Conjugate natural convection along regularly ribbed vertical surfaces: A homogenization-based study

Abstract Natural convection heat transfer from periodically ribbed vertical surfaces is targeted for upscaling, incorporating the analysis of thermal conduction through the microscale ribs. Asymptotic homogenization theory is employed, considering the steady conjugate heat transfer problem, to formulate second-order accurate effective conditions for velocity and temperature at a fictitious plane surface, beyond which the macroscale behavior of the flow is computed. This allows to avoid the numerically expensive resolution of fields within and through the microstructured corrugations. For the streamwise velocity component, the no-slip boundary condition is corrected at first order (in terms of a small parameter ϵ, ratio of microscopic to macroscopic length scales) by the classical Navier-slip condition plus a buoyancy term, while the gradient of the normal stress appears at second order together with a temperature-gradient term. The temperature at the virtual boundary deviates from the uniform value at the baseplate; the thermal slip is described via a first-order temperature-gradient term with a coefficient depending on rib geometry and thermal conductivity. Different case studies are conducted on the case of transverse square ribs, varying the density of the pattern and the rib-to-fluid thermal conductivity ratio, to provide extensive validation of the model against full feature-resolving simulations. Beyond the validation phase, a better understanding of the effects of different parameters on the heat transfer performance is pursued. The presence of ribs is found to decrease the overall heat transfer rate from the surface, and this deterioration is only partially alleviated by raising the thermal conductivity of the ribs. Increasing the number of conducting ribs on the hot surface has a complex, non-monotonic effect on the heat transfer rate, unlike the case of adiabatic ribs where the average Nusselt number decays monotonically. The performance of low-thermal-conductivity elements (e.g. wooden ribs) may considerably differ from that of perfectly adiabatic ones.


