Mebarek Belkadi

Anne Sergent
email: anne.sergent@sorbonne-universite.fr

Yann Fraigneau

Bérengère Podvin

On the role of roughness valleys in turbulent Rayleigh-Bénard convection

Keywords: Bénard convection, turbulent convection, roughness

Three-dimensional direct numerical simulations are used to characterize turbulent buoyant convection in a box-shaped Rayleigh-Bénard cavity with a rough bottom plate made of a series of square based blocks separated by valleys. The cavity is filled with water. The Rayleigh number varies over five decades up to 10 10 . As mentioned in the literature, three successive heat transfer regimes are identified: from inactive roughness (I) to a regime (III) where the heat transfer increase is larger than the one expected from the only surface increase due to roughness. The heat transfers of the transitional regime II are particularly intense. After validation against experimental and numerical data from literature, we highlight the role of the fluid retained within valleys (the inner fluid). It is shown that the heat transfer through the fluid interface between the cavity bulk and the inner fluid is strongly related to the overall heat transfer at the rough plate, with an exponent of the heat transfer scaling law close to 1/2 in the regime II. We found that this regime is active when the height of the roughness is larger than the thermal boundary layer thickness but, still, enfolded within the kinetic boundary layer. As compared to regimes I and III, regime II is characterized by larger temperature fluctuations, especially near the rough plate, and a larger friction coefficient. A fluctuating rough fluid layer overlaying both blocks and valleys appears in the regime III, in addition to the classic boundary layers formed along the plate geometry.

Introduction

The addition of wall roughness in thermal systems involving turbulent convection is a common strategy to enhance the heat transfer of industrial systems. Roughness also constitute a major factor in many turbulent flows met in nature. To explain the physical mechanisms involved in a flow interacting with roughness and at the origin of the intensification of heat transfer, many efforts have been made in the specific case of Rayleigh-Bénard (RB) convection [START_REF] Chillà | New perspectives in turbulent Rayleigh-Bénard convection[END_REF]. The classic RB convection consists in a fluid flow enclosed in a cavity heated from the bottom and cooled at the top. The corresponding flow depends on the following main control parameters: the Rayleigh number, Ra, the Prandtl number, P r, and the cavity aspect ratio, Γ , while the main response of the system can be expressed in terms of a dimensionless heat transfer i.e. by means of the Nusselt number, N u. The dependence of the Nusselt number on the control parameters (N u ∼ αRa β P r ζ ) has been widely investigated [START_REF] Ahlers | Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection[END_REF][START_REF] Chillà | New perspectives in turbulent Rayleigh-Bénard convection[END_REF], and the unifying theory of [START_REF] Grossmann | Scaling in thermal convection: a unifying theory[END_REF], 2001) [START_REF] Stevens | The unifying theory of scaling in thermal convection: the updated prefactors[END_REF] has been proposed to describe the multiple scaling laws of N u GL in the (Ra -P r) parameter space.

In the case of turbulent RB convection with rough plates, three successive heat transfer regimes have been observed as Ra is increased. It was first demonstrated experimentally by using a series of convection cavities with varying roughness aspect ratios λ, defined as the pyramid-shaped roughness height over its base [START_REF] Xie | Turbulent thermal convection over rough plates with varying roughness geometries[END_REF]. It has been shown that the two transitions delimiting the enhanced heat transfer "regime II" occur when the thicknesses of thermal, then kinematic, boundary layers are of the same size as the roughness height H p . Similar results were obtained by [START_REF] Rusaouën | Thermal transfer in Rayleigh-Bénard cell with smooth or rough boundaries[END_REF]) in a cylindrical water RB cavity, but horizontal plates were smooth at the top and roughened by rectangular shaped obstacles at the bottom. They found an increase of the scaling exponent β close to 0.5 in the regime II. A heat transfer scaling law similar to those of the smooth plate was further obtained in the regime III but with an increased prefactor. Several experimental studies describe results inside the regime II [START_REF] Roche | Observation of the 1/2 power law in Rayleigh-Bénard convection[END_REF][START_REF] Qiu | Experimental study of velocity boundary layer near a rough conducting surface in turbulent natural convection[END_REF][START_REF] Tisserand | Comparison between rough and smooth plates within the same Rayleigh-Bénard cell[END_REF][START_REF] Wei | Heat transport properties of plates with smooth and rough surfaces in turbulent thermal convection[END_REF], while other configurations correspond to the regime III [START_REF] Du | Enhanced heat transport in turbulent convection over a rough surface[END_REF][START_REF] Wei | Heat transport properties of plates with smooth and rough surfaces in turbulent thermal convection[END_REF]. In both cases, the intensification of the emission of the thermal plumes from roughness is considered to be at the origin of the heat transfer increase. By means of a quantitative analysis of the plumes [START_REF] Belkadi | Experimental and numerical shadowgraph in turbulent Rayleigh-bénard convection with a rough boundary: investigation of plumes[END_REF], it has been shown that the plume density and their velocity distribution are significantly affected by the presence of roughness, as compared to the case of a smooth plate. By introducing a critical Rayleigh number Ra c defined as the Rayleigh number for which the thermal boundary layer has the size of the roughness height, [START_REF] Rusaouën | Thermal transfer in Rayleigh-Bénard cell with smooth or rough boundaries[END_REF] succeeds to make collapsing results obtained in different asymmetric rough RB cavities over the three regimes, whatever the roughness shape.

