen/air turbulent diffusion flames[END_REF] for non-luminous flames.

In the case of soot, terms A and C evolve as the cross-correlation between temperature and soot volume fraction which was observed to be negative in the region of strong soot emission in both experimental [START_REF] Sivathanu | Transient scalar properties of strongly radiating jet flames[END_REF][START_REF] Sivathanu | Transient structure and radiation properties of strongly radiating buoyant flames[END_REF][START_REF] Murphy | Soot properties and species measurements in a two-meter diameter JP-8 pool fire[END_REF][START_REF] Zeng | The effect of oxygen depletion on soot and thermal radiation in buoyant turbulent diffusion flames[END_REF] and numerical [START_REF] Consalvi | Effects of soot absorption coefficient-Planck function correlation on radiative heat transfer in oxygen-enriched propane turbulent diffusion flame[END_REF][START_REF] Rodrigues | Analysis of radiative transfer in a turbulent sooting jet flame using a Monte Carlo method coupled to large eddy simulation[END_REF][START_REF] Kollmann | Application of a soot model to a turbulent ethylene diffusion flame[END_REF][START_REF] Nmira | Soot emission radiation-turbulence interactions in diffusion jet flames[END_REF] investigations of turbulent diffusion flames. This negative behavior is illustrated in Fig. 4 for the present flames. As a consequence, these two terms tend to reduce soot emission [START_REF] Consalvi | Effects of soot absorption coefficient-Planck function correlation on radiative heat transfer in oxygen-enriched propane turbulent diffusion flame[END_REF][START_REF] Rodrigues | Analysis of radiative transfer in a turbulent sooting jet flame using a Monte Carlo method coupled to large eddy simulation[END_REF][START_REF] Nmira | Soot emission radiation-turbulence interactions in diffusion jet flames[END_REF]. On the other hand, term B is always positive and tends to enhance soot emission. Table 7 shows that taking TRI into account slightly reduces the total soot emission, indicating that, for the present flames, terms A and C prevail over term B. Term B is expected to increase with pressure due to an increase in soot volume fraction and consequently in 〈𝜅 𝑝 〉. On the other hand, terms A and C behave as 𝑓 𝑆 ′ 𝑇 ′ ̃= 𝐶 𝑓 𝑆 𝑇 √ 𝑓 𝑆 ′2 ̃√𝑇 ′2 ̃. Figure 4 shows that, in the region of soot emission, 𝐶 𝑓 𝑆 𝑇 reaches similar levels for all the flames and, as a consequence, terms A and C are expected to become more and more negative as the pressure is enhanced owing to an increase in the fluctuation of the soot volume fraction ( √ 𝑓 𝑆 ′2 ̃). Table 7 shows that the reduction in soot emission TRI decreases slightly as the pressure increases. This suggests that term B increases with P at a slightly higher rate than that of the decrease of terms A and B.

As a consequence, TRI affects the total emission through a competition between the enhancement mechanisms, i.e., gas emission TRI and the effects of the temperature selfcorrelation on soot emission, and reducing mechanisms, i.e., the effects of terms A and C on soot emission. Soot emission plays an unimportant role in the 1-atm flame and then TRI increases significantly the total emission and consequently the net radiative loss (see Table 8).

As the pressure increases, the contribution of soot becomes higher. However, the enhancement in the total emission and net radiative loss is limited due to the inhibiting effects of terms A and B on soot emission (see Table 8).

Let us consider the local effects of TRI on the radial profiles of the divergence of radiative flux.

These effects are illustrated in Figs. 5 and6 for the 1-atm and 4-atm flames, respectively. In each figure, the radial profiles are displayed at three heights of 0.2 m, 0.4 m, and 0.6 m corresponding respectively to the lower part of the flame where soot growth dominates, the middle of the flame where the maximum soot volume fraction is encountered, and the upper part of the flame where soot is intensely oxidized (see Fig. 1). Soot radiation is negligible in the 1-atm flame and taking TRI into account enhances gas emission and, correspondingly, the divergence of radiative flux (see Fig. 5). In addition, the TRI effects are pronounced in regions where temperature fluctuations are high, i.e., near the flame wings at z = 0.2 m and 0.4 m and over the entire flame width in the upper part of the flame (z = 0.6 m). For the 4-atm flame, soot radiation becomes significant. Figure 6 shows that, in the regions where soot volume fractions are high, TRI tends to reduce the divergence of radiative flux. This means that in these regions the terms that tend to reduce emission, i.e., terms A and B in the soot emission term, dominate those that tend enhance it, i.e., the gas emission TRI and the temperature self-correlation effects on soot emission TRI. Conversely, Fig. 6 shows that, in regions where soot volume fractions are low considering TRI enhances the divergence of radiative flux. In these regions gas radiation is dominant and, as a consequence, the TRI mechanisms that tend to enhance emission and the divergence of radiative flux prevail over those that tend to suppress them.