Introduction and literature review
Natural convection is a simple, reliable, cost-effective heat transfer regime, with almost maintenance-free operation and easily employable in hostile environments under dust, moist air, and so forth.With the flow driven merely by the buoyancy force, the main problem facing the designers is the low heat transfer coefficient of these systems, relative to those adopting active heat transfer mechanisms.Optimization of the heat transfer performance of natural convection systems is essential, to cope with the trend of manufacturing compact devices with high input power, resulting in higher heat generation rates per unit volume [1,2] which should be effectively dissipated; the life and reliability of electronic systems may be severely affected by the increase in the operating temperature of a single component over the permissible limit [3].
The literature on natural convection is rich in experimental and numerical studies investigating the effectiveness of adding surface alterations to vertical plates, as a potential method to promote convective heat transfer performance.At first glance, the findings which have been reported so far appear contradictory in terms of the overall effect on the heat transfer rate [4].However, an in-depth analysis of the complex flow patterns over ribbed surfaces and their contrasting effects can provide insight into the plausible reasons for the differences between the reported trends, as outlined by Tanda [5].Thus, the study and optimization of heat transfer from corrugated/ribbed/finned surfaces have remained fertile fields for research; the major findings of some relevant studies are summarized in the following.Bhavnani and Bergles [4] carried out experiments on natural convection along an isothermal vertical surface with repeated transverse square ribs.They reported that attaching highly conducting ribs to the surface may decrease the overall heat transfer rate by more than 25%, whereas a lower deterioration, or even an enhancement, was detected with low-conductivity ribs since they interrupt the thickening of the thermal boundary layer, promoting the heat transfer process within the inter-rib regions of the baseplate.The experiments by Tanda [6] on natural convection through a vertical channel with wooden square ribs attached to the hot surface revealed similar local enhancements of the heat transfer coefficient compared with the earlier measurements by Tanda [5] using aluminum ribs, provided that the channel aspect ratio was sufficiently large.Cavazzuti and Corticelli [7] numerically investigated conjugate natural convection through a channel with one adiabatic surface and an opposite isothermal surface roughened with trapezoidal ribs; the ribbed channel cases examined were not found to outperform the smooth one, and recommendations were given to alleviate the deterioration.The numerical study by Yao [8] on natural convection along isothermal, wavy plates showed that, despite the decrease experienced in the surface-averaged heat transfer coefficient, the overall heat transfer rate was enhanced (compared with a flat surface) thanks to the increase in the surface area.Haervig and Sørensen [9] simulated the effect of adding zigzag alterations to a vertical surface exposed to free convection and found that these surface perturbations can enhance the heat transfer rate by about 12% under optimal conditions.Nishikawa et al. [10] studied conjugate natural convection along a vertical surface with minuscule square ribs of low thermal conductivity, as a simplified representation of the yarns of a textile material.Intriguingly, a minor cooling effect was detected when the minuscule ribs were arranged at an adequate pitch, sufficient to allow the cool stream to be guided onto the base surface in the inter-rib regions.Zhou et al. [11] investigated the effectiveness of attaching an array of vortex generators to the rear surface of a photovoltaic module exposed to free convection, as an economical cooling method.According to their experiments and simulations, the resulting change in the operating temperature is sensitive not only to the material thermal conductivity but also to the aerodynamic shape and spacing of the protrusions, which affect the formation of vortices and, therefore, the favorable mixing in the boundary layer.The reader is also encouraged to refer to studies on natural convection heat transfer from vertical surfaces with rippled vertical fins [12], W-type fin arrays [13], and in-line/staggered arrangements of interrupted fins [14,15].
The present study is part of an ongoing project in which different geometric and thermal conditions affecting natural convection heat transfer from ribbed vertical surfaces are extensively investigated via theoretical and numerical analyses [16,17] and experimental surveys [18].In our endeavor, we employ asymptotic homogenization to analyze buoyancy-induced flows over periodically roughened vertical surfaces with conjugate heat transfer; i.e., the conductive heat transfer through the ribs and the convective heat transfer from the ribs to the flowing fluid are both considered in the homogenization procedure.The expensive numerical resolution of the velocity and thermal fields within surface corrugations, and the resolution of the temperature distribution through the ribs, are replaced by the use of effective boundary conditions, to be enforced at a virtual smooth wall.The model is formulated proceeding along the lines of Ahmed et al. [16]; however, a general value of the thermal conductivity coefficient is set here for the material of the ribs, which allows for employing the model to study natural convection over surfaces roughened with ribs of finite conductivity, as well as perfectly conductive or adiabatic ribs as the two limiting situations.For instance, this is beneficial for the numerical study of natural convection along vertical surfaces with wooden/low-thermal-conductivity roughness elements (like those considered in Refs.[4,6,10]), to highlight the differences from the theoretical performance of perfectly adiabatic ribs.In the next section, the physical problem is described, the governing equations are given, and formal expressions of the effective boundary conditions are derived.In Section 3, the model coefficients are estimated for the case of two-dimensional ribs of square cross section, varying the rib size to pitch distance ratio and the thermal conductivity.The macroscale problem is then considered in Section 4, and the homogenization-based model is validated against standard, feature-resolving simulations under different geometric and thermal conditions; furthermore, the trends of the heat transfer performance are discussed and physical interpretations are provided.Summarizing remarks are presented in Section 5.

The dimensional governing equations
The physical problem is described in Figure 1 (left).The buoyancy-induced flow over a vertical surface of height L, periodically ribbed with roughness elements of given geometry, distribution density and thermal conductivity, is considered.The baseplate is assumed to be perfectly conducting, and thus, to remain isothermal at a temperature Tb , higher than the ambient temperature ( T1 ).After heat is supplied to the baseplate (e.g., by an electrical heater) for sufficient time, it is eventually balanced by the heat dissipated to air via natural convection; hence, a time-independent value of T b is reached, and we can adopt a steady physical model to study the problem, provided that the flow regime remains laminar.At any point in the fluid phase, the velocity vector (û i ), the temperature ( T), and the pressure ( P) are the dependent variables of interest, to be evaluated over space xi : Assuming moderate variations in fluid density due to thermal gradient, one can apply the Boussinesq approximation with a linear temperature-density relationship so that the momentum conservation equation is linearly coupled with the energy equation.The dimensional conservation equations in the fluid phase are written as follows: with P1 , g, and d ij the pressure sufficiently far away from the wall, the magnitude of the gravitational acceleration and the Kronecker delta.The fluid properties, i.e. the density q, the dynamic viscosity l, the thermal diffusivity a, and the thermal expansion coefficient b, are assumed constant in xi : Additionally, it is useful to define the Prandtl number ðPr ¼ l qa ¼ a Þ and the Grashof number ðGr ¼ gbð T b À T 1 ÞL 3 2 Þ, which control the problem.