Given its efficiency to transfer heat, many recent works have attempted to optimize the regime II and to extend its Ra-range of existence by modifying roughness geometry [START_REF] Toppaladoddi | Tailoring boundary geometry to optimize heat transport in turbulent convection[END_REF][START_REF] Toppaladoddi | Roughness as a route to the ultimate regime of thermal convection[END_REF][START_REF] Xie | Turbulent thermal convection over rough plates with varying roughness geometries[END_REF][START_REF] Jiang | Controlling heat transport and flow structures in thermal turbulence using ratchet surfaces[END_REF][START_REF] Xia | Tuning heat transport via boundary layer topographies[END_REF][START_REF] Zhu | N u ∼ Ra 1/2 scaling enabled by multiscale wall roughness in Rayleigh-Bénard turbulence[END_REF]. These studies have adopted sinusoidal-shaped roughness blocks in two-dimensional direct numerical simulations (DNS) or pyramid-shaped roughness blocks in experiments. They introduced an additional geometric parameter λ describing the roughness in terms of wavelength or pyramid aspect ratio (λ is the height over the base of the element). In all these cases, it has been demonstrated that the roughness density as well as λ increase the β scaling exponent to a value close to 1/2, at least inside a particular range of Ra. Similar trends have been obtained in the case of rectangular blocks. [START_REF] Wagner | Heat flux enhancement by regular surface roughness in turbulent thermal convection[END_REF] and [START_REF] Emran | Natural convection in cylindrical containers with isothermal ring-shaped obstacles[END_REF] performed three-dimensional DNS in cubic or cylindrical domains where the roughness is modelled respectively by large size straight or cylindrical bars. The influence of the gap width (g) between blocks and of the roughness height (H p ) on the heat transfer and on the flow structure has been documented for Rayleigh numbers up to 5•10 8 and P r ∼ 1. Bulk flow has been shown to be enhanced both by increasing H p and g, while the secondary flow circulations located inside the obstacle gap weakens as the width of the obstacle increases. This leads to an increase of N u whereas H p and g become larger than the thermal boundary layer thickness. The influence of rectangular-shaped obstacles on the flow has been previously investigated experimentally at higher Ra in a water-filled cavity [START_REF] Salort | Thermal boundary layer near roughnesses in turbulent Rayleigh-Bénard convection: Flow structure and multistability[END_REF][START_REF] Liot | Velocity fluctuations and boundary layer structure in a rough Rayleigh-Bénard cell filled with water[END_REF]. It has been shown that roughness does not clearly affect the mean flow, but enhances drastically velocity fluctuations in the whole cavity which results in a short logarithmic layer above the roughness blocks.

Two potential mechanisms are put forward: a transition to a turbulent boundary layer above the roughened plate and a plume emission increase, which relative influences may vary with the roughness shape.

These previous studies demonstrate that the roughness geometry is a crucial factor in the alteration of flow and heat transfer, illustrating the key role of flow surrounding roughness blocks. Taking advantage of the full 3D information obtained from DNS, this paper aims at describing the evolution of fluid dynamics around the roughness blocks for the three heat transfer regimes and to explain how it contributes to enhance heat transfer.

To this purpose, we simulate the flow inside an asymmetric RB cavity with a bottom plate roughened by box-shaped obstacles. This asymmetric geometry allows to study separately the smooth and rough plates in a single simulation, provided that the bulk temperature is considered [START_REF] Tisserand | Comparison between rough and smooth plates within the same Rayleigh-Bénard cell[END_REF][START_REF] Salort | Thermal boundary layer near roughnesses in turbulent Rayleigh-Bénard convection: Flow structure and multistability[END_REF]. Still, due to resolution requirements, numerical studies are usually performed with simplified geometries (macroscopic scale roughness blocks, in limited numbers, with specific symmetries or quasi-twodimensional geometry), or at moderate Rayleigh numbers [START_REF] Wagner | Heat flux enhancement by regular surface roughness in turbulent thermal convection[END_REF][START_REF] Zhu | N u ∼ Ra 1/2 scaling enabled by multiscale wall roughness in Rayleigh-Bénard turbulence[END_REF][START_REF] Emran | Natural convection in cylindrical containers with isothermal ring-shaped obstacles[END_REF]. To overcome this difficulty, we set H p at a particular value which locates the first transition between regimes I and II at a moderate Ra (here around 10 7 ). Both transition regimes are then feasible at intermediate Rayleigh numbers (less than 10 10 ) with a reasonable mesh size. This range of Rayleigh numbers corresponds to the experimental study of [START_REF] Tummers | Effect of surface roughness on heat transfer in Rayleigh-Bénard convection[END_REF]. Worth to be noted that it is also two to three decades smaller than in the previous experiments using water [START_REF] Wei | Heat transport properties of plates with smooth and rough surfaces in turbulent thermal convection[END_REF][START_REF] Xie | Turbulent thermal convection over rough plates with varying roughness geometries[END_REF][START_REF] Rusaouën | Thermal transfer in Rayleigh-Bénard cell with smooth or rough boundaries[END_REF]. Furthermore, the resulting H p leads to a spatial arrangement of box-shaped blocks in sufficient numbers to consider that the influence of the flow along the vertical walls is negligible in the cavity central part.

In this paper, we report DNS results covering five decades in Rayleigh number. The main objective is to determine whether the two successive heat transfer regime transitions can be captured numerically in a single physical set-up. Then, we investigate which physical mechanisms in the neighbourhood of the roughness blocks may explain the enhanced heat transfer of the regimes II and III. In particular, we seek to identify the respective roles of the flow above the top surface of the blocks, and of the fluid circulating within the roughness valleys ( called the inner fluid). Finally, we examine how the flow dynamics is altered by the regime changes.

The paper is organized as follows. Section 2 introduces the physical and numerical problem. Section 3 presents the roughness effect on the global heat transfer for the three heat transfer regimes and compares the DNS results to experimental data. Next, the study details the respective contributions of the roughness blocks and the inner fluid retained between them, to global heat transfer in section 4. Finally, the roughness effect on the fluid flow is described in section 5.

Physical configuration and governing equations

Physical setup

We study the fluid flow occurring in an asymmetric RB rectangular cavity with a rough bottom plate as sketched in figure 1. The geometrical aspect ratios are set at Γ x = W/H = 1 and Γ y = D/H = 0.5 where H is the height, D the depth and W the width of the cavity. The smooth cold top plate (resp. the hot bottom plate including roughness blocks) is isothermal at the constant temperature T S (resp. T R ). Vertical sidewalls are considered to be adiabatic. No-slip conditions are imposed on walls. The physical problem depends on the Rayleigh number defined as Ra = αg∆T H 3 /νκ and the Prandtl number (Pr = ν/κ), where α is the volumetric thermal expansion coefficient, g the gravity, ∆T = T R -T S the temperature difference, ν the kinematic viscosity and κ the thermal diffusivity. The Prandtl number is taken equal to 4.38, that corresponds to taking water as the working fluid at the mean temperature of 40 • C.

The roughness is modelled by a set of square-based blocks. We call valley the fluid space present between the blocks. The typical size of the blocks (width W p , depth D p and height H p ) and their horizontal distribution (D r ,W r ) have been chosen to meet two criteria: (i) A roughness height sufficiently large to obtain the first transition between regimes I and II at a Rayleigh number close to 10 7 ; (ii) A spatial distribution of roughness blocks sufficiently close to Lyon's experiments [START_REF] Salort | Thermal boundary layer near roughnesses in turbulent Rayleigh-Bénard convection: Flow structure and multistability[END_REF] to facilitate comparison.

Accordingly, we set the roughness height to H p = 0.03H. Following Rusaouën et al.

