Farid Berrahil
email: farid.berrahil@gmail.com

Abdelkader Filali

Chérifa Abid

Smail Benissaad

Rachid Bessaih

Omar Matar

Numerical investigation on natural convection of Al2O3/Water nanofluid with variable properties in an annular enclosure under magnetic field

Keywords: Natural convection, !" ! # " -water, Lorentz forces, annulus enclosure, variable properties. NOMENCLATURE

Numerical investigation of the natural convection of !" ! # " -water nanofluid is carried out in a differentially heated vertical annulus under a uniform magnetic field. An in-house Fortran code has been developed to solve the system of equations governing the magneto-hydrodynamic flow. Computations are carried out for different Rayleigh numbers (10 # ≤ '( ≤ 10 $ ), nanoparticle diameter () % = 13 and 47 /0), nanoparticle volume fraction (0 ≤ 3 ≤ 0.09), radius ratio (2 ≤ 7 ≤ 10), and different Hartmann numbers (0 ≤ 8( ≤ 100). According to the simulation data, nanoparticle size is crucial for evaluating nanofluid properties, such as viscosity and thermal conductivity. The computational results reveal that, for nanoparticles with a diameter ) % = 47 /0, the average Nusselt number 9: ;;;; & on the inner cylinder wall decreases as the nanofluid volume fraction increases. This decrease in 9: ;;;; & number is observed up to a volume fraction 3 = 0.05, after which it increases again. For the full range of volumetric fractions, it is shown that increasing '( number causes 9:

;;;; & to increase, while increasing 8( number and increasing the magnetic field causes 9: ;;;; & to decrease. Furthermore, as the 8( number increases, the heat transfer enhancement ratio =/ increases mainly when the magnetic field is oriented radially.

Finally, new correlations of 9: ;;;; & versus '(, 3, 8(, and 7 are derived for the axial and radial magnetic fields cases.

Introduction

Research studies related to natural convection in (a confined) cylindrical annular enclosures have received considerable attention and have remained a fundamental field to explain many physical phenomena. The application areas are compact heat exchangers, food preservation, solar collectors, nuclear reactors, and electronic cooling systems [START_REF] Ostrach | Natural convection in enclosures[END_REF].

The natural convection phenomena under the applied magnetic field can be found in some experimental processes, such as cooling fusion reactors and crystal growth in liquid. Various studies in the literature have been performed to characterize natural convection and heat transfer in enclosures under applied magnetic fields. Different geometrical parameters and boundary conditions were investigated [START_REF] Garandet | Buoyancy driven convection in a rectangular enclosure with a transverse magnetic field[END_REF][START_REF] Alchaar | Natural convection heat transfer in a rectangular enclosure with a transverse magnetic field[END_REF][START_REF] Al-Najem | Numerical study of laminar natural convection in tilted enclosure with transverse magnetic field[END_REF][START_REF] Gelfgat | The effect of an external magnetic field on oscillatory instability of convective flows in a rectangular cavity[END_REF][START_REF] Hossain | Buoyancy and thermo capillary driven convection flow of an electrically conducting fluid in an enclosure with heat generation[END_REF][START_REF] Sarris | MHD natural convection in a laterally and volumetrically heated square cavity[END_REF][START_REF] Ece | Natural-convection flow under a magnetic field in an inclined rectangular enclosure heated and cooled on adjacent walls[END_REF][START_REF] Pelekasis | Linear stability analysis and dynamic simulations of free convection in a differentially heated cavity in the presence of a horizontal magnetic field and a uniform heat source[END_REF][START_REF] Sathiyamoorthy | Effect of magnetic field on natural convection flow in a liquid gallium filled square cavity for linearly heated side wall(s)[END_REF][START_REF] Kherief | Effects of inclination and magnetic field on natural convection flow induced by a vertical temperature[END_REF][START_REF] Berrahil | Natural convection with volumetric heat generation and external magnetic field in differentially heated enclosure[END_REF][START_REF] Kherief | Magneto-hydrodynamic flow in a two-dimensional inclined rectangular enclosure heated and cooled on adjacent walls[END_REF]. The magnetic field direction and strength were revealed as two major factors that significantly affected the heat transfer properties and flow patterns. Similar research studies can be found in an annular cavity under a magnetic field [START_REF] Sankar | Effect of magnetic field on natural convection in a vertical cylindrical annulus[END_REF][START_REF] Sankar | Effect of magnetic field on the buoyancy and thermocapillary driven convection of an electrically conducting fluid in an annular enclosure[END_REF][START_REF] Kakarantzas | Natural convection of liquid metal in a vertical annulus with lateral and volumetric heating in the presence of a horizontal magnetic field[END_REF][START_REF] Afrand | Effect of induced electric field on magneto-natural convection in a vertical cylindrical annulus filled with liquid potassium[END_REF][START_REF] Mebarek-Oudina | Oscillatory magneto-hydrodynamic natural convection of liquid Metal between vertical coaxial cylinders[END_REF][START_REF] Kakarantzas | MHD liquid metal flow and heat transfer between vertical coaxial cylinders under horizontal magnetic field[END_REF][START_REF] Afrand | Using a magnetic field to reduce natural convection in a vertical cylindrical annulus[END_REF][START_REF] Afrand | A numerical study of natural convection in a vertical annulus filled with gallium in the presence of magnetic field[END_REF]. In most of these studies, the case of electrically conducting fluid characterized by a low thermal conductivity is considered.

Improvement of heat transfer performance is an essential aspect of saving energy. The low thermal properties of the working fluids, such as water, engine oil, and ethylene glycol, are critical constraints during the design of compact thermal systems with high efficiency. In contrast, Nanoparticles have been shown to have potential advantages in increasing the thermal conductivity of the base fluids. An exciting review on the nanofluid preparation methods reported by different investigators and describing the suitable method for preparing stable nanofluids has been published by Haddad et al. [START_REF] Haddad | A review on how the researchers prepare their nanofluids[END_REF].

The final prepared nanofluid will have specific properties, i.e., thermal conductivity and dynamic viscosity, that play an essential role during heat transfer processes [START_REF] Haddad | Natural convection of silica-water nanofluids based on experimental measured thermophysical properties: critical analysis[END_REF]. Choi [START_REF] Choi | Enhancing thermal conductivity of fluids with nanoparticles[END_REF] was the first who proposed the term nanofluid to designate the mixture of suspended solid nanoparticles in a base fluid. The resulting mixture is characterized by an improved effective thermal conductivity when compared to traditional fluids.

Several authors have considered examining the natural convection of nanofluids in a heated cavity with or without an applied magnetic field. Khanafer et al. [START_REF] Khanafer | Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids[END_REF] conducted a numerical analysis of natural convection problem in a differentially heated cavity containing a flowing nanofluid. Their results showed that 9: ;;;; number is considerably increased with rising the nanoparticle volume fraction for all Grashof number values. Putra et al. [START_REF] Putra | Natural convection of nanofluids[END_REF] analyzed the natural convection inside the horizontal cylinder experimentally. Results revealed that the presence of !" ! # " or g:# nanoparticles in water reduced the heat transfer in which both the density and concentration of the nanoparticles played a significant role in the heat transfer deterioration. Omid Abouali and Ahmad Falahatpisheh [START_REF] Abouali | Numerical investigation of natural convection of Al2O3 nanofluid in vertical annuli[END_REF] considered a vertical annulus to investigate numerically the effect of the (!" ! # " -water) nanofluid viscosity and particle diameters on the natural convection flow. Their results were validated first against the experimental results provided by Putra et al. [START_REF] Putra | Natural convection of nanofluids[END_REF].

A general correlation has been developed to predict the heat transfer rate for annuli, which depends on the viscosity model and particle diameter. The authors have advised that this correlation is also valid for the case of the square cavity. Ghasemi et al. [START_REF] Ghasemi | Magnetic field effect on natural convection in a nanofluid-filled square enclosure[END_REF] considered the natural convection phenomena for the case of (!" ! # " -water) nanofluid inside a square cavity subjected to a horizontally directed magnetic field. Their outputs showed that the computed Nu number decreases with the augment of 8( number and increases with the extension of '( number. However, the enhancement or deterioration of Nu number with the increased solid volume fraction depends on '( and 8( number values. Sheikholeslami et al. [START_REF] Sheikholeslami | Numerical investigation of MHD effects on Al2O3 water nanofluid flow and heat transfer in a semi-annulus enclosure using LBM[END_REF] applied the Lattice Boltzmann method -LBM to solve the natural convection phenomena in an annulus between a cold square and a heated elliptic cylinder. Both an applied magnetic field and the influence of Brownian motion on the effective thermal conductivity were investigated. Their outputs showed that increasing '( number and the nanoparticle volume fraction leads to an enhanced average Nu number. Increasing 8( number, on the other hand, have an opposite effect on Nu number. Later, Sheikholeslami et al. [START_REF] Sheikholeslami | Effect of a magnetic field on natural convection in an inclined half-annulus enclosure filled with Cu-water nanofluid using CVFEM[END_REF] analyzed the natural convection in a half-annulus enclosure exposed to a magnetic field. The applied thermal condition of one wall subjected to a constant heat flux has been considered, and various inclination angles have been texted. The results showed that rising the volume fraction of nanoparticles and '( number improved the Nu number. In contrast, applying a magnetic field results in an inhibited flow and temperature fluctuations by decreasing the Nu number and the flow velocity. For all investigated inclination angles and low '( numbers, increasing 8( number leads to improved heat transfer.

Computational results reported by Ashorynejad et al. [START_REF] Ashorynejad | Magnetic field effects on natural convection flow of a nanofluid in a horizontal cylindrical annulus using Lattice Boltzmann method[END_REF], for the case of the horizontal cylindrical annulus, revealed that the flow fluctuations of nanofluids for a natural convection problem could be inhibited by applying an external magnetic field. In addition, the authors concluded that higher values of the average Nu number could be achieved by increasing '( number and/or the volume fraction of nanoparticles. In contrast, increasing 8( number leads to low Nusselt number values. Sheikholeslami et al. [START_REF] Sheikholeslami | Lattice Boltzmann simulation of magneto-hydrodynamic natural convection heat transfer of Al2O3-water nanofluid in a horizontal cylindrical enclosure with an inner triangular cylinder[END_REF] carried out numerical simulations to investigate the magneto-hydrodynamic natural convection for !" ! # " -water nanofluid inside a horizontal annulus between a circular cylindrical enclosure and an internal heated triangular cylinder. It is found that the computed 9: ;;;; number is proportional to '( number and inversely proportional to 8( number. The authors established a new correlation of 9: ;;;; number as a function of the studied physical parameters. Sheikholeslami [START_REF] Sheikholeslami | Heat flux boundary condition for nanofluid filled enclosure in presence of magnetic field[END_REF] has carried out a numerical simulation on the effect of magnetic field on natural convection of nanofluid flowing inside an enclosure. An uniform temperature was imposed on the outer cylinder, while the inner cylinder was submitted to a constant heat flux. Reported results revealed that the improvement in heat transfer increases the aspect ratio and 8( number, whereas it decreases with the increasing '( number. Mebarek-Oudina and Bessaïh [START_REF] Mebarek-Oudina | Numerical simulation of natural convection heat transfer of copper-water nanofluid in a vertical cylindrical annulus with heat sources[END_REF] carried out the numerical computation of natural convection in a vertically positioned cylindrical annulus enclosure comprising g:-water nanofluid. The case study considered the impact of two discrete heat sources having different lengths and mounted on the inner enclosure wall. Results showed that the investigated parameters, including the size and the location of the heaters, the volume fraction of nanoparticles, and '( number, significantly affect the heat transfer characteristics. More precisely, when the heater is placed in the vicinity of the bottom wall, the heat transfer rate is improved, whereas it decreases with increasing the top or the bottom heater size. Furthermore, the heat transfer rate increases with the rising of the '( number and the volume fraction of nanoparticles. Mebarek-Oudina et al. [START_REF] Mebarek-Oudina | Heat transport of magnetized Newtonian nanoliquids in an annular space between porous vertical cylinders with discrete heat source[END_REF] studied numerically the magneto-hydrodynamic natural convection in a porous vertical annular enclosure containing a nanofluid whose aspect ratio and radius ratio are equal to 2. The results revealed that the increase in heat transfer rate is proportional to the augment of '( number, the concentration of nanoparticles, the Darcy number, and the porosity. Simultaneously, it is inversely proportional to the increase in the length of the heat source and the intensity of the magnetic field. They concluded that the improvement in heat transfer is better if !D nanoparticles are used compared to other nanoparticles, namely, g:, !" ! # " and @d# ! . Recently, Sadeghi et al. [START_REF] Sadeghi | Analysis of thermal behavior of magnetic buoyancy-driven flow in ferrofluid-filled wavy enclosure furnished with two circular cylinders[END_REF] conducted a numerical analysis of natural convection in an adiabatic corrugated cavity with two circular cylinders held at different temperatures using the FEM finite element method. The cavity is filled with (ij " # # -water) ferrofluid. The flow is exposed to a magnetic field where the viscosity of the ferrofluid is highly dependent on its intensity. They examined the influence of various control parameters such as '( and 8( numbers, the radiation parameter ' ) , the shape aspect of the nanoparticles, the volume fraction of the ferrofluid, and the distance between the two circular cylinders on the flow behaviour and heat transfer performance. Their findings for a platelet nanoparticle shape (0 = 5.7) suggest that the heat transfer rate and the intensity of natural convection increase monotonically with '( and ' ) but decrease at the expense of 8(. In addition, the magnetic field is more effective at low values of '(, while thermal radiation is more effective at high values of '(.

