Olivier Gicquel

J Taine

Y F Zhang

R Vicquelin
email: ronan.vicquelin@ecp.fr

Physical study of radiation effects on the boundary layer structure in a turbulent channel flow

Keywords: DNS, turbulence, gas radiation, Monte-Carlo, channel flow, temperature, wall-law Spectral quantity. Abbrevation DNS Direct Numerical Simulation. LES Large Eddy Simulation. RANS Reynolds-Averaged Navier-Stokes equations. ERM Emission Reciprocity

come

Roman Symbols c p

Thermal capacity at constant pressure [J•kg -1

Superscript

Statistical estimation or Favre mean quantities. ¯Reynolds mean quantities.

Favre mean fluctuating quantities.

+ Wall scaled quantities.

exch Exchanged quantities.

Introduction

In many industrial systems such as those dealing with combustion, conductive heat fluxes and radiative energy fluxes at walls condition the design stage and the material choice. Predicting these different fluxes with numerical simulations is therefore a great challenge that has been investigated in many works. From the fundamental understanding of boundary layers and radiation energy transfer, models have been proposed to calculate this quantities. Although improving the prediction of heat fluxes remains an active area in each field, the determination of radiative fluxes and conductive fluxes at walls is always done separately without considering any coupling between these two modes of energy transfer. In this paper, the coupling of radiation and turbulent forced convection is studied to investigate whether the wall heat flux and the boundary layer structure can be modified by radiative energy transfer.

The importance of Turbulence-Radiation Interaction (TRI) has been highlighted in several works. Experimental studies [START_REF] Gore | Spectral and total radiation properties of turbulent carbon monoxide/air diffusion flames[END_REF][START_REF] Ji | Thermal radiation properties of turbulent lean premixed methane air flames[END_REF][START_REF] Zheng | Measurements and inverse calculations of spectral radiation intensities of a turbulent ethylene/air jet flame[END_REF], theoretical analysis [START_REF] Burns | Turbulence radiation interaction modeling in hydrocarbon pool fire simulations[END_REF][START_REF] Baum | Radiation and velocity fields induced by localized temperature fluctuations[END_REF][START_REF] Soufiani | Radiation effects on turbulent heat transfer in channel flows of infrared active gases[END_REF] and numerical simulations [START_REF] Wu | Direct numerical simulation of turbulence/radiation interaction in premixed combustion systems[END_REF][START_REF] Deshmukh | Direct numerical simulation of turbulence-radiation interactions in homogeneous nonpremixed combustion systems[END_REF][START_REF] Deshmukh | Direct numerical simulation of turbulence-radiation interactions in a statistically onedimensional nonpremixed system[END_REF][START_REF] Santos | Coupled large eddy simulations of turbulent combustion and radiative heat transfer[END_REF] have been carried out to investigate the effect of TRI in different systems and a comprehensive review is available in [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]. TRI is a consequence of the highly non-linear coupling between the radiative intensity and the turbulent temperature and gas species composition fields. It consists of two parts, namely the influence of turbulence on radiation and vice-versa. Regarding the former, turbulence leads to an increase in the medium transmissivity [START_REF] Gore | Spectral and total radiation properties of turbulent carbon monoxide/air diffusion flames[END_REF][START_REF] Jeng | Radiative heat fluxes near turbulent buoyant methane diffusion flames[END_REF], the radiative power [START_REF] Coelho | Detailed numerical simulation of radiative transfer in a nonluminous turbulent jet diffusion flame[END_REF][START_REF] Tessé | Monte Carlo modeling of radiative transfer in a turbulent sooty flame[END_REF] and radiative heat loss [START_REF] Tessé | Monte Carlo modeling of radiative transfer in a turbulent sooty flame[END_REF][START_REF] Li | Importance of turbulence radiation interactions in turbulent diffusion jet flames[END_REF]. In order to isolate and quantify individual contributions to TRI in a statistically one-dimensional premixed combustion system, Wu et al. [START_REF] Wu | Direct numerical simulation of turbulence/radiation interaction in premixed combustion systems[END_REF] have performed a direct numerical simulation (DNS) coupled with a Monte-Carlo method. The results reveal that the temperature self-correlation contribution is only dominant in the case with smallest optical thickness in their study. Among the three correlation terms of the absorption coefficient, the ones with the Planck function and with the incident radiative intensity are not negligible, even in the most optically thin case, while for a case with intermediate value of optical thickness, the three correlations were all significant. Deshmukh et al. [START_REF] Deshmukh | Direct numerical simulation of turbulence-radiation interactions in homogeneous nonpremixed combustion systems[END_REF][START_REF] Deshmukh | Direct numerical simulation of turbulence-radiation interactions in a statistically onedimensional nonpremixed system[END_REF] have also studied different contributions to TRI with DNS in a statistically homogeneous isotropic nonpremixed combustion system and a one-dimensional turbulent non-premixed flame. Only the latter one was fully coupled with radiation.

Regarding the effect of radiation on the flow, radiation interacts and modifies the temperature field in non-reactive flows [START_REF] Soufiani | Radiation effects on turbulent heat transfer in channel flows of infrared active gases[END_REF][START_REF] Gupta | Large-Eddy Simulation of Turbulence-Radiation Interactions in a Turbulent Planar Channel Flow[END_REF][START_REF] Ghosh | Effects of radiative heat transfer on the structure of turbulent supersonic channel flow[END_REF] and in reactive flows [START_REF] Santos | Coupled large eddy simulations of turbulent combustion and radiative heat transfer[END_REF][START_REF] Damien | Analysis of the interaction between turbulent combustion and thermal radiation using unsteady coupled LES/DOM simulations[END_REF]. In combustion applications, the change of maximum temperature has a significant effect on NO x emission [START_REF] Barlow | Scalar profiles and no formation in laminar opposed-flow partially premixed methane/air flames[END_REF]. Influence of radiation is not restricted to the average temperature field: The intensity of the temperature fluctuations changes when radiation is taken into account [START_REF] Damien | Analysis of the interaction between turbulent combustion and thermal radiation using unsteady coupled LES/DOM simulations[END_REF][START_REF] Soufiani | Temperature turbulence spectrum for high-temperature radiating gases[END_REF]; Goncalves et al [START_REF] Santos | Coupled large eddy simulations of turbulent combustion and radiative heat transfer[END_REF] have shown an impact of radiation on turbulent flame dynamics; Ghosh et al. [START_REF] Ghosh | Effects of radiative heat transfer on the structure of turbulent supersonic channel flow[END_REF] have studied the effects of radiation in a turbulent compressible channel flow and have shown that radiation modified the Reynolds stresses in the near wall layer even for an optically thin medium. Using a low-Reynolds RANS model, Soufiani et al [START_REF] Soufiani | Radiation effects on turbulent heat transfer in channel flows of infrared active gases[END_REF] have also carried out a channel flow simulation where the wall conductive heat flux has been found to be significantly affected by radiation while, in a similar case with DNS here, Amaya et al [START_REF] Amaya | Unsteady coupling of Navier-Stokes and radiative heat transfer solvers applied to an anisothermal multicomponent turbulent channel flow[END_REF] have found the effect of radiation in the boundary layer to be weak. Using a grey gas model and varying the medium optical thickness, Gupta et al [START_REF] Gupta | Large-Eddy Simulation of Turbulence-Radiation Interactions in a Turbulent Planar Channel Flow[END_REF] have shown that the temperature profile in a turbulent channel flow can be modified by radiation.

Previous studies on the impact of radiation in a channel flow configuration [START_REF] Soufiani | Radiation effects on turbulent heat transfer in channel flows of infrared active gases[END_REF][START_REF] Gupta | Large-Eddy Simulation of Turbulence-Radiation Interactions in a Turbulent Planar Channel Flow[END_REF][START_REF] Ghosh | Effects of radiative heat transfer on the structure of turbulent supersonic channel flow[END_REF][START_REF] Amaya | Unsteady coupling of Navier-Stokes and radiative heat transfer solvers applied to an anisothermal multicomponent turbulent channel flow[END_REF][START_REF] Soufiani | Application of statistical narrow-band model to coupled radiation and convection at high temperature[END_REF] do not expose a general trend or understanding of the influence of radiation in turbulent boundary layers. In LES and RANS, wall models are necessary to predict wall heat flux in turbulent boundary layers. Among these models, standard wall laws are still widely used although they are only valid for relatively simple flows with zero-pressure gradient and constant fluid properties. In real applications with more complex conditions, standard wall laws might have a great inaccuracy. In order to extend their usage to practical systems, other physical effects need to be considered such as compressibility [START_REF] Foysi | Compressibility effects and turbulence scalings in supersonic channel flow[END_REF][START_REF] Huang | Compressible turbulent channel flows: DNS results and modelling[END_REF][START_REF] Coleman | A numerical study of turbulent supersonic isothermal-wall channel flow[END_REF], Prandtl number effect [START_REF] Kader | Temperature and concentration profiles in fully turbulent boundary layers[END_REF][START_REF] Kawamura | DNS of turbulent heat transfer in channel flow with low to medium-high Prandtl number fluid[END_REF], streamwise pressure gradient [START_REF] Huang | Law of the wall for turbulent flows in pressure gradients[END_REF][START_REF] Nickels | Inner scaling for wall-bounded flows subject to large pressure gradients[END_REF] or chemical reaction [START_REF] Cabrit | Direct simulations for wall modeling of multicomponent reacting compressible turbulent flows[END_REF].

The objective of this paper is to study the radiation effects on the temperature field in turbulent boundary layers by considering a turbulent channel flow. DNS is performed for the flow field in order to generate high-fidelity data. For radiation simulation, a reciprocal Monte Carlo method is employed. This method is generally regarded as the most accurate one and is widely used in many applications involving radiative transfer [START_REF] Wu | Direct numerical simulation of turbulence/radiation interaction in premixed combustion systems[END_REF][START_REF] Coelho | Numerical simulation of the interaction between turbulence and radiation in reactive flows[END_REF][START_REF] Tessé | Monte Carlo modeling of radiative transfer in a turbulent sooty flame[END_REF][START_REF] Zhang | Monte Carlo method of radiative transfer applied to a turbulent flame modeling with LES[END_REF]. The two solvers are fully coupled when radiation is taken into account. The gas radiative properties are determined by means of the correlated k-distribution (CK) model or its weak absorption limit [START_REF] Soufiani | High temperature gas radiative property parameters of statistical narrow-band model for H 2 O, CO 2 and CO, and correlated-K model for H 2 O and CO 2[END_REF], depending on the pressure condition. A detailed analysis of gas-gas and gas-wall radiation effect is carried out in this paper. For the gas-gas radiation, only the energy exchange between a gas cell and the surrounding gas is accounted for, while gas-wall radiation includes only radiative energy transfer between a gas cell and the walls. A priori these two parts have different effects on the gas temperature profile and, hence, the wall heat flux.

The numerical models and approaches are presented in Section 2. In Section 3, the effects of gas-gas and gas-wall radiation are studied for both large and intermediate optical thickness medium. Results associated with different wall emissivity, wall temperature and Reynolds number are discussed in Section 4.