NUMERICAL HEAT TRANSFER, PART A: APPLICATIONS
Moving to the solid phase, thermal conduction in the roughness elements is also assumed to be steady, so that only the spatial variations of the temperature ( T ) are considered.The thermal conduction is thus governed by Laplace's equation: At the base/fluid interface, the fluid temperature and velocity read Neglecting the thermal resistance between the base and the rib, we may assume that At the rib/fluid interface, the no-slip condition ûi ¼ 0 is imposed.Since it is essential to describe the thermal interaction in terms of the continuity of temperature and heat flux, we impose with j the rib-to-fluid thermal conductivity ratio (j ¼ k r k f ; the fluid thermal conductivity k f is assumed spatially constant) and n the dimensional distance along the direction normal to the rib/fluid interface at any point.

Domain decomposition
Given that the periodicity of the roughness pattern (l) and the plate height (L) are well-separated length scales, i.e., l ( L, we seek upscaling of the problem starting by decomposing the flow domain into two subregions (microscopic/macroscopic); cf. Figure 1 (right).The two newly defined problems are coupled at a matching interface whose distance from the vertical surface is to be properly specified.

The macroscopic problem
Characteristic scales are now sought for the macroscopic problem.Since the flow is driven by buoyancy, we anticipate that the inertial force (per unit volume) is of the same order of magnitude as the Boussinesq term.The characteristic velocity, U, is therefore defined as U ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bð Tb À T1 ÞgL , and the normalized velocity vector is The other dimensionless variables are defined as follows: Accordingly, the dimensionless equations governing the macroscopic problem read subject to matching conditions (defined later) at X 2 !0, while the conditions @H @X 2 ¼ @U 1 @X 2 ¼ 0 and U 2 ¼ 0 are expected to be smoothly satisfied far away from the wall, i.e., for X 2 !1:

The microscopic problem
The scale U is chosen for the velocity in the microscopic region, with the small parameter defined as ¼ l=L ( 1: This choice stems from matching the shear stress at the interface between microscopic and macroscopic domains.The following normalization is used: for the microscopic equations in the fluid phase to read with R G ¼ ffiffiffiffiffi ffi Gr p the reduced Grashof number, assumed to be of order one as the model is intended applicable at relatively large values of Gr.Steady thermal conduction within the ribs is governed by the following dimensionless equation: The following boundary conditions are imposed at the interfaces (I bf , I br , I rf ): with n ¼ n l : Matching the traction vector and the heat flux between the microscopic and the macroscopic subregions, considering the scales adopted for each, the conditions for x 2 ! 1 can be recast as follows: with S 12 , S 22 , and S 32 the macroscopic dimensionless stresses in the streamwise, normal and spanwise directions, and g the macroscopic dimensionless normal temperature gradient, i.e., Since the effective boundary conditions are sought in terms of the macroscopic velocity vector and temperature at the matching interface, coupling with the microscopic problem is required, i.e., lim

Asymptotic treatment of the microscopic problem
The microscopic dependent variables are asymptotically expanded in terms of , for the microscopic problem to be later reconstructed at the different orders; we impose i þ :::, and similarly for p, h and /: Generally, the variables in the expansions are dependent on both x i and X i : These expressions are then substituted into Eqs.(3a)-(3h), together with replacing the derivative @ @x i by @ @x i þ @ @X i as by the chain rule.

Oð 0 Þ problem
The leading-order problem in the fluid phase reads while in the roughness elements, we have with the boundary conditions Trivial solution exists for the thermal problem, i.e.
h ð0Þ ¼ 1 ðover the fluid phaseÞ, / ð0Þ ¼ 1 ðover the solid phaseÞ: Taking advantage of the linearity of the dynamic problem, a generic solution can be sought via separation of variables, that is where u ^ik , u † i , p ^k , and p † are purely microscopic variables, described over x i only.The auxiliary problem governing u ^ik and p ^k reads with The buoyant effect in the microscopic layer is described via the † variables satisfying with All auxiliary systems should be solved over a representative volume element (cf. Figure 1 (right)), enforcing periodicity of the microscopic variables along x 1 and x 3 : As the leading-order problem has revealed that h ð0Þ and / ð0Þ are constant fields, the thermal problem in the fluid and the solid phases at order can be written as follows: fluid phase : @ 2 h ð1Þ @x 2 j ¼ 0, solid phase : Thanks to linearity, a generic solution can be assumed in the form The new purely microscopic fields, h and /, solve the following Laplace equations: fluid phase: with boundary conditions Considering the dynamic problem, we have The expressions of u ð0Þ i and p ð0Þ must now be substituted into the preceding equations; the behavior of u ð1Þ i and p ð1Þ is thus described by subject to and Again, a solution is sought in a form that takes advantage of the forcing terms, i.e.: and likewise for p ð1Þ : The decoupled systems of equations arising from substituting the generic forms above into Eqs.(11a)-(11d) are similar to those given in Ref. [16].Revisiting Eq. ( 4), the macroscopic velocity and temperature at the matching interface are now available up to Oð 2 Þ and Oð 1 Þ, respectively.However, following the same lines of Ahmed et al. [16], we can find that taking the temperature to Oð 2 Þ under steady conditions leads to microscopic parameters which vanish at the matching interface, and thus, do not contribute to the effective temperature boundary condition.

