Martin David
email: martin.david@promes.cnrs.fr

Adrien Toutant
email: adrien.toutant@univ-perp.fr

Françoise Bataille
email: francoise.bataille@univ-perp.fr

Study of asymmetrically heated ows passing through gas-pressurized solar receivers using Direct Numerical Simulations

Keywords:

DNS of highly anisothermal and turbulent channel ows are performed and analyzed.

• Integral values, turbulence statistics, PDF, and spectra are investigated.

• Dierences and similarities between wall heat ux and wall friction are highlighted.

• PDF of wall heat ux and wall friction are well tted with skewed Gaussian models.

Introduction

The Concentrated Solar Power (CSP) generation was about 14.5 TWh in 2020. In the Net Zero Scenario proposed by the International Energy Agency (IEA), the power generated by this technology should reach 204.0 TWh in 2030 (IEA co, 2021). To achieve this objective, the annual average growth should be almost 31%. The global cost fell by 68% between 2010 and 2020. This is mainly due to a 50% reduction in the global weighted-average total installed costs of newly commissioned CSP plants, a 41% gain in capacity factors thanks to the use of heat storage systems, and a 33% decrease in operation and maintenance costs.

According to IRENA report (IRENA co, 2021), the Levelized Cost of Energy (LCOE) of the solar tower was lying between 0.07 e/kWh and 0.2 e/kWh in 2018-2019. The optimization of the dierent parts of the system will help pursue the decline of the CSP LCOE.

A large number of studies are investigating new working uids to permit the increase of the temperature orcid(s): 0000-0001-5213-8712 (M. David); 0000-0002-7156-1732 (A. Toutant) levels which enhance the system performances of solar tower power [START_REF] Buck | Solar tower system temperature range optimization for reduced LCOE[END_REF][START_REF] Zheng | Analysis of tubular receivers for concentrating solar tower systems with a range of working uids, in exergy-optimised ow-path congurations[END_REF][START_REF] Merchán | High temperature central tower plants for concentrated solar power: 2021 overview[END_REF]. Using air, there is no more limiting temperature due to the operating uid, the maximum is then xed with the resistance of the materials [START_REF] Kribus | A Multistage Solar Receiver:: The Route To High Temperature[END_REF]. For that reason, extremely high temperatures can be reached. The new working uids pave the way to research studies on heat transfer enhancement in solar receivers that are characterized by strong asymmetrical heating of the uid. The encountered ows are very complex, that is why upstream works are necessary to deepen the understanding of the coupling between the dynamics and the temperature eects.

Direct Numerical Simulations (DNS) are commonly used to investigate ow physics in academic congurations. Channel ows are studied because this geometry is widely encountered in industry, particularly for heat exchangers in nuclear power plants, electronic devices, and solar receivers. In those applications, the ow is often turbulent and heated from the walls of the channel. If the temperature dierence is small, the temperature eects on the uid physical properties can be neglected and the momentum balance is often decoupled from the thermal energy balance. In the literature, most of the non-isothermal DNS are performed considering the temperature as a passive scalar. Some of them are described in the rest of the paragraph.

Study of asymmetrically heated ows passing through gas-pressurized solar receivers using Direct Numerical Simulations

The following works deal with the link between the dynamic of the ow and the behavior of a passive scalar.

The transport of a scalar was investigated thanks to a DNS by Kim and Moin (Kim and Moin, 1989).

They notice that the scalar elds are strongly linked to the streamwise velocity and that the scalar uxes could be modeled similarly to the momentum uxes.

Lyons et al. [START_REF] Lyons | Direct numerical simulation of passive heat transfer in a turbulent channel ow[END_REF] studied passive heat transfer in a channel ow at a bulk Reynolds number of 2300. They found reasonably good agreements between the obtained mean temperature prole, eddy conductivity prole, and turbulent Prandtl number and experimental results. The authors notice that the dominant wall eddies, which have a spanwise scale of about 50 wall units, manage both the Reynolds stress production and turbulent heat transfer at the wall by moving hot and cold pockets of uid to and from the wall. They also highlight that in the channel center, the uctuating temperature production and dissipation are signicant. [START_REF] Kasagi | Direct Numerical Simulation of Passive Scalar Field in a Turbulent Channel Flow[END_REF] studied turbulent air ow in channel. Their results suggest that the temperature uctuations are highly correlated with the streamwise velocity uctuations, especially near the walls. Abe and Antonia [START_REF] Abe | Mean temperature calculations in a turbulent channel ow for air and mercury[END_REF] examined the turbulent Prandtl number for air and mercury in a turbulent channel ow under a constant time-averaged heat ux thermal boundary condition.

They propose laws for estimating the turbulent Prandtl number in the dierent regions of the ow. Particularly, the authors highlight the fact that the decrease in turbulent Prandtl number in the region y/δ > 0.2 is due to the unmixedness of the scalar.

The inuence of the Prandtl number is frequently investigated. Focusing on the budget of wall-normal heat ux, [START_REF] Kawamura | DNS of turbulent heat transfer in channel ow with low to mediumhigh Prandtl number uid[END_REF], 1999) notice that the temperature-pressure gradient correlation is prevailing as the destruction term for bigger Prandtl numbers while, for smaller Prandtl number, the dissipation term is. Na and Hanratty [START_REF] Na | Use of direct numerical simulation to study the eect of Prandtl number on temperature elds[END_REF] DNS of turbulent heat transfer in a channel ow was carried out by [START_REF] Kozuka | DNS of turbulent heat transfer in a channel ow with a high spatial resolution[END_REF].

The results suggest that the wall-asymptotic value of the turbulent Prandtl number is almost independent of Reynolds number for the entire tested range of Prandtl number, meaning from 0.71 to 10. The turbulence level eect has been investigated by [START_REF] Abe | Surface heat-ux uctuations in a turbulent channel ow up to Reτ=1020 with Pr=0.025 and 0.71[END_REF]. The authors performed DNS at friction Reynolds numbers of 180, 395, 640 and 1020. They highlight that, for a Prandtl number of 0.71, the surface heat-ux uctuations are very close to the streamwise wall shearstress uctuations, although dierences are observed in the large positive and negative uctuations. It appears that the large-scale structures seen in the surface heatux uctuations are mostly phenomena linked to the large-scale motions for the thermal eld of the outer layer.

In this paragraph, some works performed with temperature acting as an active scalar are presented. [START_REF] Aulery | Spectral analysis of turbulence in anisothermal channel ows[END_REF] This contribution is responsible for around 10% of the total process. [START_REF] Avellaneda | DNS of turbulent low Mach channel ow under asymmetric high temperature gradient: Eect of thermal boundary condition on turbulence statistics[END_REF] discussed the local entropy generation rate thanks to DNS performed at mean friction Reynolds from 150 to 210, with wall temperature of 293 K and 879 K. The results suggest that an increase in the wall temperature ratio leads to larger entropy generation rates, and to accentuate the asymmetry between the cold wall and the hot wall. The authors notice that the entropy generation rates is always bigger at the cold wall than at the hot wall.