Conclusions

The effects of pressure on radiative heat transfer and turbulence radiation interaction were investigated in CH4 turbulent jet diffusion flames at pressures up to 4 atm by using a wellvalidated numerical model. The following conclusions can be drawn from the results of the present study.

1) The Froude modelling preserves the flame structure in terms of flame geometry, global residence time, turbulent kinetic energy, its dissipation, turbulent mixing times, turbulent integral length scale, and turbulent fluctuations up to 4 atm. However, it becomes less accurate as the pressure further increases due to the strong increase in soot production.

2) The radiant fraction is determined by three mechanisms: I) a global residence time preserved by the Froude modeling, II) the inverse of an emission characteristic time scale, and III) a term characterizing the flame transparency that is reduced as the pressure increases. The contribution of gas emission to Mechanism II is independent of the pressure whereas the soot contribution increases significantly with pressure owing to strong enhancement in soot production. The radiant fraction in sooting flames evolves with pressure owing to a competition between Mechanisms II and III. For the present flames, the second mechanism (II) was found to dominate over the third (III), leading to an increase in the radiant fraction with increasing pressure.

3) The effects of TRI on radiative loss are governed by a competition between mechanisms that tend to enhance emission, i.e., gas emission TRI and temperature self-correlation effects on soot emission, and those that tend to reduce it, i.e., the negative correlation between the soot volume

Nomenclature

  𝐶 𝑓 𝑆 𝑇 cross-correlation between soot volume fraction and temperature [-] dF inner burner diameter [m] 𝐹 𝑟 Froude number [-] fS soot volume fraction [-] fS,EQ equivalent soot volume fraction [-] G incident radiation per unit wavenumber [W•m -1 ] ℎ enthalpy [J/kg] 𝐼 𝑏 spectrally-integrated blackbody intensity [W m -2 ] 𝐼 𝑏𝜂 blackbody intensity at wavenumber  [W m -1 ] 𝑘 turbulent kinetic energy [m 2 s -2 ] Lfl flame height [m] unit flame volume [W m -3 ] 𝑄 ̇𝑒𝑚,𝐺 ′′′ gas emission per unit flame volume [W m -3 ] 𝑄 ̇𝑒𝑚,𝑆 ′′′ soot emission per unit flame volume [W m -3 ] r radial coordinate [m] 𝑡 𝑟𝑒𝑠 residence time of the fuel in the flame envelop [s]

  mixture fraction [-]  wavenumber [cm -1 ] 𝜅 𝜂 absorption coefficient at wavenumber  [m -1 cm] 𝜅 𝑃 Planck-mean absorption coefficient [m -1 ] 𝜌 density [kg•m -3 ] 𝜌 𝑆 soot density [kg•m -3 ]

Figure 1

 1 Figure1shows the contours of mixture fraction, temperature, and soot volume fraction in the

Figure 2

 2 Figure2shows the pressure effects on the axial profiles of axial velocity, turbulent kinetic

From 2 1+𝐴 2

 22 fraction temperature that reduces soot emission. The former mechanisms dominate in the weakly sooting 1-atm flame. The latter mechanisms become increasingly important as the pressure increases which limits the increase in the global radiative loss due to TRI. In addition, model results show that, for the high-pressure flames, the TRI effects can reduce the local radiative loss in the high soot concentration regions of the flame. (B2) On the other hand, the flame intermittency is related to 𝑓 Eq. (B3), 𝜔 and 𝐴 = √ 𝑓 𝑆 ′′2 ̃𝑓𝑆 ⁄ are closely related through 𝜔 = 𝐴 . The Froude modelling is expected to preserve the turbulent flame structure and, as a result, the flame intermittency. This may explains the self-similarity of √ 𝑓 𝑆 ′′2 ̃𝑓𝑆 ⁄ observed in Fig. 2d.

Figure 1 .

 1 Figure 1. Fields of mixture fraction (a or d), temperature (b or d) and soot volume fraction (c

Figure 2 .

 2 Figure 2. Fields of mixture fraction (a or d), temperature (b or d) and soot volume fraction (c

Figure 3 .

 3 Figure 3. a) Evolution of 𝑄 ̇𝑒𝑚 ′′′ 𝜌 𝐹 ∆ℎ 𝑐 ⁄ as a function of pressure and b) 𝑄 ̇𝑒𝑚,𝑆 ′′′ as a function of

Figure 4 .

 4 Figure 4. Fields of the correlation between soot volume fraction and temperature and of soot

Figure 5 .

 5 Figure 5. Effects of pressure on the radial profiles of the divergence of radiative flux for the 1

Figure 6 .

 6 Figure 6. Effects of pressure on the radial profiles of the divergence of radiative flux for the 4

Figure 1 .Figure 2 .Figure 3 .Figure 4 .Figure 5 .Figure 6 .