So far, the survey of the previous studies shows a lack of studies dealing with the effect of nanofluid properties, namely the viscosity and the thermal conductivity, and comparing those effects for the cases of axially and radially applied magnetic field. Therefore, and to the best of the author's knowledge, no detailed contributions and established correlations were provided to show those effects combined with the nanoparticle volume fraction effects, '( and 8( numbers on the flow structure and heat transfer characteristics. Consequently, the present study aims to assess these effects numerically in a vertical cylindrical annulus containing a water-alumina. The effective thermal conductivity and viscosity of the mixture water-alumina are computed by KV (Khanafer and Vafai [START_REF] Khanafer | A critical synthesis of thermophysical characteristics of nanofluids[END_REF]) correlation. The influence of the parameters mentioned above is examined for two nanoparticle diameters ) % = 13 and 47 /0. It highlights that the consideration of variable fluid properties such as thermal conductivity and viscosity is of critical importance. This approach is more realistic compared to the one based on constant properties that, on the one hand, overestimate the heat transfer in hot and cold locations and the size of nanoparticles on the other.

Mathematical formulation

Figure 1 shows a schematic of the considered nanofluid contained between two vertical coaxial cylinders of height 8 and inner and outer cylinder's radius O & and O ( , respectively. The vertical walls are differentially heated as hot inner wall (@ * ) and cold outer wall (@ + ) whereas the horizontal walls are kept adiabatic (thermally insulated). The annular enclosure is exposed to a constant external magnetic field, and all four walls are electrically insulated boundaries.

For an appropriate formulation of the problem, the nanofluid is supposed to be homogenous, incompressible, and Newtonian, and the flow is assumed laminar and axisymmetric. The spherical !" ! # " nanoparticles and the base fluid (water) are supposed in thermal equilibrium. Induced electric current, Joule heating, radiation, and viscous dissipation are neglected. In addition, the thermophysical properties of the nanoparticles and the base fluid considered in the present study are presented in Table 1 [START_REF] Sheikholeslami | Numerical investigation of MHD effects on Al2O3 water nanofluid flow and heat transfer in a semi-annulus enclosure using LBM[END_REF].

The governing equations of the nanofluid flow and heat transfer using the Boussinesq approximation are defined as follows:

1 O k kO (O:) + kR kS = 0 (1)
kR kS pn + ([W) ,-D(@ -@ + ) + c ./1 (3) q[A % r ,-m k@ kP + 1 O k kO (O:@) + k kS (R@)n = s 1 O k kO oC ,-O k@ kO p + k kS oC ,- k@ kS pt (4)
where K, @, : and R are, respectively, the dimensional pressure, temperature, and the velocity components. c ./0 and c ./1 represent the dimensional Lorenz forces in radial and axial directions, respectively, and are computed from the equation:

H 23 = \ ,-[v × x] × x
, where x and v are, respectively, the magnetic field and the velocity vectors, computed as follows:

? = ? 0 (radial magnetic field):

z c ./0 = 0 c ./1 = -\ ,-q? ' ! rR (5a)
? = ? 1 (axial magnetic field):

z c ./0 = -\ ,-q? ' ! r: c ./1 = 0 (5b)
The nanofluid density [ ,-, thermal expansion coefficient W ,-, heat capacity A % ,-, and thermal diffusivity V ,-are defined, respectively, as:

[ ,-= (1 -3)[ -+ 3[ % (6) ([W) ,-= (1 -3)([W) -+ 3([W) % (7) q[A % r ,- = (1 -3)q[A % r - + 3q[A % r % (8) V ,-= C ,-q[A % r ,- (9)
Also, the effective electrical conductivity for the used nanofluid is expressed as follows [START_REF] Maxwell | A Treatise on Electricity and Magnetism[END_REF]:

\ ,-\ - ⁄ = 1 + 3q\ % \ - ⁄ -1r3 q\ % \ - ⁄ + 2r -q\ % \ - ⁄ -1r3 (10)
In the above equations, 3 is the nanoparticle concentration in nanofluids.

The dynamic viscosity Y ,-and the thermal conductivity C ,-of nanofluids are obtained from the below respective correlations proposed by KV [START_REF] Khanafer | A critical synthesis of thermophysical characteristics of nanofluids[END_REF] based on the various experimental results. With this model, the properties depend on temperature, volume fraction, and particle diameter:

Y ,-(AK) = -0.4491 + 28.837 @ + 0.574 3 -0.1634 3 ! + 23.053 3 ! @ ! + 0.0132 3 " -2354.735 3 @ " + 23.498

3 ! ) % ! -3.0185 3 "
) % ! [START_REF] Kherief | Effects of inclination and magnetic field on natural convection flow induced by a vertical temperature[END_REF] valid in the ranges: 1% ≤ 3 ≤ 9% , 13 /0 ≤ ) ! ≤ 131 /0 and 20 °g ≤ @ ≤ 70 °g.

C #$ C $ = 0.9843 + 0.398 3 %.'()( Å 1 ) ! (/0) Ç %.**+, Å Y #$ (@) Y $ (@) Ç %.%*(- -3.9517 3 @ + 34.034 3 * @ ( + 32.509 3 @ * (12)
valid in the ranges: 0 ≤ 3 ≤ 10%, 11 /0 ≤ ) ! ≤ 150 /0 and 20 °g ≤ @ ≤ 70 °g.

The base fluid (water) viscosity is assumed as a temperature-dependent function defined as [START_REF] Chon | Empirical correlation finding the role of temperature and particle size for nanofluid (Al2O3) thermal conductivity enhancement[END_REF]:

Y $ (@) = 2.414 × 10 .-× 10 *+'.) (01*'(.2-.2+%) ⁄ (13)
The appropriate boundary conditions are written as follows:

for O = O 5 → : = R = 0 (/) @ = @ 6 for O = O 7 → : = R = 0 (/) @ = @ 8 (14)
for S = 0 → : = R = 0 (/) k@ kS ⁄ = 0 for S = 8 → : = R = 0 (/) k@ kS ⁄ = 0

The governing equations (1)-( 5) with the applied boundary conditions are formulated in dimensionless form using the following dimensionless variables presented in equation ( 15):

^= P V $ 9 8 * , (', U) = (O, S) 1 8 , (T, Ñ) = (:, Ö) 8 V $ 9 , L = K8 * [ #$ V $ 9 * , X = @ -@ 6 @ 8 -@ 6 (15) 1 ' k k' ('T) + kJ kU = 0 (16) 1 ' k k' ('T * ) + k kU (JT) = - kL k' + Pr (1 -3) + 3 [ ! [ $ s 1 ' k k' oY' kT k' p + k kU oY kT kU p -Y T ' * t + i :;< (17) 1 ' k k' ('TJ) + k kU (J * ) = - kL kU + Pr (1 -3) + 3 [ ! [ $ s 1 ' k k' oY' kJ k' p + k kU oY kJ kU pt + m(1 -3) + 3 W ! W $ n '( Pr X + i :;= (18) 1 ' k k' ('TX) + k kU (JX) = 1 (1 -3) + 3 q[A ! r ! q[A ! r $ s 1 ' k k' oC' kX k' p + k kU oC kX kU pt (19)
Where:

Y = Y #$ Y $ 9 ; C = C #$ C $ 9 (20)
It should be noted that for the case of a pure fluid

C = 1 and Y = Y $ Y $ 9 ⁄ with Y $ 9
is the viscosity at the reference temperature, considered equal to 22 °g in the present study.

wherein equations ( 17) and ( 18), i :;< , i :;= are the radial and axial dimensionless Lorenz forces, respectively. ? = ? < :

à i :;< = 0 i :;= = -Pr(8() * J (21a)
? = ? = :

à i :;< = -Pr(8() * T i :;= = 0 (21b)
The applied boundary conditions can be formulated in dimensionless form as:

for ' = 7 !(7 -1) ⁄ → T = J = 0 (/) X = 0 for ' = 1 !(7 -1) ⁄ → T = J = 0 (/) X = 1 (22) for U = 0 → T = J = 0 (/) kX kU ⁄ = 0 for U = 1 → T = J = 0 (/) kX kU ⁄ = 0
The relevant non-dimensional parameters that characterize the flow are:

Rayleigh number '( = LM N ∆PQ 3 S N 0 U N 0 , Prandtl number Pr = V W 9 X W 9
, Hartmann number 8( = ? % 8 â

Y W Z W 9
, the enclosure aspect ratio

! = [ (< \ .< ] ) , and 7 = < \ < ]
is the radius ratio,

To evaluate the total heat transfer rate through the enclosure, the local and the average Nusselt numbers are defined on the inner and outer cylinders as:

9: = ℎ 8 C $ = -C #$ k@ kO (@ 8 -@ 6 ) 8 C $ (23)
Then, the local Nusselt number can be written in dimensionless form as:

9: = -Å C #$ C $ Ç kX k' (24)
The average Nusselt number is defined as:

9: ;;;; 7 = ä ã-Å C #$ C $ Ç kX k' å ^_ 2 `(a.2) )U 2 % (25)
(at the inner cylinder)

9: ;;;; 5 = ä ã-Å C #$ C $ Ç kX k' å ^_ a `(a.2) )U 2 % (26)
(at the outer cylinder)

Therefore, the heat balance of the pure fluid in the enclosure can be given as follows [START_REF] Choukairy | Transient behavior inside a vertical cylindrical enclosure heated from the side walls[END_REF]:

9: ;;;; 7 9: ;;;; 5 = 7 ( 27
)
The meridional dimensionless stream function _ is defined as:

T = 1 ' k_ kU , J = - 1 ' k_ k' (28)
The enhancement of heat transfer ratio between the pure fluid and the nanofluid (3 = 0.09) is defined by: =/ = 9: ;;;; 7 (3 = 0.09) -9: ;;;; 7 (ç(Qj c":d)) 9: ;;;; 7 (ç(Qj c":d))

× 100 (29)

Numerical details and model verification

The governing equations presented in section 2 are solved by the control volume-based discretization method (Patankar [START_REF] Suhas | Numerical Heat Transfer and Fluid Flow[END_REF]). The pressure and velocity of Eqs. ( 1) -( 4) are coupled using the SIMPLER algorithm. A fully implicit first-order forward Euler scheme is used for temporal discretization. The second-order central finite difference schemes spatially discretize the governing equations, then, the discretized equations for each variable are solved using the tri-diagonal matrix algorithm (TDMA).