Orlandi et al. [START_REF] Orlandi | DNS of conjugate heat transfer in presence of rough surfaces[END_REF] investigated conjugate heat transfer in channel characterized by rough surfaces. Several type of roughness elements are analyzed. The sweeps and ejections events located in the near wall region are responsible for the modication of the ow structures, and thus the temperature distribution. It seems that the amplitude and the number of these events are highly correlated with the ow generated within the surfaces, and are aecting the temperature uctuation proles. David et al. (David et al., 2021a) performed DNS of a highly turbulent channel ows under a strong asymmetric heating. This article was mainly dedicated to the study of Large-Eddy Simulations (LES), but a part deals with the investigation of the ow physics thank to two DNS. The results indicate that there is a strong coupling between the dynamic and the temperature.

The wall-normal velocity is highly dependent of the temperature prole.

The presented state of the art shows that very few studies have been conducted in anisothermal conditions considering the temperature as an active scalar.

Moreover, to the author's knowledge, the article of David et al. (David et al., 2021a) is the only paper handling the study of ow physics with DNS in the operating conditions of solar receiver, that are characterized by high Reynolds numbers, asymmetric heating of the uid, and high wall heat uxes. The present article aims to deepen the analyses presented in (David et al., 2021a) by investigating new temperature proles, proposing new results, and using dierent scalings.

A particular attention is paid to the similarities and dierences between the wall heat ux and the wall friction.

The paper is organized as follows: Section 2 gives the resolved equations. The congurations studied are detailed in Section 3. The obtained DNS results are discussed in Section 4. Lastly, a nal section concludes.

Low Mach number equations

In gas-pressurized solar receivers, the mass ow is high to improve the system performance but the ow is subsonic. Indeed, the Mach number is around 0.1.

The thermal dilatation eects have to be considered but the inuence of acoustic waves can be neglected. Thus, the Navier-Stokes equations are solved under the low Mach number approximation that is valid as long as the Mach number is below 0.3. This theory was proposed by Paolucci [START_REF] Paolucci | Filtering of sound from the Navier-Stokes equations[END_REF] and aims to narrow the gap between the incompressible and the compressible formulations of the Navier-Stokes equations. It permits suppressing the numerical constraints associated with the latter formulation. The resolved system is given by equations 1-4:

• Mass conservation equation ∂ρ ∂t + ∂ρU j ∂x j = 0, (1)
• Momentum conservation equation

∂ρU i ∂t = - ∂ρU j U i ∂x j - ∂P ∂x i + ∂Σ ij ∂x j , (2)
• Energy conservation equation

∂U j ∂x j = - 1 γP 0 (γ -1) ∂Q j ∂x j + dP 0 dt , (3)
• Ideal gas law

T = P 0 ρr , (4)
where γ is the heat capacity ratio. The pressure involved in the energy equation is the thermodynamical pressure, denoted P 0 . It represents the mean pressure in the computational domain, and only depends on the time (∂P 0 /∂x i = 0). The mechanical pressure, P , appears in the momentum conservation equation and is spatially varying.

The particular form of the energy conservation equation which involves the velocity divergence is obtained (1) by applying the ideal gas law to the energy equation written in terms of transport of temperature;

(2) by using the fact that the thermodynamical pressure is homogeneous. This formulation is also used by Nicoud [START_REF] Nicoud | Conservative High-Order Finite-Dierence Schemes for Low-Mach Number Flows[END_REF] and Dupuy et al. (Dupuy et al., 2019).

The uid is supposed to be Newtonian, leading to the following expression for the viscous stress tensor:

Σ ij (U, T ) = µ(T )( ∂U i ∂x j + ∂U j ∂x i ) - 2 3 µ(T ) ∂U k ∂x k δ ij . (5)
The conductive heat ux is given by the Fourier's law:

Q j (T ) = -λ(T ) ∂T ∂x j , (6)
where µ(T ) is the dynamic viscosity, λ(T ) is the thermal conductivity, and δ ij the Kronecker symbol.

The uid used is air. The viscosity is computed using the Sutherland's law [START_REF] Sutherland | The viscosity of gases and molecular force[END_REF]:

µ(T ) = µ 0 T T 0 3/2 T 0 + S T + S , (7)
where µ 0 = 1.716 × 10 -5 P a • s, S = 110.4 K, and T 0 = 273.15 K. The Prandtl number is assumed to be constant so as the heat capacity at constant pressure which is computed with the average of the wall temperatures. The heat capacity is

C p = 1155J • kg -1 • K -1
for all the simulations. The Prandtl number depends on the simulations and is lying between 0.70 and 0.76. The conductivity is obtained from the Prandtl number, the heat capacity at constant pressure, and the viscosity:

λ(T ) = C p Pr µ(T ). (8)
The simulations are carried out with TrioCFD code [START_REF] Calvin | An object-oriented approach to the design of uid mechanics software[END_REF]. 3. Flow analysis along a solar receiver

Study conguration

The solar absorber is composed of an irradiated wall and an insulated wall. The rst receives the concentrated solar radiation and transfers heat to the uid by conduction and to the insulated wall by radiation.

It results in uid heating from both sides of the solar receiver. As the temperature is always higher at the irradiated wall than at the insulated wall, the ow is asymmetrically heated.

In these works, the ow passing through the solar receiver is modeled with an academic bi-periodical channel ow. The domain size is L x = 4πδ, L y = 2δ, and L z = 4/3πδ with L x , L y , and L z the lengths in the streamwise, wall-normal, and spanwise directions, respectively. The half-channel height is denoted δ and is 3 mm. See Figure 1 for a schematic view of the investigated channel ow. The simulations are per-

y k = L y 1 + 1 a tanh k -1 N y -1 tanh -1 (a) (9)
where a is the mesh dilatation parameter and N y is the number of grid points in the wall-normal direction.

The mesh consists in 660 millions of cells that are distributed as follows: N x ×N y ×N z = 1152×746×768.

The normalized cell sizes of the used mesh are: ∆x + = 10.9, ∆y + (0) = 0.42, ∆y + (δ) = 5.4, ∆z + = 5.4. These values are given for the cold side of the channel which correspond to the highest friction Reynolds number.

Notice that for the simulation associated with the smallest Kolmogorov length scale gives η = ν 3 1/4 ≈ 1.5 × 10 -6 meter. The rst cell size in the wall-normal direction is slightly lower, with a length of 1.3 × 10 -6 meter, which is in agreement with the classical DNS requirement [START_REF] Kim | Turbulence statistics in fully developed channel ow at low Reynolds number[END_REF]. The Batchelor scale gives the smallest length scales of uctuations in scalar concentration that can exist before molecular diusion prevail. The smallest length scale is computed with the thermal diusivity at the cold wall and is 9.7 × 10 -6 meter which is 7.5 times bigger than the size of the rst cell in the wall direction.