 123456 Figure 1. Effect of pressure on the fields of mixture fraction (a or d), temperature (b or d) and

Table 1 .

 1 Fuel injection parameters.

	Pressure (atm) 𝑈 𝐹 (m/s) 𝑑 𝐹 (mm) Fr (-) 𝑄 ̇𝑐 (kW)
	1	133	2.5	5	21.3
	3	-	-	-	63.9
	4	-	-	-	85.2

Table 2 .

 2 Froude modelling: conservation of flame geometry and flow residence time.

	P (atm)	𝐿 𝑓𝑙 (m)	𝑉 𝑓𝑙 (m 3 )	𝑡 𝑟𝑒𝑠 (ms)
		T =1400K	T =1400K	T=1400 K
	1	0.687	3.09×10 -4	474
	3	0.693	3.37×10 -4	516
	4	0.686	3.14×10 -4	481

Table 3 .

 3 Effect of pressure on the contribution of soot to emission and radiant fraction.

	P (atm) 𝑄 ̇𝑒𝑚,𝑆 ′′′	𝑄 ̇𝑒𝑚 ′′′ ⁄	𝜒 𝑅 (-) (𝜒 𝑅,𝑃 𝜒 𝑅,𝑃=1 ⁄	)
	1	0.012	0,162 (1.00)
	3	0.14	0,165 (1.02)
	4	0.33	0,192 (1.20)

Table 4 .

 4 Effect of pressure on gas and soot emission.

	P (atm) 𝐶 𝑔𝑎𝑠 = 𝑄 ̇𝑒𝑚,𝐺 ′′′	⁄	𝜌 𝐹 ∆ℎ 𝑐	(s -1 )	𝐶 𝑠𝑜𝑜𝑡 = 𝑄 ̇𝑒𝑚,𝑆 ′′′	𝜌 𝐹 ∆ℎ 𝑐 ⁄	(s -1 )	Term II
		(𝐶 𝑔𝑎𝑠,𝑃 𝐶 𝑔𝑎𝑠,𝑃=1 ⁄	)		(𝐶 𝑠𝑜𝑜𝑡,𝑃 𝐶 𝑠𝑜𝑜𝑡,𝑃=1 ⁄	)	(II,P 𝐼𝐼 ,𝑃=1 ⁄	)
	1	0.524 (1.00)			0.006 (1.00)	0.53 (1.00)
	3	0.524 (1.00)			0.085 (13.90)	0.61 (1.15)
	4	0.566 (1.08)			0.24 (40.40)	0.82 (1.53)

Table 5 .

 5 Effect of pressure on flame optical thickness and flame transparency.

	P (atm)	𝜏 = 𝑄 ̇𝑎𝑏𝑠 ′′′ 𝑄 ̇𝑒𝑚 ′′′ ⁄ (τ,P ,𝑃=1 ⁄ )	Term III (III,P 𝐼𝐼𝐼 ,𝑃=1 ⁄	)	𝑄 ̇𝑎𝑏𝑠,𝐺 ′′′	𝑄 ̇𝑒𝑚 ′′′ ⁄	𝑄 ̇𝑎𝑏𝑠,𝑆 ′′′	⁄	𝑄 ̇𝑒𝑚 ′′′	𝑄 ̇𝑒𝑚,𝑆 ′′′	𝑄 ̇𝑒𝑚 ′′′ ⁄
	1	0.356 (1.00)	0.644 (1.00)	0.355	0		0.012
	3	0.475 (1.34) 0.525 (0.814)	0.461	0.013	0.14
	4	0.502 (1.41) 0.498 (0.773)	0.428	0.072	0.33

Table 6 .

 6 Effect of pressure on gas emission TRI.

	P (atm)	(𝑄 ̇𝑒𝑚,𝐺 ) 𝑇𝑅𝐼 (kW)	(𝑄 ̇𝑒𝑚,𝐺 ) 𝑁𝑜𝑇𝑅𝐼 (kW)	∆𝑇𝑅𝐼 (%)
	1	5.29	4.12	22.00
	3	17.20	13.4	22.20
	4	23.10	16.6	28.84

Table 7 .

 7 Effect of pressure on soot emission TRI.

	P (atm)	(𝑄 ̇𝑒𝑚,𝑆 ) 𝑇𝑅𝐼	(𝑄 ̇𝑒𝑚,𝑆 ) 𝑁𝑜𝑇𝑅𝐼	∆𝑇𝑅𝐼
		(kW)	(kW)	(%)
	1	0.082	0,086	-4.51
	3	2.81	2.88	-2.57
	4	10.1	10.2	-0.99

Table 8 .