(2018), we estimate the critical Rayleigh number (Ra c ) of the first transition equal to Ra c = 9 • 10 6 based on an approximation of the thickness of the thermal boundary layer (δ θ ) estimated from the Grossman-Lohse (GL) theory [START_REF] Stevens | The unifying theory of scaling in thermal convection: the updated prefactors[END_REF]). The retained shape and distribution of roughness blocks (W p = 0.075H, D p = 0.075H, H p = 0.03H and W r = 0.075H, D r = 0.05H, see figure 1 for definitions) is equivalent to Lyon's experiment [START_REF] Salort | Thermal boundary layer near roughnesses in turbulent Rayleigh-Bénard convection: Flow structure and multistability[END_REF], leading to attach four rows of six box-shaped blocks to the bottom plate. The resulting ratio between the heat-exchange surface (A) of the asymmetric cavity and that of a fully symmetrical smooth cavity (hereafter respectively denoted as R/S and S/S) is equal to C s = (A R + A S )/(2A S ) = 1.216, where A S (A R ) stands for the dimensionless area of the smooth and the rough plates respectively.

Governing equations and system response

We solve the Navier-Stokes equations under the Boussinesq approximation. Dimensionless equations are written in the following form considering H, ∆T and κ H √ Ra as reference scales for the length, the temperature and the velocity

∇ • u = 0 (2.1a) ∂ t u + u • ∇u = -∇P * + Pr Ra -1 2 ∇ 2 u + Pr θe z (2.1b) ∂ t θ + u • ∇θ = Ra -1 2 ∇ 2 θ (2.1c)
where u = (u, v, w) is the velocity vector, t the time, P * the dimensionless driving pressure, θ the temperature, and e z the unit vector in the vertical upward direction. The temperature of top cold plate is taken as reference, so that the dimensionless temperature θ ranges from θ S = 0 to θ R = 1. The response of the system to the temperature difference ∆T applied on the two horizontal plates is measured in terms of dimensionless heat transfer by the local Nusselt number

N u(x, t) = √ Ra × w(x, t) × θ(x, t) -∂ z θ(x, t), (2.2)
where x = (x, y, z) is the coordinate vector. We note N u R/S , the time and space average of N u(x, t) over the fluid volume contained in the upper part of the asymmetrical RB cavity for z H p . Similarly, Ra R/S refers hereafter to the Rayleigh number imposed to the asymmetric cavity.

Due to the geometrical asymmetry of the configuration, the bulk temperature θ bulk is no longer equal to the mean between smooth and rough plates temperatures, i.e. θ bulk = (θ R +θ S )/2. The temperature difference between the bulk region and the hot plate (respectively the cold plate) is called ∆θ R (resp. ∆θ S ). Following [START_REF] Tisserand | Comparison between rough and smooth plates within the same Rayleigh-Bénard cell[END_REF], we assume that the top and bottom parts of the cavity are independent. Consequently, one can define two additional Rayleigh and Nusselt numbers related to each plate as follows,

       ∆θ S = 2 × (θ bulk -θ S ) , ∆θ R = 2 × (θ R -θ bulk ) Ra S = Ra R/S × ∆θ S , Ra R = Ra R/S × ∆θ R N u S = N u R/S / ∆θ S , N u R = N u R/S / ∆θ R (2.3)
where we denote by Ra S (respectively Ra R ) and N u S (resp. N u R ) the Rayleigh and Nusselt numbers related to the smooth (resp. rough) plate. This is equivalent to taking into account different reference heat fluxes (Φ ref

Numerical methods and validation

A finite volume approach is applied to discretize the governing equations (eq. 2.1), by means of the in-house SUNFLUIDH solver. A centred scheme is used for the spatial discretization on a staggered grid and the time discretization is done by second order backward differentiation scheme. The diffusive terms are implicitly treated while convective terms are estimated by the Adams-Bashforth method. This leads to a Helmholtz-like equation for each velocity component and the temperature, which is solved by applying the Alternating Direction Implicit method. The incompressibilty constraint is ensured by using a prediction-projection method [START_REF] Goda | A multistep technique with implicit difference schemes for calculating two-or three-dimensional cavity flows[END_REF][START_REF] Guermond | An overview of projection methods for incompressible flows[END_REF]. The resulting Poisson's equation for the pressure is solved by a multi-grid method coupled to the iterative Successive Over-Relaxed algorithm [START_REF] Strang | Computational Science and Engineering[END_REF]. A domain decomposition method is implemented using MPI as well as OpenMP in order to increase the level of parallelism. In this context, the Alternating Direction Implicit method is completed by a Schur decomposition technique. Roughness blocks are not modelled, because we have defined body-fitted meshes. As a consequence, standard boundary conditions are applied to all wall boundary conditions.

SUNFLUIDH code is a general purpose solver for modelling quasi-incompressible fluid flows, like rotating flows with free interface [START_REF] Yang | Large axisymmetric surface deformation and dewetting in the flow above a rotating disk in a cylindrical tank: Spin-up and permanent regimes[END_REF], turbulent flows [START_REF] Derebail Muralidhar | Spatiotemporal proper orthogonal decomposition of turbulent channel flow[END_REF] or multi-physics studies [START_REF] Hireche | Experimental and numerical investigation of natural convection flows in two horizontal thermoacoustic cavities[END_REF].

Computations are performed for a large range of Rayleigh numbers (Ra ∈ [10 5 : 10 10 ]) in order to cover the three heat transfer regimes. Details about the test cases can be found in table 1. An irregular mesh is constructed for each test case in order to resolve 1. Computational parameters and dimensionless heat transfers: Ra R/S , Rayleigh number imposed to the cavity; Nx ×Ny ×Nz, mesh size; τ , time period used for statistics in dimensionless time units; N u R/S , the Nusselt number in the R/S cavity; (RaS, N uS), the Rayleigh and Nusselt numbers corresponding to the smooth part of the cavity; (RaR, N uR), idem for the rough part of the cavity (see eq. 2.3). (1) Note that for Ra 2 × 10 5 the flow is stationary.

the Kolmogorov microscale (η). The mesh size never exceeds 0.55η (or 0.76η) between blocks (or within the cavity bulk respectively).

The space and time convergence of statistics have been verified by computing the global Nusselt number from different formulations as proposed by [START_REF] Stevens | Radial boundary layer structure and nusselt number in Rayleigh-Bénard convection[END_REF]. This methodology remains applicable for z H p due to the adiabatic sidewalls. The obtained values converge with a deviation smaller than 1% around the mean value (N u R/S ).