∂U j ∂x j = - 1 γP 0 (γ -1) ∂Q j ∂x j -H s + dP 0 dt , ( 10
)
where H s is the volumic heat sink.

Investigated conditions

In concentrated solar systems, the uid enters the solar receiver at a relatively low temperature and is heated by the walls. The gap between the bulk temperature and the wall temperature becomes smaller as the uid progresses in the absorber [START_REF] Capeillère | Thermomechanical behavior of a plate ceramic solar receiver irradiated by concentrated sunlight[END_REF].

To deepen the understanding of the coupling between The higher the heat sink, the higher temperature gap between the uid and the walls and thus the closer from the inlet of the solar absorber. These ve DNS are listed in Table 1. The mean friction Reynolds number is dened as the average of the hot and cold friction Reynolds number given by Re τ,w = U τ,w δ/ν w , where the subscript w denotes the values computed at the wall and U τ,w is the friction velocity. For all the simulations carried out in anisothermal conditions, the mean friction Reynolds number is imposed and is about 820. Since the wall temperatures are dierent and the uid properties depend on the temperature, the cold, respectively hot, friction Reynolds number is about 1000, respectively 640. Hence, the turbulence level is higher on the cold side than on the hot side.

T h w -T c w H s T b [K ] [M W/m 3 ] [K ] isoT-
The isothermal DNS has been performed at a friction Reynolds number of 930 for the code validation with the reference DNS of Hoyas and Jiménez [START_REF] Hoyas | Reynolds number eects on the Reynolds-stress budgets in turbulent channels[END_REF]. Both simulations are in very good agreement. See David et al. (David et al., 2021a) for the comparison between the two DNS. The skin friction coecient obtained with the isothermal DNS performed in this study is C f = 4.95 × 10 -3 . This in good agreement with the skin friction coecient obtained by Hoyas and Jimenez [START_REF] Hoyas | Reynolds number eects on the Reynolds-stress budgets in turbulent channels[END_REF] since the relative error is inferior to 1.1%. The thermodynamical pressure is set to 10 bar in all the simulations. Note that the simulated times of the DNS are about 45 characteristic times, computed at the cold wall with (δ/U τ,c ). The data collection duration is 20 characteristic times.

Results and discussion

In this section, the results of the ve DNS are presented and discussed. Integral quantities as well as rst-and second-order statistics of turbulence are investigated then the focus is placed on the study of dierences and similarities between wall heat ux and wall friction.

The symbol "• + " indicates that the results are nor- malized in wall units:

x + i = x i U τ,w /ν w , U + i = U i /U τ,w , T + = (T w -T ) /T τ,w , and u i u j + = u i u j /U 2 τ,w .
The friction velocity and the friction temperature are respectively expressed as U τ,w = ν w ∂U 1 /∂x 2 , and T τ,w = φ w / (ρ w C p U τ,w ), with φ w the conductive heat ux at the wall. The symbol U i denotes the ith component of velocity, and x i denote the Cartesian coordinate in the ith direction. To shorten the notations, the subscript is sometimes removed. In this case U accounts for U 1 . The symbols ν w , µ w , and ρ w respectively account for the kinematic viscosity, the dynamic viscosity, and the density at walls. Note that since there are two dierent wall temperatures, the quantities followed by the subscript w can take two values: either the value associated with the hot wall and the value associated with the cold wall.

The Van Driest scaling (Driest, 1956) is employed for the streamwise velocity. It is given by:

U + V D = 2 P r t T τ,w /T w 1 -1 -P r t T τ,w /T w U/U τ,w . (11)
Note that the values used for the turbulent Prandtl number are those obtained by averaging wall-normal proles for each simulation.

The symbol "• * " indicates that the results are normalized with the semi-local scaling [START_REF] Patel | Semilocal scaling and turbulence modulation in variable property turbulent channel ows[END_REF][START_REF] Trettel | Mean velocity scaling for compressible wall turbulence with heat transfer[END_REF]) that takes into account the local variations of density and kinematic viscosity: . Here, the notation

x * i = x i U * τ,w /ν(y), U * i = U i /U * τ,w , T + = (T w -T ) /T * τ,w , and u i u j * = u i u j /U 2 * τ,
• mean denote the averaging of the values obtained at both walls. This scaling is well adapted to discuss the quantity variations along the entire channel height.

Integral quantities

Integral quantities permit rapidly characterizing the ow. They are frequently used in the literature and can be approximated with correlations. In this subsection, the integral quantities related to the dynamic of the ow are rstly given and discussed, then the wall heat uxes are discussed.

Table 2 provides information on integral quantities associated with the ow dynamics. The results show that for a given mean wall friction, the higher the heat sink, the higher the mass ow. When the uid temperature decreases, the velocity prole is rather attened.

Indeed, the ratio U c /U b diminishes from 1.135 in the "anisoT-S0" DNS to 1.117 in the "anisoT-S3" DNS.

The cold wall skin friction coecient is higher than the hot wall one. The lowest skin friction coecients are obtained with the simulations performed with the highest heat sinks despite the mass ow increases. This is due to the reduction of the bulk velocity observed in these DNS. The Dean correlation [START_REF] Dean | Reynolds Number Dependence of Skin Friction and Other Bulk Flow Variables in Two-Dimensional Rectangular Duct Flow[END_REF] gives a good approximation of the skin friction coefcient in the isothermal case ( =5.8%) considering the bulk Reynolds number computed with a characteristic length of 2δ. In the anisothermal case, the wall skin Simulation Mass ow

U c /U b Re b C h f C c f C mean f [kg/s] DNS Corr isoT-S0
8.50 × 10 -3 1.135 37392 4.95 × 10 -3 4.95 × 10 -3 4.95 × 10 -3 5.25 × 10 -3 5.8%

anisoT-S0

1.64 × 10 -2 1.135 29988 5.08 × 10 -3 5.47 × 10 -3 5.28 × 10 -3 5.55 × 10 -3 4.9%

anisoT-S1

1.98 × 10 -2 1.131 40928 4.85 × 10 -3 5.33 × 10 -3 5.09 × 10 -3 5.13 × 10 -3 0.8% anisoT-S2

2.44 × 10 -2 1.127 59222 4.39 × 10 -3 4.89 × 10 -3 4.64 × 10 -3 4.68 × 10 -3 0.9% anisoT-S3

2.99 × 10 -2 1.117 85568 3.68 × 10 -3 4.17 × 10 -3 3.92 × 10 -3 4.27 × 10 -3 8.1%

Table 2: Flow mean quantities relative to the dynamic. The skin friction coecient obtained with the Dean correlation [START_REF] Dean | Reynolds Number Dependence of Skin Friction and Other Bulk Flow Variables in Two-Dimensional Rectangular Duct Flow[END_REF] are indicated with the word "Corr" and are compared with the DNS results. The superscript mean indicates that the hot and cold values are averaged. Note that the denition of the mean skin friction as the mean value of the hot and cold wall value is equivalent to the denition of the skin friction which is dened by the wall-shear stress that balances the pressure gradient in the ow system.

frictions are computed at both walls then an average is performed. The correlation permits estimating the average of the wall skin friction coecients with 8.1% of error at most.