 8 Effect of pressure and TRI on the net radiative loss, 𝑄 ̇𝑛𝑒𝑡 = 𝑄 ̇𝑒𝑚 -𝑄 ̇𝑎𝑏𝑠 .

	P (atm)	(𝑄 ̇𝑛𝑒𝑡 ) 𝑇𝑅𝐼 (kW)	(𝑄 ̇𝑛𝑒𝑡 ) 𝑁𝑜𝑇𝑅𝐼 (kW)	∆𝑇𝑅𝐼 (%)
	1	3.45	2.61	24,10
	3	10.50	8.45	19.50
	4	16.60	13.5	18.20

Appendix A. Froude modeling

This appendix discusses how the flow properties evolve with pressure in Froude modelling.

The analysis is based on the assumption that the flames considered in the present study are in the momentum-driven regime. This simplifies the analysis since the characteristic flow length, LC, and velocity, UC, scales are the nozzle diameter (see Eq. 2) and the fuel injection velocity, 𝑢 𝐹 :

The Froude modeling preserves these two quantities as the pressure is scaled up. The characteristic flow time scale is then also independent of pressure:

The turbulent kinetic energy, dissipation of turbulent kinetic energy, turbulent mixing time and integral length scale are also independent of pressure:

The flame width can be estimated by assuming that the time scale for fuel convection in the axial direction is of the same order of magnitude as the turbulent diffusion time scale of fuel in the radial direction:

Equations A1, A2, A4 and A5 show that 𝑊 𝑓𝑙 is preserved by the Froude modelling. As a consequence, the flame volume, 𝑉 𝑓𝑙 ∝ 𝐿 𝑓𝑙 𝑊 𝑓𝑙 2 , and the flow residence time (Term I in Eq. 1),

scale with 𝑃 0 .

The maximum soot volume fraction scales as [START_REF] Ris | Fire radiation modelling[END_REF]:

where 𝑡 𝑆 and 𝜔̇𝑆 𝐹 ′′′ are the flow time for soot production and the soot formation rate, respectively. In turbulent flames, several studies suggested that the Kolmogorov microscale time, 𝑡 𝐾𝑜𝑙 = √𝜇 𝑓𝑙 𝐿 𝑓𝑙 𝜌 𝑓𝑙 𝑢 𝐹 3 ⁄ ∝ 𝑃 -1/2 is the flow time for soot formation [START_REF] Kollmann | Application of a soot model to a turbulent ethylene diffusion flame[END_REF]. On the other hand, soot formation and oxidation rates are enhanced as P increases, leading to 𝜔̇𝑆 𝐹 ′′′ ∝ 𝑃 𝑛 with n > 0 and ranging typically from 1 to 3 [START_REF] Karataş | Soot formation in high pressure laminar diffusion flames[END_REF]. As a consequence, 𝑓 𝑆,𝑚𝑎𝑥 ∝ 𝑃 𝑛-1/2 and soot production is not preserved by the Froude modelling.

Appendix B.

Due to flame intermittency, the PDF of soot volume fraction is expected to behave as a double- distribution with a ''nonsooting'' mode and a ''sooting'' mode [START_REF] Mueller | Large eddy simulation subfilter modeling of soot-turbulence interactions[END_REF][45]:

where 𝜔 is the soot intermittency and 𝑓 𝑆 * is the soot volume fraction in the sooting mode.

𝑓 𝑆 * can be related to the mean soot volume fraction and the soot intermittency as:


R Hackert

L D Maes

M Herbin

P A Libourel

A Abourachid

LIMB PREFERENCE IN THE GALLOP OF DOGS AND THE HALF-BOUND OF PIKAS ON FLAT GROUND

Keywords: handedness, quadruped, locomotion, laterality, asymmetrical gaits, Carnivora, Lagomorpha, stability

During fast locomotion -gallop, half bound -of quadruped mammals, the ground contact of the limbs in each pair do not alternate symmetrically. Animals using such asymmetrical gait thus choose whether the left or the right limb will contact the ground first, and this gives rise to limb preference. Here, we report that dogs (Mammalia, Carnivora) and pikas (Mammalia, Lagomorpha) prefer one forelimb as trailing limb and use it as such almost twice as often as the other. We also show that this choice depends on the individual and is not a characteristic of the species, and that the strength of the preference was not dependent on the animal's running speed.

INTRODUCTION

The expression of brain lateralization has been observed in behaviours as different as feeding, manipulation, and communication in several families of vertebrates. Limb preference is a subset of this expression and has been documented in birds and in most families of tetrapods including toads, anurans and mammals (for review in [START_REF] Vallortigara | Possible evolutionary origins of cognitive brain lateralization[END_REF]. In some species, sex was identified as a determinant of the direction of the preference at population level (in dogs [START_REF] Wells | Lateralised behaviour in the domestic dog, Canis familiaris[END_REF][START_REF] Quaranta | Paw preference in dogs : relations between lateralised behaviour and immunity[END_REF][START_REF] Poyser | Dog paw preference shows lability and sex differences[END_REF]in cats: Tan et al., 1990) but not of the strength of preference. The strength of preference is thought to be task dependent and increases as tasks require more cognitive function [START_REF] Fagot | Manual laterality in nonhuman primates: a distinction between handedness and manual specialization[END_REF] and coordination [START_REF] Hopkins | Hand preferences for a coordinated bimanual task in 110 chimpanzees: Cross-sectional analysis[END_REF].