The parameters of interest
We are only interested here in the closure parameters that contribute to the effective velocity and temperature boundary conditions at the matching interface (x 2 ¼ y 1 ).Fortunately, many of the auxiliary systems defined in Subsection 2.3 yield microscopic variables that vanish at the matching plane.Following the procedure by Ahmed et al. [16], we specify the following parameters of interest: u 233 , and u 0 1 : The distributions of these variables are to be sought by numerically solving the corresponding auxiliary problems through a representative periodic unit cell in the microscopic domain, bounded by the solid/solid and solid/fluid interfaces from one side and by the matching interface on the other.The outer boundary of the microscopic problems should be located far enough away from the physical surface for the fields to become uniform in x 1 and x 3 : The following step is to vary the location of the matching interface and to accordingly estimate the values of the closure variables at the outer x 2 -edge of the domain.For convenience, the values of the closure variables computed at different y 1 are later extrapolated to a virtual matching plane chosen to go through the tips of the ribs, at x 2 ¼ 0; cf. Figure 1 (right).At this specific location, the model coefficients may be defined as follows, taking advantage of the equalities found in Ref. [16]: The Navier-slip coefficients (k x , k z ) and surface permeability coefficients (m 12 , m 32 ) are only dependent on the geometric parameters of the ribbed surface, while the thermal-slip coefficient (k h ) and the velocity-flux sensitivity (B) depend also on the rib-to-fluid thermal conductivity ratio (j), which is the novel contribution of the present work.If the ribs were assumed to be perfectly conducting, the thermal-slip coefficient would be identical to k z [16].One of the main contributions of the present work consists in the following set of second-order macroscopic boundary conditions:

The case of transverse square ribs
In this work, the case of transverse (x 3 -elongated) ribs of square cross section is considered so that the auxiliary systems can be considerably simplified, with the derivatives @=@x 3 set to zero.Only two-dimensional ðx 1 , x 2 Þ Stokes-/Laplace-/Poisson-like problems are to be solved, enforcing periodicity along x 1 : A microscopic representative unit cell is sketched in Figure 2.
The numerical results of sample auxiliary systems, with the matching interface located at x 2 ¼ y 1 ¼ 5, are presented as Supplementary Material, available online alongside this article.The behaviors of the model coefficients, resulting from extrapolation to x 2 ¼ 0, are then sought.The trends of the merely geometry-dependent parameters (k x , k z , m 12 , m 32 ) with variations in e l , displayed in Figure 3 (left), are identical to those in Ref. [16].As k h explicitly depends on j, and B is implicitly dependent on it, it is useful to plot them against e l for different values of j; cf. Figure 3 (right).The cases of adiabatic ribs and isothermal ribs are chosen as the two limiting situations, while the finite value j % 4 is chosen as a representative case, corresponding to airflow over wooden ribs.For e l !0, the surface tends to be flat and isothermal; thus, the thermal-slip coefficient k h !0: In general, the value of k h increases with the decrease of j at a given value of e l , with the maximum k h corresponding to the case of adiabatic ribs.This effect becomes more pronounced for relatively large-sized ribs.For e l ! 1, the surface becomes flat again as the inter-rib spaces vanish and the ribs become contiguous; the thermal-slip coefficient approaches zero for isothermal ribs and 1 for adiabatic ribs.