Grid sensitivity tests are performed, allowing the best grid's characterization to reduce computational time with a high degree of accuracy. The different uniform grid sizes are tested from 52 × 52 to 202 × 202 for '( = 10 -, Pr = 6.2, 8( = 50 (radial magnetic field), 3 = 0.06 and ) ! = 13 /0. Table 2 illustrates the effect of the different tested grids on the average Nu number of the inner cylinder, the maximum velocities, and the maximum stream function. The grid independence tests indicated that a grid size of 102 × 102 ensures the best compromise between computational effort and required accuracy. The selected convergence criterion for the performed computations is:

é `# -`#.2 `# é < ê = 10 .- (30)
In the above equation, ` represents the different computed variables T, J or X and the / represents the iteration number.

The current computational results are compared with those available in the literature to validate the accuracy of the present modeling. A first validation is carried out with the results of Choukairy et al. [START_REF] Choukairy | Transient behavior inside a vertical cylindrical enclosure heated from the side walls[END_REF] for the case of natural convection (Pr = 0.7) in an annular cavity differentially heated for different radii ratios. It should be noted that in the case of a square cavity (! = 1), the radii ratio tends to 1 (7 → 1). The results reported in figure 4 show an excellent agreement. The second validation is performed against the results of Sankar et al. [START_REF] Sankar | Effect of magnetic field on natural convection in a vertical cylindrical annulus[END_REF] for the case of magneto-hydrodynamic natural convection in enclosures. A similar configuration was analyzed for the case of the applied magnetic field and a low Prandtl number (Pr = 0.054) for ! = 1 and 7 = 2. Two cases corresponding to '( = 10 + and 8( = 40 (radial magnetic field) and '( = 10 , and 8( = 100 (axial magnetic field) are considered. As presented in figure 4 in the form of isotherms and stream functions, the obtained results are in good agreement with those of Sankar et al. [START_REF] Sankar | Effect of magnetic field on natural convection in a vertical cylindrical annulus[END_REF].

Final numerical validation was carried out for the case of the MHD natural convection of a nanofluid (g:# -water) within a square enclosure that is differentially heated. The models of Brinkman [START_REF] Brinkman | The viscosity of concentrated suspensions and solutions[END_REF] and Maxwell-Garnett [START_REF] Maxwell | A treatise on electricity and magnetism[END_REF] are used respectively to determine the nanofluid's dynamic viscosity and thermal conductivity. Table (3) demonstrates the quantitative comparison of the present work with those computed by Ghasemi et al. [START_REF] Ghasemi | Magnetic field effect on natural convection in a nanofluid-filled square enclosure[END_REF] with a difference of less than 0.02%. Results presented in figure 4 show a perfect agreement between the present isotherms and streamlines with those presented in reference [START_REF] Ghasemi | Magnetic field effect on natural convection in a nanofluid-filled square enclosure[END_REF]. This allows us to confirm that the developed FORTRAN code is validated successfully, and it can accurately solve the flow field and temperature variation in the cavity.

Results and discussion

The consequence of adding !" * # ( nanoparticles of different diameters () ! = 13 and 47 /0) in the base fluid on the isotherms and streamlines for various Rayleigh numbers '( = 10 + , 10 -and 10 , is illustrated in figure 5. This figure obviously shows the effect of increasing '( number on the nanofluid flow (3 = 0.09) and the pure water (3 = 0) with the important effect of the nanoparticle's diameter on the flow structure and the thermal field.

For the streamlines, there are significant variations in the central region, notably for high values of '( number ('( ≥ 10 -) for both cases of diameters. The addition of nanoparticles of small diameter () ! = 13 /0) increases the strength of the streamlines, particularly in the central region. Unlike the case of ) ! = 47 /0, the weakening of the flow intensity is noted relative to the flow of the base fluid. However, near the isothermal walls and small Rayleigh number values, the stream function magnitude variation is minor.

Comparatively significant variations in the central region and near the upper and lower walls for high Rayleigh number values could be observed for isotherms. The heat transfer near the isothermal walls of the small nanoparticle nanofluid is slightly higher than that of pure water. However, the difference increases with increasing Rayleigh number, suggesting an improved heat transfer when the working fluid is the nanofluid. On the other hand, the opposite situation is happening for the nanofluid with large nanoparticles, and therefore, the most important heat transfer occurs in water. Thus, the isotherms for ) ! = 13 /0 are more distorted compared to the case of ) ! = 47 /0. For a better understanding of these results, the thermo-physical properties, namely viscosity, and conductivity are strongly linked to the temperature, the volume fraction, and the nanoparticle's diameter. To better understand these results, the thermo-physical properties, namely viscosity and conductivity, are strongly linked to the temperature, volume fraction, and nanoparticle's diameter.

The increase in temperature leads to a decreased dimensionless viscosity. Also, this decrease is more pronounced for smaller nanoparticle's diameters. Indeed, the impact of the high viscosity of the nanofluid for ) ! = 47 /0 is complemented by an increased heat transfer anticipated by the nanoparticle's elevated thermal conductivity, although this improvement is minor compared to the drop favored by the viscosity. This behavior leads to the nanofluid flow's intensification, whose diameter is smaller than the base fluid flow and deceleration of the nanofluid flow whose diameter is larger (see the maximum stream function). This behavior leads to the nanofluid flow's intensification, whose diameter of the nanoparticles is smaller than the base fluid flow and deceleration of the nanofluid flow whose diameter of the nanoparticles is larger (see the maximum stream function).

Figure 6 presents the variation of the local Nu number along the surface of the inner (hot) and outer (cold) cylinders at various Rayleigh numbers for 3 = 0,09 and ) ! = 13 and 47 /0. In all cases, the local Nu number near the hot wall rises to a maximum local value at the bottom and subsequently declines along with the inner cylinder to a minimum value at the top. However, the opposite effect occurs with a different magnitude order for the local Nu number near the cold wall due to the curvature effect. However, for the local Nu number near the cold wall, the opposite effect occurs with a different magnitude order due to the curvature effect. It should be noted that the maximum heat transfer rate near the isothermal walls is mostly due to the distribution of the maximum axial velocity, which characterizes, on the one hand, the upward or downward movement of the nanofluid and on the other hand, the presence of a large radial temperature gradient in these regions. In the case of ) ! = 47 /0, it is remarkable that an increase in the nanoparticles volume fraction (3 = 0.09) indicates a declined local Nu number compared to that of pure water. On the other hand, for ) ! = 13 /0, an increase in the Nu number is observed. This result can be explained through Figure 7, which shows the axial velocity and temperature profiles as a function of the radius ' at mid-height of the enclosure for various '( numbers and for 3 = 0 and 0,09 () ! = 13 and 47 /0). In addition, the axial velocity exhibits a parabolic variation in the vicinity of the isothermal walls due to the presence of the buoyancy force in the enclosure. The axial velocity and temperature profiles are very sensitive between the base fluid and the nanofluid and the nanoparticles' diameters. This is caused by the changed nanofluid conductivity and viscosity where the formulas (Eqs. 11 and 12) are simply sensitive to the volume fraction, the temperature, and the nanoparticle's diameter. However, the axial velocity of the nanofluid for ) ! = 13 /0 is higher than that of pure fluid in the vicinity of the hot region and lower on the cold part. For ) ! = 47 /0, the axial velocity is lower than that of the base fluid on both sides. This means that the suspension and the nanoparticles' diameter affect the flow field in which the velocity tends to zero around the center of the cavity. This area widens as the Rayleigh number increases.

The axial velocity profiles also give an idea of the flow rotation direction. Regarding the temperature profiles, the temperature gradients for the particles of diameter ) ! = 47 /0 are more important near the isothermal walls compared to the case of ) ! = 13 /0 and to the base fluid. On the other side, the temperature of the case ) ! = 13 /0 is higher than the case ) ! = 47 /0 and the base fluid far from the walls for the values of '( ≥ 10 -. We can relate this result to the distribution of the conductivity and dimensionless viscosity in these regions.

To shed more light on our results and highlight what we have found in the present study compared to previously published data, Table [START_REF] Al-Najem | Numerical study of laminar natural convection in tilted enclosure with transverse magnetic field[END_REF] presents an overview of some previously published works for natural convection of nanofluids in cylinder/annulus enclosure with and without applied magnetic field. Details of the used methods, models, type of nanofluids and the range of the investigated parameters are also provided. In general, the present results agree well with previous findings from various investigations and the discussion on the compared results for the different investigated parameters are reported trough the following texts. Figure 8 presents the computed average Nu number against the volume fraction of the nanoparticles for various '( number values, for 8( = 0 and ) ! = 13 and 47 /0. First, it is observed that the convection becomes more pronounced with increasing '( number, and therefore an increased average Nu number. In the absence of the magnetic field, and for all '( number values, the average Nu number rises slightly as the nanoparticles' volume fraction increases from 0 to 1%. This is due to the nanoparticles' presence with a low concentration, which does not affect the nanofluid's viscosity. Similar observation has been reported by Abu-Nada et al. [START_REF] Jang | The role of Brownian motion in the enhanced thermal conductivity of nanofluids[END_REF].

In the case of ) ! = 47 /0, it is observed that an increase of the volume fraction of nanoparticles leads to a declined average Nu number. On the other hand, for ) ! = 13 /0, it is observed that, for a low concentration of nanoparticles (volume fraction < 5%), a decrease in the average Nu number occurs to a minimum value. However, this trend is inversed for higher concentration (volume fraction > 5%) where an increase in the average Nu number was observed. Generally, the consideration of nanoparticles in the base fluid has two opposite effects on the computed heat transfer rate: a positive and desirable effect motivated by the existence of nanoparticles characterized by high thermal conductivity and a negative, undesirable effect favoured by the increased fluid viscosity caused by the presence of nanoparticles. Thus, nanoparticles with large diameters will produce a high viscous nanofluid, which will reduce the intensity of convection and consequently decrease the average Nu number. However, the consideration of nanoparticles with small diameters will produce a less viscous nanofluid, which will improve the convection mechanism. This behavior is also observed in the numerical results of Arani et al. [START_REF] Arani | Numerical study of different conduction models for Al2O3-water nanofluid with properties inside a trapezoidal enclosure[END_REF] which indicate a strong dependence of the mean Nusselt number on the diameter of the nanoparticles ) ! where the mean Nusselt number is a decreasing function at a range of ) ! between 30 and 90 /0.

In addition, the improvement in the heat transfer of the nanofluid (3 = 0.09) in comparison with the base fluid is more noticeable for an increased Rayleigh number (20.49% for '( = 10 , , 18.29% for '( = 10 -and 15.37% for '( = 10 + ), while deterioration is observed for ) ! = 47 /0. Referring to Table [START_REF] Al-Najem | Numerical study of laminar natural convection in tilted enclosure with transverse magnetic field[END_REF], our results agree well with the findings reported by the experimental study of Putra [START_REF] Putra | Natural convection of nanofluids[END_REF] and the numerical study of Omid Abouali and Ahmad Falahatpisheh [START_REF] Abouali | Numerical investigation of natural convection of Al2O3 nanofluid in vertical annuli[END_REF] concerning large particles ) ! = 131.2 /0 and 36 /0, respectively. In fact, the improvement in heat transfer was influenced in the opposite way where detorioration took place with increasing volume fraction of nanoparticles. Regarding the thermal field, the increase in the radius ratio leads to an increased density of the isothermal lines in the vicinity of the hot internal wall, indicating a strong temperature gradient. However, the opposite behaviour occurs in the vicinity of the cold external wall, where the temperature gradient is very low. Therefore, this effect leads to the reduction of the temperature in the chamber core. This result agrees well with Satya Sai et al. [START_REF] Satya Sai | Finite element analysis of the effect of radius ratio on natural convection in an annular cavity[END_REF] and Woon Shing Yeung [START_REF] Yeung | Analysis of natural convection in a closed vertical annulus[END_REF]. In the presence of the nanoparticles, it is evident that by enhancing the volume fraction of the flowing nanoparticles, the isotherms become less distorted than the pure fluid case and show an almost similar tendency as they approach the hot wall and move away from the hot wall to the cold wall.

