Morgane Bellec
email: morgane.bellec@gresille.org

Alain Girard
email: alain.girard@cea.fr

Guillaume Balarac

Ulrich Bieder

Nicolas Luchier

Effect of turbulence on the wavefront of an ultra high intensity laser beam

Ultra high intensity lasers face thermal management issues that limit their repetition rates. The key challenge is to efficiently evacuate the heat deposited in the amplifier by the optical pumping without impacting the output laser beam quality. The amplifier can have a multi-slab geometry where the laser beam crosses successive amplifying slabs and the cooling channels that separate them. The present work investigates numerically how a cryogenic cooling of the multi-slab amplifier by turbulent channel flows may affect the wavefront of the laser beam. To this end, Large Eddy Simulations representative of the amplifier cooling are performed using TrioCFD, a code developed by the CEA. First, validation simulations are carried out for heated channel flows, allowing comparisons to Direct Numerical Simulation results from the literature. Then Large Eddy Simu-

INTRODUCTION

There is nowadays in Europe a strong interest in the field of high power lasers. Several multipetawatt lasers are currently in operation, among which the three ELI projects in eastern Europe [START_REF] Wills | ELI: Open for Business[END_REF] and Apollon in France [START_REF] Papadopoulos | First commissioning results of the Apollon laser on the 1 PW beam line[END_REF]. When designing the solid-state amplifier of a high power laser, thermal management is a critical element to attain a high repetition rate [START_REF] Garrec | Laser-diode and Flash Lamp Pumped Solid-State Lasers[END_REF]. The solid-state laser amplifier DiPOLE100 (105 J per pulse) demonstrated stable operation at a repetition rate of 10 Hz using cryogenically cooled multi-slab amplifiers [START_REF] Navratil | Characterization of Bivoj/DiPOLE 100: HiLASE 100-J/10-Hz diode pumped solid state laser[END_REF]. Cooling the amplifying material at cryogenic temperatures, i.e. below 150 K, presents advantages both from the thermal and the optical point of view, as reviewed by Brown et al. [START_REF] Brown | The Application of Cryogenic Laser Physics to the Development of High Average Power Ultra-Short Pulse Lasers[END_REF]. Compared to room temperature, the thermal conductivity of amplifying materials is higher at low temperature. A more homogeneous temperature is therefore expected in the amplifying material, reducing the thermo-mechanical stresses and the geometric deformations. The optical performances are also enhanced, especially for ytterbium-doped crystals: from quasi-three level laser materials at 300 K they become four level materials at 77 K, thus increasing the overall laser efficiency [START_REF] Brown | The promise of cryogenic solid-state lasers[END_REF].

In a cryogenic multi-slab amplifier, the amplifying material is split into several slabs, which allows to increase the heat exchange surface [START_REF] Mason | Scalable design for a high energy cryogenic gas cooled diode pumped laser amplifier[END_REF]. The slabs are separated by cooling channel flows of cryogenic gaseous helium. In this configuration, the laser beam goes perpendicularly through the channel flows. The homogeneity of the refractive index in the media crossed by the laser beam is crucial to ensure the output beam coherence. As stated by the Gladstone-Dale law [START_REF] Gladstone | XIV. Researches on the refraction, dispersion, and sensitiveness of liquids[END_REF], the refractive index of a fluid varies with its density and thus with its temperature. Helium has the desired optical properties for this application: its thermo-optic coefficient at 100 K is (dn/dT ) f = 4.6 × 10 -6 K -1 [START_REF] Arp | Thermophysical Properties of Helium-4 from 0.8 to 1500 K with Pressures to[END_REF]. By comparison, the thermo-optic coefficient of 1%-doped Yb:YAG (ytterbium-doped crystal of yttrium aluminum garnet) at 100 K and λ = 1034nm is (dn/dT ) s = 10 -6 K -1 [START_REF] Cardinali | Mat ériaux lasers dop és à l'ion ytterbium: Performances lasers en pompage par diodes lasers et étude des propri ét és thermo-optiques à des temp ératures cryog éniques[END_REF]. Due to the low kinematic viscosity of cryogenic helium (ν f = 3.10 -6 m 2 /s at 80 K and 5 bar [START_REF] Arp | Thermophysical Properties of Helium-4 from 0.8 to 1500 K with Pressures to[END_REF]), the cooling channel flows will be turbulent at the mass flow rates needed to evacuate the heating in the amplifier. The velocity fluctuations induce temperature fluctuations in the fluid. The consequent refractive index inhomogeneities, however small given the fluid thermo-optic coefficient, may not be negligible. Their impact on the laser beam quality must be assessed.