Overview of the conditions considered and numerical procedure
The effective boundary conditions formulated in the previous section are to be validated here for the laminar, steady buoyancy-driven flow over transverse square ribs.Different case studies are analyzed at a Prandtl number of 0.712 (air) and a plate Grashof number of about 2:78 Â 10 8 (this corresponds, for instance, to a vertical surface of 0.4 m height with a baseplate temperature roughly 40 C above the ambient temperature), considering changes in thermal conductivity and size-to-pitch ratio of the ribs.
The indicated conditions raise concerns about the validity of the physical model adopted, in particular the Boussinesq approximation and the laminar-flow assumption.First, the Boussinesq approximation is applicable provided that bð Tb À T 1 Þ ( 1 (cf.[9]), and this applies here since, for example, if Tb À T1 ¼ 40 C and bðairÞ % 0:0034 K À1 , the value of bð T b À T 1 Þ is equal to 0.136.Second, steady natural convection is considered at a plate Rayleigh number (Ra ¼ Gr Â Pr) of about 2 Â 10 8 : A critical value of Gr ¼ 10 9 is typically indicated in the literature for turbulent transition over a vertical smooth surface [19,20]; this corresponds to Ra % 7 Â 10 8 with air as the convective fluid.Since the upper threshold of the laminar flow range may differ when the surface is ribbed, an experimental campaign has been carried out to investigate this issue [18], employing the Schlieren imaging technique to study natural convection from a vertical surface ribbed with continuous transverse elements of square cross section, varying rib size and pitch, at values of the plate Rayleigh number up to about 5 Â 10 8 : Interestingly, a steady thermal boundary layer has been optically captured, and this was confirmed via measurements of the time history of temperature at different normal distances from the wall, where the intensity of air temperature fluctuations in the boundary layer has been found to be far below the ranges reported in the literature for the onset of transitional effects (cf.[21]).Furthermore, a discussion on the effects of external disturbances, present in practice, on the heat transfer performance is provided in Ref. [18].The dimensionless equations governing the macroscale problem, i.e., Eqs.(2a)-(2c), are solved beyond the fictitious surface passing through the outer rims of the ribs.The effective boundary conditions at X 2 ¼ 0, i.e., Eqs.(12a)-(12d), can be further simplified as validation of the model is targeted at the steady-state solution of a two-dimensional laminar flow; the gradients in the spanwise direction can thus be neglected.In addition, since the ribbed surface is impermeable, the local transpiration velocity (U 2 at X 2 ¼ 0) is zero on average and its inclusion has been found to be irrelevant under laminar flow conditions.Eventually, the effective boundary conditions simplify to the slip velocity in the streamwise direction U 1 j X 2 ¼0 and the effective temperature Hj X 2 ¼0 : The validation of the homogenization-based model requires conducting reliable feature-resolving simulations, in which the full domain is considered and the velocity and thermal fields within the inter-rib regions are fully resolved; cf. Figure 4 for a clearer view of the differences between the two categories of numerical simulations.The fine-grained fields can contribute to our understanding of the associated phenomena; nevertheless, they cannot be used directly to validate the results of the homogenized model.For this purpose, the running averages of the full simulation results should be defined.For instance, the running-average value of the dimensionless temperature (H) at the arbitrary point X 1 ¼ a, X 2 ¼ b is calculated as follows: HðX 1 , bÞ dX 1 : For each one of the cases studied, the local results of the full simulations will be first described, and the running-average fields will then be used to validate the homogenization-based solutions.
The mesh was generated using quadrilateral cells, where structured layers were created in the vicinity of the rough surface (or, in macroscopic simulations, the matching interface).To get a preliminary estimation of the maximum thickness of the thermal boundary layer ( dmax ), i.e., the thickness reached at x1 ¼ L, the classical Squire-Eckert theoretical prediction (refer to [22]) was used, that is Accordingly, to ensure that the boundary layer is well resolved, a near-wall region of interest for mesh refinement, of thickness % 0:08 L, was specified.In addition, attention should be focused on the region near the inlet of the domain, where the thickness of the thermal boundary layer approaches zero as X 1 !0 and the local heat transfer coefficient tends to 1: To handle this issue, avoiding extra refinement of the mesh near the inlet, a correlation proposed in Ref. [23], and assumed to be valid over the full laminar-flow range, was employed to calculate the average Nusselt number over the range 0 X where 278 and Pr ¼ 0:712, we obtain Nu d % 1:023: The value of d should be much smaller than the distance from the leading edge to the first rib.For more information about the grid structure and the mesh-dependence levels for the macroscopic and the feature-resolving simulations, the reader is referred to Ref. [16].However, in the current work, an additional complexity appears because of thermal conduction through the roughness elements, so that the solid domain is also discretized, increasing the total mesh requirements for the full feature-resolving simulations.For example, 35 Â 35 square cells (finest mesh) were used to discretize each rib in the cases discussed in Subsection 4.3, where the number of ribs was N ribs ¼ 20 and e l ¼ 0:25: In all simulations, the finite volume method was applied to discretize the two-dimensional transport equations, using Simcenter STAR-CCM þ multiphysics software (version 16.02.009-R8).The numerical procedure is similar to that employed in Ref. [16], except for the spatial discretization which has been upgraded via the hybrid MUSCL 3rd-order/central-differencing scheme (refer to [24,25]).The hybrid Gauss-least squares method was used for the computation of gradients, with the Venkatakrishnan gradient limiter activated (refer to [26]).The pressure-velocity coupling was based on the SIMPLE scheme.The average Nusselt number (Nu) was monitored and used as the convergence criterion; the reported values of Nu are, in general, considered accurate within about 62% on account of the degree of convergence achieved and the numerical errors involved in the simulations.Throughout this article, the average Nusselt number is evaluated by integrating the local Nusselt number (Nu) over the area of the surface of interest, while the projected surface area (area of a flat surface of same height) is used as weight such that the changes in Nu directly reflect the adverse/favorable trends of the heat transfer rate [9,16]; the expressions of Nu and Nu over a vertical surface of height L roughened with two-dimensional square ribs of size e and pitch l are where s ¼ ŝ L , with ŝ a dimensional distance that goes along the ribbed surface capturing its details (as opposed to the vertical distance x1 ).Conversely, Nu and Nu over the vertical virtual interface (for the simulations employing the effective conditions) are simply evaluated as