Models and numerical approaches

Flow simulation

Under the low-Mach number approximation, the mass, momentum, energy balance equations and the ideal gas equation of state write respectively, in tensorial notations

∂ρ ∂t + ∂(ρu i ) ∂x i = 0, (1)
∂(ρu i ) ∂t + ∂(ρu i u j ) ∂x j = - ∂p ∂x i + ∂τ ij ∂x j + S i , (2)
∂(ρh) ∂t + ∂(ρu j h) ∂x j = ∂p ∂t - ∂q cd j ∂x j + P R (3)
p = ρ r T, (4)
where ρ, p and u i are the fluid mass density, pressure and velocity components, respectively. The enthalpy per unit mass h is expressed: h = ∆h 0 + T T 0 c p (T ) dT , where c p is the mixture thermal capacity at constant pressure, T the temperature, T 0 a reference temperature and ∆h 0 the corresponding standard formation enthalpy. The viscous shear stress tensor τ ij and the conductive flux vector q cd i write

τ ij = µ ∂u i ∂x j + ∂u j ∂u i - 2µ 3
∂u k ∂x k δ ij , (5)
q cd i = -λ ∂T ∂x i , (6)
where µ is the dynamic viscosity and λ the thermal conductivity. S i is a forcing source term required in the following channel flow computations. It is uniform and is not null only in the streamwise direction. It plays the same role as the pressure gradient to drive the flow against viscous forces. P R is the radiative power per unit volume.

A finite-volume method for massively parallel computations on complex grids, suitable for variable density flow (code YALES2 [START_REF] Moureau | Design of a massively parallel CFD code for complex geometries[END_REF][START_REF] Moureau | From Large-Eddy Simulation to Direct Numerical Simulation of a lean premixed swirl flame: Filtered laminar flame-PDF modeling[END_REF]) is used to solve equations 1-4 with associated boundary conditions. The code is a low Mach-number solver: the pressure p is split into a uniform thermodynamic pressure p 0 and a hydrodynamic one p 1 . p 0 is used in the equation of state to compute the gas density ρ, while p 1 is obtained by solving a Poisson equation to enforce mass conservation. Spatial gradients are calculated with a centered fourth order accurate scheme. Advancement in time is done with a fourthorder Runge Kutta scheme for velocity. For scalars (here enthalpy), the latter temporal scheme is blended with another fourth-order two-step scheme based on Taylor expansion [START_REF] Kraushaar | Application of the compressible and low-mach number approaches to large-eddy simulation of turbulent flows in aero-engines[END_REF] that is more dissipative. The blending factor is set as small as possible and yet high enough to avoid spurious oscillations in the scalar field with centered schemes. In this study, a typical value of 0.05 is retained.

Non-dimensional quantities in channel flows

For the channel flow defined in Fig. 1, bulk, practical and friction Reynolds numbers Re b , Re D h and Re τ are respectively defined by

Re b = ρ b u b δ µ b , Re D h = 4Re b , Re τ = ρ w u τ δ µ w (7)
with

ρ b = 2δ 0 ρdY 2δ , u b = 2δ 0 ρudY 2δ 0 ρdY , µ b = µ(T b ) ; (8)
The bulk enthalpy per mass unit and the friction velocity, based on the wall viscous stress τ w , by

h b = 2δ 0 ρuhdY 2δ 0 ρudY , u τ = τ w ρ w 1/2 , (9)
where indices b and w are related to bulk and wall quantities, respectively, and • refers to average quantities. The bulk temperature T b is evaluated from the bulk enthalpy : h b = h(T b ). When the channel flow case is not symmetrical (T w,c = T w,h ), the friction velocity is different on both walls: u τ,c = u τ,h and, therefore: Re τ,c = Re τ,h . The non-dimensional distance y + , based on the distance to a wall y, the non-dimensional streamwise velocity u + , and the non-dimensional temperature T + are written

y + = ρ w y u τ µ w , u + = u u τ , T + = |T -T w | T τ , (10)
where T τ is the friction temperature defined by

T τ = |q cd w | ρ w c pw u τ , (11)
where q cd w is the wall conductive flux, chosen positive in the Y direction. According to this definition and Fig. 1,q cd w is negative on both walls since T w,c ≤ T w,h .

Validation of the flow simulation

In order to test the suitability of the code for direct numerical simulations, the channel flow case of Kim and Moin [START_REF] Kim | Turbulence statistics in fully developed channel flow at low Reynolds number[END_REF][START_REF] Kim | Transport of passive scalars in a turbulent channel flow[END_REF] is computed and taken as reference for cases where temperature acts as a passive scalar. This case, named C0, is defined in Tab. 1. The discretization of the domain, defined in Tab. 2, is uniform along X and Z. ∆X, the cell length in the X direction, is expressed in wall units:

∆X + = ρ w ∆Xu τ µ w . (12)
∆Y + and ∆Z + , associated with the cell lengths ∆Y and ∆Z in the Y and Z directions respectively, are defined similarly. The mesh is refined along the Y direction close to the wall where ∆Y + = 0.8. The size of the domain was chosen large enough for results to be independent of this parameter. For this case C0 only, density and gas properties are uniform and T w,c = T w,h . Three temperature fields associated with three values of Prandtl numbers Pr 1 = 0.1, Pr 2 = 0.71 and Pr 3 = 2.0 are simulated. In this case, the temperature behaves like a passive scalar and non-dimensional results do not depend on the actual value of T w,c . A uniform source term is added in the energy equation as in Ref. [START_REF] Kim | Transport of passive scalars in a turbulent channel flow[END_REF] to counterbalance conductive heat fluxes at the walls.

The non-dimensional mean velocity u + and the scalar T + are plotted in Fig. 2 and compared to numerical results of Kim and Moin [START_REF] Kim | Turbulence statistics in fully developed channel flow at low Reynolds number[END_REF][START_REF] Kim | Transport of passive scalars in a turbulent channel flow[END_REF], and to the log-law obtained by these authors: u + = 2.5 lny + + 5.5, in Fig. 2 a and to the log-law of Kasagi et al. [START_REF] Kasagi | Direct numerical simulation of passive scalar field in a turbulent channel flow[END_REF]: T + = 2.78 lny + + 2.09, in Fig. 2 b. Good agreement is obtained between our results and those of the previous authors, demonstrating the accuracy of the chosen numerical set up to perform DNS of channel flows.

Re b Re D h T w,c (K) T w,h (

Radiation simulation

The general organization of the radiation model, based on a reciprocal Monte Carlo approach, has been detailed by Tesse et al. [START_REF] Tessé | Radiative transfer in real gases using reciprocal and forward monte carlo methods and a correlated-k approach[END_REF]. This model has been previously applied to combusting media involving radiation [START_REF] Tessé | Monte Carlo modeling of radiative transfer in a turbulent sooty flame[END_REF] and improved in [START_REF] Zhang | Optimized emission-based reciprocity monte carlo method to speed up computation in complex systems[END_REF]. Only the principles of the method are briefly summarized here.

In the approach of Ref. [START_REF] Tessé | Radiative transfer in real gases using reciprocal and forward monte carlo methods and a correlated-k approach[END_REF], the radiation computational domain is discretized into N v and N f isothermal finite cells of volume V i and faces of area S i , respectively. The radiative power in any cell i is written as the sum of the exchange powers P exch ij between i and all the other cells j, i.e.

P i = Nv+N f j=1 P exch ij = - Nv+N f j=1 P exch ji . ( 13
)
For volume cells, for instance, P exch ij is given by

P exch ij = +∞ 0 κ ν (T i )[I • ν (T j ) -I • ν (T i )] V i 4π A ij ν dΩ i dV i dν, (14)
where I • ν (T ) is the equilibrium spectral intensity and κ ν (T i ) the spectral absorption coefficient relative to the cell i. dΩ is an elementary solid angle and A ij ν accounts for all the paths between emission from any point of the cell i and absorption in any point of the cell j, after transmission, scattering and possible wall reflections along the paths. Its expression and similar expressions for exchanges between a volume cell and a surface cell or between two surface cells are detailed in Ref. [START_REF] Tessé | Radiative transfer in real gases using reciprocal and forward monte carlo methods and a correlated-k approach[END_REF]. In the reciprocity Monte Carlo method, a huge number of optical shots are issued from the cells. summing the contributions of the N ij shots that connect i and j, i.e.

P exch ij = N ij n=1 P exch ijn ν . (15)
In order to increase the computational efficiency, emission is here only carried out from arbitrary small spheres around grid points instead of using finite grid cells that are neither isothermal nor homogeneous. In this condition, the small emission spheres are isothermal, homogeneous and optically thin. A consequence of this choice is that the Absorption-based Reciprocity Method (ARM), which requires emission from finite cell, is not suitable. Only Emission-based Reciprocity Methods (ERM) can then be used. In order to overcome some drawbacks of ERM compared to ARM in cold regions of the medium, the Optimized Emission Based Reciprocity Method (OERM), proposed by Zhang et al. [START_REF] Zhang | Optimized emission-based reciprocity monte carlo method to speed up computation in complex systems[END_REF], is here used for the radiation frequency treatment.

An other advantage of ERM and OERM method is to allow the Monte Carlo convergence to be locally controlled. In all the present simulations a radiative power standard deviation of 3% of radiative power maximum value has been imposed at any grid point (as shown in Fig. 3 for instance).

R1 R2 R3 R4

1 (cold wall) 0.8 0.3 0.1 0.1 2 (hot wall) 0.8 0.3 0.1 1.0 Table 3: Wall emissivities in radiative conditions R1, R2, R3 and R4.

Coupled simulation

For all cases considered in the paper, a non-reacting CO 2 -H 2 O-N 2 gas mixture flows, in developed turbulent regime, through a plane channel in different conditions. The molar fractions of CO 2 , H 2 O and N 2 are 0.116, 0.155 and 0.729. Cases involving radiation cannot be entirely characterized by non-dimensional numbers, contrary to cases without radiation. The value of δ, the channel half-width, is then given here: δ = 0.1 m.

The dynamic viscosity µ is computed as a function of temperature from the CHEMKIN package [START_REF] Kee | A Fortran Computer Code Package For The Evaluation Of Gas-Phase, Multicomponent Transport Properties[END_REF][START_REF] Kee | CHEMKIN-II: A FORTRAN Chemical Kinetics Package for the Analysis of Gas-Phase Chemical Kinetics[END_REF] for the chosen mixture composition. The thermal conductivity λ is computed from a Prandtl number Pr. In all considered configurations, the Prandtl number is very close to the chosen value 0.71.

The flow computational cases, called Cn (n= 1 to 4), are defined in Table 1 by a set of bulk Reynolds number Re b , pressure and wall temperatures. Similarly, cases which include radiative energy transfer are called Rm (m= 1 to 4) and are defined by the emissivities (ε 1 , ε 2 ) of the opaque walls given in Table 3. Consequently, a computation case without radiation in conditions n is called Cn and a computation that accounts for radiation effects, in conditions n and m, CnRm. The associated spatial discretizations are defined in Table 2.