Table 3 provides information relative to the wall heat uxes. The results point out that the simulation "anisoT-S0" has the same conductive wall heat uxes at both walls. This was expected since the ow is periodic and fully developed. When the heat sink is involved, the wall heat uxes are no more identical for a given simulation. The maximum asymmetry between the two walls is obtained for the simulation "anisoT-S1" (|φ h /φ c | = 4.1). The highest heat ux is obtained at the hot wall of the simulation "anisoT-S3". The Nusselt number is a widely used dimensionless number in the turbulent heat and mass transfer between a uid and a solid wall. The heat transfer correlation of David et al. (David et al., 2021c) dedicated to asymmetrical heating conditions is assessed with carried out DNS. The correlation gives very good estimations of the wall heat uxes with a maximum relative error of 4.1%.

First-order statistics

In this subsection, rst-order statistics are investigated. Figure 2 plots the proles of • o scaled streamwise velocity, wall-normal velocity, and temperature as functions of the channel height. Focusing on the top-left graph, the results of anisothermal simulations show that the streamwise velocity decreases as the heat sink grows. To maintain a constant mean wall friction, the streamwise velocity has to be reduced when the uid temperature drop. The normalized velocity is higher in the isothermal case than in anisothermal congurations due to the scaling with the mean friction velocity. The maximum streamwise velocity is slightly shifted toward the hot side of the channel due to uid property variations. The wall-normal velocity is induced by temperature variations. Indeed, it is null in the isothermal case and the peak amplitude grows with the increase of the gap between the bulk temperature and the wall temperature. In the case without a heat sink (anisoT-S0), the prole exhibits a "u" shape because the temperature gradient induces a wall-normal velocity ux from the hot wall to the cold wall. For the other DNS, the uid is heated from both walls. Thermal expansion forces it to move away from the walls, which leads to wall-normal velocities oriented toward the channel center line. The peaks are opposed (positive on the hot side and negative on the cold side) and their amplitudes are higher on the hot side than on the cold side due to the bigger hot wall heat ux. The peaks are localized in the log-layer at y/(2δ) ≈ 0.022 corresponding to a distance from the wall of y + ≈ 45. The higher the asymmetry of the temperature gradient between the two walls is, the closer to the cold wall the wall-normal velocity crosses the zero line: around y/(2δ) = 0.2 for the simulation "anisoT-S1" whereas it appears around y/(2δ) = 0.4 for the simulation "anisoT-S3". The normalized wallnormal velocity is equal to -0.007 in wall unit for all the simulations at the normalized distance y/(2δ) = 0.48.

For the DNS performed without a heat sink, the temperature prole indicates that the uid temperature is equal to the average of the wall temperature at the location y/(2δ) = 0.53, meaning on the hot side. For the other DNS, the weaker the heat sink, the closer to the cold wall the minimum temperature. Indeed, for the simulation "anisoT-S1", "anisoT-S2", and "anisoT-S3" the minimum of temperature are respectively obtained for y/(2δ) = 0.21, 0.35 and 0.40. The density linked to the temperature by the expression ρ ∝ 1/T because of the use of the ideal gas law and the low Mach Mach number approximation. For that reason, the density prole presents reverse tendencies when compared to the temperature prole. The kinematic viscosity is dened with the density and th dynamic viscosity, which is linked to the temperature by the Sutherland's law, see Eq. 7. The same behaviors are observed for the temperature and the viscosity proles.

When the heat sink grows, the uid temperature is The proles of wall scaled streamwise velocity and temperature are plotted in Figure 4. For both quantities, the curves overlap until y + = 10, then signicant discrepancies arise. With this scaling, the streamwise proles have the almost the same slope in the region 50 < y + < 300 but dierent osets. For the temperature prole, the same observations can be made but in the range 50 < y + < 100. The cold side proles are higher than the hot side ones. In this region, the simulations performed with the highest wall heat uxes are characterized by the lowest proles and the gap between the two sides of a given simulation decreases with the increase of the Reynolds number. The higher the wall heat uxes, the more signicant the distance between the law of the wall and the velocity proles.

φ h φ c |φ h /φ c | N u h N u c [kW/m 2 ] [kW
In the isothermal case, the wall functions give a good approximation of the streamwise velocity prole in the range y * < 5 for f 1 and 30 < y * < 400 for f 2 . They are dened by: The wall functions associated with the temperature proles are given for information, they are discussed with the semi-local scaling in Figure 5.

f 1 (y * ) = y * , (12) and f 2 (y * ) = 1 κ ln(y * ) + C, (13)
The rst-order statistics normalized with the semilocal scaling are presented in Figure 5. This normalization permits analyzing the results without the mean density and viscosity inuence. The streamwise velocity proles are very close then they split up after y * = 10, as noticed in [START_REF] Bellec | Large Eddy Simulations of thermal boundary layer developments in a turbulent channel ow under asymmetrical heating[END_REF]. Increasing the heat sink results in increasing the normalized streamwise velocity in the center of the channel. The prole asymmetry is highlighted with the scaling. The streamwise velocity on the hot side is higher than the cold side one.

The slope of the logarithmic law exhibits a downward trend with lowering wall heat ux. The proles can be quite well approximated with the laws of the wall in the regions y * < 5 for f 1 and 30 < y * < 400 for f 2 .

Nevertheless, the results show that there are combined eects of the wall-normal velocity and the turbulence level that aects the streamwise velocity. Indeed, the prole slopes in the logarithmic and outer regions are modied when compared to the isothermal case. The region are modied by the heating. When the heat uxes are important, the slope is signicantly altered.

When the wall heat uxes are high, the wall-normal velocity moves a large amount of uid toward the center of the channel which is probably responsible for the streamwise velocity increases in the outer layer.

Nicoud and Bradshaw [START_REF] Nicoud | A velocity transformation for heat and mass transfer[END_REF] express a mathematical analogy between the Van Driest transformation for compressible boundary layers with heat transfer and the Simpson transformation of the velocity prole for incompressible turbulent wall ows with uniform injection [START_REF] Simpson | Characteristics of turbulent boundary layers at low Reynolds numbers with and without transpiration[END_REF]. This analogy leads to the following expression of the equivalent injection velocity used in the Van Driest scaling:

V inj = P r t T τ,w /T w U τ,w (16)
The velocity injection is compared to the wall-normal velocity proles on the right graph. Both velocities are in very good agreement. Toutant and Bataille [START_REF] Toutant | Turbulence statistics in a fully developed channel ow submitted to a high temperature gradient[END_REF] and Bellec et al. [START_REF] Bellec | Large Eddy Simulations of thermal boundary layer developments in a turbulent channel ow under asymmetrical heating[END_REF] also found a good accordance between the equivalent injection velocity and the wall-normal velocity at lower Reynolds numbers and wall heat uxes.