For a better analysis of the radius ratio effect on the heat transfer in the annular enclosure, the average Nu number is presented in Figure 11 for three Rayleigh number values and four values of the radius ratios. First, it is revealed that the rate of heat transfer is improved as the Rayleigh number increases. The increasing radius ratio results in a declined thickness of the thermal boundary layer at the inner cylinder's hot wall. Reducing this layer causes temperature gradients to increase, which tends to increase the Nu number accordingly. Generally, the average Nu number's variation against the volume fraction presents a similar appearance to that observed in Figure 7. Therefore, the same reasoning is also applied regarding the behaviour of the average Nu versus the volume fraction for 7 = 4, but for 7 ≥ 6 the results revealed that, Nu number is higher than that of pure fluid for all volume fractions. In such a case, the effect of increased thermal conductivity is more pronounced and predominates over the impact of increased viscosity, especially for a high-volume fraction.

The heat transfer enhancement ratio "=/" caused by the introduced nanoparticles (3 = 0.09) is presented in Figure 12 for several values of '( and 7. The results show that the ratio =/ evolves in a monotonically increasing way when the radius ratio rises. In parallel, this figure also indicates that increasing '( number causes an increased heat transfer enhancement ratio due to a strong convective heat transfer effect.

The effect of the orientation and strength of the magnetic field on the streamlines and isotherms are shown in Figure 13 for '( = 10 , , 3 = 0.09, ) ! = 13 and 47 /0. The flow is characterized by the primary cell with two secondary cells located at mid-height near the inner and outer cylinders for no applied magnetic field case. The application of the radial magnetic field causes noticeable changes in the flow field. Once the magnetic field's strength increases, the flow becomes a single cell with a significant concentration of streamlines at the top of the enclosure caused by the axial Lorentz force's action. Smaller changes in the flow structure are observed for the two cases ) ! = 13 et 47 /0. It is interesting for the axially oriented magnetic field to note two secondary cells inside the primary cell with a concentration of streamlines at the isothermal walls. This phenomenon can be interpreted as the consequence of the action of the resulting Lorentz radial force. Moreover, it can be observed that the centers of these secondary vortices continue to move downward in the vicinity of the hot wall and upward in the vicinity of the cold wall due to the increased magnetic field strength of a side and on the other side, the effect of the curvature of the annular enclosure.

Table 5 shows the flow characteristics such as the maximum radial and axial velocities and the maximum stream function for 3 = 0.09, ) ! = 13 /0 and for different numbers of 8( and îO. Usually, when the magnetic field strength rises, the annulus' flow movement is damped, and the maximum stream function value decreases significantly. Additionally, this decrease is more significant when the magnetic field is oriented radially. However, the rates of decline are different for different Rayleigh numbers. For example, for the Rayleigh numbers of '( = 10 + , 10 -and 10 , , if we vary 8( = 0 to 25, the maximum stream function values decline respectively by 32.4%, 61.7%, and 79.3% for a radial magnetic field and 32.1%, 62.3%, and 89.6% for an axial magnetic field. Similarly, Ashorynejad et al. [START_REF] Ashorynejad | Magnetic field effects on natural convection flow of a nanofluid in a horizontal cylindrical annulus using Lattice Boltzmann method[END_REF] and Sheikholeslami et al. [START_REF] Sheikholeslami | Lattice Boltzmann simulation of magneto-hydrodynamic natural convection heat transfer of Al2O3-water nanofluid in a horizontal cylindrical enclosure with an inner triangular cylinder[END_REF] had already observed this decrease in ï_ #$ ï bcd as 8( increases. Hence, the trend indicates that the magnetic field's influence on fluid flow increases when the Rayleigh number is increased.

Regarding the thermal field, conduction is more pronounced compared to convection for intense magnetic fields. The isotherms become less and less deformed, especially near isothermal walls when the magnetic field is oriented radially.

However, at the heart of the annular space, these isotherms are parallel and inclined concerning the case 8( = 0, where the lines are aligned horizontally. For the case of the axially oriented magnetic field, the isotherms' straightening and alignment are noticed in the vicinity of the isothermal walls and likewise close to the heart of the annular space with a very significant inclination.

Figure 14 presents the computed average Nu number versus Hartmann number for '( = 10 , , 3 = 0, 0.03, 0.06 and 0.09, ) ! = 13 and 47 /0 and for two orientations of the radial and axial magnetic field. Based on the results of [START_REF] Sheikholeslami | Numerical investigation of MHD effects on Al2O3 water nanofluid flow and heat transfer in a semi-annulus enclosure using LBM[END_REF][START_REF] Ashorynejad | Magnetic field effects on natural convection flow of a nanofluid in a horizontal cylindrical annulus using Lattice Boltzmann method[END_REF][START_REF] Sheikholeslami | Lattice Boltzmann simulation of magneto-hydrodynamic natural convection heat transfer of Al2O3-water nanofluid in a horizontal cylindrical enclosure with an inner triangular cylinder[END_REF][START_REF] Sheikholeslami | Heat flux boundary condition for nanofluid filled enclosure in presence of magnetic field[END_REF] indicating the effect of the magnetic field on the heat transfer which is accompanied in all cases by a decrease in the average Nu number with the increase of 8(. This result is due to the flow's deceleration (see Table 5) by the Lorentz force's effect, which acts in the opposite flow direction. Thus, reducing the flow velocity induces an enhanced thickness of the thermal boundary layer. This layer's evolution produces a diminishing temperature gradient at the heated wall, leading to a decreased Nu number.

The magnetic field orientation has also been observed and is more pronounced in the radial direction. This is because the resulting axial Lorentz force significantly damps the motion of the flow induced by the buoyancy force. This force becomes stronger with strong magnetic fields. The effect of nanoparticle diameter is most evident at an intense magnetic field where the curves of 9: ;;;; 7 at a different volume fractions become tighter for ) ! = 47 /0. On the other hand, for ) ! = 13 /0, the difference is relatively large. This difference can be attributed to the effect of high viscosity for the large diameter of the nanoparticles. For '( = 10 + , the effect of the magnetic field orientation on the variation of the Nu number is insignificant because of the impact of convection, which is relatively weak on the one hand and the dominance of the conduction regime on the other hand from 8( = 50. Furthermore, this variation of 9: ;;;; 7 decreases with increasing the strength of the magnetic field before reaching an asymptotic value. The increased volume fraction of the nanoparticles results in a further increase in this asymptotic value. At high values of the Hartmann number, conduction is dominant, and the effective thermal conductivity of the nanofluid contributes to an enhanced Nu number relative to those of the pure fluid in all cases. However, for high Rayleigh numbers, say at '( = 10 , , the Nu number for the pure fluid is relatively higher than that of a nanofluid for ) ! = 47 /0 and 3 = 0.09, which results in the detrimental effect of the viscosity, which counterbalanced the favourable impact of thermal conductivity on one side and the other side the weak effect of the magnetic field axial direction compared to that of the radial direction.

Figure 15 shows the impact of Hartmann number 8( and the radius ratio 7 on the computed average Nu number for '( = 10 , and 3 = 0.09 for both radial and axial magnetic field directions. Once the strength of the magnetic field increases, then 9: ;;;; 7 decreases due to the dominance of conduction. This figure also reveals that the decrease in the 9: ;;;; 7 is relatively large when the magnetic field is directed radially for all the ratios 7. Moreover, the rate of decrease of the average Nusselt number (9: ;;;; 7 (8( = 0) -9: ;;;; 7 (8( = 100) 9: ;;;; 7 (8( = 0) ⁄ ) is even stronger when the radius ratio decreases. For example, in the case of a radial magnetic field, the rate of decrease of 9: ;;;; 7 for 7 = 2 when Ha varies from 0 to 100 is 57.25%, while for 7 = 10, the rate of decrease is 52.28 % but for the case axially applied magnetic field, the decrease rates for 7 = 2 and 10 are respectively 41.78% and 30.27%.

From the above-achieved results, it is evident that the average Nu number can be correlated as a function of '( and 8( numbers, the volume fraction 3, and the radius ratio 7 as 9: ;;;; 7 = f ('(, 3, 8(, 7). Two original developed correlations provided in equations ( 31) and ( 32) are proposed and derived for the radial and axial magnetic fields cases. The coefficients are obtained using the least square regressions method in which R 2 of 0.983 and 0.978 are obtained for the radial and axial magnetic fields cases, respectively. It is essential to mention that these two correlations correspond to the case of small nanoparticles diameter ) ! = 13 /0 in which an important heat transfer enhancement is obtained.

Furthermore, the derived correlations are valid within the range of the investigated parameters of the present study. Within these limits, the correlations can accurately capture the overall variation of the average Nu number. These derived correlations are:

• For the radial magnetic field, ? < :

In the above correlations, '( * = log('(), 8( * = 0.18(, 7 * = 0.17 and 3 * = 103 .

The heat transfer enhancement ratio "=/" computed for the case of existing nanoparticles (3 = 0.09) for various 8( and '( values is presented in Figure 16. When the magnetic field is absent, it is observed that the effect of adding nanoparticles is more evident at high '( number values compared to the case of low '( number. However, an inverse behavior is noticed when the magnetic field is applied. For 8( = 0, the enhancement ratio increases with increasing '( number. As explained previously, this result is due to the increased convection mechanism. However, for high values of 8(, this ratio increases with the decrease of '( number due to the dominant conduction in the heat transfer mechanism. Thus, the consideration of nanoparticles with high thermal conductivity results in an enhanced conduction process and improves the heat transfer enhancement ratio as demonstrated by the numerical work of Sheikholeslami et al. [29.33] indicating that =/ is an increasing function with 8(. For high '( number values, the strong magnetic field effectively reduces convective motion, and consequently, the conduction becomes the prevalent mechanism; hence, the enhancement ratio increases. The magnetic field orientation's influence has also been observed. Most interestingly, the magnetic field is more effective when directed radially, which results in the high values of =/ in this case compared to the axial magnetic field.

Conclusion

In the present study, the finite volume method-FVM is used to solve the system of equations governing the flow and heat transfer by natural convection in differentially heated vertical annulus subject to an uniform magnetic field, oriented in the axial and radial directions. The !" $ # % -water nanofluid is used, and the flow field and heat transfer results have been presented in the form of streamlines, isotherms, and Nusselt number. Several combinations of parameters in the range of Rayleigh number 10 & ≤ '( ≤ 10 ' , Hartmann number 0 ≤ 8( ≤ 100, nanoparticle volume fraction 0 ≤ 3 ≤ 0.09, radius ratio 2 ≤ 7 ≤ 10, and nanoparticle diameter ) ( = 13 and 47 /0 have been considered.

A detailed validation of the numerical model was carried out, and the main conclusions that can be drawn from this study can be summarized as follows:

• In the absence of a magnetic field, increasing the Rayleigh number improves the average Nusselt number over the full range of the parameter set combinations. On the other side and for an applied magnetic field, the average Nu number decreases with the increasing of the Hartmann number.

• For radius ratio 7 = 2, increment of the volume fraction of nanoparticles led to the continuous deterioration of the average Nu number at ) ( = 47 /0, however, for ) ( = 13 /0, the average Nu number decreases to a minimum value at 3 = 0.05, then it increases to reach its maximum value at 3 = 0.09. This behaviour of the average Nu number for ) ( = 13 /0 remains the same for 7 = 4. However, for 7 ≥ 6, the average Nu number is greater than that of pure fluid for all volume fractions. In this case, the effect of increased thermal conductivity is more pronounced and predominates over the impact of increased viscosity, especially for a high-volume fraction.

• The influence of the magnetic field is more pronounced in the radial direction. This is because the resulting axial Lorentz force significantly damps the motion of the flow induced by the buoyancy force. This force becomes greater for strong magnetic fields.