Gas radiative properties are treated in a correlated manner by the CK approach for atmospheric pressure condition, and by using the weak absorption limit for high pressure cases [START_REF] Soufiani | High temperature gas radiative property parameters of statistical narrow-band model for H 2 O, CO 2 and CO, and correlated-K model for H 2 O and CO 2[END_REF]. In the directions X and Z, periodicity conditions are also used for radiation simulation, i.e., if a shot exits the domain, for instance, at the point (L X , Y, Z), it will enter at the point (0, Y, Z) with the same propagation direction. The grid of the radiation model is three times coarser in X direction and two times in Y and Z directions than the corresponding grid of the flow model. In the chosen DNS conditions, all averaged fluxes are uniform in directions X and Z. The averaged energy balance equation then writes, from Eq. 3,

d dY [ρ v h (Y ) + q cd (Y ) + q R (Y )] = d dY q tot = 0, (16)
where ρ v h , q cd , and q R and q tot are the averaged turbulent convective heat flux, conductive flux, radiative energy flux and total flux, respectively. This total flux is also uniform along Y . Note that, the radiative energy flux can be split into two parts :

q R (Y ) = q R * (Y ) + q R ww , (17)
where q R ww is the flux exchanged between the walls through the whole gaseous medium, that does not participate to the fluid energy balance and is zero if the walls are at the same temperature, as encountered in many applications. In the studied configurations, it is always uniform. Then Equation 16writes

ρ v h (Y ) + q cd (Y ) + q R * (Y ) = q w , (18)
where q w = q tot is the total flux exchanged between a wall and the gaseous mixture without the wall-wall radiation contribution. At the wall Y = δ, for instance, q w writes

q w = q cd (Y = δ) + q R * (Y = δ) = q cd w + q R * w . ( 19
)
The balance of these three terms in Eq. 18 is presented in Fig. 4 for the case C2 without radiation and the related case C2R3 which accounts for radiative energy transfer. This illustrates that radiation modifies the balance of the terms in Eq. 19 and hence, the wall conductive flux, given in Tab. 4. These complex coupling effects are progressively detailed in the next sections.

Effects of Gas-Gas and Gas-Wall radiative interactions

The effects of radiation are due to different coupled phenomena, in particular Gas-Gas (GG) and Gas-Wall (GW) interactions, but also possible multiple reflections. In this section, the effects of gas-gas and gas-wall interactions are separately considered. In gas-gas interaction, a given gas volume only exchanges radiative energy with all other gas volumes, while the walls only partially reflect radiation. Emission by the walls is not accounted for in the GG calculations. On the contrary, in gas-wall interaction, a given gas volume only exchanges energy with walls, while the other gas volumes only transmit energy. Emission by the other gas volumes is not accounted for in GW calculations. A coupled computational case that is limited to gas-gas (respectively gas-wall) radiative interactions is called CnRm GG (respectively CnRm GW). The obtained results are then compared with the corresponding cases CnRm, which includes all the radiation interactions, and Cn, which does not take into account radiative energy transfer.

Large optical thickness medium

In the case C1R1, defined in Tables 1 to 3 and characterized by a high pressure, the optical thickness of the gaseous mixture is large: The global medium Hotell's transmissivity at 1000 K is equal to 0.271. This allows the effects of gas-gas and gas-wall radiative interactions to be studied with a weak radiative coupling between the two walls.

Due to the small wall temperature difference, the variation in mass density in cases C1 and C1R1 is small. The profiles of the mean non-dimensional streamwise velocity u + , not shown here, are not different from the one of case C0.

The profiles of the mean temperature T associated with the cases C1 and C1R1 are compared in Fig. 5 a. The corresponding T + profiles, for both the cold and the hot sides, are plotted in Fig. 5 b. In the case C1, without radiation, the distribution of T is practically antisymmetric and the T + profiles are identical for the two sides. These results agree well with the results of Kim and Moin [START_REF] Kim | Transport of passive scalars in a turbulent channel flow[END_REF], obtained under the assumptions that the temperature is a passive scalar. When only gas-wall radiation is considered, in the C1R1 GW case, the temperature gradient is smaller in the vicinity of a wall than in C1 case: The associated wall conductive fluxes presented in Table 4 are two to three times smaller than in C1 case. Indeed, the wall tends to impose its temperature to the fluid. Consequently, the temperature variation is higher in the core of the flow than in case C1. The conductive flux variations are more important at the hot side than at the cold one, as gas-wall radiative interactions strongly increase with the temperature.

On the other hand, in the case C1R1 GG, the gas-gas radiative transfer is a supplementary transfer that homogenizes the temperature field within the gas by comparison with the case C1, without radiation, as shown in Figs. 5 a and b . Consequently the temperature gradients and the conductive fluxes strongly increase at the two walls, as shown in Table 4. 4: Wall fluxes (in W/m 2 ) q cd w and q R* w for different cases at cold and hot walls (see Eqs. 16 and 17). For each case CnRm, the relative variation of q cd w compared to the case Cn without radiation is put between parentheses. Flux values are rounded, typical errors are within 2-3 %. When all the radiative effects are accounted for, in the case C1R1, the gas-gas and gas-wall interactions, that have opposite effects, are coupled. As shown in Fig. 3, in the vicinity of the wall, the amplitude of the radiative power associated with gas-gas interaction is much larger than the one associated with gas-wall interactions. Consequently, in the present conditions, the wall temperature gradients and the conductive fluxes increase, as in the gasgas case, at the two walls by comparison with the case C1 without radiation, as shown in Tab. 4 and Fig. 5 a.

Re τ,c Re τ,h q cd w,c q cd w,h q R* w,c q R* w,h (cold) (hot) (cold) (hot) (cold) (hot)
An important result is that, in this case, no log-law can be clearly identified.

Intermediate optical thickness medium

All the previous study has been again carried out for the case C2R1, characterized by a fluid at atmospheric pressure instead of 40 atm, but with the same data as for C1R1 (Re b , δ, T 1 , T 2 , ε 1 , ε 2 ). In these new conditions, the optical thickness of the medium is much smaller than previously. The global Hotell's transmissivity at 1000 K of the channel is equal to 0.811. Consequently, interaction phenomena between the two walls now occur, due to multiple reflections. Results are shown in Figs. 6 a, b andc. The same analysis as previously can be drawn, but the effects related to gas-gas and gaswall interactions are smaller than at high-pressure. As shown in Tab. 4 , the conductive fluxes obtained by only accounting for gas-gas interaction (case C2R1 GG) for the two sides are larger than the conductive fluxes associated with the case C2, without radiation. On the contrary, the conductive fluxes obtained in case C2 and by only accounting for gas-wall interaction (case C2R1 GW) are close, especially at the cold side. Finally, the conductive flux associated with all radiation effects (C2R1) is larger than in the C2 case at the two walls as in the optically thick case. Moreover, the temperature fields obtained by accounting for all radiation effects differ from the fields computed without radiation. Once again the profiles associated with the case C2 agree with the temperature profile of [START_REF] Kim | Transport of passive scalars in a turbulent channel flow[END_REF] while it is not any more valid when radiation is accounted for.

As for the case C1R1, the usual thermal log-law is not valid in this case.

Influences of different parameters

The radiation intensity field and, consequently, the radiative power and the temperature fields within the gaseous medium are strongly influenced by the wall emissivities, the wall temperatures and the bulk Reynolds number of the flow. The roles of these three quantities are studied in this Section.

Influence of wall emissivity

Three other couples of wall emissivities are now considered, in the conditions C2 of the flow at atmospheric pressure: These radiative conditions R2, R3 and R4 are defined in Tab. 3. From a practical point of view, an emissivity of about 0.8 typically corresponds to walls made of oxides, an emissivity of about 0.3 to polished metals. Emissivities of 0.1 and 1 are extreme cases.

The mean temperature fields associated with C2, C2R1, C2R2 and C2R3 cases are plotted in Fig. 7 a. When the wall emissivity decreases, gasgas effects become more and more important and the temperature profile becomes more uniform in the flow center part while the temperature gradient near the wall increases, as shown in Tab. 4. Similarly, T + decreases on both sides when the wall emissivity decreases. Only the cold side results are shown in Fig. 7 b. The antagonist gas-gas and gas-wall effects also appear on the average radiative power field shown in Fig. 8 a. On both walls, the radiative power magnitude increases when wall emissivities decrease. This global effect can be analyzed from gas-gas and gas-wall exact contributions to the total radiative power in cases C2Rm. These contributions are given in Figs. 8 b andc. and C2R3 ( : DNS results of a passive scalar of Kim and Moin [START_REF] Kim | Turbulence statistics in fully developed channel flow at low Reynolds number[END_REF] (with Pr=0.71); : C2; : C2R1;

• : C2R2; • • • : C2R3).
When wall emissivity decreases, the reflected flux and possibly the number of reflections increase. Consequently, the gas-gas interaction effects increase as seen in Fig. 8 b . On the other hand, the flux exchanged between the gas and the walls and, hence, the gas-wall interaction effects decrease as seen in Fig. 8 c. The shape of the gas-wall contribution to the total radiative power is explained by splitting the gas-wall interaction into a gas-cold-wall and a gas-hot-wall interactions. These latter contributions are given in Fig. 8 d. Finally, the global radiative power is the sum of all contributions and follows the same trend as the gas-gas interaction which is significantly larger than the gas-wall interaction in these cases.

The radiative conditions R4 (see Tab. 3 ) corresponds to two extreme cases: Very reflecting cold wall of emissivity 0.1 and black hot wall. The previous effects on the averaged temperature field are here amplified, as shown in Fig. 9 a, by comparing the cases C2, C2R3 and C2R4. For case C2R4, the hot black wall strongly imposes its temperature to the close gaseous layer. In the other hand, the reflecting cold wall strongly increase gas-gas radiative interactions and homogenizes the fluid temperature. The cumulation of these effects leads to: i) A much smaller temperature gradient and conductive flux at the hot wall than in case C2R3 associated with an emissivity equal to 0.1 (see Tab. 4 ); ii) A weak averaged temperature variation in the core of the gaseous medium; iii) Consequently the temperature gradient at the cold wall and the associated conductive flux are much larger than in case C2R3, as shown in Tab. 4 .

The radiative power fields of the two extreme cases C2R3 and C2R4 differ in the vicinity of both walls as shown in Fig. 9 b. It can be explained by the previously discussed cumulative effects on the temperature profile : On the hot side, the gas temperature is close to the hot wall temperature, which decreases the magnitude of the radiative power; On the cold side, both the black hot wall and the large hot region contribute to increase the radiative power.

Figure 10 shows the radiative power field in the hot half part of the channel. It appears that the radiative power in case C2R3 is dominated by gas-gas interaction. On the contrary, in the case C2R4, gas-wall interaction overcomes gas-gas interaction except for the close vicinity of the hot wall. For intermediate emissivity value, intermediate radiative power fields are encountered.

Note that, in the present simple case as in most of the previous ones, both gas-wall and gas-gas radiative interactions are strongly modified by the wall reflection, as the optical thickness of the mediums is weak. Gasgas an gas-wall interactions are not isolated phenomena, as in case C1Rm at high-pressure, characterized by a large global optical thickness. The wall reflection law and the medium optical thickness have important effects on the temperature fields as they can enhance or reduce the gas-gas or the gas-wall contributions.