Light propagation through a turbulent medium has first been studied by astronomers: the atmospheric turbulence usually degrades the resolution of a telescope [START_REF] Roggemann | Imaging through Turbulence[END_REF]. Various techniques, including adaptive optics, were developed to mitigate those effects. Turbulence also affects the performance of airborne laser systems [START_REF] Gordeyev | Far-field laser intensity drop-outs caused by turbulent boundary layers[END_REF]. In this case the flow is compressible. The laser beam propagates through a medium with an inhomogeneous refractive index because of the density variations. Different parts of its wavefront travel at different speeds, causing wavefront aberrations. To predict the wavefront aberrations, Sutton [START_REF] Sutton | Aero-optical foundations and applications[END_REF] developed a model linking the wavefront phase distortion with the variance and the correlation length of the fluctuating density. One of the assumptions made is that the turbulent flow is homogeneous and isotropic. This assumption is met neither in the airborne applications where the considered flow is a compressible boundary layer, nor in the case of the cooling channels with developing thermal boundary layers. Tromeur et al. [START_REF] Tromeur | Analysis of the Sutton Model for Aero-Optical Properties of Compressible Boundary Layers[END_REF][START_REF] Tromeur | Large-eddy simulation of aero-optical effects in a spatially developing turbulent boundary layer[END_REF] tested the Sutton model in a turbulent boundary layer flow by comparing it to Large Eddy Simulations (LES) and to experimental results. They recommend the use of LES to correctly capture the aero-optic effects as they found that the Sutton model was leading to significantly different wavefront aberrations compared to experiments. However, Wang and Wang [START_REF] Wang | Aero-optics of subsonic turbulent boundary layers[END_REF] questioned Tromeur et al.'s formulation of the Sutton model, pointing this as the cause for their unsatisfactory results. They found good agreement between the Sutton model predictions and the directly integrated LES results by defining appropriately the density length scale.

Truman and Lee [START_REF] Truman | Effects of organized turbulence structures on the phase distortion in a coherent optical beam propagating through a turbulent shear flow[END_REF] studied the propagation of an optical beam through a uniform shear turbulent flow without wall boundary. They used the instantaneous passive-scalar field results obtained by Direct Numerical Simulation (DNS) of such a flow to represent the fluctuating refractive index field.

They found a significant phase distortion of the optical beam that can be minimized by adjusting the angle of the beam so that the direction of propagation is approximately normal to the turbulent structures. To the best of the authors' knowledge, there exists no such study in an actual wall-bounded flow with developing thermal boundary layers. Most aero-optics studies of flows with heat exchange are experimental [START_REF] Gordeyev | Aero-optical measurements in a subsonic, turbulent boundary layer with non-adiabatic walls[END_REF]. The closest case was conducted again by Truman [START_REF] Truman | The influence of turbulent structure on optical phase distortion through turbulent shear flows[END_REF] in a periodic turbulent channel flow at low Reynolds number (Re τ = 180) by DNS. He showed that the phase distortion is mainly due to the large scale turbulent structures of the flow. Filtering out the small scale fluctuations (arbitrarily chosen in his case as smaller than one root mean square) permitted to retrieve 85% of the phase distortion. This advocates for the usage of LES in aero-optics studies.

The aim of this work is to analyze the effect of the turbulent gaseous cooling of a laser amplifier on the wavefront of the laser beam using TrioCFD, a code developed by the CEA [START_REF] Angeli | Overview of the TrioCFD code: Main features, V&V procedures and typical applications to nuclear engineering[END_REF]. The mean phase distortion could be compensated by adaptive optics, but not the fluctuations. It is therefore crucial to assess their impact on the laser beam quality in order to provide some guidance for future design and operation. As the statistical approach (i.e. Reynolds-Averaged Navier-Stokes simulations) is inadequate to investigate the phase distortion fluctuations, Large Eddy Simulations are carried out. To represent the cryogenic cooling of a high power and high repetition rate laser, a plane turbulent open channel flow is first simulated. The fluid, gaseous helium at 80 K, cools two solid slabs made of Yb:YAG amplifying crystal. The resulting temperature distribution is analyzed from the optical point of view. The paper is organized as follows: the section 2 describes the numerical methods, both for the thermo-hydraulics and the optics aspects; the section 2.2 assesses the numerical model by comparing dynamical and thermal profiles to the literature; the section 3.1 presents the temperature field in the amplifier and the section 3.2 investigates its effect on the laser beam; finally the section 3.3 deals with the consequences of an increased optical heating of the slabs.