The code SUNFLUIDH has been validated in classic RB configuration beforehand. For this purpose, simulations have been performed in a fully smooth cavity (called S/S) of aspect ratio Γ y = 0.5 filled with water for Rayleigh numbers up to Ra = 2 • 10 9 , in order to compare with [START_REF] Kaczorowski | Turbulent flow in the bulk of Rayleigh-Bénard convection: aspect-ratio dependence of the small-scale properties[END_REF] data. A very good agreement is obtained for the compensated Nusselt number (N uRa -1/3 ), as shown in figure 2.

3. Roughness effect on the global heat transfer 220

Global heat transfer measured in the asymmetric cavity

The influence of roughness on the heat transfer is first brought to light by comparing the responses of a fully smooth cavity (S/S) and of the asymmetric cavity (R/S). The effect of the roughness on heat transfer due to the increase of the heat exchange surface area (C s × N u S/S ) is plotted as an indication. comparing with N u S/S is observed at low Rayleigh numbers for one decade in the range Ra 10 6 . This phenomenon has already been described experimentally [START_REF] Tisserand | Comparison between rough and smooth plates within the same Rayleigh-Bénard cell[END_REF] or using two-dimensional simulations [START_REF] Zhang | How surface roughness reduces heat transport for small roughness heights in turbulent Rayleigh-Bénard convection[END_REF]). (ii) For Ra 10 8 , an increase of the Nusselt number N u R/S compared to the S/S cavity is obtained, that exceeds the relative increase due to additional surface induced by roughness blocks as reported in previous works (Tisserand et al. 2011). (iii) In-between, a transitional regime is present, corresponding to an enhancement of the heat transfer.

The following section aims to verify whether the effect of roughness on the heat transfer of each of the two plates, whether smooth or rough, is the same as that observed in the asymmetrical cavity.

Analysis of the scaling laws based on the approach of plates separation

We use the plate separation method to identify respective behaviours of the smooth and rough plates, as proposed by [START_REF] Tisserand | Comparison between rough and smooth plates within the same Rayleigh-Bénard cell[END_REF]. The resulting (Ra S , N u S ) and (Ra R , N u R ) (see equations 2.3 and table 1) are plotted in figure 3. First, we note that the heat transfer on the smooth plate (N u S ) follows a single N u-Ra scaling law. Conversely, the heat transfer on the rough plate (N u R ) clearly presents two changes in the scaling law, around Ra R ∼ 3 • 10 6 and 1.2 • 10 8 . We note that the critical Rayleigh number (Ra c = 9 • 10 6 ) is in between, which is explainable by the increase of the heat transfer in the presence of roughness.

As a result, three heat transfer regimes can be identified on the rough plate, in agreement with previous experimental studies [START_REF] Xie | Turbulent thermal convection over rough plates with varying roughness geometries[END_REF][START_REF] Rusaouën | Thermal transfer in Rayleigh-Bénard cell with smooth or rough boundaries[END_REF][START_REF] Tummers | Effect of surface roughness on heat transfer in Rayleigh-Bénard convection[END_REF]. In the regime I, no difference between the rough and the smooth plate are distinguishable. In the regime III, the rough plate presents a scaling law exponent β almost similar to the regime I and close to the classic value 1/3. But the prefactor α is smaller for regime I than for regime III. In contrast, the regime II corresponds to an exponent of β ∼ 0.42, indicating an intensified heat transfer. Figure 3-c is used as a basis for setting the limits for regime II. The range of Ra numbers thereby determined, will be shaded on all the figures in the rest of the article.

To summarize, roughness enhances the heat transfer either by increasing the exponent β (regime II) or the prefactor α of the (N u -Ra) scaling law in the regime III for simulations up to Ra = 10 10 . This suggests that the regime II can be seen as a transitional regime, after which the flow would revert to a classic organization. However, it is worthy noted that in Regime III, the overall heat transfer N u R/S is larger than the simple additional contribution due to the increase in exchange surface area.

Results obtained in the S/S cavity have been added to the figure 3-b for comparison with the smooth plate (S). Generally speaking, a similar behaviour is observed for N u S/S and N u S . This result is consistent with previous experimental observations [START_REF] Tisserand | Comparison between rough and smooth plates within the same Rayleigh-Bénard cell[END_REF][START_REF] Wei | Heat transport properties of plates with smooth and rough surfaces in turbulent thermal convection[END_REF]). But we note in regime I that the heat transfer is slighly reduced by the addition of roughness in the asymmetric cavity when compared to the S/S cavity, as previously shown by [START_REF] Tisserand | Comparison between rough and smooth plates within the same Rayleigh-Bénard cell[END_REF][START_REF] Zhang | How surface roughness reduces heat transport for small roughness heights in turbulent Rayleigh-Bénard convection[END_REF]). An opposite effect is observed in the regime III, where N u S is slightly larger than N u S/S . A potential interpretation is that not only the thermal boundary layers are altered by the roughness, but also the dynamics of the bulk flow, as suggested by [START_REF] Wei | Heat transport properties of plates with smooth and rough surfaces in turbulent thermal convection[END_REF].

Comparison with experimental data

Recently, [START_REF] Rusaouën | Thermal transfer in Rayleigh-Bénard cell with smooth or rough boundaries[END_REF] proposed to make use of the critical Rayleigh number (Ra c ) to bring out the effect of roughness on the heat transfer. It is based on the idea that the transition to the enhanced heat transfer regime (II) occurs when the thermal boundary layer thickness δ θ becomes of the same size as roughness blocks. The authors obtained collapsed data, showing the same trend from the reduced heat transfer regime (I) for Ra R < Ra c to an increased regime (III), when applied to experiments performed in asymmetric RB cavities. Despite a gap of three Ra-decades between our physical configuration and the displayed experiments, the present DNS results agree well with this physical representation (figure 4). The figure 4-b retains the reduced variables (N u R /N u GL ; Ra R /Ra c ) to compare our DNS data with experiments performed in water. The shape of the experimental containers are either cylindrical [START_REF] Tisserand | Comparison between rough and smooth plates within the same Rayleigh-Bénard cell[END_REF][START_REF] Wei | Heat transport properties of plates with smooth and rough surfaces in turbulent thermal convection[END_REF][START_REF] Rusaouën | Thermal transfer in Rayleigh-Bénard cell with smooth or rough boundaries[END_REF] or rectangular [START_REF] Salort | Thermal boundary layer near roughnesses in turbulent Rayleigh-Bénard convection: Flow structure and multistability[END_REF]. The roughness is made by square-based blocks, except the [START_REF] Wei | Heat transport properties of plates with smooth and rough surfaces in turbulent thermal convection[END_REF] set-up, where pyramid-shape blocks are used.