Influence of the temperature

In order to study the influence of both the temperature level and the temperature difference, the C3 and C3R1 cases considered. They are characterized by wall temperatures equal to 950K and 2050K. On the cold side, these data correspond to classical combustion applications at high pressure. The associated results are compared with those of C1 and C1R1 cases.

Contrary to the previous cases, the gaseous medium density strongly varies for C3 and C3R1 cases and variations in the averaged velocity profiles are now observed in Fig. 11 a. Similarly, differences between the u + profiles are observed in Fig. 11 b, showing that the classical velocity log-law is not valid any more. However, if the Van Driest transformation, defined by

u + V D = u + 0 ρ ρ w du + , (20)
is used to account for variable density effects, all velocity profiles collapse to the usual log-law (see Fig. 11 c). The averaged temperature profile of C3R1 case is compared in Fig. 12 a with those of C3, C1 and C1R1. As the temperature of the hot wall is much higher in C3R1 case than in C1R1 case, the radiative transfer is much stronger in this case (see q R * w in Tab. 4) and the opposite effects of gas-gas and gas-wall radiative interactions shown in this latter case are amplified. The same analysis as for the case C1R1 can be achieved (see Sec. 3.1) and the results are similar, with an amplification effect for the conductive fluxes, as given in Tab. 4. Consequently, the averaged temperature is much higher than in C3 case. Figure 12 b shows that the usual log-law is not valid when radiation is accounted for. Moreover, no temperature log-law even appears between y + = 30 and y + = 200.

Influence of the Reynolds number

The results of the C4 and C4R1 cases, characterized by a higher Reynolds number than previously (see Tab. 1), are here compared with cases C1 and C1R1 in order to study the influence of the Reynolds number. The temperature profiles are plotted in Fig. 13. The curve T + (y + ) for case C4R1 in Fig. 13 b lies between the ones where radiation is not accounted for and the one of case C1R1. Increasing the Reynolds number between cases C1R1 and C4R1 has then moved the obtained wall-law closer to the usual one. There are two different explanations for this behavior. First, by increasing the Reynolds number, turbulent transport has been enhanced and its weight compared to the other energy transfer mechanisms is increased. This is shown in Fig. 14 a where turbulent transport is twice larger in case C4R1 than in case C1R1. One effect of the Reynolds number is therefore to relax the curve T + (y + ) obtained when radiation effects are considered towards the usual law of the wall which should be retrieved when radiative energy transfer is negligible. In addition to this first effect, there is a second one that deals with a modification of the radiative energy transfer. In this specific configuration, the gas-gas and gas-wall radiative contributions to the total radiative power field in case C4R1 are modified as shown in Fig. 15 where they are compared to results in case C1R1. Magnitude of the gas-gas contribution decreases for the larger Reynolds number case while the magnitude of the gas-wall contribution increases. Since a larger Reynolds number leads to a larger gradient of the mean temperature close to the wall, the gas temperature at a given position gets closer to the average one in the core of the channel when the Reynolds number increases, while its difference with the near wall temperature gets larger. This explains the observed trends for the gas-gas and gas-wall radiative contributions. Consequently, the gas-gas effects on the wall conductive heat flux and the wall law dwindle while the gas-wall effects grow. This is seen in Tab. 4 where the conductive heat flux increase on the cold wall due to radiation is less in case C4R1 than in case C1R1. On the hot wall, the conductive heat flux even decreases between cases C4 and C4R1, showing that gas-wall effects overcome gas-gas effects there. Finally, the modification of the gas-gas and gas-wall contributions make the observed wall law go up compared to case C1R1.

Among these two effects of the Reynolds number, the latter one is dominant in the present configuration where the radiative flux is much larger than turbulent transport (see Fig. 14). Would radiation remain dominant, increasing the Reynolds number even more could move the curve T + (y + ) above the usual wall law. This could not be check here due to the limitations of DNS on computational ressources with increasing Reynolds numbers.

Conclusion

In practical conditions of coupling between turbulent convection and radiation, there is no simple way for accurately predicting the averaged tem- When the radiative flux within a turbulent gaseous medium is of the same order of magnitude or higher than the averaged turbulent convective flux, or the wall conductive flux, strong coupling effects occur within all the medium and at the walls. It was shown that the usual temperature profile and its corresponding log-law are generally no more valid within a turbulent boundary layer. The couplings between conduction, turbulent convection and radiation are complex and first strongly depend on both gas-gas and gas-wall radiative interactions. The global radiation effects are not easily predictable as gas-gas and gas-wall radiative interactions bring contributions of opposite sign to the wall conductive flux. Moreover, depending on the transversal optical thickness of the gaseous medium, complex effects, that are linked to multiple wall reflections, can also occur and strongly modify both the averaged temperature profiles and the conductive wall fluxes. Finally, the averaged temperature profiles and wall conductive fluxes were shown to also strongly depend on: i) the wall emissivities, that rule the wall reflection effects; ii) the temperature level, that controls the non linear radiative fluxes; iii) the Reynolds number that controls the weight of turbulence transport and the balance of gas-gas and gas-wall radiative effects.

The development of simpler models for the determination of the temperature profile in the turbulent boundary layer is necessary.

Figure 1 :

 1 Figure 1: Computational domain of channel flow cases. X, Y and Z are the streamwise, wall normal and spanwise directions. L X , L Y and L Z are the dimensions of the channel case in each direction. δ is the channel half-width. The lower wall (resp. upper wall) is at temperature T w,c (resp. T w,h ; T w,h ≥ T w,c ). Periodic boundary conditions are applied along X and Z.

Figure 2 :

 2 Figure2: Mean profiles of non-dimensional streamwise velocity (a):[START_REF] Kim | Turbulence statistics in fully developed channel flow at low Reynolds number[END_REF]; and nondimensional temperature in wall units (b): (Pr = 0.1), (Pr = 0.71), (Pr = 2)[START_REF] Kim | Transport of passive scalars in a turbulent channel flow[END_REF];: present results.

Figure 3 :

 3 Figure 3: Comparison of mean radiative power of C1R1, C1R1 GG and C1R1 GW ( : C1R1; : C1R1 GW; : C1R1 GG; color : Cold side; color : Hot side; error bars represent the standard deviation).

Figure 4 :

 4 Figure 4: Mean heat flux distribution for C2 (a), C2R3 (b) and comparison of these two cases (c) ( : Turbulent convective heat flux; : Conductive heat flux; • • • : Radiative energy flux; • : total heat flux; black lines: C2; gray lines: C2R3).

Figure 5 :

 5 Figure 5: Mean temperature profiles in global coordinates (a) and in wall units (b): DNS results (Pr= 0.71) [38]; : C1; : C1R1 GW; : C1R1 GG; : C1R1; in (b), : Cold side; color : Hot side.

Figure 6 :

 6 Figure 6: Comparison of mean temperature profile (a), T + profile (b,c) of C2, C2R1 GW, C2R1 GG and C2R1 ( : DNS results of a passive scalar of Kim and Moin [37] (with Pr=0.71); : C2; : C2R1 GW; : C2R1 GW; : C2R1; in (b), color : Cold side; color : Hot side).

Figure 7 :

 7 Figure 7: Mean profile of temperature (a), T + on the cold side (b) for C2, C2R1, C2R2

Figure 8 :

 8 Figure 8: Mean profile of total radiative power (a), gas-gas radiative power (b), gas-wall radiative power (c) and gas-cold wall and gas-hot wall radiative power (d) of C2R1, C2R2 and C2R3 ( : C2R1; • : C2R2; • • • : C2R3; in (a) and (b), color : Cold side; color : Hot side; error bars in (a) represents standard deviation).

Figure 9 :

 9 Figure 9: Mean profile of temperature of C2, C2R3 and C2R4 (a) and radiative power of C2R3 and C2R4 (b) ( : C2; • • • : C2R3; •• : C2R4; in (b), color : Cold side; color : Hot side; error bars in (b) represent standard deviation).

Figure 10 :

 10 Figure 10: Mean profile of radiative power for the channel hot side for C2R3 (a) and C2R4 (b) ( ••: Total; * : Gas-wall; : Gas-gas).

Figure 11 :

 11 Figure 11: Mean profile of velocity (a), u + (b) and u + V D (c) of C3, C3R1 and C1 ( • : Wall function of Ref. [37]; : C3; •• : C3R1; : C1; in (b) and (c), color : Cold side; color : Hot side).

Figure 12 :

 12 Figure 12: Mean profile of mean temperature(a) and T + (b) of C3, C3R1, C1 and C1R1 ( • : Wall function of Ref. [39]; : C3; •• : C3R1; : C1; : C1R1; in (b), color : Cold side; color : Hot side ).

Figure 13 :

 13 Figure 13: Mean temperature profile (a) and T + of C4, C4R1, C1 and C1R1 (b) ( : C4; •• : C4R1; : C1; : C1R1; in (b), color : Cold side; color : Hot side).

Figure 14 :

 14 Figure 14: Comparison of turbulent convective heat flux (a) and radiative energy flux (b) of C4R1 and C1R1 ( •• : C4R1; : C1R1).

Figure 15 :

 15 Figure 15: Mean radiative power profile of C4R1 and C1R1 on the cold (left) and hot (right) sides ( : Total; : Gas-wall; : Gas-gas; black lines : C4R1; gray lines : C1R1; the error bars represent the standard deviation).

  perature profile and wall conductive flux, without undertaking a complete coupled computation. Such a computation has been achieved in this work, in typical combustion conditions, by coupling a direct numerical simulation of a turbulent channel flow with a radiative transfer model based on a Monte Carlo simulation, an optimized emission-based reciprocity and a CK or k gaseous radiative model.

Table 1 :

 1 Channel flow parameters. In case C0, non-dimensional passive scalars are considered. Reynolds numbers are given with an accuracy of 0.5 %.

	K) p(atm)

n X × n Y × n Z ∆X + ∆Y + ∆Z + hot/cold hot/cold C0 120 × 100 × 120 17.85/17.85 [0.8-8.0] 8.92/8.92 C1,C2 110 × 135 × 110 16.9/21.3 [0.8-8.0] 8.4/10.6 C3 160 × 163 × 160 8.9/22.4 [0.8-8.0] 4.4/11.2 C4 200 × 230 × 200 17.3/21.8 [0.8-8.0] 8.6/10.8 Table 2: Discretization of the simulated cases: n X , respectively n Y and n Z , is the number of points in the X direction, respectively Y and Z direction. Domain size: (4πδ, 2δ, 2πδ) for C0 and (2πδ, 2δ, πδ) for C1-C4. ∆X + and ∆Z + are given at the cold and hot sides for C1-C4.

Acknowledgements

The first author is supported by a doctoral fellowship from China Scholarship Council (CSC). This work was granted access to the HPC resources of [CCRT/CINES/IDRIS] under the allocation 2012-2b0164 made by GENCI (Grand Equipement National de Calcul Intensif). We also acknowledge Dr. Vincent Moureau and the SUCCESS scientific group for providing the code YALES2 and support.