Curiously, limb preference during locomotion has not been extensively explored, although locomotion is the primary function of limbs. [START_REF] Malashichev | One-Sided Limb Preference Is Linked to Alternating-Limb Locomotion in Anuran Amphibians[END_REF] observed five species of anuran and found a limb preference in those that exhibit an "alternated limb locomotion or other unilateral limb activity". He suggested that the degree of lateralization in the motor response depends on the mode of locomotion used by a species. In the asymmetrical gaits of mammals-i.e. gallop, bound, and half-bound -the fore and hind pairs of legs make contact with the ground alternately. During gallops or half-bounds, the motions of the two limbs of a pair are not symmetrically alternated, such that one limb of each pair touches the ground first and is called the trailing limb, the other one is called leading limb. The trailing and leading limb of each pair cannot be systematically assigned to the left or right limb because during locomotion animals switch trailing and leading limbs from time to time. In their study of the gallop of four race horses, [START_REF] Deuel | Laterality in the gallop gait of horses[END_REF] compared the kinematics of trailing and leading limbs. All the horses preferred the right limb as trailing limb. Moreover, the duration of the stance phase of the trailing forelimb differed significantly according to whether it was the right or the left limb. [START_REF] Walter | Ground forces applied by galloping dogs[END_REF] observed that four of the six dogs in their study preferred one limb as the trailing limb during gallop. Finally, [START_REF] Hook | Leading-limb preference in marmosets (Callithrix jacchus): Walking, leaping and landing?[END_REF] found that marmosets land preferentially with the right limb and that the choice of this limb is correlated with the leading limb used during leaping and walking.

There have been few quantitative studies of the direction or the strength of limb preference during animal locomotion. We therefore investigated this issue and changes in laterality with speed in dogs and pikas, two mammals with different morphologies, behaviours and gaits. Dogs are medium-sized digitigrades that trot over a large range of speeds before switching to galloping at high speeds. In contrast, pikas are small plantigrades that switch to half-bound locomotion for all speeds faster than walking [START_REF] Gambaryan | How mammals run: anatomical adaptations[END_REF][START_REF] Fischer | Application of cineradiography for the metric and kinematic study of in-phase gait during locomotion of the pika (Ochotona rufescens, Mammalia, Lagomorpha[END_REF].

ANIMALS AND METHODS

We studied five dogs (Canis familiaris, Mammalia: Carnivora) and four pikas (Ochotona rufescens, Mammalia: Lagomorpha). The dogs were male Belgian shepherd malinois of similar size (withers height=0.61±0.05 m, BW=28.3±2.0 kg); they belonged to the French army and were thus healthy, obedient, and used to energetic exercise. Pikas (withers height=0.05 m, body length=0.16 m, BW=150-200 g) were reared at the IPBS (CNRS, Toulouse) by A. Puget and were kept at the Motion Lab, University of Iena, Germany; they were reared and used in conformity with German animal welfare regulations.

Methods

Dogs were studied while running outdoors along a 1 m x 12 m long carpet graduated each 10 cm for stride length calculations. They were filmed with a high speed camera (Basler A504K, highland Illinois, USA) positioned 8 m from the left-hand side of the carpet and driven at 200 Hz. The camera was also driven to maintain the dog in the field of view (about 3 m). Under outdoor conditions -in contrast to treadmill -the animal's speed is not controlled and the collected locomotor cycles depend largely on the performance and motivation of the dog. The dog handler ran the distance at five different speeds to allow the dogs to gallop at different speeds. Dogs were all lead on a leash by the handler on the right-hand side. The two meters long leash was light and long enough to avoid interfering with the dog's locomotion. To study speeds faster than human running speed in a second series of records, dogs were also encouraged to run freely towards a ball at the opposite end of the same carpet. We used this second series of records to test whether the handler had an influence on the direction of the preference. The direction of preference for each dog was the same in trials with and without the leash. Trials where the leash was bent or where the animal looked at the handler (very rare because galloping quadruped mammals lock the degree of freedom of the neck) were excluded from our data set. Animal speed was calculated from stride length and stride duration. Speeds were between 2.2 m/s and 10.3 m/s and each sequence contained 4-8 locomotor cycles according to speed. However, each dog did not cover exactly the same range of speeds.