Normalization by the respective Ra c for each data set allows to bring together most results including our numerical data, which fit a similar trend of N u increase, especially with Lyon's data. In particular, the agreement is remarkable during the regime II. This result was expected, as we use comparable shape and distribution of roughness blocks.

In contrast, a clear decreasing N u for Ra R /Ra c 10 2 is reported by [START_REF] Wei | Heat transport properties of plates with smooth and rough surfaces in turbulent thermal convection[END_REF], when using pyramid-shape roughness. This illustrates the potential influence of the 3D flow dynamics around roughness blocks on the global heat transfer.

Contribution of the inner fluid to the heat transfer

The heat transfer regime depends strongly on the pair (Rayleigh number ; height of roughness blocks), as shown by the unifying aspect of Ra c . In the R/S cavity, the vertical heat flux (N u) below the roughness height is smaller than its global value (N u (z < H p ) < N u R/S ) due to the horizontal contribution originated from the vertical surfaces of the roughness blocks. Conversely, N u(z H p ) = N u R/S , because of the adiabaticity of the

N u R | Hp = N u R | solid Hp + N u cd R | f luid Hp + N u cv R | f luid Hp (4.1) with N u R | solid Hp = 1 ∆θ R A S A solid -∂ z θ A solid ds; (4.2) N u cd R | f luid Hp = 1 ∆θ R A S A f luid -∂ z θ A f luid ds; (4.3) N u cv R | f luid Hp = 1 ∆θ R A S A f luid Ra R/S wθ A f luid ds, (4.4)
where A solid (A f luid ) is the total area of the top surface of all blocks (of the fluid interface at z = H p respectively), i.e. A S = A solid + A f luid . The notations φ A and φ stand for the space average over the horizontal surface area A and the time average of the variable φ, respectively. Hp is hardly modified by the successive regimes. It can be roughly associated with a single scaling exponent close to β ≈ 2/7 as expected in the hard turbulence regime, that supposes diffusive thermal boundary layers [START_REF] Castaing | Scaling of hard thermal turbulence in Rayleigh-Bénard convection[END_REF].

N u R Ra R N u cv R | f luid H * p N u cd R | f
As a consequence, the physical mechanisms responsible of the two transitions between the successive heat transfer regimes appears being mainly driven by the fluid dynamics occurring within the valleys. This finding is consistent with manipulating the scaling laws of heat transfer through roughness wavelength modification (Toppaladoddi et

Contributions of conduction and convection to the rough heat flux

The heat transfer through the fluid interface depends both on the temperature field for its conductive part (N u cd R | f luid Hp ) and on the temperature and velocity fields for its convective part (N u cv R | f luid Hp ) (see equation 4.1). The figure 6-a illustrates this dividing.

First, we observe that the three successive heat transfer regimes do not appear clearly with Ra increasing on this figure. But surprisingly, the critical Rayleigh number (Ra c ) seems to be becoming a significant parameter: when Ra < Ra c , the conduction mode is dominant and the convection mode through the fluid interface negligible, while this is the opposite when Ra > Ra c . A negligible convective heat transfer through the fluid interface does not mean that the fluid within valleys is at rest, but that the mass exchange between the valleys and the bulk is negligible. Around Ra c , the three contributions to N u R are on the same order. Besides, some specific features can be identified in the regimes I and III, the intermediate regime II appearing as transitional with athe competition between the conductive and convective modes at the fluid interface. In the regime I, N u R | solid ). This is also the case for the bulk temperature that saturates around θ bulk ∼ 0.6 in the regime III on the figure 6-b. This figure also demonstrates the up-down symmetry breaking of the temperature field in regimes II and III.

The dominance of convection and the saturation of θ bulk and N u cd R | f luid Hp towards constant values suggest that the fluid is well mixed in the regime III, within the cavity bulk but also within the valleys. Considering uniform diffusive boundary layers of similar thickness covering the top and bottom walls, a simple thermal balance between the top and bottom walls gives an estimate of the bulk temperature as

θ * bulk = A R θ R + A S θ S A R + A S . (4.5)
The asterisk ( * ) marks the theoretical estimate of variable. The above formula gives a good estimate of θ * bulk 0.59, when compared to the asymptotic value of the figure 6-b. The bulk temperature in the regime III is thus only determined by the roughness geometry.

We now explain that it is also the case for N u cd R | f luid Hp . At z = H p , we cannot consider that the temperature is equal to θ bulk due to the inhomogeneity imposed by the alternating of blocks and fluid interfaces. A fluid layer at an intermediate temperature (θ * i ) results from mixing processes occurring above the roughness height. We refer hereafter to this layer as the fluctuating rough fluid layer.

Additionally, the inner fluid retained inside the valleys can be seen to act as small, well-mixed RB cells with a bulk temperature equal to the mixing temperature of the fluctuating rough layer (θ * i ). This enables to define a film temperature of the inner boundary layer of valleys, which goes along the bottom wall, as θ * f = 1 2 (θ * i + θ R ). As a consequence, the conductive heat flux through the fluid interface exchanges heat between the fluctuating rough fluid layer at θ * i and the inner boundary layer at θ * f , as

N u * cd R | f luid Hp = A f luid A S ∆θ R θ * f -θ * i H p with θ * i = A f luid A S θ * bulk + A solid A S θ R . (4.6)
This reasoning leads to a theoretical estimate of N u * cd R | f luid Hp 4.5 for the regime III, which is in good agreement with the DNS result. It suggests that, in the regime III, the global thermal organization of the cavity is fixed by the geometry, with a diffusive and viscous boundary layers following the geometry of the roughness and a thicker fluctuating rough fluid layer overlaying roughness, the rest of the cavity being well mixed including the inner fluid.

Effect of roughness on the flow structure

Boundary layers along the plate centre

In this section, we focus on a restrictive volume of the cavity far from the vertical side-walls, in order to describe the mean boundary layers developing along the top solid surface of the blocks, or within the valleys and above. To do this, we retain the spatial division methodology of the previous section (in terms of solid or fluid zones), but only considering eight of the twenty-four blocks located in the centre of the bottom plate, or their direct fluid neighbourhood. Before focusing on the evolution of the boundary layer (BL) thicknesses with the heat transfer regimes, we present the space and time averaged vertical profiles of the temperature and horizontal velocity fields close to the top and bottom plates. The horizontal velocity is defined as

U = √ u 2 + v 2 .
For clarity reason, we consider three particular Ra belonging to the three regimes, as shown in the figures 7 and 8.