Olivier Gicquel

J Taine

Y F Zhang

R Vicquelin
email: ronan.vicquelin@ecp.fr

A wall model for LES accounting for radiation effects

Keywords: Wall model, LES, radiation, temperature, channel flow Simulation. RANS Reynolds-Averaged Numerical Simulations OERM Optimized Emission

published or not. The documents may come

A wall model for LES accounting for radiation effects

Roman Symbols c p

Thermal capacity at constant pressure [J•kg -1 •K -1 ]. e

Optical thickness [-]. h Enthalpy per unit mass [J•kg -1 ].

I

Radiative intensity [W• m -2 Dynamic viscosity [kg•s -1 •m -1 ] or cosine of polar angle [-].

Superscript

• Filtered quantities.

•

Mass-weighted filtered quantities.

Introduction

Fully resolved Large Eddy Simulation (LES) of the inner layer of a wall-bounded turbulent flow requires highly resolved grids since the integral length scale becomes of the same order of magnitude as viscous scales in the close vicinity of the wall. The computational cost is then proportional to Re 2.4 [START_REF] Piomelli | Wall-layer models for large-eddy simulations[END_REF]. Hence, fully resolved LES is impracticable for wall-bounded flows at high Reynolds number, encountered in most of engineering applications, due to the prohibitive cost. Several kinds of approaches are commonly used in order to alleviate these difficulties: A wall model prescribes the correct wall shear stress to the LES that is too poorly resolved close to the wall to estimate it accurately.

In hybrid RANS/LES, the simulation is switched from RANS in the inner layer to LES in the outer layer by the modification of the length scales [START_REF] Spalart | Comments of feasibility of les for wings, and on a hybrid RANS/LES approach[END_REF][START_REF] Nikitin | An approach to wall modeling in large-eddy simulations[END_REF] or the use of a blending function [START_REF] Hamba | A hybrid rans/les simulation of turbulent channel flow[END_REF][START_REF] Shur | A hybrid RANS-LES approach with delayed-DES and wall-modelled LES capabilities[END_REF] in the turbulent transport model. In other wall models for LES, the wall stress is estimated by using an algebraic wall function or by locally solving a simplified RANS equation. These approaches correspond to equilibrium-stress model and Two-Layer Model (TLM), respectively.

The equilibrium-stress model has been firstly proposed by Deardroff [START_REF] Deardorff | A numerical study of three-dimensional turbulent channel flow at large reynolds numbers[END_REF]. It has then been successfully applied to turbulent channel flows and annuli flows [START_REF] Schumann | Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli[END_REF]. However, this model is restricted to simple flows, since it implies the existence of a logarithmic layer. In order to widen its use to more complex flows, the original equilibrium model has been modified by considering 3 inclination of the elongated structure in near wall region [START_REF] Piomelli | New approximate boundary conditions for large eddy simulations of wall-bounded flows[END_REF], pressure gradient [START_REF] Wang | LES with wall models for trailing edge aeroacoustics[END_REF][START_REF] Duprat | A wall-layer model for large-eddy simulations of turbulent flows with/out pressure gradient[END_REF], buoyancy [START_REF] Moeng | A large-eddy-simulation model for the study of planetary boundary-layer turbulence[END_REF] or chemistry [START_REF] Cabrit | Direct simulations for wall modeling of multicomponent reacting compressible turbulent flows[END_REF].

In two-layer models, turbulent boundary layer equations are resolved on a local embedded grid [START_REF] Balaras | Two-layer approximate boundary conditions for large-eddy simulations[END_REF]. This approach has been extensively applied and assessed in different configurations [START_REF] Cabot | Approximate wall boundary conditions in the large-eddy simulation of high reynolds number flow[END_REF][START_REF] Wang | Dynamic wall modeling for large-eddy simulation of complex turbulent flows[END_REF][START_REF] Gungor | Direct simulation of subgrid turbulence in a high-Re wall-bounded flow[END_REF]. Moreover, since all wall modeled LES unavoidably suffers from the numerical and sub-grid error at the first grid point close to a wall [START_REF] Cabot | Approximate wall boundary conditions in the large-eddy simulation of high reynolds number flow[END_REF][START_REF] Nicoud | Large eddy simulation wall-modeling based on suboptimal control theory and linear stochastic estimation[END_REF], an effective strategy has recently been proposed by Kawai et al. [START_REF] Kawai | Wall-modeling in large eddy simulation: Length scales, grid resolution, and accuracy[END_REF] to increase the accuracy of the information transmitted from LES in the outer layer to the inner thin turbulent boundary layer equations.

A more detailed description of wall models for velocity is given in Refs. [START_REF] Piomelli | Wall-layer models for large-eddy simulations[END_REF][START_REF] Piomelli | Wall-layer models for large-eddy simulations[END_REF][START_REF] Cabot | Approximate wall boundary conditions in the large-eddy simulation of high reynolds number flow[END_REF]. In order to deal with turbulent heat transfer and predict wall heat fluxes accurately, these wall models have to be extended to describe the thermal boundary layer as in [START_REF] Benarafa | Temperature wall modelling for large-eddy simulation in a heated turbulent plane channel flow[END_REF][START_REF] Rani | Boundary-Layer Equation-Based Wall Model for Large-Eddy Simulation of Turbulent Flows with Wall Heat Transfer[END_REF]. To the best of our knowledge, no wall model for LES has accounted for radiation effects, although radiation strongly modifies the temperature field in many applications, particularly in combustion processes at high pressure [START_REF] Gupta | Large-Eddy Simulation of Turbulence-Radiation Interactions in a Turbulent Planar Channel Flow[END_REF][START_REF] Soufiani | Radiation effects on turbulent heat transfer in channel flows of infrared active gases[END_REF][START_REF] Ghosh | Effects of radiative heat transfer on the structure of turbulent supersonic channel flow[END_REF]. It has been recently shown [START_REF] Zhang | Physical study of radiation effects on the boundary layer structure in a turbulent channel flow[END_REF] in coupled DNS-Monte Carlo simulations that radiation can significantly influence the temperature wall-law and the corresponding wall conductive heat flux. The temperature law is very different from the usual logarithmic law for strong radiation effects and has been observed to differ significantly under different radiative conditions. It is therefore unrealistic to hope for a general algebraic wall-law to account for these effects and a two-layer approach is then chosen. Besides, in order to predict the radiative field outside of the inner boundary layer, a reciprocal Monte-Carlo method is considered. The method is accurate and can be applied to complex geometries so that the proposed wall-model and its coupling with LES and the Monte-Carlo method remain general.

The objective of this study is to account for radiation effects in the inner layer wall model to accurately predict wall stress and heat flux. Here, a two-layer model is retained where, in the outer layer, LES is coupled to a radiation Monte Carlo method as in Ref. [START_REF] Zhang | Physical study of radiation effects on the boundary layer structure in a turbulent channel flow[END_REF]. Coupled DNS-Monte Carlo results of Ref. [START_REF] Zhang | Physical study of radiation effects on the boundary layer structure in a turbulent channel flow[END_REF] are considered to validate the proposed LES wall model. The fluid and radiation models in both layers are detailed in Sec. 2, followed by a description of coupling between the inner and outer layers. Separate validations of the different model components are presented in Sec. 3. Finally, in Sec. 4, fully coupled results assess the model accuracy.

Wall-modeled LES coupled to radiation

In all fluid simulations, LES is here carried out in the outer layer and the boundary inner layer is modeled by solving 1D balance equations. For radiation, a reciprocal Monte Carlo approach is implemented to estimate the radiative power at all LES grid points and an analytical radiative 1D model is developed for the inner layer. For both radiation and fluid models, a particular care is brought to the boundary conditions, especially between the inner and outer layers.

Fluid model

As shown in Fig. 1, an embedded grid is used in the inner layer. The inner layer model uses the velocity ũy w and temperature Ty w values computed by the LES model at a particular point characterized by the wall distance y w . The wall stress τ w and conductive heat flux q cd w computed by the inner layer model are then sent back to the LES solver.

Inner layer fluid model

As explained in Ref. [START_REF] Piomelli | Wall-layer models for large-eddy simulations[END_REF], filtered equations in the inner layer are similar to averaged Navier-Stokes equations. Then, treating the unresolved inner layer [0, y w ] as a thin equilibrium boundary layer [START_REF] Wang | Dynamic wall modeling for large-eddy simulation of complex turbulent flows[END_REF][START_REF] Cabot | Approximate wall boundary conditions in the large-eddy simulation of high reynolds number flow[END_REF]

+ P R = 0, (1)
where the angled brackets • and curly brackets {•} denote Reynolds-averaged and Favreaveraged values respectively. u || is the tangential velocity, parallel to the wall, and y is the distance to the wall. T is the temperature, P R is the radiative power per unit volume. As in Ref. [START_REF] Zhang | Physical study of radiation effects on the boundary layer structure in a turbulent channel flow[END_REF], µ, the dynamic viscosity, and c p , the thermal capacity at constant pressure, are functions of temperature, while the molecular Prandtl number Pr is set to 0.71. µ t and Pr t are the turbulent viscosity and the turbulent Prandtl number, respectively. µ t is computed using a mixing-length model [START_REF] Cabot | Near-wall models in large-eddy simulations of flow behind a backward-facing step[END_REF][START_REF] Cabot | Approximate wall boundary conditions in the large-eddy simulation of high reynolds number flow[END_REF],

µ t = ρ κy τ w / ρ (1 -exp(-y + /A + )) 2 , (2)
where κ = 0.4 and A + = 17 are here chosen in order to retrieve the velocity logarithmic law in the low-Reynolds DNS cases that are considered for validation. ρ is the gas density and τ w the shear stress at the wall. The wall coordinate y + is defined as

y + = ρ w u τ y µ w , u τ = τ w ρ w , (3)
where gas properties at the wall, ρ w and µ w , are used. In order to take into account the nonuniformity of the gas properties, an alternative semi-local coordinate y * [START_REF] Huang | Compressible turbulent channel flows: DNS results and modelling[END_REF] can also be used in Eq. ( 2) instead of y + , where local gas properties are considered

y * = ρ u * τ y µ , u * τ = τ w ρ , (4)
The mixing-length models based on wall coordinate y + and semi-local coordinate y * will subsequently be referred as standard and semi-local mixing-length models, respectively. Three models for the turbulent Prandtl number are considered in the following. First, a constant value Pr t = 0.9 which is a usual crude assumption in wall models. Secondly, in order to account for variation of Pr t in the boundary layer, two different formula proposed in Ref. [START_REF] Kays | Turbulent Prandtl number. Where are we?[END_REF], both fitted from experimental data, are considered

Pr t = 2.0 Pr ( µ t / µ ) + 0.85, (5)
and Pr t = 1 0.5882 + 0.228( µ t / µ ) -0.0441( µ t / µ ) 2 (1 -exp( -5.165 µ t / µ ))

. ( 6
)

Outer layer LES

A low-Mach code, YALES2 [START_REF] Moureau | From Large-Eddy Simulation to Direct Numerical Simulation of a lean premixed swirl flame: Filtered laminar flame-PDF modeling[END_REF][START_REF] Moureau | Design of a massively parallel CFD code for complex geometries[END_REF], is used for the Large Eddy Simulation (LES) of the main flow. The same numerical set-up as in Ref. [START_REF] Zhang | Physical study of radiation effects on the boundary layer structure in a turbulent channel flow[END_REF] is retained: A 4th-order central difference finite-volume scheme with a 4th-order time integration scheme.