Second-order statistics

The velocity -velocity, velocity -temperature, and temperature -temperature uctuations as well as the coecient correlations are studied in this part. regions. Note that, the maximum amplitude is about 1.2 times higher on the cold side than on the hot side for a given simulation due to the lower turbulence level on the last side. For a normalized distance lying between y + = 50 and y + = 300, a reduction in the slope is noted. This is due to the large-scale structures that are associated with high turbulence level (Hutchins and Marusic, 2007b). Hence, in the studied conditions, and more signicantly for the simulation "anisoT-S3", the energetic peak due to the viscous-scaled near-wall cycle (located at y + = 15) is contested by much larger outerscaled events in the logarithmic region (lying between y + = 50 and y + = 300). It is well known that the eects of these last structures become more signicant at higher turbulence levels which requires very intensive computational resources (Hutchins and Marusic, 2007a). Here, the presented results show that the second energy peak, which is revealed by the reduction of the slope in the logarithmic region, becomes more important when the gap between the bulk and the wall temperature is high. This can probably be explained by the increase of the bulk Reynolds number when the heat sink grows. Thus, turbulence production spreads out on a signicant wall unit range. The cross-correlation exhibits peaks of opposite signs and cancels out on the center line of the channel for all the simulations. The temperature eects only impact the prole amplitude.

The four presented graphs point out that the heat transfer direction at the cold wall has no signicant inuence on the uctuations aside from the eects due to the reduction of the uid temperature. S3" is directly linked to the increase of the streamwise velocity in the outer layer of the ow (see Figure 5).

The proles of wall-normal velocity uctuations are signicantly dierent. As observed in Figure 7, the cold side uctuations are higher than those observed in the hot side. After removing the eects of the mean density and viscosity variations, it appears that the highest uctuations are obtained for the simulation "anisoT-S3" which is characterized by the highest Reynolds number. The uctuation proles of the hot and cold sides do not collapse, which suggests that the turbulence and the temperature gradient interactions are more complex than a simple inuence of the uid property variations. The same observations and conclusions can be made for the spanwise velocity uctuations

(not presented in this paper).

The correlations of velocity -temperature and temperature -temperature are plotted in Figure 9. Study of asymmetrically heated ows passing through gas-pressurized solar receivers using Direct Numerical Simulations

Moin (Kim and Moin, 1989) have found a ratio of 0.38 studying the transport of passive scalar in turbulent channel ow at a friction Reynolds number of 180. Note that the non-monotonic trend of the uctuations of wall friction with respect to the symmetrical might be explained by the insuciency of ensemble averaging.

Figure 11 plots the proles of velocity -temperature coecient correlations. The two presented graphs highlight the fact that the prole on the cold side is signicantly modied by the temperature distribution.

On the left graph, when the cold wall heats the uid, the term u θ is negative. As the uid temperature grows, the observed bell shape proles get bigger and become closer to the prole performed without a heat sink. Thus, the top of the bell shape is shifted toward the cold wall as the gap between the uid temperature and the cold wall temperature is reduced. A singularity in the coecient sign at the cold wall should appear when the wall heat ux becomes null accordingly to what has been observed by Kuznetsov and Nield on the Nusselt number [START_REF] Kuznetsov | Forced convection in a channel partly occupied by a bidisperse porous medium: Asymmetric case[END_REF]. The amplitudes of the correlation coecient are close to 1 in the vicinity of the walls, meaning that the scalar eld is highly correlated with the streamwise velocity in the viscous wall region. This is in good agreement with Lyons et al. [START_REF] Lyons | Direct numerical simulation of passive heat transfer in a turbulent channel ow[END_REF] observations. Regarding the right graph, it appears that the coecient correlation involving the wall-normal velocity is lower than the one involving the streamwise velocity. This was also observed by Kim and Moin (Kim and Moin, 1989) and Orlandi et al. [START_REF] Orlandi | DNS of conjugate heat transfer in presence of rough surfaces[END_REF] in passive scalar transport conditions. The coecient cancels out closer to the cold wall when the heat sink is weaker. The asymmetry of the coupling between the dynamic and the temperature is then increased for small temperature dierences between the cold wall and the uid. This is in agreement with the observations carried out on heat ux uctuations, see Table 4. On the hot side, the proles are similar between all the simulations, meaning that the correlation between velocity and temperature is poorly inuenced by the hot wall heat ux.

The turbulent Prandtl number is dened by:

P r t = ν t α t = u v ∂ T /∂y v θ ∂ U /∂y , (17)
with α t the turbulent diusivity for heat. Figure 12 shows the turbulent Prandtl number proles. The proles are rather scattered because of the strong variations due to the streamwise velocity and temperature derivatives that cancel out in the center area of the channel. The results indicate that it can be relatively well approximated by 0.9. This is conforming with Kays [START_REF] Kays | Turbulent Prandtl NumberWhere Are We?[END_REF]. The turbulent Prandtl number proles display peaks between y + = 30 and y + = 50, also observed in [START_REF] Toutant | Turbulence statistics in a fully developed channel ow submitted to a high temperature gradient[END_REF].

Singularities are observed near the center of the channel due to the null streamwise velocity gradient.

The proles of the turbulent kinetic energy production (P rod = u v ∂U/∂y) normalized with the semilocal scaling are given by P rod * = P rod/ U 4, * τ,w /ν(y) .

(18)

They are plotted on Figure 13. For a given side, the production proles are almost merged. They exhibits peaks in the near wall region, as classically observed in the isothermal case. The cold peak is higher than the hot one. It is due to the dierent wall temperatures that induce variations in the uid properties and aect the turbulence production. This term explain the higher turbulence levels observed on the cold side than on the hot side.

Instantaneous elds

In this subsection, the coupling between the wall heat ux and the wall friction is rstly visualized with heat maps then it is quantied thanks to probability density functions (PDF) and spectral analyzes to provide a deep understanding of these interactions.

In Figure 14, the quantities are normalized with the associated standard deviation. On the left graphs, respectively right graphs, the results are those of the hot wall, respectively cold wall. The legend is adapted for each instantaneous eld to facilitate the reading:

X w,maxplotted X w,rms = X w,mean X w,rms + 3.9

where X is either the wall heat ux or the wall friction.