• The rate of decrease in the average Nu number caused by the application of the magnetic field is greater as the radius ratio 7 decreases.

• As the Rayleigh number and the radius ratio 7 are increased, the heat transfer enhancement ratio =/ increases with the insertion of nanoparticles (3 = 0.09) of small diameters.

• The heat transfer enhancement ratio =/ increases as the Hartmann number is increased, and for high values of 8(, this ratio increases further at the expense of the Rayleigh number. Axial magnetic field, B z Table 1. Thermo-physical properties of water and alumina [START_REF] Abouali | Numerical investigation of natural convection of Al2O3 nanofluid in vertical annuli[END_REF]. -67 8888 decreases with the increase of 1 independently of the Grashof number, the aspect ratio (* ≥ 1) and the radius ratio of the ring following the correlation: 67 8888 = 67 8888 ' (1 -;1) where the constant c depends on the particle diameter dp and how the viscosity 5 &' varies with 1.

ùû ü ( †°¢ C ⁄ ) £ D ( § †°• ⁄ ) † ( ¶ ¢• ⁄ ) ß × õú E@ (õ • ⁄ ) ® (õ ©. ¢ ⁄ )
-The model of Jang and Choi (3 &' ) agrees with the experience of Putra et al. [START_REF] Putra | Natural convection of nanofluids[END_REF] -For AB ≥ 10 ( and 1 > 0.05, 67 8888 is reduced by increasing 1.

-at AB = 10 $ , 67 8888 increases with increasing 1. -For AB ≥ 10 ( , the difference between the prediction of MG and that of the model of Chon et al. is weak. However, there is a deviation at AB = 10 $ and this deviation becomes more significant at high \ varphi.

-The Nguyen and Brinkman models give completely different predictions for AB ≥ 10 ( where the difference of 67 8888 exceeds 30%. However, this difference reduces to less than 10% at AB = 10 $ . -67 8888 is an increasing function of Ra, and of the radius ratio H while it is a decreasing function of +B.

-For H = 2, 67 8888 is a decreasing function of 1 for ! ! = 47 &'.

-For ! ! = 13 &' and H = 2 and 4, 67 8888 decreases to 1 = 0.05 and then increases to its maximum at 1 = 0.09.

-For ! ! = 13 &' and H ≥ 6, the heat transfer rate of the nanofluid is greater than that of the base fluid for all values of 1.

-The influence of the magnetic field is more pronounced in the radial direction.

-With the magnetic field, the rate of decrease in 67 8888 is greater as H decreases.

-As AB, +B and H are increased, the heat transfer enhancement ratio J& increases and for high values of +B, J& increases further at the expense of AB.

  velocities in the O and S directions,

Figure 9

 9 Figure 9 describes the impact of the radius ratio on flow and thermal behavior. Two radius ratios 7 = 4 and 10 are considered for two values of Rayleigh number '( = 10 -and 10 , and for different volume fractions 3. It is observed that

Fig. 1 .

 1 Fig. 1. Schematic of the problem.

Fig. 2 .

 2 Fig. 2. Comparison of 9:;;;; " numbers with[START_REF] Sadeghi | Analysis of thermal behavior of magnetic buoyancy-driven flow in ferrofluid-filled wavy enclosure furnished with two circular cylinders[END_REF].

Fig. 3 .

 3 Fig. 3. Comparison of streamlines and isotherms with [14] for Pr = 0.054, ! =1 and 7 = 2 at (a) '( = 10 & , 8( = 100 (radial magnetic field) and (b) '( = 10 ' , 8( = 100 (axial magnetic field).

)Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .

 4567 Fig. 4. Comparison of streamlines and isotherms with [26] for '( = 10 ' and 3 = 0.03 at (a) 8( = 0, (b) 8( = 30,

Fig. 8 .Fig. 9 .Fig. 10 .

 8910 Fig. 8. Variation of 9: ;;;; " with 3 for different '( at 7 = 2.

Fig. 11 .Fig. 12 .Fig. 13 .Fig. 14 .Fig. 15 .Fig. 16 .

 111213141516 Fig. 11. Variation of 9: ;;;; " with 3 for different '( and 7 at ) ( = 13 /0.

Table 2 .

 2 Characteristics of the flow for different grids at: '( = 10 < , Pr = 6.2, 8( = 50, 3 = 0.06 and )K = 13 /0.

	µ¨≤∂	∑∏ × ∑∏ ≥π∂≠∫	õú∏ × õú∏ ≥π∂≠∫	∏ú∏ × ∏ú∏ ≥π∂≠∫
	ª; ;;; G	3.574	3.550	3.542
	º H8I	17.528	17.490	17.490
	¶ H8I	21.563	21.649	21.651
	|Ω| H8I	5.896	5.887	5.889

Table 3 .

 3 Comparison of 9: ;;;; " with[START_REF] Putra | Natural convection of nanofluids[END_REF] for different '( and 3 at 8( = 30.

		ae = ú	ae = ú. ú∏	ae = ú. úø	ae = ú. ú¿
	ôö = õú C
	Ghasemi et al.[26]	1.002	1.060	1.121	1.184
	Present work	1.002	1.060	1.120	1.183
	ôö = õú ?
	Ghasemi et al.[26]	1.183	1.212	1.249	1.291
	Present work	1.182	1.212	1.248	1.290
	ôö = õú @
	Ghasemi et al.[26]	3.150	3.138	3.124	3.108
	Present work	3.149	3.137	3.123	3.107
	ôö = õú A
	Ghasemi et al.[26]	7.907	7.979	8.042	8.098
	Present work	7.907	7.978	8.042	8.097
	ôö = õú J
	Ghasemi et al.[26]	16.929	17.197	17.449	17.688
	Present work	16.926	17.194	17.447	17.687

Table 4 .

 4 Natural convection of nanofluid in cylinder/annulus enclosure with and without applied magnetic field

	Authors	Method	NF	Case study	Magnitic field	Parameters	Models	Conclusions
	Abouali and Falahatpisheh [27]	FVM	Al2O3	Natural convection in vertical annuli differentially heated	/	! ! = 36 &' 1 ≤ * = + , " -, # 10 $ ≤ 0, ≤ 10 % 0 ≤ 1 ≤ 0.06 ≤ 5	3 &' : model of Jang and Choi [44] 5 &' : model of Nguyen et al. [45]

  and shows a deterioration in heat transfer.

							3 &' :
							1. model of Chon et al.
	Abu-Nada [46]	FVM	Al2O3	Natural convection in horizontal annuli	/	! ! = 47 &' 0.2 ≤ , " -, # ≤ 0.8 , # 10 $ ≤ AB ≤ 10 % 0 ≤ 1 ≤ 0.09	[47] 2. model of Maxwell-Garnett [43] 5 &' : 1. Model of Nguyen et al. [48]
							2. Model of Brinkman
							[42]

  -67 8888 is an increasing function of 1 and Ra, while it is a decreasing function of Ha. increasing function of 1 and AB, while it is a decreasing function of +B.-EF &' E *+, increases as 1 increases or AB increases, while it decreases as +B increases.-67 8888 is an increasing function of AB, while it is a decreasing function of +B.-EF &' E

	MHD	natural
	convection in a
	Sheikholeslami et al. [29] Ashorynejad et al. [31] et al. [32] Sheikholeslami et al. [33] 8888 is an Sheikholeslami LBM Al2O3 MHD Natural convection in a concentric annulus between a cold square and heated elliptic cylinders / ! ! = 47 &' 10 ( ≤ AB ≤ 10 ) 0 ≤ +B ≤ 100 0 ≤ 1 ≤ 0.04 3 &' and 5 &' : Correlation of Koo-Kleinstreuer-Li [50,52] LBM Ag MHD Natural convection heat transfer in a horizontal cylindrical annulus enclosure Radial magnetic field , " [42] 0 ≤ 1 ≤ 0.06 Model of Brinkman 0 ≤ +B ≤ 60 5 &' : 10 $ ≤ AB ≤ 10 % Garnett [43] , # = 3 3 &' : Model of Maxwell--67 LBM Al2O3 horizontal annulus with heated triangular inner cylinder and couled outer circular cylinder Horizontal magnetic field 3 &' and 5 &' : ! ! = 47 &' 10 $ ≤ AB ≤ 10 % 0 ≤ +B ≤ 100 0 ≤ 1 ≤ 0.04 Correlation of Koo-Kleinstreuer-Li [50,52] *+, CVFEM Al2O3 MHD natural convection in a horizontal annulus with cold outer cylinder, and the inner circular wall is under constant heat flux Horizontal magnetic field ! ! = 47 &' 0.2 ≤ , #& G ≤ 0.4 10 $ ≤ AB ≤ 10 % 0 ≤ +B ≤ 100 0 ≤ 1 ≤ 0.04 3 &' and 5 &' : Correlation of Koo-Kleinstreuer-Li [50,52] 67 8888 is an increasing function of 1, AB, and of the aspect decreases as +B increases. ratio ( -!" . ) while it is a decreasing function of +B Present work FVM Al2O3 MHD natural convection in vertical annuli differentially heated Radial and horizontal magnetic fields ! ! = 13 and 47 &' 2 ≤ H = , " ≤ 10 3 &' and 5 &' : , # Correlation of 10 ( ≤ AB ≤ 10 ) Khanafer and Vafai 0 ≤ +B ≤ 100 0 ≤ 1 ≤ 0.09 [37]

Table 5 .

 5 Effect of !" on the characteristics of the flow for different #" at $ = 0.09 and ) ! = 13 -..

			!" = 0	!" = 25	!" = 50	!" = 75	!" = 100
				2 "	2 #	2 "	2 #	2 "	2 #	2 "	2 #
		3 %&'	22.804	7.779	7.296	3.099	2.562	1.737	1.241	1.141	0.729
	#" = 10 $	4 %&'	32.245	9.349	12.518	2.790	5.030	1.300	2.836	0.747	1.883
		|6| %&'	8.956	2.907	2.877	0.886	0.904	0.409	0.425	0.234	0.246
		3 %&'	66.697	37.912	35.515	23.659	23.276	15.847	13.743	11.151	8.219
	#" = 10 (	4 %&'	122.369	66.780	88.579	28.391	52.932	14.075	31.409	8.069	20.204
		|6| %&'	20.517	12.659	12.781	6.596	6.904	3.700	3.915	2.276	2.399
		3 %&'	190.033	127.715 125.880	91.429	90.477	72.371	78.542	59.301	67.020
	#" = 10 )	4 %&'	406.671	311.433 375.323 198.026 317.074 123.813 262.889	80.495	214.288
		|6| %&'	37.357	29.649	33.479	21.347	26.491	15.953	20.596	12.230	16.050


Paulina Leiva-Padilla
email: paulina.leiva-padill@univ-eiffel.fr

Juliette Blanc
email: juliette.blanc@univ-eiffel.fr

Aitor Salgado
email: aitor.salgado@idiada.com

Ferhat Hammoum
email: ferhat.hammoum@univ-eiffel.fr

Pierre Hornych
email: pierre.hornych@univ-eiffel.fr

Fatigue life predictions for a European pavement test section subjected to individual and platoon truck configurations

Keywords: individual truck, platoon, fatigue life, pavement

Truck platooning for the transportation of loads is a strategy recently proposed by the automotive sector to cope with traffic congestion, fuel consumption, and operational costs. This new form to configure trucks changes the typical solicitations the pavements structures are used to experience. In this sense, the research efforts of the pavement sector should be aligned with the automotive sector to propose road-friendly platoon configurations. This is one of the objectives of the European project ENSEMBLE. ENSEMBLE, as indicated by its acronym, works on Enabling SafE Multi-Brand pLatooning for Europe. In this context, the present study presents a real scale test done in the Applus IDIADA facilities to evaluate the fatigue behavior of a pavement structure subjected to individual and platoon truck configurations. The effects of parameters as traffic distribution along the year and along the time of the day, percentage of platoons, truck loads, number of trucks in platoon configuration, lateral wandering, and inter-truck distances were evaluated. The study's findings revealed that the reduced rest times between trucks in the platoon configuration reduce the recovery time of the asphalt layers, increasing the fatigue damage of the pavement at high temperature conditions. This underlines the need for further research to allow the proper implementation of truck platoons. For example, research is needed to define strategies to make truck platoon configurations more pavement-friendly and analyze the costs associated with the changes in the required road maintenance/rehabilitation treatments, among others.