Under the low-Mach number approximation, the spatially filtered instantaneous mass, momentum and energy balance equations write

∂ρ ∂t + ∂(ρ u i ) ∂x i = 0, (7)
∂(ρ u i ) ∂t + ∂(ρ u i u j ) ∂x j = ∂τ SGS i j ∂x j - ∂p ∂x i + ∂τ i j ∂x j + S i , (8)
∂(ρ h) ∂t + ∂(ρ u j h) ∂x j = - ∂q SGS j ∂x j + ∂p ∂t - ∂q cd j ∂x j + P R , (9)
where p is the pressure, u i the velocity components and • and • denote filtered and mass-weighted filtered quantities in the context of LES. The enthalpy per unit mass h is expressed as h = ∆h 0 + T T 0 c p (T ′ ) dT ′ , where c p is the mixture thermal capacity at constant pressure, T 0 a reference temperature and ∆h 0 the corresponding standard formation enthalpy. τ i j and q cd j are the viscous shear stress tensor and the conductive heat flux vector respectively. S i is a driving force source term to obtain the intended bulk Reynolds number in channel flow simulations.

Based on the Sub-Grid Scale (SGS) eddy-viscosity concept, the SGS stress tensor τ SGS i j = -ρ( u i u j -u i u j ) and the SGS heat flux q SGS j = ρ( hu j -h u j ) are modeled as

τ SGS i j - 1 3 δ i j τ SGS kk = 2ρν SGS ( S i j - 1 3 δ i j S kk ), ( 10
)
q SGS j = -λ SGS ∂ T ∂x j , (11)
where S i j is the filtered shear stress tensor. The Sigma model [START_REF] Nicoud | Using singular values to build a subgrid-scale model for large eddy simulations[END_REF] is used for modeling the SGS kinematic viscosity ν SGS . The SGS thermal conductivity λ SGS is computed from the SGS Prandtl number Pr SGS which is set to 0.9

λ SGS = ρ c p ν SGS Pr SGS . ( 12
)

Radiation model

For the radiation field, a Monte-Carlo method is used to calculate the radiative power field in the outer layer, i.e. at all the LES grid points. For the inner layer, the radiation field is analytically obtained from a one-dimensional model that uses the intensity field obtained from the Monte Carlo approach as a boundary condition for the point with wall distance y w , where information from the LES grid is fed back to the wall model.

Monte Carlo approach

The general organization of the radiation model, based on a reciprocal Monte Carlo approach, has been detailed by Tessé et al. [START_REF] Tessé | Radiative transfer in real gases using reciprocal and forward Monte Carlo methods and a correlated-k approach[END_REF]. The precise approach used here is the Optimized Emissionbased Reciprocity Monte Carlo Method (OERM) [START_REF] Zhang | Optimized Emission-based Reciprocity Monte Carlo Method to speed up computation in complex systems[END_REF], as in Ref. [START_REF] Zhang | Physical study of radiation effects on the boundary layer structure in a turbulent channel flow[END_REF]. This method allows the convergence to be locally controlled while it overcomes the drawback of the original Emissionbased Reciprocity Method [START_REF] Tessé | Radiative transfer in real gases using reciprocal and forward Monte Carlo methods and a correlated-k approach[END_REF] in the cold region and greatly increases the computational efficiency. This method also allows the spectral anisotropic radiation intensity field I - ν (y w , µ) at any point of abscissa y w of the LES grid, required by the inner layer radiation model, to be determined, as detailed in Sec. 2.2.2 and Appendix. The one-dimensional configuration for radiation in the inner layer is shown in Fig. 2. The opaque wall is isothermal at temperature T w and its emissivity ǫ ν is assumed isotropic. At any point y of the inner layer, the radiative power per unit volume P R (y), difference between absorbed and emitted powers, writes P R (y) = P a (y) -P e (y) ( 13)

Analytical radiation model in the inner

I - ν (y, µ 2 ) θ 1 θ 2 T w ǫ ν
P e (y) = 4π +∞ 0 κ ν (y)I • ν (y)dν ( 14
)
P a (y) = +∞ 0 κ ν (y) 0 -1 I - ν (y, µ)2πdµ + 1 0 I + ν (y, µ)2πdµ dν ( 15
)
where ν is the radiation wave number, µ is the cosine of the angle θ defined in Fig. 2, and I • ν (y) is the equilibrium spectral intensity at the temperature T (y) associated with the current position y. The local spectral intensity in a positive y direction I + ν (y, µ), associated with µ > 0, and the spectral intensity in a negative direction I - ν (y, µ), associated with µ < 0 are given by

I + ν (y, µ) = τ ν0y (µ)I + ν (0) + y 0 κ ν (y ′ )I • ν (y ′ )τ νy ′ y (µ)
dy ′ µ with: y ′ < y; dy ′ > 0; µ > 0, ( 16)

I - ν (y, µ) = τ νy w y (µ)I - ν (y w , µ) + y y w κ ν (y ′ )I • ν (y ′ )τ νy ′ y (µ) dy ′ µ with: y ′ > y; dy ′ < 0; µ < 0, (17)
where

I + ν (0) = ǫ ν I • ν (T w ) -2 (1 -ǫ ν ) 0 -1 I - ν (0, µ) µdµ, (18)
and τ ′ νy ′ y is the directional spectral transmissivity between y ′ and y given by

τ νy ′ y (µ) = exp[-e ν (y ′ , y)/µ] with : e ν (y ′ , y) = y y ′ κ ν (y ′′ )dy ′′ . (19)
Here e ν (y ′ , y)/µ > 0 is the spectral optical thickness between y ′ and y in the direction µ while κ ν is the spectral absorption coefficient of the medium. Note that: i) dy ′ and µ in Eqs. ( 16) and [START_REF] Nicoud | Large eddy simulation wall-modeling based on suboptimal control theory and linear stochastic estimation[END_REF], and e ν (y ′ , y) and µ in Eq. ( 19) are simultaneously positive or negative; ii) As the wall reflection is assumed diffuse, the intensity leaving a wall I + ν (0) is isotropic, whereas I - ν (y w , µ) the intensity entering the inner layer at abscissa y w is anisotropic.

By introducing the exponential integral function, detailed in Ref. [START_REF] Modest | Radiative Heat Transfer[END_REF], i.e.

E n (X) = 1 0 µ n-2 exp(-X/µ)dµ [START_REF] Benarafa | Temperature wall modelling for large-eddy simulation in a heated turbulent plane channel flow[END_REF] and the generalized incomplete function defined by

E n (X, [µ j , µ j+1 ]) = µ j+1 µ j µ n-2 exp(-X/µ)dµ, (21)
a discretized expression of the radiative power P R (y) is simply given by

P R (y) ≈ 2π +∞ 0 κ ν (y)          N µ j=1 E 2 (e ν (y, y w ), [µ j , µ j+1 ]) I - ν (y w , [µ j , µ j+1 ])          dν +2π +∞ 0 κ ν (y) E 2 (e ν (0, y)) I + ν (0) dν +2π +∞ 0 κ ν (y) y w 0 κ ν (y ′ )I • ν (y ′ ) E 1 (e ν (y ′ , y)) dy ′ dν, -4π +∞ 0 κ ν (y) I • ν (y) dν. ( 22
)
where N µ is the number of angular sectors [µ j , µ j+1 ] used to discretize the 2π steradians associated with the incoming intensity at a point y w .

I - ν (y w , [µ j , µ j+1 ]) is the average value of I - ν (y w , µ) over the range [µ j , µ j+1 ] 1 .
The mean anisotropic spectral incoming intensity field I ν (y w , [µ j , µ j+1 ]) at any grid point y w is determined by the Monte Carlo method, as detailed in Appendix.

In this analytical radiation model in the boundary inner layer, the radiative power P R (x) is a function of the temperature field within the fluid inner layer model given by Eq. ( 1). It is then worth noticing that the effects of turbulence fluctuations on the radiative power are not accounted for within the inner layer only. In fact, these effects can indeed be neglected in channel flows with non-reacting gases where fluctuations of temperature remain moderate, as shown by Ref. [START_REF] Gupta | Large-Eddy Simulation of Turbulence-Radiation Interactions in a Turbulent Planar Channel Flow[END_REF] and by post-processing of DNS results in Ref. [START_REF] Zhang | Physical study of radiation effects on the boundary layer structure in a turbulent channel flow[END_REF] .

Coupling of inner and outer layers radiation and turbulence models

The purpose of wall-modeled LES is to overcome the under-resolution of the boundary inner layer, which leads to erroneous estimations of wall temperature and velocity gradients. For each LES grid point on the wall and each time step, the set of equations ( 1) combined with the analytical radiation model is solved with an iterative procedure and provides an accurate estimation of the wall stress τ w and wall conductive heat flux q cd w . The procedure used to couple LES, Monte-Carlo model and the radiation and turbulence wall models is schematized in Fig. 3.

For each grid point on the wall, LES provides velocity u y w and temperature T y w at a distance y w from the wall to the 1D turbulence model. It also provides the resolved temperature field T LES to the Monte-Carlo method.

The Monte-Carlo method calculates both the radiative power P R MC , source term of the energy balance equation in LES, and the incoming intensities I ν (y w , [µ j , µ j+1 ]) required by the 1D radiation model.

Note that in the Monte Carlo method, when a ray enters the inner layer, the 1D-model temperature field from the previous iteration is used to determine the exchanged energy between the point initiating the ray and the inner layer. The accuracy of the results is then improved when compared to results associated only with the LES temperature field (much coarser in the near wall region).

Finally, the effect of delayed information between LES and both the wall model and radiation solver is here not investigated. Therefore, the coupling among LES, the Monte-Carlo method and the 1D wall model is carried out for each iteration of the LES solver. During the iterative procedure of the wall model between the 1D equations for the fluid fields and the 1D radiation model, the incoming intensity I ν (y w , [µ j , µ j+1 ]) specified by the Monte-Carlo method is fixed.

Separate validation of the turbulence and radiation wall model components

The previously described 1D turbulence wall model is a priori validated against DNS data from Ref. [START_REF] Zhang | Physical study of radiation effects on the boundary layer structure in a turbulent channel flow[END_REF] accounting or not for radiation. This a priori study consists in comparing 1D profiles of averaged quantities predicted by the wall-model while taking some information from DNS cases.

LES

Monte-Carlo

T LES P R MC τ w , q cd w u y w , T y w T 1D I - ν (y w , [µ j , µ j+1 ]) 1D Wall Model Fluid/Turbulence Radiation P R 1D T 1D
Figure 3: Scheme of coupling between LES, the Monte-Carlo method and the 1D wall models located at each point of the walls. T 1D and P R 1D are the inner layer temperature and radiative power fields on the embedded grids.