The patterns observed at the cold wall are similar to those of the hot wall but they are smaller. This is due to the lower turbulence level on the hot side that is explained by the higher temperature of the hot wall, and thus, the higher kinematic viscosity. Similar results are observed in the works of [START_REF] Kawamura | DNS of turbulent heat transfer in channel ow with respect to Reynolds and Prandtl number eects[END_REF] and Abe et al. [START_REF] Abe | Surface heat-ux uctuations in a turbulent channel ow up to Reτ=1020 with Pr=0.025 and 0.71[END_REF], which deal with the study of passive scalar in turbulent channel ows. High wall heat uxes are associated with small circular structures whereas low wall heat uxes are characterized by elongated structures in the streamwise direction. The regions of high wall heat ux coincide with those of high-speed uid whereas the regions covered by the low-speed streaks are associated with low heat transfer or low wall friction. This is conforming to the results of Kim and Moin (Kim and Moin, 1989). The instantaneous elds of wall friction are close to those of wall heat ux but they exhibit some discrepancies. Indeed, the results indicate that small circular regions of very low wall friction appear in elongated structures which are not observed in the wall heat ux elds. The bottom graphics show areas of higher normalized wall heat ux that are either circular or elongated in the spanwise direction.

φ h rms φ c rms φ h rms /φ h φ c rms /φ c τ h rms τ c rms τ h rms /τ h τ c rms /τ c [kW/m 2 ] [kW/m 2 ] [P a] [P a]
where X is either the wall heat ux or the wall friction.

All the simulations exhibit similar proles of PDF.

However, the minimum wall heat ux is 0.23φ mean while the minimum wall friction is 0. highlight that all the peaks of PDF are located at φ/φ mean -1 = -0.34 and around τ /τ mean -1 = -0.28.

They correspond to the most frequent wall heat uxes or wall frictions. It seems that the hot peaks of wall friction are very slightly shifted toward 0 when compared to the cold peaks. The PDF peak of wall heat ux is mitigated when the heat transfer is increased. For a given wall heat ux, the PDF peak is lower when the turbulence level is higher (see curves corresponding to hot and cold sides of the simulation "anisoT-S0"). This is because, the maximum wall heat ux grows with the friction Reynolds number. No clear behavior appears when regarding the amplitude of the wall friction PDF.

The obtained density probability function can be well interpolated with skewed Gaussian distribution, see dashed lines in the Figure 15, which expression is given by:

f (x) = A β √ 2π e - (x-µ) 2 2β 2 1 + erf γ(x -µ) β √ 2 (21)
with erf the error function, A the amplitude parameter, µ the location parameter, β the scale parameter, and γ the shape parameter. The results of the interpolation are presented in Figure 15 with dashed lines.

They have been obtained with the parameters listed in Table 5. The turbulent kinetic energy spectra are given by:

+∞ 0 E(k xi )dk xi = 1 2 u 2 rms + v 2 rms + w 2 rms , ( 22
)
where k xi is either k x or k z . In Figure 17, the turbulent kinetic energy spectra in the streamwise direction are investigated at a normalized distance of y + = 40.

The spectra diminish on the whole studied range of Figure 18 presents the turbulent kinetic energy spectra in the spanwise direction. On these spectra, a maximum of energy is observed at k + z = 0.0020 for the cold side and k + z = 0.0035 for the hot side. Hence, the highest values are obtained for wider turbulent structures when the turbulence level is higher. The energy levels are lower than those observed in Figure 17.

The inertial range is thinner for 1D spectra in the spanwise direction than for 1D spectra in the spanwise direction. The dissipation range is characterized by a small slope when compared to the results of 1D spectra in the spanwise direction. The slopes in the dissipation zone approximately follow a -6 power law which is consistent with the results commonly observed in the litterature, see Refs. [START_REF] Kim | Turbulence statistics in fully developed channel ow at low Reynolds number[END_REF][START_REF] Moser | Direct numerical simulation of turbulent channel ow up to Reτ=590[END_REF]Vreman et al., 2014). ~k-20 ~k-5/3 The power and the friction spectra in the streamwise direction are respectively dened by:

E(k x )/(U τ 2 δ) k + x isoT-S0 anisoT-S0, h anisoT-S1, h anisoT-S2, h anisoT-S3, h isoT-S0 anisoT-S0, c anisoT-S1, c anisoT-S2, c anisoT-S3, c
+∞ 0 P (k xi )dk xi = 1 2 φ 2 rms (23) and +∞ 0 F (k xi )dk xi = 1 2 τ 2 rms , (24)
where k xi is either k x or k z .

Spectral analysis of the wall heat uxes and the wall frictions permits identifying the size of the turbulent structures that have to be enhanced to maximize the heat transfer while minimizing the wall friction.

Thus it permits adjusting the pattern of textured surface [START_REF] Daguenet-Frick | Experimental Analysis of the Turbulent Flow Behavior of a Textured Surface Proposed for Asymmetric Heat Exchangers[END_REF] or the frequency of oscillating walls [START_REF] Rittidech | Experimental study of the performance of a solar collector by closed-end oscillating heat pipe (CEOHP)[END_REF] to optimize solar receivers. To the author's knowledge, there are very few spectral analyzes of these quantities in the literature. x . The same turbulence level and wall heat ux eects are observed in Figure 17. When observing the ratio between the power and friction spectra it appears that for low wavenumber the wall heat ux spectra are slightly lower than the wall friction spectra at the cold wall.

A peak is observed in the range 0.03 < k +

x < 0.05, corresponding to 126 < x + < 210. It seems that the heat transfer is signicantly favored in this range for all the studied congurations. The higher the wall heat ux, the bigger the wavenumber associated with the peak. When the wavenumber is higher than k + x = 0.08 for the cold wall, and lower than k + x = 0.1 for the hot wall, the normalized power spectra are lower than the normalized friction spectra. For the last region, which corresponds to the highest wavenumber values for each simulation, the results have to be interpreted with caution because of the eects of the numerical setup. Sharp oscillations are noted. For the simulation "anisoT-S0", the normalized wall heat ux spectrum [START_REF] Abe | Surface heat-ux uctuations in a turbulent channel ow up to Reτ=1020 with Pr=0.025 and 0.71[END_REF]. Regarding the right graph, it appears that there is no evident structure size that should be forced to enhance heat transfer while keeping low-pressure losses. Nevertheless, it seems that, globally, largest structures in the spanwise direction should be promoted. This is in agreement with the elongated red stains observed on the bottom graph of Figure 14. Once again, a sharp drop in the ratio is noted. The decrease occurs for smaller normalized wavenumbers at the cold wall (k + z ≈ 0.03) than at the hot wall (k + z ≈ 0.08).