NUMERICAL METHODS

Large Eddy Simulation

The purpose of this work is to investigate the cryogenic cooling of amplifiers by a gaseous helium flow for high power lasers. The characteristics of the studied case were chosen so as to be as representative as possible of an actual multi-slab amplifier, while accounting for computational cost considerations. Figure 1 shows a schematic view of the computational domains. Only a portion of one of the cooling channels is simulated. The dimensions of the main channel are

0.1 m × 0.1 m [21]
), but the crosswise dimensions are respected. The flow is assumed to be fully developed, with a bulk velocity of 10 m/s. That corresponds to a bulk Reynolds number of Re b = 14000. To generate the velocity inlet, a classical technique (see for example [START_REF] Sagaut | Large Eddy Simulation for Incompressible Flows[END_REF]) is used: a so-called precursor channel is simulated. It is an isothermal bi-periodic channel whose spanwise and crosswise dimensions are the same as the main channel ones, but with half the length.

The properties of the fluid and the solid are summarized in Tab. 1. They are conventional values for a cryogenically cooled amplifier. The fluid is helium at 80 K and 5 bar, whose properties are taken from the database Hepak [START_REF] Arp | Thermophysical Properties of Helium-4 from 0.8 to 1500 K with Pressures to[END_REF]. The properties are taken constant and the flow is considered incompressible. The solid properties are given for 1%-doped Yb:YAG at 80 K, as measured by Cardinali [START_REF] Cardinali | Mat ériaux lasers dop és à l'ion ytterbium: Performances lasers en pompage par diodes lasers et étude des propri ét és thermo-optiques à des temp ératures cryog éniques[END_REF]. A bulk heating of 9.2 W is deposited homogeneously in the two solid volumes.

The fluid evacuates it via conjugate heat transfer with the solid. This heat deposit is proportional in volume to a typical heat deposit in a laser amplifier [START_REF] Divoky | Conceptual design of 100 J cryogenically-cooled multi-slab laser for fusion research[END_REF]. In an actual amplifier however, the heat is deposited at discrete times corresponding to the pumping pulses. Typical pulses last a few nanoseconds, and they have a repetition rate of about 10 Hz. A thermal balance calculation shows that the solid temperature decreases by about 1 K, i.e. about 1%, between each pulse at thermal equilibrium. In this work, the heating is constant in time, and the results presented are taken at thermal equilibrium. This is justified below by the different time scales encountered.

The governing equations are the incompressible Navier-Stokes equations for the fluid, and the energy conservation equations for the fluid and the solid. In Large Eddy Simulation, or LES, (see e.g. [START_REF] Sagaut | Large Eddy Simulation for Incompressible Flows[END_REF]) the large scales of turbulence are resolved while the effect of the small scales (smaller than the grid characteristic size) is modeled. The sub-grid scale model used in this work is the walladapting local eddy-viscosity (WALE) model of Nicoud and Ducros [START_REF] Nicoud | Subgrid-scale stress modelling based on the square of the velocity gradient tensor[END_REF]. The governing equations are written below, where ∼ denotes the LES filtering.

                                               ∂ ∼ u i ∂x i = 0 ∂ ∼ u i ∂t + ∂ ∼ u i ∼ u j ∂x j = - 1 ρ f ∂ ∼ P ∂x i + ν ∂ ∂x j ∂ ∼ u i ∂x j + ∂ ∼ u j ∂x i - ∂τ ij ∂x j ∂ ∼ T ∂t + ∂ ∼ u i ∼ T ∂x i = α f ∂ 2 ∼ T ∂x 2 i - ∂θ i ∂x i ∂T ∂t = α s ∂ 2 T ∂x 2 i + 1 C ps ρ s Q (1)
Here, u i are the velocity components in the x i directions, P is the pressure, T is the temperature.

The material properties -the kinematic viscosity ν, the density ρ, the thermal diffusivity α and the heat capacity C p -have the subscript s or f referring to solid or fluid respectively. Q is the volumetric heat deposit. Finally,

τ ij = ∼ u i u j - ∼ u i ∼ u j and θ i = ∼ u i T - ∼ u i ∼
T are respectively the sub-grid scale stress tensor and the sub-grid scale temperature flux to model in LES.

The simulations were performed using TrioCFD [START_REF] Angeli | Overview of the TrioCFD code: Main features, V&V procedures and typical applications to nuclear engineering[END_REF], an open-source Computational Fluid Dynamics code supported by the CEA, formerly known as Trio U [START_REF] Calvin | An object-oriented approach to the design of fluid mechanics software[END_REF]. We capitalize on the work performed over the past years with TrioCFD, to determine the suitable parameters for the simu- following a tangential hyperbolic growing law. These sizes are typical resolutions for a LES, which is thus expected to correctly capture the aero-optical effects according to Mani et. al. analysis of grid requirements [START_REF] Mani | Resolution requirements for aero-optical simulations[END_REF]. Each solid plate mesh consists of 120 × 30 × 240 cells homogeneously distributed. Although we are confident that such dimensions allow to obtain satisfactory results, we will perform some validation in section 2.2.