DNS cases from Ref. [START_REF] Zhang | Physical study of radiation effects on the boundary layer structure in a turbulent channel flow[END_REF] that are retained in this study are defined in Table 1. For cases A, B, C and D, the temperature of the two walls T w,h and T w,c are different, as shown in Fig. 4, and radiation is not accounted for. For cases A, B and C, the corresponding cases with radiation are named A RAD, B RAD and C RAD respectively and opaque wall emissivities are set to 0.8. Two different cases at one atmosphere, named D RAD03 and D RAD08, are considered for case D with wall emissivities of 0.3 and 0.8 respectively. The gas radiative properties are modeled by using the CK model [START_REF] Soufiani | High temperature gas radiative property parameters of statistical narrow-band model for H 2 O, CO 2 and CO, and correlated-K model for H 2 O and CO 2[END_REF] for atmospheric pressure cases and its weak absorption limit for cases at 40 atm.

The domain is defined in Fig. 4 with : δ = 0.1m, L X = 2πδ, L Y = 2δ, L Z = πδ. A nonreacting CO 2 -H 2 O-N 2 gas mixture is considered and the molar fractions of CO 2 , H 2 O and N 2 are set to 0.116, 0.155 and 0.729 to mimic combustion exhaust gases. Similarly to Ref. [START_REF] Zhang | Physical study of radiation effects on the boundary layer structure in a turbulent channel flow[END_REF], the dynamic viscosity µ and thermal capacity c p are computed as functions of temperature from the CHEMKIN package [START_REF] Kee | A Fortran Computer Code Package For The Evaluation Of Gas-Phase, Multicomponent Transport Properties[END_REF][START_REF] Kee | CHEMKIN-II: A Fortran Chemical Kinetics Package for the Analysis of Gas-Phase Chemical Kinetics[END_REF] for the chosen mixture composition. The thermal conductivity λ is computed from the Prandtl number Pr= 0.71.

In the following sections, different wall-model components are separately validated.

Validation of the turbulence wall model without radiation

An integration of Eq. ( 1), in which the wall friction stress and wall heat flux results of DNS are imposed as boundary conditions, leads to velocity and temperature profiles in the near wall region. Since the proposed mixing-length model is only valid in the inner layer, only results within the near wall region (0, 0.2δ) are presented. The corresponding y + values at y = 0.2δ in wall coordinate are given in Tab. 2 for the different studied cases.

Velocity profile of case B that presents the most important wall temperature and gas properties variations is shown in Fig. 5. The velocity profile is under-predicted by the wall model based on standard mixing-length model, Eq. ( 3). The semi-local model, Eq. ( 4), enables to correct this Figure 4: Computational domain of channel flow cases. X, Y and Z are the streamwise, wall-normal and spanwise directions. L X , L Y and L Z are the dimensions of the channel case in each direction. δ is the channel half-width. The lower wall (resp. upper wall) is at temperature T w,c (resp. T w,h ; T w,h ≥ T w,c ). Periodic boundary conditions are applied along X and Z.

Re b Re D h T w,c [K] T w,h [K] p [atm]

A

B

Ref. [START_REF] Hoyas | Scaling of the velocity fluctuations in turbulent channels up to Re τ = 2003[END_REF] Figure 5: Mean velocity profile in wall units, i.e. {u}/u τ , of case B (only the cold side) and Ref. [START_REF] Hoyas | Scaling of the velocity fluctuations in turbulent channels up to Re τ = 2003[END_REF] (circles: DNS reference data; black line: Semi-local mixing-length model; gray line: Standard wall mixing-length model). For the sake of clarity, curves of the different cases are translated. effect. When temperature variations are negligible, both models are identical and perform accurately as shown in Fig. 5 for the DNS case from Ref. [START_REF] Hoyas | Scaling of the velocity fluctuations in turbulent channels up to Re τ = 2003[END_REF] characterized by constant temperature and a larger Reynolds number.

Regarding the temperature profile, the two mixing-length models and the three models of turbulent Prandtl number defined in Sec. 2.1.1 are investigated in Fig. 6. Best results are obtained using the semi-local mixing-length model and Eq. ( 6) for the turbulent Prandtl number.

These modeling conditions are henceforth retained. Velocity and temperature profiles for cases A and C are shown in Fig. 7. Very good agreement with DNS data is observed.

Validation of the turbulence wall model with prescribed radiative power field

In this validation case, the reference radiative power issued from the radiation Monte Carlo method coupled to DNS is also used in addition to the DNS wall friction stress and wall heat flux in Eq. [START_REF] Piomelli | Wall-layer models for large-eddy simulations[END_REF].

Figure 8 shows that the complete wall model (semi-local mixing-length model, Eq. ( 6) for Pr t and radiative power source term) accurately predicts the temperature field for the two near wall regions of the three cases A RAD, B RAD and C RAD. Note that the results of wall models that do not account for radiative power source term strongly deviate from the DNS corresponding results, which indicates that in these three cases, radiation strongly modifies the mean temperature field within the inner layer.

Validation of the radiation wall model with prescribed temperature field

In order to validate the radiative analytical wall model, Eq. ( 22) is solved using the mean temperature profile obtained from DNS data in Ref. [START_REF] Zhang | Physical study of radiation effects on the boundary layer structure in a turbulent channel flow[END_REF] and N µ spectral anisotropic incoming intensity values issued from the reference cases post-processing as explained in Appendix. The influence of the number of angular sectors N µ and of the wall distance y w where anisotropic intensities are calculated is investigated. Results are shown in Fig. 9. The radiative power is accurately predicted with N µ = 2 at each value of y w between 0.05δ and 0.2δ. N µ = 2 is therefore used in the following. Note that results associated with N µ = 1 are practically acceptable. 5) for Pr t ; plain line: Eq. ( 6) for Pr t ).

Results of the turbulence and radiation wall models fully coupled with LES and Monte Carlo method

In this section, the turbulence and radiation wall models are fully coupled to LES and Monte Carlo models for the same cases A to D as in the previous section without prior knowledge from the reference DNS cases. For all these cases, the LES grid consists of 36 × 36 × 36 points and the first off-wall point locates at y + = 15, while all the other grid spacings are uniform. The wall model, defined by Eqs. [START_REF] Piomelli | Wall-layer models for large-eddy simulations[END_REF][START_REF] Gupta | Large-Eddy Simulation of Turbulence-Radiation Interactions in a Turbulent Planar Channel Flow[END_REF], is resolved at each point on the wall on a local embedded grid (see Fig. 1). Given first guesses of the wall shear stress and wall heat flux, this set of equations is iteratively solved while ensuring grid convergence.

Cases without radiation

When coupling LES and wall models, numerical and sub-grid model errors at the first few points near the wall induce perturbations within the LES domain [START_REF] Cabot | Approximate wall boundary conditions in the large-eddy simulation of high reynolds number flow[END_REF][START_REF] Nicoud | Large eddy simulation wall-modeling based on suboptimal control theory and linear stochastic estimation[END_REF], leading to the socalled "logarithmic law mismatch" [START_REF] Piomelli | Wall-layer models for large-eddy simulations[END_REF]. The term "logarithmic law mismatch" specifically refers to cases where a logarithmic law is found for the velocity profile. A similar mismatch between the wall modeled LES and the reference results can also be observed for the temperature profile of the different cases studied here, where the temperature law is not logarithmic as soon as radiation is accounted for.

In order to reduce this error, Kawai et al. [START_REF] Kawai | Wall-modeling in large eddy simulation: Length scales, grid resolution, and accuracy[END_REF] have proposed to place the feed back point of the wall model, placed at the wall distance y w , further away from the wall than the first off-wall LES grid point. Following this approach, different y w positions are compared, i.e. y w located at the first point, second and third off-wall points, denoted case Y1, Y2 and Y3 respectively. The corresponding embedded grids, which are stretched along the wall normal direction, contain 30, 60 and 100 points.

Figure 10 shows that, for case A, the case Y3 presents the best agreement for both u + and T + profiles as expected. This trend is retrieved for wall friction stress τ w and conductive heat flux 14 q cd w values shown in Tab. 3. The feed back point position does not affect the root mean square (rms) profiles of u, v, w and T (see Fig. 11). Henceforward, the third off-wall point is retained as a feed back point.

Results for cases B and C are presented in Fig. 12 and Tab. 3 demonstrating the relevance of this choice.

Cases involving radiation at high pressure

The finally proposed wall model based on semi-local mixing-length model, Eq. ( 6) for Pr t , the radiation analytical model described in Sec. 2.2.2 and using the third off-wall point for coupling (case Y3) will be denoted as "new" wall model. In order to demonstrate the accuracy of this model, it is compared to a "standard" wall model based on standard mixing-length model and constant turbulent Prandtl number, that does not account for radiation effects within the inner layer. Note that this "standard" model is also coupled at the third off-wall point and that radiation is treated with the Monte-Carlo method in the outer layer. Results without any wall model are also presented.

DNS results for profiles of u + and T + in cases A RAD, B RAD and C RAD are compared in Fig. 13 to those predicted by wall-modeled LES with the "standard" and "new" approaches. The velocity field is similarly reproduced by both models whereas the "new" model shows a significant improvement of the predicted temperature profile. Values of τ w and q cd w for the different cases are given in Tab. 4 where the same conclusion is retrieved. The wall radiative flux q R * w , without its passive wall-wall component that accounts for radiative energy exchanged between walls while considering the transmissivity of the fluid region, is also given for the case A RAD only, showing a fair agreement between the value predicted by the proposed wall model and the one from DNS results.

Case involving radiation at 1 atm

Cases A to C correspond to optically thick media at 40 atm, in which the effects of radiation are important. Case D, considered in this paragraph and defined in Tab. 1, deals with media at 0 0. Table 3: Comparison of mean wall shear stress τ w and wall conductive heat flux q cd w for cases A, B and C between wall-modeled LES (Y1, Y2, Y3) and DNS results from Ref. [START_REF] Zhang | Physical study of radiation effects on the boundary layer structure in a turbulent channel flow[END_REF]. For case A, Y1, Y2 and Y3 correspond to wall-normal distances of 0.05δ, 0.098δ and 0.156δ respectively. atmospheric pressure. LES of the two cases D RAD03 and D RAD08, with wall emissivities set to 0.3 and 0.8 respectively, with the "new" and "standard" wall models are compared with DNS data from Ref. [START_REF] Zhang | Physical study of radiation effects on the boundary layer structure in a turbulent channel flow[END_REF].

Figure 14 (a) compares the u + profiles that are only shown on the cold side (similar on the hot side) and demonstrates that both models predict the correct result as in the high pressure cases. The same behavior is observed for the wall shear stress in Tab. [START_REF] Hamba | A hybrid rans/les simulation of turbulent channel flow[END_REF].

Regarding the T + profiles (Figs. 14.b and 14.c) and wall conductive fluxes given in Tab. 4, the "new" model leads to the best agreement with DNS-Monte Carlo results compared to the "standard" model. However, since the radiation effects are weaker at 1 atm than at 40 atm, the difference between the results of the two wall-modeled LES are smaller than those at high pressure, especially on the cold side.

Conclusion

A wall model for LES has been proposed to take into account the effects of radiation within the turbulent boundary inner layer. The model follows a two-layer approach with an embedded grid for each point at the wall. It is composed of 1D equilibrium thin boundary layer equations that are solved on each embedded grid along with mixing-length and turbulent Prandtl models and an analytical expression of the radiative source term within the inner layer.