The basic validation case
In pursuit of basic validation of the theory, it is wise to consider the two limiting values of the thermal conductivity ratio, j, i.e., j ¼ 0 (adiabatic ribs) and j ! 1 (isothermal ribs).Since the model is validated in detail for isothermal ribs in Ref. [16], our basic validation here will be limited to the case of adiabatic ribs.Moreover, we start by analyzing an intensively ribbed surface (N ribs ¼ 100), to ensure that the value of the parameter ¼ 1 N ribs is sufficiently small, and the model is therefore applied within its validity range.The reader is referred to Ref. [16], in which a parameter C ¼ 2 ffiffiffiffiffi ffi Gr p was identified as a reliable accuracy indicator, with the homogenized model yielding acceptable results for C Շ 40: Given that the Grashof number is here equal to 2:78 Â 10 8 , we have a value of C % 1:67, well within the range of applicability of the model.In addition, a value of 0.25 is specified for the rib size-to-pitch ratio ( e l ).
Distributions of the thermal and velocity fields in vicinity of the ribbed wall are displayed in Figure 5.The role of adiabatic ribs in interrupting the development of the thermal boundary layer can be noticed, in terms of the temperature drop experienced at the outer rims.The fully featured patterns of U 1 and H along the plane at X 2 ¼ 0 are presented in Figure 6.The running-average values of these fields can be directly used to validate the homogenization-based results of the effective velocity and temperature boundary conditions as shown in the figure; reasonably good agreement can be observed, particularly for the temperature (H).
It is also important to assess the accuracy of the model predictions for the behavior of the velocity and thermal boundary layers.For this purpose, the model results of the profiles of U 1 and H across a section at the middle of the plate are compared in Figure 7 with the corresponding running-average results of the full simulation.It can be noticed that the results are quantitatively in perfect agreement.It is worth noting that, since the adiabatic ribs attenuate the temperature increase over the plate (and consequently the buoyant force), the peak velocity is lower than in the case of an isothermal flat surface.
Analyzing the model results of the temperature profiles across different normal sections along the plate, we can state that the normal temperature gradients at X 2 ¼ 0, i.e., @H @X 2 j X 2 ¼0 , directly indicate the model prediction of the macroscopic Nusselt number distribution along the plate.Conversely, the fully featured results of the Nusselt number, those directly obtained from the feature-resolving simulation, are calculated based on the temperature gradient in the direction normal to the physical surface at each point on the baseplate and on the rims of the square ribs; the running-average behavior should be therefore evaluated to validate the model results (cf. Figure 8).One should take into account that the evaluation of the running-average Nusselt number is done in a similar way to Eq. ( 13), yet the local values of the Nusselt number are to be integrated along the physical surface in a unit cell.Furthermore, the average Nusselt number (based on the projected area), Nu, was calculated.It is noticeable that the predicted value from the model (Nu ¼ 52:28) deviates from the result of the full simulation (Nu ¼ 52:68) by less than 1%: Both are below the corresponding value for the case of an isothermal flat surface (Nu ¼ 63:24).