Time marching is explicit and relies on a third order Runge-Kutta scheme. Second order centered schemes are used for the energy and momentum diffusion, as well as the dynamic convection.

The thermal convection uses a third order QUICK scheme (Quadratic Upstream Interpolation for Convective Kinematics). The time step varies between 7 × 10 -7 s and 9 × 10 -7 s, it is chosen so as to ensure the stability condition: Courant number of 1. The simulations were run on the HPC resources of CINES, and 252 cores were used in parallel.

A major challenge presented by this simulation comes from the different characteristic times of the phenomena investigated: the solid conduction is slow compared to the turbulent convection. A heat balance leads to a thermal characteristic time of τ th = ρ s C ps L s h exch = 2.6 s, where L s is the solid plate thickness, and h exch is the exchange coefficient evaluated by the classical Dittus-Boelter correlation [START_REF] Dittus | Heat transfer in automobile radiators of the tubular type[END_REF]. This long time constant of the amplifier with respect to the repetition time (0.1 s)

justifies the assumption of a constant heating. By comparison, the turbulence characteristic time is of three orders of magnitude smaller: τ f l = h u τ = 3.5 × 10 -3 s where h is the channel half-height

Model validation

In the first place, the numerical choices are assessed by comparisons with the literature in cases as close as possible to ours. their normalized form, using the classical scaling [START_REF] Sagaut | Large Eddy Simulation for Incompressible Flows[END_REF].

y + = y u τ ν U + i = u u τ u + irms = u irms u τ (2)
with u τ is the friction velocity, defined by (see e.g. [START_REF] Pope | Turbulent Flows[END_REF])

u τ = ν d u dy w , (3)
where the subscript w denotes the wall (y = 0 or y = 2h).

The mean longitudinal velocity profile, plotted Fig. 2, shows a good agreement between the present LES and the reference DNS in the viscous sub-layer close to the wall. The standardized mean velocity is however under-estimated in the center of the channel by 10% at most. The It is indeed expected that a finer mesh leads to better results, as pointed out again by Sanchez et al. [START_REF] Sanchez | Large Eddy Simulations of Thermal Boundary Layer Spatial Development in a Turbulent Channel Flow[END_REF] in a similar configuration at a lower Reynolds number. The present mesh refinement thus represents a good trade-off between accuracy and computational time. The velocity results could also be improved by adopting a higher order scheme, at the expense of a higher numerical cost. It seems however unnecessary as the mean velocity under-estimation appears to have little impact on the mean temperature (see Fig. 4 discussed below).

In the second place, the temperatures are also assessed by comparison with the literature. To this end, the calculation has been conducted in a case similar to the one presented in section 2 except that the channel was here streamwise periodic. The precursor channel was therefore not needed, and the outlet velocity and temperature fields were used as inlet boundary conditions.

All the numerical and physical characteristics of the plates and of the fluid were unchanged. A negative volumetric thermal source term was added in the fluid in order to exactly compensate the heat deposit in the plates so that the case would reach thermal equilibrium. Figures 4 and 5 present respectively the standardized mean temperature and the temperature root mean square.

The results are compared to Kawamura et al. [START_REF] Kawamura | DNS of turbulent heat transfer in channel flow[END_REF] DNS results. The classical scaling [START_REF] Sagaut | Large Eddy Simulation for Incompressible Flows[END_REF] used is

T + = T w -T T τ T + rms = T rms T τ , (4)
with T w the wall temperature and T τ the friction temperature, calculated from the wall normal gradient of the mean temperature and the friction velocity.

T τ = λ d T dy w ρC p u τ (5)
The mean temperature profile plotted on Fig. 4 2) is therefore expected to be slightly underestimated as well.

Optical calculations in the fluid

The propagation of light is described as an electromagnetic wave following the Maxwell equations, where the medium is non-conducting and with a constant magnetic susceptibility. The speed of light being about 8 order of magnitude higher than the fluid velocity, the turbulence is considered as "frozen" from the laser beam point of view. The refractive index n = c 0 c , with c the speed of light in the medium and c 0 the speed of light in free space, is taken constant in time. The propagation equation of the electric field E is then