First results have shown that: i) A semi-local scaling of the mixing-length is necessary to account for the variations of gas properties, such as mass density and dynamic viscosity, within the inner layer; ii) A non-constant turbulent Prandtl number, here Eq. ( 6), gives better results; iii) In the studied cases, the mean incoming intensity at the outer boundary of the wall model has required the discretization of the corresponding solid angle into at least two angular sectors to accurately account for anisotropy; iv) The point where information is transmitted from LES to the wall model has been moved away from the wall while remaining in the validity range of the wall-model. Here, the third grid point off the wall has been used in order to decrease the mismatch in the obtained wall laws due to numerical and sub-grid errors in the wall vicinity.

The LES is coupled to a reciprocal Monte-Carlo method in such a way that it could be done in any configuration to tackle radiation and turbulent flows. Here, the combination of the proposed wall model and LES/Monte-Carlo method is compared to DNS/Monte-Carlo results on turbulent channel flows. Coupled results are compared to those obtained with a standard wall model which does not account for radiation effects within the inner layer. Note that this standard model with LES and Monte-Carlo simulations is already quite advanced to accurately predict wall heat transfer in the presence of radiation. Nonetheless, coupled results for high pressure cases show that the new wall-model greatly improves the accuracy of the predicted results compared to a standard wall model. The same trend is retrieved in atmospheric cases even though the magnitude of difference is smaller than the one at high pressure.

Finally, it has been shown that accounting for radiation effects within the modeled boundary inner layer is necessary in several conditions, especially in high pressure applications, as soon as the radiative energy transfer is strong enough to modify the temperature wall law. Deeper investigation is nonetheless still necessary to fully characterize when such effects can be expected and therefore when the proposed wall model should be applied.

to the ranges [µ j , µ j+1 ] as a boundary condition. The emission-based reciprocity Monte Carlo method defined in Ref. [START_REF] Zhang | Physical study of radiation effects on the boundary layer structure in a turbulent channel flow[END_REF] is used in this paper. Consequently, the mean spectral intensities can easily be deduced from reciprocal emission phenomena issued from all the other cells of the system.

The precise used procedure described in Fig. 15 is the following: i) The grid point at wall distance y w is assumed to be a virtual opaque wall of temperature T (y w ) and of arbitrary wall emissivity ǫ = 1; ii) For each range [µ j , µ j+1 ], a large number of shots are emitted in the Monte Carlo method from this virtual wall in a randomly determined direction with the cosine of polar angle within the range [µ j , µ j+1 ], and in any elementary spectral range ∆ν ; iii) The reciprocity method allows Φ ∆ν (y w , [µ j , µ j+1 ]), the contribution to the radiative flux at the virtual wall associated with both [µ j , µ j+1 ] and ∆ν, to be determined; vi) The absorbed flux at the wall Φ a ∆ν (y w , [µ j , µ j+1 ]), and consequently the mean spectral absorbed intensity I a ν (y w , [µ j , µ j+1 ]), associated with [µ j , µ j+1 ] and ∆ν are then calculated as 4: Comparison of mean wall shear stress τ w , wall conductive heat flux q cd w and wall radiative flux q R* w between wall-modeled LES (new, standard and no wall model) and DNS results from Ref. [START_REF] Zhang | Physical study of radiation effects on the boundary layer structure in a turbulent channel flow[END_REF] (the wall-wall radiative flux is not included in q R* w ).

Φ

Figure 1 :

 1 Figure 1: Scheme of wall model grids embedded within the LES grid. In this example, the wall distance y w for the coupling point corresponds to the first LES grid point off the wall.

Figure 2 :

 2 Figure 2: Definition of forward and backward intensities I + ν and I - ν (µ 1 and µ 2 are the cosine of polar angle θ 1 and θ 2 respectively; y = 0 and y = y w are the two boundaries of the 1D model).

Figure 6 :

 6 Figure 6: Mean temperature profile in wall units, i.e. |{T } -T w |/T τ with T τ = |q cd w |/( ρ w c pw u τ ), on the cold side of case B (circles: DNS data from Ref. [25]; black lines: Semi-local mixing-length model; gray lines: Standard mixinglength model; dashed-dotted line: Constant Pr t ; dashed line: Eq. (5) for Pr t ; plain line: Eq. (6) for Pr t ).

Figure 7 :Figure 8 :

 78 Figure 7: Mean velocity profile (a) and mean temperature profile (b) in wall units on the cold side of cases A and C (circles: DNS data from[START_REF] Zhang | Physical study of radiation effects on the boundary layer structure in a turbulent channel flow[END_REF]; plain line: Semi-local mixing-length model + Eq. (6) for Pr t ). For the sake of clarity, curves of the different cases are translated.

Figure 9 :Figure 10 :

 910 Figure 9: Radiative power in W.m -3 on the cold side of case B with y w = 0.05δ (a), y w = 0.1δ (b) and y w = 0.2δ (c) (circles: DNS data from Ref. [25]; dashed-dotted line: N µ = 1; dashed line: N µ = 2; plain line: N µ = 3).

Figure 11 :

 11 Figure 11: Rms of velocity components u (a), v (b), w (c) and temperature (d) in wall units on the cold side of case A (circles: DNS data from [25]; dashed-dotted line: Y1; dashed line: Y2; plain line: Y3).

Figure 12 :Figure 13 :

 1213 Figure 12: Mean velocity (a) and mean temperature (b) profile in wall units on the cold side of cases B and C (circles: DNS data from[START_REF] Zhang | Physical study of radiation effects on the boundary layer structure in a turbulent channel flow[END_REF]; plain line: Y3). For the sake of clarity, curves of the different cases are translated.

Figure 14 :

 14 Figure 14: Mean velocity profile in wall units on the cold side (a), mean temperature profile in wall units on the cold side (b) and hot side (c) of cases D RAD03 and D RAD08 (circles: DNS data from [25]; dashed-dotted line: No wall model; dashed line: Standard wall model; plain line: New wall model). Curves of the different cases are translated.

  • sr -1 ].

	L	Length [m].
	p	Pressure [Pa].
	Pr	Prandtl number [-].
	P	Power per unit volume [W•m -3 ].
	q	Heat flux [W•m -2 ].
	Re	Reynolds number [-].
	S i	Momentum source term [N•m -3 ].
	t	Time [s].
	T	Temperature [K].
	u	Streamwise velocity component [m•s -1 ].
	u i , u j	Velocity vector [m•s -1 ].
	X, Y, Z	Cartesian coordinates [m].
	x Greek Symbols
	δ	Channel half-width [m].
	δ ij	Kronecker delta operator [-].

i Coordinate vector (tensorial) [m]. y Distance to a wall [m]. κ Spectral absorption coefficient [m -1 ]. λ Thermal conductivity [W•K -1 •m -1 ]. ρ Gas mass density [kg•m -3 ]. τ ij Viscous shear stress tensor [N•m -2 ]. Ω Solid angle [sr]. ν Radiation wave number [cm -1 ] or kinematic viscosity [m 2 /s]. θ Polar angle [sr]. µ

Table 1 :

 1 Channel flow parameters: Bulk Reynolds number Re b , Reynolds number based on the hydraulic diameter D

			B	5850	23400	950 950	1150 2050	40.0 40.0
			C	11750	47000	950	1150	40.0
			D	5850	23400	950	1150	1.0
										C1, C3, C4 and C2
	in Ref. [25] respectively, and radiative cases A RAD, B RAD, C RAD, D RAD03 and D RAD08 to cases C1R1, C3R1,
	C4R1, C2R2 and C2R1 respectively.
		A	A RAD	B	B RAD	C	C RAD	D	D RAD03	D RAD08
	Cold side 77.3	78.2	115.7	133.6	143.8	144.6	77.3	77.3	77.3
	Hot side	61.1	66.7	45.9	52.1	111.4	113.4	61.0	61.3	61.4

h = 4δ, wall temperatures and pressure. Cases without radiation A, B, C and D correspond to cases

Table 2 :

 2 Wall normal distance y + in wall units corresponding to 0.2δ on both sides in each case.

  a ∆ν (y w , [µ j , µ j+1 ]) = Φ ∆ν (y w , [µ j , µ j+1 ]) + Φ e ∆ν (y w , [µ j , µ j+1 ]), (23)I a ν (y w , [µ j , µ j+1 ]) = Φ a ∆ν (y w , [µ j , µ j+1 ]) 2π , [µ j , µ j+1 ]), as the virtual wall absorptivity has been set to unity.

			side	DNS	new model	standard model	no model
		τ w	cold hot	1.67E-3 1.59E-3	1.64E-3 1.63E-3	1.68E-3 1.61E-3	1.0E-3 1.09E-3
	A RAD	q cd w	cold hot	1230 960	1270 1030		918 631	443 363
	µ j+1 µ j 6970 µ dµ ∆ν 7240 v) The mean spectral incoming intensity I cold q R* w hot cold 5.0E-3 4.8E-3 = 6630 7060 τ w hot 4.6E-3 4.6E-3 B RAD cold 16260 16041 q cd w hot 8720 8911	Φ ∆ν (y w , [µ j , µ j+1 ]) 2π µ j+1 µ j µ dµ ∆ν --4.82E-3 4.37E-3 10985 3738	--ν (y w ); + I 0 3.1E-3 3.1E-3 6583 3303	(24)
	C RAD	τ w q cd w	cold hot cold hot	5.7E-3 5.5E-3 1650 1290	5.6E-3 5.4E-3 1666 1358		5.71E-3 5.37E-3 1288 925	3.24E-3 3.51E-3 660 569
	D RAD03	τ w q cd w	cold hot cold hot	6.6E-3 6.5E-3 1220 1100	6.3E-3 6.3E-3 1180 1055		6.5E-3 6.4E-3 1105 955	4.0E-3 4.4E-3 677 622
	D RAD08	τ w q cd w	cold hot cold hot	6.6E-3 6.5E-3 1070 930	6.4E-3 6.6E-3 1040 926		6.6E-3 6.3E-3 1018 821	4.1E-3 4.5E-3 773 750
	Table

ν (y w , [µ j , µ j+1 ]) required by the wall model is then equal to I a ν (y w

If I - ν (y w , µ) were isotropic the first term of the second member of Eq. (22) would be similar to the classical result of the second one.

Acknowledgments

We thank the China Scholarship Council (CSC) for granting a doctoral fellowship to the first author. This work was granted access to the HPC resources of [CCRT/CINES/IDRIS] under the allocations 2012-2b0164 and 2013-2b0164 made by GENCI (Grand Equipement National de Calcul Intensif). We are grateful to Dr. Philippe Riviere for providing the database of the Correlated-K model. We also acknowledge Dr. Vincent Moureau and the SUCCESS scientific group for providing the code YALES2 and support.

Appendix : Determination of the incoming intensity field for the wall model physical wall virtual wall ), boundary condition of the wall model, is computed from the reciprocal Monte-Carlo method by using a virtual wall. µ j and µ j+1 correspond to the cosine of angles θ j and θ j+1 respectively.

As described in Sec. 2.2.2, the proposed wall model requires mean anisotropic spectral incoming intensities I ν (y w , [µ j , µ j+1 ]) at a grid point y w averaged over the solid angles corresponding