3 Pikas ran on a treadmill and running speed was imposed on the animals. They were habituated to the manipulation and to the running environment for 15 minutes per day for two weeks. High speed video films were recorded with a CAMSYS camera driven at 400 Hz and equipped with a Fujinon zoom-lens (1.2/75) adjusted to cover one body length. Sequences of 4s were recorded at speeds between 1.2 m/s and 2.2 m/s, a range of speeds comfortably achieved by the four animals. The speed was increased by 0.05 m/s between consecutive sequences. Each sequence included between 15 and 30 locomotor cycles. Animal were allowed rest for 3 minutes between the running sequences.

Processing films and statistics

We noted the limb used as the trailing limb in each locomotor cycle recorded on each film sequence; films were each scored twice independently and readings compared. There was no disagreement associated with the determination of the trailing limb.

However, assuming that the left/right distribution converges on a particular value as the number of cycles analysed increased, the truncation of data acquisition after any given number of cycles would limit accuracy.

The power analysis module of statistical software (STATISTICA v.6, Statsoft) was used to determine the accuracy according to the sample size.

We grouped our data into three classes of speed: slow (S), medium (M) and fast (F) gallop (or half-bound). As the range of speed covered depended on the motivation of the dogs to perform the exercise, speed classes were adapted to the performance of each dog by dividing the range of speed into three equal intervals for each animal; thus, the classes differed between individuals. The limits of the classes are given in Fig. 1. For pikas, the three classes each had the same width [1.2-1.5] m/s, [1.5-1.8] m/s and [1.8-2.2] m/s; these classes have biological relevance as pikas introduce a first aerial phase at about 1.5-1.6 m/s and then a second at around 1.8-1.9 m/s [START_REF] Fischer | Application of cineradiography for the metric and kinematic study of in-phase gait during locomotion of the pika (Ochotona rufescens, Mammalia, Lagomorpha[END_REF][START_REF] Hackert | Dynamics of the pikas' half-Bound -Spinal flexion contributes to dynamic stability[END_REF].

The significance of our binary (right/left) data was tested for each speed class using a Chi 2 test (1 degree of freedom) to detect a significant difference (p<0.05) from a 50:50 distribution. Classes were then compared two by two using a Chi 2 test (p<0.05, H 0 = "distributions are different").

A handedness index (HI) was calculated by subtracting the total number of left-trailing touch-downs (L) from the total number of the right-trailing touch-downs (R) divided by the total number of trailing touch-downs: (R -L)/(R + L). The resulting values, ranging from -1.0 to 1.0, provide a score for each animal's hand preference on a continuum from strongly left-handed to strongly right-handed. The absolute value of the HI (ABS-HI) indicates the strength of hand preference, irrespective of its direction (see [START_REF] Hopkins | Hand preferences for a coordinated bimanual task in 110 chimpanzees: Cross-sectional analysis[END_REF].

RESULTS

Left-and right-handed individuals were found among dogs and pikas. In each speed class the 10 animals significantly preferred (p<0.05) one limb, except pika 4 and the slow classes of pika 1 and of dog 3 (fig. 1).

The direction of preference remained the same in the three speed classes for each animal. Comparing the strength of limb preference of speed classes two by two, significant (p<0.05) differences were observed only between the slow class of pika 1 and the medium and fast classes, and between the fast class of pika 2 and the slow and medium classes. Thus, we observed a constant strength of limb preference in 8 of 10 animals, including both dogs and pikas. The strength of preference varied between individuals and was generally stronger in dogs (ABS-HI from 0.31 to 1, mean ABS-HI= 0.65) than in pikas (ABS-HI from 0 to 0.46, mean ABS-HI= 0.3).

DISCUSSION

Lateralized behaviour in dogs has been described [START_REF] Tan | Paw preferences in dogs[END_REF][START_REF] Wells | Lateralised behaviour in the domestic dog, Canis familiaris[END_REF][START_REF] Quaranta | Paw preference in dogs : relations between lateralised behaviour and immunity[END_REF][START_REF] Van Alphen | Paw Preference Correlates to Task Performance in Dogs[END_REF][START_REF] Poyser | Dog paw preference shows lability and sex differences[END_REF][START_REF] Branson | Relationship between paw preference strength and noise phobia in Canis familiaris[END_REF] but not as concerns locomotion. Our study shows that paw preference is present during asymmetrical locomotion in two quadrupeds: dogs and, for the first time, a lagomorphian.

Previous studies report an influence of sex on the direction of the lateralization at the population level in dogs [START_REF] Quaranta | Paw preference in dogs : relations between lateralised behaviour and immunity[END_REF] but not on the strength of the lateralization [START_REF] Wells | Lateralised behaviour in the domestic dog, Canis familiaris[END_REF]. Our group only included adult males and was equally divided into right-and left-pawed dogs, but the small number of subjects studied prevents conclusions at the population level.