∇E 2 = n 2 c 0 2 ∂ 2 E ∂t 2 (6)
Adopting the classical approach used in aero-optics [START_REF] Sutton | Aero-optical foundations and applications[END_REF], the light field is modeled as a monochromatic sinusoidal plane wave propagating from y = 0 to y = L. L is the distance traveled by the light wave. After the crossing of the full height of the channel, L = 2h. The assumption is here that the wavelength is small compared to the smallest turbulent structures and all diffraction effects are neglected. Equation. ( 6) then has a solution,

E(r, t) = E 0 e i(ϕ(r,k)-ωt) , (7)
where r = (x, y, z) is the position, t is the time, E 0 is the constant field amplitude, ϕ is the wave phase at r, k = 2π λ is the wave number and ω is the wave frequency. The wave phase may vary as the laser beam crosses the turbulent medium, and it is written as the sum

ϕ = ϕ 0 + ∆ϕ, (8)
where ϕ 0 is the non-perturbed phase variation and ∆ϕ is the cumulative phase distortion from y = 0 to y = L. ∆ϕ is related to the refractive index perturbation ∆n = n -n 0 along the optical path, where ∆n is the difference to the refractive index in a non-perturbed medium n 0 .

∆ϕ = k L 0 ∆n dy (9)
The wave number k = 2π λ depends on the monochromatic laser beam wavelength. In this work, the wavelength value was λ = 1.034 × 10 -6 m, which is typical for Yb:YAG.

As already mentioned, the refractive index in a gas varies with the density following the Gladstone-Dale law [START_REF] Gladstone | XIV. Researches on the refraction, dispersion, and sensitiveness of liquids[END_REF]. For the present flow calculation, the density varies at most by a few 10 -3 % with the pressure, and at most by a few 10% with the temperature. The variations with pressure are therefore neglected, and the refractive index depends only on the temperature, as

n = 1 + G(λ) × ρ(T ) = 1 + 0.044 T , (10)
where G = 2.25 × 10 -4 m 3 /kg is the Gladstone-Dale constant.

To sum up, the laser beam wavefront deformation after crossing the turbulent channel can be computed from the LES instantaneous temperature field using equations 10 and 9. This method was applied to obtained the results presented below in section 3.2 and 3.3.

The Sutton model [START_REF] Sutton | Aero-optical foundations and applications[END_REF] links the phase distortion fluctuations to the density fluctuations. It originally assumes a homogeneous and isotropic turbulence, but it has been used in different flows including turbulent boundary layers by Wang et. al. [START_REF] Wang | Aero-optics of subsonic turbulent boundary layers[END_REF]. The general form of the linking equation [START_REF] Havener | Optical wave front variance -A study on analytic models in use today[END_REF] is

∆ϕ rms 2 = G 2 k 2 L 0 L 0 R ρ ρ (y, y ) dy dy, (11)
where R ρ ρ is the two-point density correlation in the direction of the light propagation. The difficulty arises from attempted modelings of the right-hand term. The more widely used formulation of the Sutton model bases on the density correlation length Λ. It is equivalent to the previous formulation provided that R ρ ρ and Λ are defined consistently [START_REF] Wang | Aero-optics of subsonic turbulent boundary layers[END_REF].

∆ϕ rms 2 = 2G 2 k 2 L 0 ρ rms 2 (y)Λ(y) dy (12)
The Sutton model alone does not allow to avoid conducting a Large Eddy Simulation. Indeed, the knowledge of the instantaneous densities is still needed in Eqn. [START_REF] Gordeyev | Far-field laser intensity drop-outs caused by turbulent boundary layers[END_REF] to calculate the variance of density fluctuations and the density correlation length. Yet it would be very interesting to be able to predict the wavefront deformation from mean quantities only, such as the one obtained with RANS simulations at a much cheaper numerical cost than LES. In order to investigate this possibility, a modeling of the density rms and the density correlation length was tested. The approach proposed by Tromeur et. al. [START_REF] Tromeur | Analysis of the Sutton Model for Aero-Optical Properties of Compressible Boundary Layers[END_REF] was here adapted to a thermal boundary layer. They assume Λ and ρ rms homogeneous over the direction of light propagation y and roughly approximate them. The density correlation length is modeled by Λ = 0.1δ where δ is the boundary layer thickness. The density rms is modeled by ρ rms = A(ρ w -ρ ∞ ) where A is a parameter variable between 0.1 and 0.2, ρ w is the wall density and ρ ∞ is the freestream density, here the centerline density. The modeled phase distortion variance is then given by The value of T bl is chosen so that T w -T bl = 99%(T w -T f ) on average along the channel, where T f = 80 K is the fluid temperature at the channel centerline and T w is the wall temperature. It is reminded that the flow is fully developed, and there is no viscous boundary layer. In the slabs, a temperature gradient is created between the entry corner near the fluid and the opposite corner.