As expected, we observe that the thermal BL located along the block top surface and the smooth plate becomes thinner with Ra. But surprisingly, in the regimes II and III, the temperature profiles above blocks and close to the smooth plate appears to be similar in near wall region, although the bulk temperature value is not equal to the mean temperature of plates ((θ S + θ R )/2). The temperature profile in the fluid zone is more complicated. In the regime I, a slow decrease of the temperature is observed in the valleys. In the regime II, the decrease is more pronounced, but with a change of slope as z passes through H p . This slope change illustrates the onset of the convective heat transfers between the bulk of the cavity and the inner fluid of valleys. In the regime III, the temperature shows a quasi-plateau in the centre of valleys (see figure 7-c), confirming the presence of a kind of secondary well-mixed cells within valleys. Similar interpretations can be drawn for the viscous BL. In particular, within the valleys, the horizontal velocity increases with the heat transfer regime, up to presenting a plateau in the regime III that we can liken to a mean wind (figure 8-c).

As seen above, the temperature distribution and the fluid flow within valleys do not present a classic BL shape. As a consequence, we consider the displacement thickness definition, to take into account of inhomogeneity of the temperature and velocity fields, especially within the valleys. The definitions of the thermal and viscous BL thicknesses (δ θ and δ U ) are as follows,

δ θ = 0.5 0 θ A (z) -θ bulk θ R -θ bulk dz (5.1) δ U = z0 0 1 - U A (z) U 0 dz with (5.2) U 0 = max( U A (z) : 0 z 0.5) and z 0 = z( U A = U 0 )
The BL thicknesses can be measured over the smooth plate, but also separately above the block top surfaces and the fluid interfaces, following the methodology proposed at the beginning of the present section. Their Ra-dependences are plotted in the figure 9.

First, we note that a single law (given in the caption of the figure 9) is sufficient to describe the δ θ and δ U decreases with Ra over the three heat transfer regimes, once the spatial division (solid/fluid) is applied. The similarity between the smooth and solid BL, previously described for three particular Ra in figures 7 and 8, is confirmed. In the fluid zone, the decrease of both BL thicknesses (δ θ and δ U ) is much slower, although it always remains larger than the BL thicknesses above the solid and smooth zones. However, it is noteworthy that the regime II begins with the crossing of δ f luid θ with H p . This is in good agreement with previous experimental investigations made with different roughness shapes such as [START_REF] Du | Turbulent thermal convection in a cell with ordered rough boundaries[END_REF] using pyramids, [START_REF] Tisserand | Comparison between rough and smooth plates within the same Rayleigh-Bénard cell[END_REF][START_REF] Salort | Thermal boundary layer near roughnesses in turbulent Rayleigh-Bénard convection: Flow structure and multistability[END_REF][START_REF] Xie | Turbulent thermal convection over rough plates with varying roughness geometries[END_REF]) using square based parallelepipeds or with the numerical study of [START_REF] Stringano | Turbulent thermal convection over grooved plates[END_REF] using grooved plates. Additionally, we observe that the regime II ends when δ f luid U becomes smaller than H p , which was also observed experimentally [START_REF] Xie | Turbulent thermal convection over rough plates with varying roughness geometries[END_REF].

A second measure of the boundary layer thicknesses (noted δ rms ) considers the distance from the wall to the peak of the temperature or horizontal velocity rms-fluctuations. The figure 10 illustrates their evolution with Ra. Once again, we divide the rough cavity part into two parts, the solid zone above the roughness blocks and the f luid zone above the valleys. As already observed, the smooth and solid δ rms θ and δ rms U follow a similar trend, with a single scaling law describing the BL thickness decrease whatever the regime. We note that the thermal BL along the smooth wall remains always slightly thicker than the solid one above the roughness blocks. The fluid BL behaves in a different way. After becoming thinner with Ra in the regime I and II, δ rms θ and δ rms U tend towards a plateau

Global flow structure

The effect of roughness on flow structure is first investigated by considering temperature fluctuations around roughness (see figure 11). We observe that a turbulent layer develops around roughness in all cases. But, while in the regime I, this layer remains mainly above roughness, it fills almost entirely the valleys in the regime II. In the regime III, a less fluctuating small flow takes place within the valleys, with a BL along the bottom plate and a turbulent layer around z ∼ H p , illustrating interactions between the valley flow and the large scale circulation (LSC). These two layers are responsible of the two peaks observed in the rms-fluctuations used to define the BL thicknesses displayed in figure 10. Additionally, it is noticeable that the temperature fluctuations are particularly intense in the regime II, when comparing with regimes I and III.

The figure 12 illustrates how the change in the heat transfer regime modifies plume organization. A qualitative overview of the iso-contours of instantaneous temperature shows a number of large hot plumes within the cavity bulk in the regime II (figure 12-b), while plumes appear more altered by the LSC in the regime III (figures 12-c,d). Moreover, the asymmetry of the flow seems to appear for regimes II and III.

A more global point of view can be obtained by considering the spatial average of rms temperature over the volume of half a cavity (V = V R or V S ), as a function of Ra. As plotted in figure 13-a, the asymmetry of temperature rms-fluctuations only occurs in regime II, where larger values are present on the rough part of the cavity rather than on the smooth part. But for each half cavity, the maximum value of fluctuations is reached around Ra c . Unlike regime II, the intensity of θ rms is similar on both parts of the cavity in regimes I and III. These observations are in agreement with the statement of [START_REF] Du | Enhanced heat transport in turbulent convection over a rough surface[END_REF], that interactions between roughness and LSC enhances the detachment of the thermal boundary layer leading to extra thermal plumes, but only in the regime II in our case.

Mechanical interactions of the LSC with roughness can be quantified considering the friction coefficient expressed as N uRa/(Re 3 U P r 2 ), as suggested by [START_REF] Chavanne | Turbulent Rayleigh-Bénard convection in gaseous and liquid He[END_REF]. The figure 13-b presents its evolution as a function of Re U , that is the Reynolds number based on the maximum of the horizontal velocity ( U ). In a similar manner as N u and Ra, Re U can be estimated separately for the rough or smooth parts of the cavity. It is seen that the friction coefficient fits well with the expected Re -1/2 power law, but surprisingly with a higher prefactor during the regime II by comparing with regime I and III. For very large Ra ( 5 • 10 9 ), the friction coefficient tends to revert to the scaling law of the regime I. This can be interpreted as stronger interactions between the flow coming from the roughness region and the LSC, when the block top is sandwiched between the thermal and kinetic BL (δ θ H p δ U ).