Conclusions

In this study, four DNS of anisothermal channel ow

  studied the eect of the Prandtl and the Schmidt numbers at a friction Reynolds number of 150. Their results show that the momentum and scalar transport analogy is not applicable to dene the limiting behavior of turbulent diusivity in the vicinity of the wall. Moreover, since the turbulent transport is weak in the conductive sublayer, this limiting relation cannot be used to calculate the concentration or temperature prole for large values of Prandtl or Schmidt numbers. The next studies are related to the inuence of the Reynolds number. Three DNS at friction Reynolds number of 500, 1000 and 2000, with medium to low Prandtl number, were performed by Alcántara-Ávila et al. (Alcántara-Ávila et al., 2018). They notice that the logarithmic prole of the thermal ow appears for Prandtl number above 0.3 because of the thinner viscous thermal layer generated as the Prandtl number is increased. Alcántara-Ávila et al. (Alcántara-Ávila and Hoyas, 2021) supplement the previous work with medium to high Prandtl number. They characterize the thermal eld in those new conditions and notice that the near-wall maximum of temperature uctuations is independent of the Reynolds number for suciently high Prandtl numbers, meaning above 7.

  carried out DNS at mean friction Reynolds of 180 and 395, with wall temperature of 293 K and 596 K. They establish the turbulent kinetic energy spectral equation and decompose them in three terms: production, nonlinear transfer and viscous eects. The authors discuss the inuence of each terms in both isothermal and anisothermal conditions. It was pointed out that the purely anisothermal contribution transfers the energy from the hot side of the channel toward the cold side.

Figure 1 :

 1 Figure 1: Investigated channel ow.

  the dynamics and the temperature, four anisothermal ow congurations are investigated. In addition, a DNS in isothermal conditions is studied to highlight the temperature eects of the anisothermal simulations. For the anisothermal simulations, the wall temperatures are xed to 900 K and 1300 K to be representative of the temperature levels reached by the irradiated and insulated walls of the solar receivers. The heat sink values are chosen to obtain several uid temperature that are encountered in the operating conditions of solar receivers. The heat sink permits investigating ows of various axial locations in the solar receiver.

  w . The fric- tion velocity and the friction temperature are respectively expressed as U * τ,w = µ w /ρ(y)∂U 1 /∂x 2 , and T * τ,w = φ w / ρ(y)C p U * τ,w . Note that, the semi- local scaling computes the wall friction with the wall viscosity but the quantities are expressed locally. The wall friction velocity is inuenced by the local variations of the density. The symbol "• o " signies that the ow parameters are normalized with the mean friction velocity or temperature: U o i = U i /U mean τ,w , T o = (T w -T )/T mean τ,w . The velocity -temperature and temperature -temperature are similarly scaled u i θ o = u i θ / U mean τ,w T mean τ,w and θ θ o = θ θ / T mean2 τ,w

Figure 3

 3 Figure3shows the proles of density and kinematic viscosity as functions of the normalized channel height.

Figure 2 :

 2 Figure 2: Proles of scaled streamwise velocity, wall-normal velocity, and temperature as functions of the normalized channel height.

Figure 3 :Figure 4 :

 34 Figure 3: Proles of density and kinematic viscosity as functions of the normalized channel height.

Figure 5 :Figure 6 :

 56 Figure 6 presents the streamwise velocity scaled with the Van Driest transformation as well as wallnormal velocity normalized with the wall scaling. The Van Driest scaling permits almost merging the streamwise velocity proles. The slopes in the logarithmic

Figure 7

 7 Figure 7 exhibits the streamwise, spanwise, wallnormal, and cross velocity correlations. Similar behaviors are noticed: uctuation peaks are higher on the cold side. This is because, the increase of viscosity on the hot side tends to dissipate turbulent kinetic energy. Peak amplitudes are mitigated and located

Figure 8

 8 Figure 8 displays the proles of streamwise and wallnormal velocity correlations normalized with the semilocal scaling. The results indicate that the streamwise velocity uctuations peaks are closer from each other with this scaling. Signicant dierences are noticed after y + = 50. The second energy peak clearly increases with the heat sink and thus the Reynolds number. The

FocusingFigure 7 :

 7 Figure 7: Proles of velocity uctuations as functions of the channel height normalized in wall unit.

Figure 8 :

 8 Figure 8: Proles of velocity uctuations as functions of the channel height normalized with the semi-local scaling.

Figure 10 Figure 9 :

 109 Figure 10 presents the proles of streamwise velocity -temperature correlation and temperature uctuations normalized with the semi-local scaling. As observed in Figure 8, the semi-local scaling brings closer the uctuation peaks. The strong asymmetry of the simulation "anisoT-S1" is highlighted on the left graph.Regarding the temperature uctuations, the results show that the hot and cold proles of the simulation

Figure 10 :

 10 Figure 10: Proles of streamwise velocity -temperature correlation and temperature uctuations normalized with the semi-local scaling.

Figure 11 :

 11 Figure 11: Proles of velocity -temperature coecient correlations as functions of the normalized channel height.

Figure 12 :

 12 Figure 12: Proles turbulent Prandtl number.

Figure 13 :

 13 Figure 13: Proles of the turbulent kinetic energy production as functions of the normalized channel height. The results are normalized with the semi-local scaling.

Figure 14 :

 14 Figure 14: Instantaneous elds of normalized wall heat ux and wall friction as well as the ratio between the two quantities. The presented results are those of the DNS "anisoT-S3". The left legend is associated with the graphics of the two rst lines and the right legend is associated with the graphics of the last line. The symbol X A either denotes the wall heat ux or the wall friction depending on the observed line of graph.

Figure 16 Figure 15 :Figure 16 :

 161516 Figure 16 is obtained by extracting data along the center line in the streamwise direction of Figure 14 for the top graph and along the center line in the spanwise direction of Figure 14 for the bottom graph. It permits performing a quantitative analysis of the spatial distribution of the wall heat ux and the wall

  wavenumber. The curves are divided into two groups, each one corresponding to a side. The isothermal simulation presents similar results to the cold wall turbulent kinetic energy spectra. It appears that the turbulence level is clearly impacting those results. The transition between the inertial zone and the energy dissipation zone is observed for lower wavenumber at high friction Reynolds numbers. It seems that, for a given wall and wavenumber, the higher the wall heat uxes are, the wider the inertial range and the bigger the energy level in the energy dissipation range. The obtained slope in the dissipation zone is too sharp. Vreman et al. (Vreman et al., 2014) compared DNS energy spectra obtained with nite dierence algorithms and spectral algorithms. They show that the energy spectrum in the streamwise direction falls o too sharply when the nite dierence algorithm is used. This may be due to the nite dierence operator in the convective terms that behaves like an implicit lter over the nonlinear transfer to the highest wavenumbers. Notice that this error only aects the streamwise energy spectra and is limited to the smallest scales. The integrals of the energy spectra remain valid.

Figure 17 :

 17 Figure 17: Turbulent kinetic energy spectra in the streamwise direction at y + ≈ 40.

Figure 18 :

 18 Figure 18: Turbulent kinetic energy spectra in the spanwise direction at y + ≈ 40.

Figure 19

 19 Figure 19 plots the power and friction spectra as well as the ratio between the two quantities at hot and cold walls for all the performed anisothermal simulations, excepted "anisoT-S2" to facilitate the reading. The power and friction spectra exhibit similar tendencies that the classically studied turbulent kinetic energy spectra. The slope of the energy transfer is proportional to k -5/3 x in the inertial zone then energy is dissipated with a slope proportional to k -7

Figure 19 :Figure 20 :

 1920 Figure 19: Power and friction spectra in the streamwise direction at the walls.

  Figure15presents the PDF of wall heat uxes and wall frictions. Each curve is approximated thanks to a skewed Gaussian distribution. The used parameters are given in Table5.