INTRODUCTION

Partially or fully self-driven trucks in platoon configurations are part of the most recent and innovative advances presented by automotive companies in the last decade [START_REF] Noorvand | Autonomous vehicles: Assessment of the implications of truck positioning on flexible pavement performance and design[END_REF]- [START_REF] Mascalchi | Specifications for multi-brand truck platooning[END_REF]. These technologies seem to be capable of providing benefits in terms of reducing congestion for a better traffic flow, improving the braking/acceleration abilities of the vehicles, reducing fuel consumption, and more generally reducing operating costs of the vehicles and enhancing road safety [START_REF] Noorvand | Autonomous vehicles: Assessment of the implications of truck positioning on flexible pavement performance and design[END_REF], [START_REF] Ladino | Cross-Platform Simulation Architecture with application to truck platooning impact assessment[END_REF], [START_REF] Gungor | One for all: Decentralized optimization of lateral position of autonomous trucks in a platoon to improve roadway infrastructure sustainability[END_REF]- [START_REF] Gungor | All for one: Centralized optimization of truck platoons to improve roadway infrastructure sustainability[END_REF]. However, platooning trucks introduce new traffic multi-load configurations with the following two characteristics: [START_REF] Noorvand | Autonomous vehicles: Assessment of the implications of truck positioning on flexible pavement performance and design[END_REF] reduced deviation of the lateral position of the vehicles forming the platoon and therefore load channelization [START_REF] Noorvand | Autonomous vehicles: Assessment of the implications of truck positioning on flexible pavement performance and design[END_REF], [START_REF] Gungor | One for all: Decentralized optimization of lateral position of autonomous trucks in a platoon to improve roadway infrastructure sustainability[END_REF], [START_REF] Gungor | All for one: Centralized optimization of truck platoons to improve roadway infrastructure sustainability[END_REF]- [START_REF] Marsac | Optimization of truck platoon wander patterns based on thermoviscoelastic simulations to mitigate the damage effects on road structures[END_REF] and (2) reduced inter-truck distances between the trucks in the platoon, which may hinder the self-healing capacity of asphalt concrete materials [START_REF] Gungor | One for all: Decentralized optimization of lateral position of autonomous trucks in a platoon to improve roadway infrastructure sustainability[END_REF], [START_REF] Gungor | All for one: Centralized optimization of truck platoons to improve roadway infrastructure sustainability[END_REF], [START_REF] Gungor | Wander 2D: a flexible pavement design framework for autonomous and connected trucks[END_REF]. In this sense, a truck platoon deployment without precaution could accelerate pavement damage in terms of lower fatigue cracking/permanent deformation life [START_REF] Noorvand | Autonomous vehicles: Assessment of the implications of truck positioning on flexible pavement performance and design[END_REF], [START_REF] Gungor | One for all: Decentralized optimization of lateral position of autonomous trucks in a platoon to improve roadway infrastructure sustainability[END_REF], [START_REF] Gungor | All for one: Centralized optimization of truck platoons to improve roadway infrastructure sustainability[END_REF], [START_REF] Song | Organization of autonomous truck platoon considering energy saving and pavement fatigue[END_REF], [START_REF] Zhou | Optimization of Lateral Wandering of Automated Vehicles to Reduce Hydroplaning Potential and to Improve Pavement Life[END_REF], [START_REF] Rana | Simulation of autonomous truck for minimizing asphalt pavement distresses[END_REF], [START_REF] Marsac | Optimization of truck platoon wander patterns based on thermoviscoelastic simulations to mitigate the damage effects on road structures[END_REF] and lead to earlier rehabilitation/maintenance treatments [START_REF] Chen | Assess the impacts of different autonomous trucks' lateral control modes on asphalt pavement performance[END_REF], [START_REF] Rana | Simulation of autonomous truck for minimizing asphalt pavement distresses[END_REF]. In this context, since 2018, the European Union has been developing a research project called ENSEMBLE. ENSEMBLE's objectives are to pave the way for adopting multi-brand truck platooning in Europe to improve fuel economy, traffic safety, and throughput [START_REF] Ladino | Cross-Platform Simulation Architecture with application to truck platooning impact assessment[END_REF], [START_REF] Mascalchi | Specifications for multi-brand truck platooning[END_REF]. ENSEMBLE is a European project co-funded under the Horizon2020 Research and Innovation Programme, grant agreement No 769115. This project is coordinated by the TNO (The Netherlands Organization), and associates main European trucks manufacturers (DAF, DAIMLER Truck, IVECO, MAN, SCANIA, VOLVO Group), the European Association of Automotive Suppliers (CLEPA), ERTICO (European Road Transport Telematics Implementation Coordination Organization, which is a link with the European Truck Platooning Community), and several research organizations: IDIADA, Gustave Eiffel University, KTH, and VU Brussel. ENSEMBLE is composed of five work packages that integrate the different sectors related to the truck platoon development tasks in Europe. WP1 is related to Management, WP2 to Specification of a generic solution, WP3 to Platooning In-Vehicle Technology, and WP4, from which is part of this document, to Infrastructure, Logistics, and Impact analysis. In this regard, the objective of this paper was to assess the effect that individual and platoon truck configurations can produce on the fatigue service life of a real scale pavement section located in the test track facilities of Applus IDIADA in Tarragona, Spain, which are commonly used for vehicle testing and development activities. The approach used in this project is based on an original fatigue model, developed by Homsi et al. [START_REF] Homsi | Multiple axle loadings: Shape parameters and their effect on the fatigue life of asphalt mixtures[END_REF]- [START_REF] Homsi | A multi-linear fatigue life model of flexible pavements under multiple axle loadings[END_REF] to consider the effects of multiple axle loads. This model was used to evaluate the fatigue life of the pavement structure for three potential scenarios of truck platooning, using a mechanistic-empirical approach based on the cumulative damage concept.

1.1.Cumulative damage

Miner's rule is the most popular and widely used damage model for materials [START_REF] Miner | Cumulative Damage in Fatigue[END_REF]. As shown in Equation 1, Miner's rule states that damage (𝐷) can be predicted by the linear accumulation of fatigue damage fractions due to each individual cycle, until failure occurs.

𝐷 = ∑

𝑛 𝑖 𝑁 𝑓𝑖 ≥ 1 𝑇 𝑖=1

(1) With 𝑛 𝑖 : the number of cycles at each stress amplitude level, 𝑁 𝑓𝑖 : the number of cycles to failure for the stress amplitude of interest, and 𝑇: total number of load sequences. For asphalt materials, different test protocols are used to define fatigue equations to determine the number of cycles to failure according to different strain/stress levels. In Europe, several test protocols are standardized by the European Committee for Standardization (CEN) under the EN 12697-24. New protocols have been recently defined to try to extend these fatigue equations to more complex loading conditions (non-sinusoidal signals, multiple loads). In the model proposed by Homsi et al. [START_REF] Chen | Assess the impacts of different autonomous trucks' lateral control modes on asphalt pavement performance[END_REF], new parameters are introduced, based on the shape of the strain signal obtained by the passage of multi-axle configurations.

1.2.Fatigue test protocols for asphalt materials in Europe

Based on EN 12697-24, there are six fatigue tests used in Europe to study the fatigue behavior of asphalt materials. Table 1 summarizes the test conditions applied in these six tests, as well as the fatigue equation proposed to fit the results. The different fatigue models are expressed by relationships between the applied cyclic strain or stress and the logarithm of the number of cycles to failure when specific temperature and frequency conditions are applied. Failure criteria: energy coefficient based in the dissipated energy approach. Considering that the equations described in Table 1 do not consider the effect produced by multiple axle configurations, Homsi et al. [START_REF] Homsi | Multiple axle loadings: Shape parameters and their effect on the fatigue life of asphalt mixtures[END_REF]- [START_REF] Homsi | A multi-linear fatigue life model of flexible pavements under multiple axle loadings[END_REF], [START_REF] Homsi | Estimating truck aggressiveness using a multilinear fatigue model and a viscoelastic pavement model[END_REF] proposed the fatigue model shown in Equation 2. Homsi's model is based on the laboratory reproduction of 12 synthetic strain signals obtained from the real strain signals obtained from the Accelerated Pavement Testing facility of IFSTTAR-Gustave Eiffel University (from the French acronym: French Institute of Science and Technology for Transport, Development, and Networks) under single, tandem and tridem axles. The laboratory test used to reproduce the 12 synthetic signals was the 2-point bending fatigue test, on trapezoidal specimens conditioned at 20°C (common European fatigue test standardized as UNE EN 12697-24 [START_REF] Gungor | Wander 2D: a flexible pavement design framework for autonomous and connected trucks[END_REF]), which was adapted to apply frequencies ranging from 8.33 Hz to 40 Hz and strain levels ranging from 47 µm/m to 550 µm/m. The fatigue failure criterion used to calibrate the model considers the point when there is a 50% reduction in the initial stiffness value of the material.

[25]- [START_REF] Homsi | A multi-linear fatigue life model of flexible pavements under multiple axle loadings[END_REF], [START_REF] Homsi | Estimating truck aggressiveness using a multilinear fatigue model and a viscoelastic pavement model[END_REF] Where (see Figure 1), Ɛ: strain intensity (peak strain level) produced by the passage of the reference axle (tridem axle in the example of Figure 1), 𝑁𝑝: number of peaks of the strain signal (1 for single axles, 2 for double axles and 3 for tridem axles), 𝐴 ̂𝑛: positive area under the loading signal in the transversal/longitudinal direction divided by the peak strain and its duration, and 𝐷 ̅ : Duration of the loading signal divided by the number of peaks (in seconds).

FULL SCALE EXPERIMENT 2.1. Pavement section and instrumentation

The test section used in the study was located in the facilities of the automotive company IDIADA, in Tarragona, Spain. The pavement structure consisted of 3 asphalt layers: a 4 cm thick wearing course, a 6 cm thick binder course, and a 15 cm thick base course. According to the standard UNE EN 13108-1, the types of asphalt mixture for each layer are respectively: asphalt concrete mixture with 11 mm maximum particle size used for surface layers (AC11 surf (D12)), semi-dense graded asphalt concrete mixture with 20 mm maximum particle size (AC 22 S (S-20)), and coarse graded asphalt concrete mixture with 22 mm maximum particle size (AC 22 G (G-20)). The mixtures were manufactured with a polymer-modified binder type PMB 45-80/65 according to the UNE-EN 14023, reaching densities of 2.38 to 2.39 g/cm 3 and air voids of 4.7% to 6.8%. As shown in Figure 2, the section was instrumented with an array of 24 strain gauges type KM-100HAS Tokyo Sokki Kenkyujo, placed at 0.85m, 1.05m, and 1.25 from the edge of the right lane. 12 strain gauges (6 longitudinal and 6 transversal), were placed at the bottom of the binder course and 12 other gauges at the bottom of the base course. Thermocouples were also installed at the bottom of each asphalt layer, to measure the temperature conditions during the tests.

Test protocol

The objective of the tests was to compare the pavement response under individual trucks and trucks in platoon configuration. Two test campaigns were performed, one in the winter, and one in the summer, to cover both cold and hot temperature conditions. The different test conditions are summarized in Table 2 and Figure 3, and include: i. two test campaigns, one done in the winter and the other in the summer, ii. 20 cm of lateral deviation (wandering) for the first four strain signals and addition of 4 extra strain signals with a lateral deviation (wandering) increased to 40 cm during the summer campaign (see Figure 3b), iii. lateral offset of 0 cm from the centerline of the lane (see Figure 3b), inter-truck distances adjusted to a time gap of 0.8s, v. four truck speed values, vi.

individual and platoon truck load configurations. Values measured with the laser system shown in Figure 4. Lateral offset 0 cm from the center line of the lane.