(∆ϕ rms 2 ) mdl = 2G 2 k 2 0.1δ 2 A 2 (ρ w -ρ ∞ ) 2 (13)

RESULTS AND DISCUSSION

Temperatures in the open channel

The heat transfer is higher at the beginning of the channel where the temperature gradients are stronger, than at the end of the channel where the boundary layer is thicker. The temperature difference between the two slab corners is 3 K. The elongation coefficient for undoped crystal YAG at 100 K is α = 2 × 10 -6 K -1 [START_REF] Aggarwal | Measurement of thermooptic properties of Y3Al5O12, Lu3Al5O12, YAIO3, LiYF4, LiLuF4, BaY2F8, KGd(WO4)2, and KY(WO4)2 laser crystals in the 80-300K temperature range[END_REF], which corresponds to a small elongation: ∆L L = 6 × 10 -4 % for 3 K. The thermo-optic coefficient of 1%-doped Yb:YAG at 100 K is dn dT = 10 -6 K -1 [START_REF] Cardinali | Mat ériaux lasers dop és à l'ion ytterbium: Performances lasers en pompage par diodes lasers et étude des propri ét és thermo-optiques à des temp ératures cryog éniques[END_REF].

Helium thermo-optic coefficient is not much bigger: it varies between 4.6 × 10 -6 K -1 at 100 K while it reaches ∆n f,max = 1.2 × 10 -4 in the fluid.

Optical outcome

The refractive index is computed from the temperature field as detailed in Eqn. [START_REF] Cardinali | Mat ériaux lasers dop és à l'ion ytterbium: Performances lasers en pompage par diodes lasers et étude des propri ét és thermo-optiques à des temp ératures cryog éniques[END_REF]. All the usual turbulence statistics (mean, fluctuations, correlations...) can be computed for this quantity. It should be noted that n differs from ∆n, as used in Eqn. ( 9): the former is the difference to the mean refractive index, while the latter is the difference to the refractive index for an unperturbed medium. ∆n includes the perturbation due to the mean thermal boundary layer, whereas n as plotted on Fig. 8 represents only the perturbation due to the temperature fluctuations. Figure 8 thus highlights the turbulence effect on the refractive index. Consistently with the dynamical and the thermal results, the refractive index fluctuations are located close to the walls, and their area By integrating a three-dimensional instantaneous refractive index field along the y direction as written in the Eqn. ( 9), the wavefront deformation map presented in Fig. 9 was computed. In Fig. 9, ∆ϕ is the difference to the phase at x/h = 0, where the fluid is uniformly at 80 K. The fluctuations in the refractive index reflects on the instantaneous wavefront map, with fluctuations increasing along the channel. On average, the phase gets smaller along the channel, i.e. its absolute value increases, as the temperature increases and thus the refractive index decreases. This mean tendency is shown in Fig. 10. The mean wavefront deformation reaches ∆ϕ = -0.24 rad at x/h = 12, the end of the simulated channel. The instantaneous wavefront deformation map Fig. 11: Root mean square of phase distortion at z/h = 3 and along the channel, calculated directly from the instantaneous phase distortions (Eqn. ( 9)), using the Sutton model with the instantaneous densities (Eqn. [START_REF] Roggemann | Imaging through Turbulence[END_REF]) and using the Sutton model with the modelled density correlation length and rms (Eqn. [START_REF] Gordeyev | Far-field laser intensity drop-outs caused by turbulent boundary layers[END_REF] with A = 0.12)

presents higher values, up to ∆ϕ max = -0.45 rad locally. A phase distortion could start to negatively impact the laser beam coherence from around ∆ϕ = π rad as it corresponds to an optical path difference of half the beam wavefront λ/2. It is however worth noting that this wavefront map represents only the effect of one cooling channel, and only about a quarter of it. In a typical multi-slabs amplifier, there could be about ten cooling channels crossed several times by the laser beam, which could then generate significant perturbations if the wavefront deformations caused by each channel add up. 12)) allows a satisfactory agreement with the direct computation results. But this is only made possible by the adjustment of the variable A of the model to this end. At this point, it is therefore hazardous to rely on the model alone.

Effect of a stronger heating of the slabs

To assess the effect of the optical heating of the solid on the laser beam, two additional simulations were conducted with heat sources three and six times higher than the heat source in the original simulation. The same procedure was adopted to attain the thermal equilibrium: the initial conditions were obtained from RANS calculations, and the solid heat capacity was reduced during the transient state. Figure 12 presents the mean temperatures for the three simulations. To compare them more conveniently, the temperatures for the three simulations are plotted on the same graph on a vertical profile close to the end of the channel (x/h = 11). The left flat part of the profiles corresponds to the temperatures in one half-slab while the right curved part corresponds to half of the channel, with the temperatures reaching 80 K for y/h = 1 in the center of the channel.