Role of the ribs' thermal conductivity
We now discuss the effects of the rib-to-fluid thermal conductivity ratio (j) on the characteristics of the buoyancy-driven flow and the heat transfer phenomena.Three distinct situations are   considered for analysis via both the fine-grained simulations and the homogenization-based model, specifically the case of adiabatic ribs (j ¼ 0), isothermal ribs (j !1), and ribs of finite thermal conductivity (j % 4).The latter represents a situation that is frequently encountered in experiments, where wooden roughness elements (with k r % 0:1 W=mK) are used and the flowing fluid is air (with k f % 0:025 À 0:028 W=mK).We consider a roughness pattern consisting of 20 square ribs, i.e., ¼ 1 N ribs ¼ 1  20 , with e l ¼ 0:25 and with the flow conditions described by Pr ¼ 0:712 and Gr % 2:78 Â 10 8 :

The feature-resolving cases
As a starting point, results of the full feature-resolving simulations are presented and discussed.Contours of the temperature H and the streamwise velocity U 1 are displayed in Figure 9, with the aid of streamlines to detect the complex flow structures within the inter-rib regions.This provides a clear view of the expensive mesh requirements of the full simulations and thus highlights the advantages of employing the homogenized model.The fields are displayed in the figure close to the leading edge (X 1 ¼ 0:1 to 0.25) and near the end of the plate (X 1 ¼ 0:85 to 1), to indicate the development of the thermal and the viscous boundary layers along the vertical direction and the associated variations in flow patterns within the corrugations.Proceeding along the plate, we can realize that the mainstream reattachment to the baseplate is gradually delayed and eventually lost, with a large, primary eddy separating the mainstream from the baseplate.Secondary eddies are also resolved in the figure.In-depth inspection of Figure 9 reveals that the development of the thermal boundary layer is significantly interrupted when ribs of low thermal conductivity (or adiabatic ribs) are used, and therefore, the temperature levels in the adjacent fluid layers are lower compared to the case of isothermal ribs.This effect becomes more pronounced as we proceed along the plate (far from the leading edge).The patterns of the temperature and the streamwise velocity along the fictitious interface at X 2 ¼ 0 are plotted in Figure 10 (top).Exclusively for isothermal ribs, the temperature at the outer rims of the square elements is equal to the baseplate temperature (H ¼ 1).The level of the streamwise velocity is adversely affected by the decrease of the thermal conductivity of the ribs, as the buoyancy force is limited.
The consequent effects of the thermal conductivity variations on the distribution of the local Nusselt number along the ribbed surface are illustrated in Figure 10 (bottom).The Nusselt number experiences a quasi-periodic behavior while proceeding along the vertical plate (for the three cases considered), which agrees with the findings in Refs.[5,10] with highly conducting ribs and in Refs.[6,10] with poorly conducting ribs.It is evident that the interruption of the thermal boundary layer development over adiabatic, or low-conductivity, ribs enhances the heat transfer performance of the isothermal baseplate in the inter-rib regions, consistent with the results reported in Refs.[4,6].Conversely, the heat transfer from the outer rims of the ribs plays a significant role in the case of isothermal elements (j !1), while this contribution is reduced as j decreases, until j ¼ 0 is reached where no heat transfer takes place from the surfaces of the adiabatic ribs.It is shown later that, on average, the highly conducting ribs outperform the low-conductivity ones (contrary to Ref. [4]), which implies that the latter contribution (heat transfer from ribs) is dominant.