Table 1 :

 1 Investigated thermal conditions.

	S0	293 -293	0	293
	anisoT-S0	1300 -900	0	1093
	anisoT-S1	1300 -900	55	883
	anisoT-S2	1300 -900	109	690
	anisoT-S3	1300 -900	164	525

Table 3 :

 3 Mean ow quantities relative to temperature eects. "nr" signies non relevant and corresponds to operating conditions that are beyond the applicability domain of the correlation ofDavid et al. (David et al., 2021c). The correlation results are associated with the word "Corr".

			/m	2 ]		DNS	Corr		DNS	Corr
	anisoT-S0	98.4	98.4		1.0	88.6	87.2	-1.6%	117.7	nr	-
	anisoT-S1	262.5	-63.7		4.1	117.1	112.3	-4.1%	876.5	nr	-
	anisoT-S2	441.0	-244.4	1.8	134.4	132.8	-1.2%	269.0	272.0	1.1%
	anisoT-S3	587.1	-410.2	1.4	140.9	146.5	4.0%	253.2	245.0	-3.3%

Table 4 :

 4 Wall heat ux and wall friction uctuations.

	anisoT-S0	42.0	42.0	0.43	0.43	17.6	19.7	0.40	0.41
	anisoT-S1	112.3	27.1	0.43	0.43	17.6	19.5	0.41	0.41
	anisoT-S2	194.8	104.4	0.44	0.43	17.5	20.2	0.42	0.43
	anisoT-S3	262.8	187.5	0.45	0.46	16.7	19.5	0.42	0.44

Table 5 .

 5 Simulation Parameter P DF (φ h ) P DF (φ c ) P DF (τ h ) P DF (τ c ) anisoT-S0 A 3.77 × 10 -02 3.99 × 10 -02 2.96 × 10 -02 4.34 × 10 -02 µ -5.55 × 10 -01 -5.62 × 10 -01 -4.99 × 10 -01 -5.25 × 10 -01 σ 6.54 × 10 -01 6.63 × 10 -01 6.11 × 10 -01 6.30 × 10 -01 γ 6.39 × 10 +00 6.37 × 10 +00 4.23 × 10 +00 4.71 × 10 +00 anisoT-S1 A 3.46 × 10 -02 3.83 × 10 -02 3.63 × 10 -024.29 × 10 -02 µ -5.55 × 10 -01 -5.44 × 10 -01 -5.10 × 10 -01 -5.21 × 10 -01 10 -01 -5.60 × 10 -01 -5.06 × 10 -01 -5.39 × 10 -01 σ 6.29 × 10 -01 6.29 × 10 -01 6.09 × 10 -01 6.25 × 10 -01 γ 6.43 × 10 +00 6.58 × 10 +00 4.12 × 10 +00 4.58 × 10 +00

		σ	6.55 × 10 -01	6.47 × 10 -01	6.18 × 10 -01	6.24 × 10 -01
		γ	6.70 × 10 +00	6.39 × 10 +00	4.36 × 10 +00	4.55 × 10 +00
		A	4.06 × 10 -02	3.31 × 10 -02	3.84 × 10 -02	4.30 × 10 -02
	anisoT-S2	µ σ	-5.54 × 10 -01 -5.48 × 10 -01 -5.04 × 10 -01 -5.26 × 10 -01 6.43 × 10 -01 6.39 × 10 -01 6.09 × 10 -01 6.21 × 10 -01
		γ	6.52 × 10 +00	6.49 × 10 +00	4.10 × 10 +00	4.51 × 10 +00
		A	3.85 × 10 -02	4.43 × 10 -02	3.35 × 10 -02	4.67 × 10 -02
	anisoT-S3	µ	-5.55 ×

Table 5 :

 5 Parameters used to t the wall heat ux and wall friction PDF with skewed Gaussian distributions. D. Dupuy, A. Toutant, and F. Bataille. A posteriori tests of subgrid-scale models in strongly anisothermal turbulent ows. Physics of Fluids, 31:(6): 065113, June 2019. A. W. Vreman and J. G. M. Kuerten Comparison of direct numerical simulation databases of turbulent channel ow at

Acknowledgments

This work was granted access to the HPC resources of CINES under the allocation 2022-A0112A05099 made by GENCI. The authors gratefully acknowledge the CEA for the development of the TRUST platform.

CRediT authorship contribution statement

Martin David: Conceptualization, Methodology, Investigation, Data acquisition, Writing. Adrien Toutant: Work guiding, Reviewing. Françoise Bataille: Work guiding, Reviewing.


Context:

Recently, surface temperatures in urban areas of Nantes reached more than 12°C higher than in rural areas [START_REF] Auran | 40°C à l'ombre : faut-il craindre de vivre dans un climat plus chaud ? Les synthèses de l[END_REF]. This phenomenon has been identified as urban heat island (UHI) [START_REF]Recommandation pour lutter contre l'effet d'îlot de chaleur urbain à destination des collectivités territoriales[END_REF]. UHI reducing pavements (cool pavements), have gained an increasing attention from city officials and researchers to mitigate UHI effect. This poster presents a literature review followed by finite element analysis as a first step of the co-tutelle PhD (2022-26) between University of New Hampshire (USA) and Univ. Eiffel (France). This PhD aims to improve the durability of UHI reducing pavements that currently exist and are under development by focusing on their structural optimization (interface performance, layer arrangements, geometric extents).

Preliminary FEM Exploration: Effects of bilayer UHI pavement properties on global response

Conclusions/perspectives: A first literature review has highlighted 4 different families of "cool pavements" used to reduce UHI. UHI reducing multi-material system durability requires high performance of interfaces. To optimize such composite structures, there is an urgent need to adopt and develop fundamental characterization of multi-material interfaces [START_REF] Buttlar | Mechanisms of Cracking and Debonding in Asphalt and Composite Pavements[END_REF]. To this aim, preliminary finite element modeling has been conducted to evaluate effects of thermal and mechanical properties variations on global response using the cohesive zone model (CZM) in ABAQUS®. This work will be followed by experimental testing of interfaces under thermal and moisture loadings, more finite element analysis of the interface behavior using CZM, and life cycle assessment to optimize the UHI reducing pavements.

Cohesive Zone Model

Figure 2 -Figure 1 -Figure 3 -

 213 Figure 2 -Surface temperatures in Nantes [3] Figure 1 -Temperatures in Nantes [1] Figure 3 -Geographical locations of the collected papers

Layer 1 : 1 Layer 2 :

 112 E 1 =5400 MPa, n 1 = 0.35, 𝜶 E 2 =12300 MPa, n 2 = 0.35, 𝜶 𝟐 Urban temperature > Rural temperature sadek.tormos@univ-eiffel.fr