As expected, the slabs are hotter when the heating is stronger with an unchanged helium flow.

Consequently, the temperature difference in the channel is increased: from ∆T = 22 K between Following the same procedure as for the original simulation, the refractive indexes were computed for the two additional simulations using Eqn. ( 10) and ( 9). Instantaneous and mean wavefront deformation maps similar to those plotted on Fig. 9 and 10 were obtained. To compare them more conveniently, they are plotted on profiles along the channel at z/h = 3. deposit. At x/h = 11 for instance, the mean phase distortion is 2.8 times higher at 3 times higher heat deposit, and 4.7 times higher at 6 times higher heat deposit. The mean phase distortion can be compensated by adaptive optics, unlike the phase distortion fluctuations.

Figure 15 presents the phase distortion rms for the three simulations. Again as expected, the phase distortion rms (plotted by lines) is higher at higher heat deposit but not proportionally. At

x/h = 11, the phase distortion rms is 2.6 times higher when the heat deposit is multiplied by 3, and 4.3 times higher when it is multiplied by 6. It is reminded here that these two additional simulations present a major limitation to their representativeness by lacking the increase of the fluid velocity that would accompany the increase of the slabs heating in a real case. Consequently, the increase of temperature fluctuations linked with a higher Reynolds number of the cooling flow is here overlooked.

For the sake of clarity, the Sutton model results based on LES instantaneous densities are not plotted on Fig. 15 as they collapse exactly with the direct computation results. But similarly to Fig. 11, the Sutton model results based on the modeled density rms ρ rms = A(ρ w -ρ ∞ ) and the modeled correlation length Λ = 0.1δ following Eqn. ( 12) are plotted with markers on Fig. 15. The same value A = 0.12 was used for the three cases as it has allowed a satisfactory agreement with the directly computed results in the original case (see section 3.2). The model does not reproduce well the increase of phase distortion rms with the heating: it underestimates by 10 % the phase distortion rms when the heat deposit is multiplied by 3, and by up to 30 % for the heat deposit multiplied by 6. This outcome confirms the need to carry fine simulations such as LES to study the turbulence effect of a cooling flow on the laser beam. Indeed, even though the general form of the Sutton model has proven effective in this configuration, the need for the knowledge of the density fluctuations limits its applicability. The proposed modeling of the density rms as ρ rms = A(ρ w -ρ ∞ )

does not lead to satisfactory results, and as long as an efficient alternative is not found, LES will remain mandatory for this application.

CONCLUSIONS

This work aimed at analyzing the effect of the turbulent cryogenic cooling of a laser amplifier on the output laser beam quality. To this end, Large Eddy Simulations of an open turbulent channel flow cooling two slabs at different optical heating were carried out. The phase distortions, mean and fluctuations, induced by an inhomogeneous and turbulent temperature field were investigated.

A moderate although non-negligible effect of the turbulence on the laser wavefront was found. It increases when the slabs heating increases. The phase distortion rms are of prime importance as they cannot be compensated by adaptive optics. Even though the Sutton model linking phase distortion rms to density rms was found theoretically accurate in this configuration, its application using modeled density rms was not. Resort to Large Eddy Simulations thus remains necessary to investigate aero-optical effects in such laser amplifiers. Similar studies can be carried out for different amplifier designs and for different coolants. Liquid cryogenic fluids (e.g. nitrogen) are of special interest: although the cooling is more efficient than with gaseous helium, the refractive index varies more. It will then be unavoidable to pay attention to the induced phase distortion rms.

Fig. 1 :

 1 Fig. 1: Computational domains with precursor flow channel, open flow channel and heated solid slabs. h = 0.002 m

  lation [26]. The spatial discretization used in this work is a "Finite Difference-Volume" method for hexahedral structured grid where the scalar unknowns are computed at the center of the cells while the velocities are computed at the center of the faces. The mesh comprises 60 × 192 × 240 cells for the precursor channel and 120 × 192 × 240 cells for the main open channel. With a Reynolds number based on the friction velocity (see Eqn. (3)) of Re τ = 400, the streamwise and spanwise dimensionless sizes are respectively ∆x + = 40 and ∆z + = 10. The crosswise dimensionless size varies between ∆y + = 1 at the wall and ∆y + = 15 in the center of the channel,

Fig. 2 :

 2 Fig.2: Mean longitudinal velocity standardized by the friction velocity, compared to the DNS results of Moser et al.[START_REF] Moser | Direct numerical simulation of turbulent channel flow up to Re T= 590[END_REF] and to the LES results of Serra et al.[START_REF] Serra | Thermal Large Eddy Simulation in a Very Simplified Geometry of a Solar Receiver[END_REF]