The source of asymmetry in gallop and half-bound locomotion corresponds to a functional differentiation between the trailing and leading limbs of a pair. Indeed, [START_REF] Walter | Ground forces applied by galloping dogs[END_REF] found significant differences in the peak vertical forces of galloping dogs in the hind but not in the fore pair, a larger decelerating impulse in the leading than trailing forelimb and a more accelerating impulse in the leading than trailing hind limb. Hence, the specific exercises of each limb may lead to a specific muscular fatigue. To minimize this muscular fatigue, animals often switch between trailing and leading limbs when galloping and half-bounding. The strength of preference observed thus depends on the number of switches and the number of consecutive cycles recorded in a locomotor sequence. In our study of dogs, sequences included 4-8 consecutive locomotor cycles; for pikas locomotor sequences involved 15-30 consecutive cycles, potentially giving rise to more fatigue and consequently more switches. This methodological difference may have contributed to the lower strength of preference observed in pikas.

Ecological factors certainly influence the strength of preference during gallop in a species. Using the data published by [START_REF] Deuel | Laterality in the gallop gait of horses[END_REF] concerning a group of four ridden race horses, all left footed (15 cycles recorded per horses), we calculated a mean HI of 0.32 (HI from 0.12 to 0.50). This is slightly higher than the value we obtained for pikas but lower than that for dogs. This indicates that the strength of preference does not correlate strongly with animal size. Possibly, the type of locomotion influences the strength of lateralization: pikas, a model for many small mammals, do not plan their trajectory and therefore discover the relief at the last moment. Manoeuvrability, i.e. the ability to suddenly turn left and right, and reactivity are most important for prey species like pikas. In contrast, the locomotion of dogs and other carnivores is goal directed because they gallop when pursuing prey; their locomotion is more cognitive. These two types of locomotion correspond to two different tasks -escaping or hunting -and define a gradient of cognitive involvement [START_REF] Fagot | Manual laterality in nonhuman primates: a distinction between handedness and manual specialization[END_REF]. On this gradient, horses like pikas are prey, but with a much larger field of view and so there is a larger cognitive input. Indeed, comparing species involves defining more precisely the types of locomotion they use, and thus the types of task they perform.

In their study of manual laterality in non human primates, [START_REF] Fagot | Manual laterality in nonhuman primates: a distinction between handedness and manual specialization[END_REF] developed a framework to classify tasks. They defined a gradient for the strength of lateralization according to the complexity of the task. Complexity is here defined in terms of spatiotemporal precision of the movement and novelty of the situation (inducing cognitive functions). They thus opposed complex high level tasks to low level tasks including familiar practised activities that should require less cognitive involvement.

Different tasks have been given to dogs to explore the strength of their paw preference: for example retrieving a chocolate from a can (mean ABS-HI=0.39; [START_REF] Wells | Lateralised behaviour in the domestic dog, Canis familiaris[END_REF], and holding a hollow cylinder filled with food (Kong test; [START_REF] Branson | Relationship between paw preference strength and noise phobia in Canis familiaris[END_REF]. In the latter experiment the ABS-HI values were between 0 and 0.8, with most falling between 0.1 and 0.5; we estimate that the mean ABS-HI value was around 0.35-0.40, a value in accordance with the 0.39 found by Wells for the other task. The slightly lower value found for removing an adhesive tape from the nose (mean ABS-HI about 0.30; [START_REF] Quaranta | Paw preference in dogs : relations between lateralised behaviour and immunity[END_REF]) appears natural since the movement is directed towards a body part -the nose -and is thus more stereotyped i.e. familiar comparable in some extent to cleaning movement. In contrast, removing a blanket from head required handhead coordination, and is thus a more complex task and this HI was found higher (mean ABS-HI=0.5; [START_REF] Wells | Lateralised behaviour in the domestic dog, Canis familiaris[END_REF]. The mean ABS-HI values for all these idiomotive tasks falls between 0.3 and 0.5 despite large variations at the individual level and notable differences in the composition of the groups of animals. For paw lifting on command, Wells found, by contrast, a very high degree of lateralization (mean ABS-HI=0.8) but this may have been due to learning reinforcement effects affecting performance. Thus, asymmetrical locomotion in dogs (mean ABS-HI=0.65) seems to be subject to a strong preference, although there are methodological issues to be considered when interpreting this finding.