Inter-truck distance

Adjusted to a time gap of 0.8 s between trucks.

Values measured with the laser system shown in Figure 4.

Campaigns and truck configurations

Winter:

• 48 ton, fully loaded semitrailer truck limits defined by the EU (see details per axle type in Figure 3a). Truck speed 40, 60, 70 and 80 km/h (EU speed limits defined for heavy trucks circulation).

Values measured with the laser system shown in Figure 3c.

Number of passages

Winter:

• Individual configuration: 4 speeds x 2 repetitions x 3 trucks = 24 passages.

• Platoon configuration: 4 speeds x 5 repetitions = 20 passages.

• Total passages = 44. Summer:

• Individual configuration: 4 speeds x 2 repetitions x 3 trucks = 24 passages.

• Platoon configuration: 4 speeds x 5 repetitions = 20 passages.

• Total passages = 44

Three 5-axle human-driven semi-trailer trucks, fully loaded at their maximum legal load in Europe (40 tons), were used for each test campaign. The main characteristics of the vehicles used are shown in Figure 3a. The use of human-driven trucks following platoon truck configurations is the cause of the variation in the loads per truck and test campaign.

DATA TREATMENT -CALCULATION OF PAVEMENT DAMAGE 4.1 Definition of fatigue damage and calculation of the Coefficient of Aggressiveness

The concept of Coefficient of Aggressiveness (CA) of a vehicle, used in the French pavement design method, is introduced to evaluate and compare the fatigue damage produced by the different vehicle configurations. In the case of fatigue damage, the Coefficient of Aggressiveness of a vehicle is defined as the 𝛼factor (in percentage) of the ratio between the fatigue damage produced by a given truck and the damage produced by the equivalent standard axle (ESAL), which is a single axle with dual wheels, loaded at 130 kN in the French pavement design method. The value of the 𝛼-factor corresponds to the accepted level of damage corresponding to the end of service life of the pavement (20% in this paper case study). According to this definition, the CA of a truck can be defined by Equation 3. Where, 𝑑 𝑡𝑟𝑢𝑐𝑘 is the fatigue damage produced by the whole truck, 𝑑 𝐸𝑆𝐴𝐿 is the fatigue damage produced by the equivalent standard axle (130 kN axle), 𝑑 𝐴𝑥𝑙𝑒 𝑖 is the damage produced by the 𝑖 𝑡ℎ axle of the truck and 𝑛 is the number of axles of the truck. The concept of cumulative damage is used for the calculation of the fatigue damage 𝑑 produced by a single load. With this approach, the elementary damage produced by one load can be simply defined by Equation 4.

𝐶𝐴 𝑡𝑟𝑢𝑐𝑘 =  * 𝑑 𝑡𝑟𝑢𝑐𝑘 𝑑 𝐸𝑆𝐴𝐿 =  ∑ ( 𝑑 𝐴𝑥𝑙𝑒 𝑖 𝑑 𝐸𝑆𝐴𝐿 ) 𝑛 𝑖=1 (3)
𝑑 = 1 𝑁 𝑓 (4)
Where, 𝑁 𝑓 is the number of load cycles leading to failure, defined using the fatigue model. In this study, in order to consider complex strain signals, the number of cycles to failure 𝑁 𝑓 due to each truck axle (steer axle, driven axle, and trailer tridem) was calculated using the fatigue model for multiple axle loads proposed by Homsi (Equation 2). This equation is based on the shape parameters of the strain signals registered from the passage of isolated single, tandem and tridem axles in the Accelerated Pavement Testing (APT) facilities of Gustave Eiffel University. In the case of the strain signal corresponding to the reference 130 KN axle, as this signal was not measured, it was obtained by modeling, using the multilayered elastic software ALIZE, for the same pavement structure, same temperature, and loading speed conditions. From all the experimental data, only four-strain signals where the lateral wandering values between passages for the individual trucks and platoon configuration was less than 20 cm for both test campaigns were considered. In the case of the summer campaign, four additional strain signals with an increased lateral wandering (40 cm) were added to compare the effect of this increased lateral wandering. Finally, to evaluate the fatigue life and fatigue damage corresponding to different traffic scenarios, the approach of the French pavement design method was followed. In this approach, after determining the Coefficient of Aggressiveness of each truck (𝐶𝐴 𝑡𝑟𝑢𝑐𝑘 ), according to equation 4, a mean coefficient of aggressiveness corresponding to the total cumulative traffic called 𝐶𝐴𝑀is determined according to Equation 5.

𝐶𝐴𝑀 = 1 𝑁 𝑡𝑟𝑢𝑐𝑘 ∑ 𝐶𝐴 𝑡𝑟𝑢𝑐𝑘 𝑗 𝑁 𝑡𝑟𝑢𝑐𝑘 𝑗=𝑖
(5) Where, 𝑁 𝑡𝑟𝑢𝑐𝑘 is the total number of trucks considered, and 𝐶𝐴 𝑡𝑟𝑢𝑐𝑘 𝑗 the coefficient of aggressiveness of truck 𝑗. With this definition, the 𝐶𝐴𝑀value can also be defined as the coefficient which is used to convert the number of trucks in a traffic 𝑁 𝑡𝑟𝑢𝑐𝑘 into the corresponding number of ESALs, designated by 𝑁𝐸 Equation 6). 𝑁𝐸 = 𝐶𝐴𝑀 × 𝑁 𝑡𝑟𝑢𝑐𝑘 (6)

Traffic scenarios for the calculation of pavement fatigue life.

To evaluate the impact of different proportions of platoons in the heavy vehicle traffic, five different traffic scenarios (shown in Figure 6) have been considered:  The reference scenario represents traffic with no platoons and with a lateral wandering of the vehicles of 20 cm  Scenario 1 represents traffic with 100% of platoons in the winter and no platoons in the summer, with the same lateral wandering of 20 cm.  Scenario 2 represents traffic with 100% of platoons in the summer and no platoons in the winter, with the same lateral wandering of 20 cm.  Scenarios 3 represents traffic with 100% of platoons during the whole year, with the same lateral wandering of 20 cm.  Scenario 4 represents traffic with 100% of platoons during the whole year, but with an increased lateral wandering of 40 cm FIGURE 6 Scenarios of analysis. These traffic scenarios have been applied to calculate the fatigue life of the pavement structure of the full-scale experiment (Figure 2). For these fatigue life calculations, typical data corresponding to heavy traffic roads, category T00 according to EU definitions were used:

-An Average Daily truck traffic (ADT, for the weighing highways in Spain where the experiment took place) of 15951 for winter and 28587 for summer was used, corresponding to a two-way road with light and heavy vehicles, according to the traffic database of the Spanish Road and Highway Administration [32]. -The corresponding cumulative traffic was then calculated using Equation 7.

𝑁 = 𝑇 * 𝐶 * 𝐿 * 𝐷 * 365 * 𝐴𝐷𝑇 (7) With, 𝑇: Percentage of trucks in the ADT, January: 18% and August: 13% [32]. 𝐶: Annual traffic growth factor (Equation 4, for 𝑔: annual traffic growth rate, January: 3,3% and August: 3,1% [32]). 𝐷 and 𝐿: Direction Distribution Factor and Lane Distribution Factor, which were considered 50% and 85% for a two-way road with three lanes by traffic direction, according the Spanish Order FOM/3460/2003 [START_REF] De Fomento | Instrucción de carreteras. Norma 6.1 IC: Secciones de firme[END_REF].

𝐶 =

(1 + 𝑔) 𝑝 -1 𝑔 𝑓𝑜𝑟𝑔 ≠ 0 (8)

PAVEMENT DAMAGE AND PAVEMENT FATIGUE LIFE RESULTS

Winter tests campaign

As described before, the equation proposed by Homsi [START_REF] Homsi | Multiple axle loadings: Shape parameters and their effect on the fatigue life of asphalt mixtures[END_REF]- [START_REF] Homsi | A multi-linear fatigue life model of flexible pavements under multiple axle loadings[END_REF], [START_REF] Homsi | Estimating truck aggressiveness using a multilinear fatigue model and a viscoelastic pavement model[END_REF] for multiple axle configurations was used to determine the number of cycles to fatigue for each truck under each load configuration and test campaign. The different parameters of the fatigue model obtained for each axle (Axle 1: single wheels, Axle 2: dual wheels, and Axle 3: tridem with single tires), for each truck, either in single truck configuration or in platoon configuration, are shown in Figure 7. For this winter test campaign, there is no significant difference between the model parameters obtained for the single trucks and the platoon, probably because for the temperature range corresponding to this campaign (between about 6 and 11 °C), the behavior of the pavement is relatively elastic, and there is no significant effect due to the application of multiple truck loads in a short time interval. [START_REF] Homsi | Multiple axle loadings: Shape parameters and their effect on the fatigue life of asphalt mixtures[END_REF]- [START_REF] Homsi | A multi-linear fatigue life model of flexible pavements under multiple axle loadings[END_REF], [START_REF] Homsi | Estimating truck aggressiveness using a multilinear fatigue model and a viscoelastic pavement model[END_REF]. Single: steer axle, Double: driven axle and Tridem: trailer tridem axle.

FIGURE 7 Fatigue parameters of the strain signals for the winter campaign (tension is considered positive). Fatigue coefficients obtained from

The relationship between the damage obtained from each axle and the equivalent standard axle (130 kN) (which represents the aggressiveness of each individual axle) is shown in Figure 8. This figure shows that the tridem axles cause the highest damage, as was expected. The corresponding coefficients of aggressiveness for entire trucks (𝐶𝐴 𝑡𝑟𝑢𝑐𝑘 , Equation 3), calculated for a level of damage of 20% of the pavement, are also shown in the same figure. The results indicate that:

 The 𝐶𝐴 𝑡𝑟𝑢𝑐𝑘 values are a little higher for truck 2 and truck 3 (for the individual trucks), which can be due to probably the natural variability of the test (temperature, position of the wheels, differences in axle loads).  There is no clear difference between the 𝐶𝐴 𝑡𝑟𝑢𝑐𝑘 values obtained for the individual trucks and for the platoons. In some cases, for trucks 2 and 3, the individual trucks are more aggressive than in the platoon. This means that for cold temperatures (here around 10°C), platooning does not seem to create more fatigue damage than individual trucks (for the configurations considered in this experiment).

Fatigue life for the different traffic scenarios

Using the 𝐶𝐴 𝑡𝑟𝑢𝑐𝑘 values displayed in the previous sections, it is possible to predict the corresponding number of ESALs by each half-year for the pavement test section under study. The accumulation of this value compared to the number of cycles to fatigue projected by the use of the Homsi fatigue equation, represents the fatigue damage for the structure. In this sense, considering reaching 20% of fatigue damage for each traffic scenario, Figure 11 shows the corresponding remaining fatigue lives expressed in years. In comparison to the reference scenario, which represents the passage of fully loaded 5-axle semi-trailers, in individual configurations, the following trends are observed:  The first scenario, corresponding to a traffic distribution with 100% of platoons during the winter and no platoons during the summer, does not change the remaining fatigue life of the pavement structure.  On the contrary, scenarios 2 and 3, corresponding to a traffic distribution with 100% of platoons just during summer or all along the year, significantly decrease the remaining fatigue life, with a maximum reduction of -68% at 70 km/h.  However, as shown in scenario 4, this effect can be reduced by increasing the lateral wander of the platoons, which is also supported by other results from the literature [START_REF] Gungor | One for all: Decentralized optimization of lateral position of autonomous trucks in a platoon to improve roadway infrastructure sustainability[END_REF], [START_REF] Gungor | All for one: Centralized optimization of truck platoons to improve roadway infrastructure sustainability[END_REF], [START_REF] Chen | Assess the impacts of different autonomous trucks' lateral control modes on asphalt pavement performance[END_REF],

[20]- [START_REF] Gungor | Wander 2D: a flexible pavement design framework for autonomous and connected trucks[END_REF], [START_REF] Marsac | Optimization of truck platoon wander patterns based on thermoviscoelastic simulations to mitigate the damage effects on road structures[END_REF]. Considering that these measurements were done with fully loaded trucks, and assuming 100% platoon penetration, the negative impact of platoons could be reduced by varying these parameters, as well as by changing the inter-truck distances [START_REF] Gungor | All for one: Centralized optimization of truck platoons to improve roadway infrastructure sustainability[END_REF] the truck speeds, and also the time of the day [START_REF] Rana | Simulation of autonomous truck for minimizing asphalt pavement distresses[END_REF] at which platoons are allowed to circulate. It is important to add that the CA values determined in this study are only valid for this pavement structure. Therefore, for different conditions, the corresponding values should be appropriately calibrated.