  Fig.3: Velocities root mean square standardized by the friction velocity, compared to the DNS results of Moser et al.[START_REF] Moser | Direct numerical simulation of turbulent channel flow up to Re T= 590[END_REF] and to the LES results of Serra et al.[START_REF] Serra | High-temperature gradient effect on a turbulent channel flow using thermal large-eddy simulation in physical and spectral spaces[END_REF]

Fig. 4 :

 4 Fig.4: Mean temperature standardized by the friction temperature and compared to the DNS results of Kawamura et al.[START_REF] Kawamura | DNS of turbulent heat transfer in channel flow[END_REF]

Fig. 5 :

 5 Fig.5: Temperature root mean square standardized by the friction temperature and compared to the DNS results of Kawamura et al.[START_REF] Kawamura | DNS of turbulent heat transfer in channel flow[END_REF]

  shows an excellent agreement between the present LES and Kawamura et. al. DNS. The temperature fluctuations plotted on Fig. 5 are however underestimated except close to the wall. The biggest discrepancy is reached at the fluctuation peak where the LES captured fluctuations are 12% lower than the DNS profile. Any following optical effect of the turbulent temperature field (see section 3.

Figure 6 3 (Fig. 6 :Fig. 7 :

 6367 Figure 6 presents the mean temperature at thermal equilibrium on a (xy) surface at z/h = 3 (the spanwise middle of the simulated domain), in the open channel of helium (in blue) and in the two half slabs (in red). The helium flow enters the channel from the left side and it is heated up as

Fig. 8 :

 8 Fig. 8: Instantaneous refractive index fluctuations n , mapped on a streamwise plane in the channel flow

Figure 7

 7 Figure 7 presents the temperature root mean square (rms) at thermal equilibrium on a (xy) surface z/h = 3 in the open channel. The contour line at T rms = 1 K shows that the temperature fluctuates mostly in the thermal boundary layer, where the temperatures are higher. The peak of temperature fluctuations is situated close to the wall, where the velocity fluctuations are also the strongest. The temperature rms reaches there 3.2 K. The temperature rms are not presented in the solid as they are in the range of 10 -2 K, which is small compared to the fluid. These small temperature fluctuations in the solid are consistent with its properties: Yb:YAG has a large thermal diffusivity α s = λ s ρ s C ps = 76 mm 2 /s, when the fluid thermal diffusivity is α f = 4 mm 2 /s.

Figure 8

 8 Figure 8 is one example of refractive index fluctuations n = n -n on a (xy) surface at z/h = 3.

Fig. 9 :Fig. 10 :

 910 Fig. 9: Instantaneous wavefront deformation map on a spanwise plane after crossing the channel flow

Figure 11 presentsFig. 12 :

 1112 Figure 11 presents the phase distortion rms after one crossing of the channel in the spanwise middle and at different streamwise positions. Consistently with the instantaneous data observed on Fig.8 and 9, the phase distortion rms increases along the channel as the temperature field gets more turbulent, reaching ∆ϕ rms = 0.9 rad at the end of the simulated channel. It should be reminded here that the phase distortion rms is probably underestimated since the temperature rms were underestimated in the validation case (see section 2.2). The estimations of the phase distortion rms using the Sutton model are also plotted on Fig.11. The general form of the linking equation (Eqn.[START_REF] Roggemann | Imaging through Turbulence[END_REF]) when using the LES results yields excellent agreement with the direct com-

Fig. 13 :

 13 Fig. 13: Temperature root mean square on a vertical line in half of the channel, at x/h = 11 and at z/h = 3, for different heat deposits in the slabs

Figure 14 Fig. 14 .Fig. 14 :Fig. 15 :

 14141415 Fig.[START_REF] Tromeur | Analysis of the Sutton Model for Aero-Optical Properties of Compressible Boundary Layers[END_REF]. As expected, the absolute mean phase distortion is higher at stronger heating. Coherently with the Gladstone-Dale law, the mean phase rise is not however proportional with the heat

Table 1 :

 1 Physical properties of the amplifying material and of the gaseous helium at 80 K and 5 bar

	Fluid	Solid

ACKNOWLEDGMENT

This work was granted access to the HPC resources of CINES under the allocations 2019-A0052A10626 and 2020-A0072A10626 attributed by GENCI (Grand Équipement National de Calcul Intensif).

FUNDING

We acknowledge the financial support of the Cross-Disciplinary Program on Numerical Simulation of CEA, the French Alternative Energies and Atomic Energy Commission.

LIST OF FIGURES