Conclusion

In this paper, we present DNS results of turbulent Rayleigh-Bénard convection in an asymmetric rough water-filled cavity for five decades in Rayleigh number (Ra ∈ [10 5 -10 10 ]). The study case has been dimensioned in order to obtain a moderate value of the critical roughness-height-based Rayleigh number (Ra c = 9 • 10 6 ). The main objective is to determine whether particular physical mechanisms in the block surroundings can explain the enhanced heat transfer in regimes II and III. To this purpose, it is proposed to use a single physical set-up to capture the two successive heat transfer regime transitions.

First, a global description of the heat transfer in the asymmetric cavity R/S is discussed. As expected, we have identified three successive regimes of heat transfer: (i) a thermally resistant regime I where the global Nusselt number (N u R/S ) is reduced comparing with the heat transfer in a perfectly smooth cavity (N u S/S ), (ii) a transitional regime II where the heat transfer is particularly intense, and (iii) a regime III in which the increase of N u R/S is larger than the relative increase of surface due to roughness.

When the smooth and rough plates are separated, only one scaling exponent can describe the heat transfer on the smooth plate, whereas two scaling exponents stand for the rough plate (N u R ∼ Ra β R ): in regimes I and III, β R ∼ 1/3 is found and it increases to β R = 0.42 in regime II.

In order to highlight the role of the inner fluid retained within roughness valleys, the horizontal plane at the roughness height is divided into two parts, which enables to define two distinct heat transfer contributions: a first one coming from the solid top surface of roughness blocks, and a second one crossing the fluid interface between the cavity bulk and the inner fluid. Whatever the heat transfer regime, a unique scaling law for the heat transfer on the solid top surface is found. In contrast, the fluid interface appears to drive the complete heat transfer along the rough plate (N u R ). Concerning N u R | f luid Hp , the β exponent of the (N u -Ra) scaling law is approximately β F ∼ 1/3 in regimes I and III and it increases to β F ∼ 0.5 in regime II.

By using the decomposition of the rough Nusselt number (N u R ) into conductive and convective parts, it has been shown that conduction is the dominant heat transfer mode in regime I. Convection contribution becomes sufficiently large in regime II to compete against conduction. In regime III, convection becomes the dominant mode, while heat transfer by conduction saturates at a specific value. This value as well as the bulk temperature are shown to depend only on geometric parameters. It suggests that the flow in regime III is organized in the form of boundary layers flowing along the geometry of the top and bottom plates and of a separate fluctuating rough fluid layer overlaying roughness. The remainder of the fluid volume can be considered to be well mixed, including the small fluid volumes within the roughness valleys.

Since the heat transfer is initiated inside the boundary layers, the Ra-evolution of their thicknesses has been analysed by adopting the previous spatial division (viz. smooth plate / the solid zone above the top surface of blocks / the fluid zone located inside the valleys and above). Considering the displacement boundary layer thicknesses, the boundary layers above valleys are detected to mark out the limits of the regime II. As proposed by [START_REF] Xie | Turbulent thermal convection over rough plates with varying roughness geometries[END_REF], this regime starts when the thermal boundary layer is thinner than the roughness height and ends when the kinetic boundary layer is thinner than the roughness height. Concurrently, the boundary layer above blocks behaves like the smooth boundary layer, becoming thinner than the roughness height early in the regime I. A measure of the fluctuating rough fluid layer thickness has been obtained by tracking the peaks of temperature rms-fluctuations. This measure points out the presence of this fluid layer only in the regime III.

Finally, as could be seen by adopting a global point of view, the top and bottom half-cavities display the same level of thermal fluctuations, except in regime II where more intense fluctuations are present in the rough part. This is in agreement with the [START_REF] Du | Enhanced heat transport in turbulent convection over a rough surface[END_REF] interpretation of extra plumes emissions by roughness, but only for the regime II. More surprisingly, the global friction coefficient increases in both halfcavities during regime II, suggesting larger interactions between roughness and large scale circulation.

However, these conclusions pertain to a range of moderate Rayleigh numbers due to the value of the set critical Rayleigh number. Further investigations should be performed to clarify the interplays between roughness, large scale circulation and higher turbulence level.

Acknowledgments

This study was granted access to the HPC resources of GENCI-IDRIS under allocation 2a0326 made by GENCI. This work has benefited from the financial support of the INSIS CNRS (PEPS Energie 2018). We are grateful to Francesca Chillà and Julien Salort for valuable discussions.

Figure 1 .

 1 Figure 1. (a) Asymmetric Rayleigh-Bénard cavity (R/S) with a rough bottom plate. (b) Characteristic lengths of the block spatial arrangement.

R=

  A S ∆θ R and Φ ref S = A S ∆θ S , here expressed dimensionless).

Figure 3 .

 3 Figure 3. (a) Comparison of the Rayleigh-scaling of the Nusselt numbers for the asymmetric R/S cavity, the smooth S/S cavity or for the rough R and the smooth S plates. (b,c) Compensated Nusselt numbers. The solid lines correspond to the least-squares fits of the R plate results for the three regimes: regime I: N uR ∼ 0.078Ra 0.34 R , regime II: N uR ∼ 0.024Ra 0.42 R and regime III: N uR ∼ 0.136Ra 0.33 R . The shaded area marks the Ra-range of the regime II.

Figure 4 .

 4 Figure 4. Comparison of the normalized heat transfer on the rough plate with experimental data. N uR is normalized by the Grossmann -Lohse model (N uGL) estimated from (Stevens et al. 2013). DNS data are plotted with red open circles (•). Symbols correspond to experimental data: Hp = 2mm in the small cavity ( ), or the tall R/S cylindrical cavity (••) from Tisserand et al. (2011); Hp = 4mm in the small cavity ( ) or the tall R/S cylindrical cavity (•••) from Rusaouën et al. (2018); Hp = 3mm R/S (⊗), Hp = 8mm S/R (⊕) and Hp = 8mm R/S ( * ) in a cylindrical cavity with pyramid-shaped roughness blocks from Wei et al. (2014) ; Hp = 2mm in a R/S rectangular cavity ( ) from Salort et al. (2014).

Figure 5 .

 5 Figure 5. (a) Sketch of the geometric division of the horizontal plane at z = Hp. (b) Separation of rough heat flux at z = Hp (noted N uR|H p ) into contribution from the fluid interface (N uR| f luid Hp ) and from the solid surface (N uR| solid Hp ). The solid lines correspond to the least-squares fits of the results on the fluid interface for the three regimes: regime I: N uR| f luid Hp

Figure 6 .