Where does asymmetrical locomotion fall on the gradient defined by Fagot and Vauclair? Cyclic locomotion on flat terrain is by nature the most stereotyped and practised movement and consequently belongs, a priori, to the class of low level tasks. However, galloping stably sets constraints on the animal's motion: touching down at the appropriate instant with the trailing limb improves the dynamic stability of running [START_REF] Seyfarth | Swing-leg retraction: a simple control model for stable running[END_REF][START_REF] Hackert | Dynamics of the pikas' half-Bound -Spinal flexion contributes to dynamic stability[END_REF][START_REF] Fischer | The tri-segmented limbs of therian mammals: kinematics, dynamics, self-stabilizationa review[END_REF]. The instant of touch-down of the limbs is tuned precisely by the so-called pre-stance retraction of the trailing limb, i.e. the movement of rotation of the trailing limb toward the ground before contact with it. This movement of rotation is a general characteristic of the kinematics of bouncing and running: during the last instants of the swing phase, the shoulder blade initiates a back rotation of the forelimb in many quadruped mammals [START_REF] Fischer | Basic limb kinematics of small therian mammals[END_REF]. Because of the requirement for precision in the motion of the single limb, asymmetrical locomotion could be considered to be a high level task.

Moreover, spatiotemporal coordination between the limbs of the pair (and between the pairs) makes asymmetrical locomotion closer to bimanual manipulative tasks than unimanual task. The mean values of ABS-HI in non human primates during bimanual manipulation are higher (ABS-HI=0.80 in a male group of capuchin monkeys, Cebus apella, and 0.76 in a mixed group of white-faced capuchin monkeys, Cebus capucinus) than during unimanual task, for example a reaching movement [mean ABS-HI=0.4 in [START_REF] Spinozzi | Hand preferences in unimanual and coordinated bimanual tasks by tufted capuchin monkeys (Cebus apella)[END_REF] and 0.31 in [START_REF] Meunier | Hand preferences on unimanual and bimanual tasks in white-faced capuchins (Cebus capucinus)[END_REF]; note that the value for reaching movements in primates and dogs are comparable]. Involving the familiar and repetitive aspects of galloping but also a requirement for spatiotemporal precision and limb coordination, asymmetrical locomotion comes between low and high level tasks as defined by [START_REF] Fagot | Manual laterality in nonhuman primates: a distinction between handedness and manual specialization[END_REF]. Moreover, although a dynamic stable gallop requires a threshold speed, an increase of speed only weakly enhances dynamic stability [START_REF] Seyfarth | A movement criterion for running[END_REF]. In Bernstein's view, a fast gallop is as complex as a slow gallop, at least in the context of advancing in a straight line on even terrain, because the number of degrees of freedom of the musculo-skeletal system to be controlled is the same [START_REF] Bernstein | Some emergent problems of the regulation of motor acts: In Human motor actions[END_REF]. Consistent with this view, we found that the strengths of preference were independent of speed during galloping.

Asymmetrical gaits are a quasi exclusivity of mammals. The comparison of the morphology of the oldest mammalian fossils with the morphology of extant small mammals indicates that the potential for such asymmetrical locomotion was already present in early mammals [START_REF] Luo | The earliest eutherian mammal[END_REF]. The introduction of asymmetrical locomotion with a shift between forelimbs (gallop, half-bound) minimized the vertical displacement of the centre of mass and thus led to more economic locomotion [START_REF] Hackert | Dynamics of the pikas' half-Bound -Spinal flexion contributes to dynamic stability[END_REF]. Moreover, the functional differentiation between trailing and leading limb during asymmetrical locomotion gives rises to a limb preference that may also reduce energetic cost. Indeed, torques that act in the preferred arm during manipulative movements in humans are less than those in the other arm, as both arms performed the same task by using the same trajectory [START_REF] Bagesteiro | Handedness: Dominant Arm Advantages in Control of Limb Dynamics[END_REF]. These observations lead us to suggest that limb preference was present in the locomotion of early mammalian species. The ability to perform precise and/or powerful manipulations may have sustained the lateralization of brain organization through evolution [START_REF] Vallortigara | Possible evolutionary origins of cognitive brain lateralization[END_REF][START_REF] Rogers | Comparative vertebrate lateralization: The evolution of brain lateralization[END_REF][START_REF] Vallortigara | How ancient is brain lateralization? In Comparative vertebrate lateralization The evolution of brain lateralization[END_REF][START_REF] Bradshaw | The Evolution of lateral asymmetries, Language, Tool Use and Intellect[END_REF], so it is plausible that the benefits of an asymmetrical gait for efficient locomotion may also have sustained lateralization.

Acknowledgments

Observations of pikas were funded by the DFG (Deutsche Forschungsgemeinschaft) at the college of innovation "Motion Systems" at the Institute of Systematic Zoology of the University of Iena. Observations of dogs were funded by the French ANR (Agence Nationale pour la Recherche) as part of the LOCOMO project (ANR 06 BLAN 0132-02). We are indebted to Dr. Alain Puget (IPBS), Toulouse, for providing us with pikas and to the 132e BCAT of the French Army for its collaboration in the study; in particular we thank Lt.-Cl.

Deuwel, Cdt Lamour, Cpl.-Ch. Séné, Grardel, Debiasi and Lievens, 1 st Cl. Noroy, Baron and Baud.