CONCLUSIONS

This paper presents the results of a study on the effect of platoons on the fatigue life predictions of a European pavement test section. The test protocol and the data treatment proposed in this document are based on an original fatigue model for multiple axle loads, which considers the maximum tensile strain, the number of peaks of the strain signal, the area, and the duration of the strain signal. This approach was applied to strain signals measured by strain gauges installed in an experimental pavement section but can also be used with strain signals obtained by modeling. The following conclusions can be drawn from this study:

• The strain measurements performed on the pavement structure, under 5-axle semi-trailer truck loading, indicated that at the bottom of the asphalt layers, the transversal strains were significantly higher than the longitudinal strains. Strain accumulation effects were also more important for transversal strains, especially at high temperatures (during the summer test campaign).

• With the fatigue model used, which considers the shape of the strain signals, it was clearly shown that for the same loading conditions (same axle loads, speed, and lateral wander), the fatigue damage produced by platoons (compared with single trucks) varies significantly with temperature. At low temperatures (around 10 °C), there was no difference between the damage induced by individual trucks and by platoons. However, at higher temperatures (around 27°C), the damage induced by platoons was significantly higher due to strain accumulation under multiple loads. • Despite the good predictions obtained with the new fatigue model, further research is carried out, based on laboratory studies, to consider better the most significant effects related to platoon loadings: influence of rest periods and accumulated deformations. • For the case study presented in this document, truck platoon configurations showed considerably higher fatigue life reductions during summer. In this sense, limiting platooning along this season, or potentially during hours with the highest temperatures, could be a possible management strategy to limit/avoid early pavement fatigue damage. It was also found that the lateral wandering of the trucks has a significant effect and that increasing the lateral wandering could also help limit fatigue damage. • Following these results, the ENSEMBLE work package related to infrastructure is currently developing further research studies based on parametric studies with a pavement modeling program. The objective is to make a more general analysis of the effects: (1) traffic distribution along the year and along the time of the day, (2) percentage of platoon penetration in the daily and annual traffic, (3) level of loading of the trucks, number of trucks in platoon configuration, (4) lateral wandering, (5) inter-truck distances and (6) representative existing pavement structures. From these simulations, recommendations to limit the impact of platoons will be proposed.

  log(𝑁 𝑓 ) = 𝑎 log(Ɛ) + 𝑏 * log(𝑁𝑝) + 𝑐 * 𝐴 ̂𝑛 + 𝑑 * 𝐷 ̅ + 𝑒

FIGURE 1

 1 FIGURE 1 Example of parameters used to characterize the transversal/longitudinal strain signals of a tridem axle (reprinted from[START_REF] Homsi | Multiple axle loadings: Shape parameters and their effect on the fatigue life of asphalt mixtures[END_REF]).

FIGURE 2

 2 FIGURE 2 Schema of the pavement structure and instrumentation (not to scale).

  11 of January 2020: o Configuration: Individual trucks o Temperature: 10.3°C and 11.3°C respectively at 25cm and 10cm depth from the pavement surface. • 12 of January 2020: o Configuration: Platoon. o Temperature: 8.3°C and 6.1°C respectively at 25cm and 10cm depth from the pavement surface. Summer: • 29 of August 2020: o Configuration: Individual trucks o Temperature: 27.5°C and 25.9°C respectively at 25cm and 10cm depth from the pavement surface. • 30 of August 2020: o Configuration: Platoon. o Temperature: 25.7°C and 25.0°C respectively at 25cm and 10cm depth from the pavement surface. Truck load

  Figure 4 shows the laser system used to measure the ability of the human drivers to follow platoon truck configurations. The parameters measured by the laser system were the truck speed, lateral deviation (wandering), and inter-truck distances. The tests were performed first with each individual truck and then with the platoon configuration for each test condition. The longitudinal/transversal strains were measured at the bottom of the binder course and base course during each truck passage. Four strain signals per truck speed were selected to analyze the fatigue behavior in the first lateral wandering interval. And four additional strain signals were added to the analysis of the summer campaign to study the effect of increasing the wandering. (a) Truck loads (b) lateral deviation (wandering) FIGURE 3 Truck-loads, lateral deviation (wandering) and laser system.

FIGURE 5

 5 FIGURE 4 Laser system for monitoring truck speeds, and longitudinal and lateral positions

  Reference Campaign: Winter, Wandering = 20 cm, Conf: Individual (0% platoon penetration) 𝐴𝐷𝑇 = 𝐴𝐷𝑇 𝐽𝑎𝑛 Campaign: Summer, Wandering = 20 cm, Conf: Individual (0% platoon penetration) 𝐴𝐷𝑇 = 𝐴𝐷𝑇 𝐴𝑢𝑔 Scenario 1 Campaign: Winter, Wandering = 20 cm, Conf: Platoon (100% platoon penetration) 𝐴𝐷𝑇 = 𝐴𝐷𝑇 𝐽𝑎𝑛 Campaign: Summer, Wandering = 20 cm, Conf: Individual (0% platoon penetration) 𝐴𝐷𝑇 = 𝐴𝐷𝑇 𝐴𝑢𝑔 Scenario 2 Campaign: Winter, Wandering = 20 cm, Conf: Individual (0% platoon penetration) 𝐴𝐷𝑇 = 𝐴𝐷𝑇 𝐽𝑎𝑛 Campaign: Summer, Wandering = 20 cm, Conf: Platoon (100% platoon penetration) 𝐴𝐷𝑇 = 𝐴𝐷𝑇 𝐴𝑢𝑔 Scenario 3 Campaign: Winter, Wandering = 20 cm, Conf: Platoon (100% platoon penetration) 𝐴𝐷𝑇 = 𝐴𝐷𝑇 𝐽𝑎𝑛 Campaign: Summer, Wandering = 20 cm, Conf: Platoon (100% platoon penetration) 𝐴𝐷𝑇 = 𝐴𝐷𝑇 𝐴𝑢𝑔 Scenario 4 Campaign: Winter, Wandering = 40 cm, Conf: Platoon (100% platoon penetration) 𝐴𝐷𝑇 = 𝐴𝐷𝑇 𝐽𝑎𝑛 Campaign: Summer, Wandering = 40 cm, Conf: Platoon (100% platoon penetration) 𝐴𝐷𝑇 = 𝐴𝐷𝑇 𝐴𝑢𝑔

  log(𝑁 𝑓 ) = -𝟒, 𝟓𝟖 log(Ɛ) -0,84 * log(𝑁𝑝) + 𝟏, 𝟑𝟏 * 𝐴 ̂𝑛 + 𝟏, 𝟕𝟔 * 𝐷 ̅ + 15,22

FIGURE 8 FIGURE 9

 89 FIGURE 8 Values of fatigue damage induced by each truck axle (relatively to the reference 130 kN axle), and 𝑪𝑨 𝒕𝒓𝒖𝒄𝒌 values of the different trucks, for the winter test campaign. 5.2. Summer campaign Figure 9 shows, for the summer campaign, the values of the parameters of the fatigue model of Homsi, for each truck and each axle, under both individual and platoon load configurations. Understanding that the positive or negative sign of each coefficient in the multi-axle fatigue equation indicates an increase or decrease in the number of cycles to fatigue, the results indicate

Figure 10 shows

 10 Figure 10 shows the damage ratios of each axle, relatively to the reference 130 kN axle. The highest damage values are obtained for the tridem axles, in the platoon truck configuration. The aggressiveness values of each truck, 𝐶𝐴 𝑡𝑟𝑢𝑐𝑘 are also shown in this figure. It can be seen that:

FIGURE 11

 11 FIGURE 11 Calculated pavement fatigue lives, for 20% of damage, for different platoon traffic scenarios.

TABLE 1 Some standard protocols of testing in Europe according to EN 12697-24 [29].

 1

	Fatigue Test	Fatigue equation	Variables	Test conditions
	Two points bending	𝑙𝑜𝑔(𝑁) = 𝑎 + ( 1 𝑏 ) 𝑙𝑜𝑔(𝜀)	𝑁: loading cycles. 𝜀: relative maximum strain.	Load mode of testing: displacement (3 strain levels).
	fatigue test		𝑎: intercept obtained by	Frequency: 25±1 Hz, sinusoidal in the upper
	on		regression.	part of the specimen with an amplitude = ±5µm.
	trapezoidal shaped		1 𝑏	: fatigue curve slope.	Temperature: 10±1°C. Failure criteria: conventional (stiffness = 50%
	specimen.				of initial value).
	Two points	𝑙𝑛(𝑁 𝑖𝑗 ) = 𝐴 0 + 𝐴 1 𝑙𝑛 (𝜎 𝑗𝑚𝑎𝑥 )	𝑁 𝑖𝑗 : fatigue life specimen 𝑖 for	Load mode of testing: load (3 tension levels).
	bending fatigue test on prismatic shaped specimen.		the tension level 𝜎 𝑗𝑚𝑎𝑥 . 𝐴 0 : intercept obtained by regression. 𝐴 1 : fatigue curve slope. 𝜎 𝑗𝑚𝑎𝑥 : maximum relative tension.	Frequency: 25±1 Hz, sinusoidal in the upper part of the specimen. Temperature: not specified, can be -20 to 30±1°C. Failure criteria: once reached a displacement = 280 µm.
	Three points bending fatigue test on prismatic	𝜀 = 𝑘 1 * 𝑁 𝑘 2	𝜀: instantaneous strain or half of the cyclic amplitude for the strain function at cycle 100. 𝑘 1 , 𝑘 Load mode of testing: displacement (3 displacement levels).
	shaped
	specimen.

2 : fatigue law coefficients obtained by regression. 𝑁: total number of cycles. Frequency: 10 Hz, sinusoidal with a total amplitude 2𝐷 0 =80µm to 350µm. Temperature: 20±1°C. Failure criteria: amplitude for cycle N is equal to half the amplitude of the cyclic load calculated for cycle 100, (1/2𝜀 𝑐 (100)).

TABLE 2 Parameters evaluated in the experimental program.

 2 strain signals for winter/summer and individual/platoon, at each test speed. 40 cm: additional 4 strain signals (8 strain signals in total) for platoon and summer at each test speed. Lateral deviation (wandering) between platoons and between trucks.

	Parameter	Characteristics
	Pavement responses:	Depth of measurement:
	horizontal strains	• 10 cm (4cm of a wearing course + 6cm of a binder course).
		• 25 cm (4cm of a wearing course + 6cm of a binder course + 15 cm of a
		base course).
		Direction of measurement:
		• Longitudinal.
		• Transversal.
	Lateral deviation	20 cm: 4
	(wandering)

AUTHOR CONTRIBUTIONS

The authors confirm contribution to the paper as follows: study conception and design: P. Leiva-Padilla, J. Blanc, F. Hammoum, P. Hornych; data collection: J. Blanc, A. Salgado; analysis and interpretation of results: P. Leiva-Padilla, J. Blanc, F. Hammoum, P. Hornych; draft manuscript preparation: P. Leiva-Padilla. All authors reviewed the results and approved the final version of the manuscript.

ACKNOWLEDGES

The research presented in this paper is part of the European project ENSEMBLE, which is cofunded by the EU under the Horizon2020 Research and Innovation Program, grant agreement No 769115.