 6 Figure 6. (a) Comparison of the different contributions from the solid surface (N u cd R | solid Hp ) and the fluid interface (N u cd R | f luid Hp and N u cv R | f luid Hp ) to the rough heat flux N uR at z = Hp as a function of the rough Rayleigh number (RaR). The vertical red line marks the critical Rayleigh number (Rac). (b) Bulk temperature (θ bulk ) as a function of Ra. The shaded area marks the Ra-range of the regime II.

4. 1 .

 1 Contributions from solid and fluid zones to the rough heat flux The figure 5-b compares the evolution of the global rough Nusselt number N u R as a function of Ra, with Nusselt numbers originated from the top solid surface of the blocks and from the fluid interface. First, it is shown that the three regimes of heat transfer observed on the (Ra R -N u R ) scaling law come mainly from a change of N u R | f luid Hp . The three power-law fittings for N u R | f luid Hp are given in the caption of the figure 5. In agreement with previous 2D DNS or experiments with pyramid-shaped roughness (see for example Roche et al. (2001); Qiu et al. (2005); Toppaladoddi et al. (2017); Zhu et al. (2017)), we obtain a scaling exponent for the regime II close to 1/2 (β = 0.49). In contrast, N u R | solid

  trend (in particular the exponent β of the scaling law in Ra), that is compatible with a diffusive boundary layer covering the top of blocks. In the regime III, the dominance of N u cv R | f luid Hp on N u R reveals an intensification of the mass exchange through the fluid interface. Concurrently, we observe a saturation of the conductive part of the heat transfer through the fluid interface, that forms a plateau around a constant value (N u cd R | f luid Hp ≈ 3.7

Figure 7 .Figure 8 .

 78 Figure 7. Space and time averaged vertical temperature profiles in the solid, fluid and smooth zones for three particular Ra: (a) Ra = 2 • 10 6 (regime I) ; (b) Ra = 5 • 10 7 (regime II) ; (c) Ra = 10 9 (regime III). The profile for the smooth zone (in green) has been reflected in (1 -z; 1 -θ) allowing the comparison with rough zone profiles. A profile offset by the distance Hp is plotted for the solid zone with a dashed black line. The red line marks the roughness height Hp.(a)

Figure 9 .Figure 10 .

 910 Figure 9. Displacement boundary layer thicknesses as a function of Ra and for each space-averaging zone (smooth, solid and fluid). (a) Thermal thickness δ θ , (b) Kinetic thickness δU . The shaded area illustrates the Ra-range of the regime II. The red line marks the roughness height (Hp). The black and blue solid lines correspond to the least-squares fits of the results for the solid and fluid zones δ solid θ

Figure 11 .Figure 12 .

 1112 Figure 11. Temperature fluctuation rms field (θrms) on vertical planes in the vicinity of the rough plate for three particular Ra: (a,d) Ra = 2 • 10 6 (regime I) ; (b,e) Ra = 5 • 10 7 (regime II) ; (c,f) Ra = 10 9 (regime III). (a,b,c) between two rows of roughness blocks y = 0.3125 ; (d,e,f) at mid-depth of the cavity y = 0.25).

Figure 13 .

 13 Figure 13. (a) Integral of rms temperature fields over the fluid volume in half a cavity (VR volume of the rough half cavity ; VS volume of the smooth half cavity). The vertical red line represents the critical Rayleigh number Rac. (b) Friction coefficient normalized by dissipation vs Reynolds number. The dashed lines display Re -0.5 U power laws. The shaded area illustrates the Ra-range of the regime II.

  .20 • 10 9 71.7 8.02 • 107.1 2 • 10 9 768 × 384 × 768 200 107.0 2.40 • 10 9 89.2 1.60 • 133.7 5 • 10 9 768 × 384 × 768 200 144.6 5.98 • 10 9 120.9 4.02 • 179.8 10 10 1024 × 512 × 1024 155 179.5 1.20 • 10 10 149.6 8.00 • 224.4

	Ra R/S	Nx × Ny × Nz	τ	N u R/S	RaS	N uS	RaR	N uR
	10 5 320 × 200 × 320	/ (1)	3.8 9.96 • 10 4	3.8 1.00 •	3.8
	2 • 10 5 320 × 200 × 320	/ (1)	4.8 2.00 • 10 5	4.8 2.00 •	4.8
	5 • 10 5 320 × 200 × 320	300	6.3 5.03 • 10 5	6.3 4.97 •	6.4
	10 6 320 × 200 × 320	400	8.0 1.01 • 10 6	7.9 9.90 •	8.0
	2 • 10 6 320 × 200 × 320	500	10.0 2.04 • 10 6	9.9 1.96 • 10.2
	5 • 10 6 320 × 200 × 320	460	13.5 5.23 • 10 6 12.9 4.77 • 14.2
	10 7 320 × 200 × 320	300	17.9 1.06 • 10 7 16.8 9.36 • 19.1
	2 • 10 7 320 × 200 × 320	300	23.1 2.19 • 10 7 21.0 1.81 • 25.5
	5 • 10 7 512 × 256 × 512	450	31.9 5.68 • 10 7 28.1 4.32 • 36.9
	10 8 512 × 256 × 512	430	40.6 1.16 • 10 8 35.1 8.42 • 48.2
	2 • 10 8 512 × 256 × 512	450	51.0 2.36 • 10 8 43.2 1.64 • 62.1
	5 • 10 8 768 × 384 × 768	287	68.4 5.97 • 10 8 57.3 4.03 • 84.9
	10 9 768 × 384 × 768 85.9 1Table 220

  Three regimes of heat transfer clearly

		0.11 0.11
	N u Ra -1/3	0.08 0.09 0.10 0.08 0.09 0.10	×C s	Kaczorowski et al. (2014)
		0.07 0.07
		0.06 0.06
		10 5 10 5	10 6 10 6	10 7 10 7	10 8 10 8	10 9 10 9	10 10 10 10
				Ra

(Ra R/S , N u R/S ) (Ra S/S , N u S/S ) (Ra S/S , C s × N u S/S ) Figure

2

. Compensated Nusselt number as a function of Rayleigh number for the fully smooth cavity (S/S) and the asymmetric cavity (R/S). The blue line corresponds to the Nusselt number increased by the factor Cs (corresponding to the relative increase of the heat exchange surface in the (R/S) cavity). Red points refer to DNS results from

[START_REF] Kaczorowski | Turbulent flow in the bulk of Rayleigh-Bénard convection: aspect-ratio dependence of the small-scale properties[END_REF]

.

  al. 2015; Xie & Xia 2017; Zhu et al. 2019).

appear for the R/S cavity in the figure 2. (i) A reduction of the Nusselt number N u R/S