Validation of the model results
Predictions of the homogenization-based simulations are now compared with the corresponding running-average results of the detail-resolving simulations, for accuracy estimation and assessment of possible weak points in the model.It is anticipated that the model is tested here close to its expected limit of applicability (cf.[16]) as C ¼ 2 ffiffiffiffiffi ffi Gr p is now equal to 41.6, close to the threshold (C % 40).
The running-average distributions of the Nusselt number (calculated based on the fully resolved patterns given in Figure 10 (bottom)) are plotted against X 1 in Figure 11.The heat transfer performance of the ribbed surfaces is lower than the performance of an isothermal smooth surface; surface roughening deteriorates heat transfer under the present conditions.The deterioration becomes more pronounced with the decrease of the ribs' thermal conductivity.
Considering the model results of the macroscopic distribution of the Nusselt numbers, we observe that they fit well with the reference running-average values beyond some distance from the leading edge of the plate; deviations along almost the first 20% of the plate length are significant.In order to alleviate this problem, a length, indicated here as the compensation distance (X c ), is specified near the leading edge, where the model results of the Nusselt number are to be excluded and replaced by the corresponding results over an isothermal flat surface when evaluating the average Nusselt number (Nu) over the plate.This distance is chosen here such that the compensation region extends over the first two pitch distances (X c ¼ 2) if the rib size is negligible, while it should be gradually decreased with the increase of the rib size as the contribution of the ribs to the physical phenomenon acquires prominence.Given that A r is the surface area of the ribs and  A t is the total surface area (including the ribs and the baseplate portions in the inter-rib regions), the following expression of X c is proposed: For the case of transverse square ribs of size e and pitch l, the previous equation reads: For l=e ¼ 4, the compensation distance is equal to (the first pitch distance).The hypothesis stated here contributes to the calculations of Nu throughout this article (including results in Subsection 4.2); accordingly, the resulting values of Nu are indicated as the corrected homogenized-model predictions, while the uncorrected, pure results of the model are presented occasionally for comparison.
The effect of the thermal conductivity ratio j on the average Nusselt number (Nu) is graphically displayed in Figure 12.Predictions of the homogenized model are interestingly consistent with results of the full simulations, with the maximum deviation below 3:4%: The average Nusselt number for the case of isothermal ribs (j ! 1) is 56.22 based on the homogenized model, and 56.00 based on the full simulation.This value declines gradually with the decrease of j until reaching the case of adiabatic ribs j ¼ 0, in which the heat transfer takes place only through the inter-rib regions (Nu % 35:4).As ribs are, by definition, of small size relative to the plate height, thermal conduction across them is of significance even for relatively low values of the thermal conductivity coefficient.It can be realized that the average Nusselt number for values of j տ 30 is very close to its value for the isothermal case.With air as the moving fluid, this range corresponds to ribs' thermal conductivity k r տ 0:8 W=mK, which can be easily satisfied in practice (for instance, aluminum ribs are of thermal conductivity far beyond this threshold, especially when they are press-fit into the baseplate, as in Ref. [4], or machined integral with the surface, as in Ref. [5], to avoid contact resistance).Even for wooden ribs (j % 4), the average Nusselt number is about 47, which is intermediate between the values for adiabatic and isothermal ribs; this implies that it is difficult to experimentally approach the adiabatic limit, and that modeling wooden ribs (or low-conductivity ribs in general) as adiabatic elements in numerical simulations may lead to questionable results.
For a clearer view of the significance of correcting the model predictions for Nu, based on Eq. ( 19), the uncorrected homogenized-model results and their percentage deviations from the reference results of the full feature-resolving simulations are given in Table 1; the errors in the pure predictions of the model are considerably large at relatively low values of j: For replication of the macroscopic simulations, typical values of the model coefficients (k h , B), corresponding to e l ¼ 0:25 and the specific considered values of j, are also presented in the table.Being dependent only on e l , the coefficients k x and m 12 are constant regardless of variations in j; their values (cf. Figure 3) are 0.0398 and 0.0023, respectively.

Considerations on the density of the roughness pattern
The aim of this section is to investigate how the behavior of the buoyancy-driven flow and the Nusselt number are affected when the density of the roughness pattern is varied, i.e., the number of ribs attached to the surface is changed.We consider here buoyancy-induced flows (Gr % 2:78 Â 10 8 and Pr % 0:712) over a vertical plate roughened with two-dimensional square elements which are either perfectly conducting (j !1), perfectly adiabatic (j ¼ 0), or of low thermal conductivity (j ¼ 4).The rib size to plate height ratio is fixed ( e L ¼ 1 80 ).The maximum possible number of ribs to be attached to the surface corresponds to the case in which the roughness elements become contiguous; at this point the pitch distance (l) becomes equal to the rib size (e) such that e L ¼ l L ¼ ¼ 1 80 , hence, the maximum value of N ribs is 80. Sample results of the full feature-resolving simulations will be displayed first before moving to validation of the homogenized predictions.

Results of the feature-resolving simulations
Figure 13 describes the behavior of the surface-averaged Nusselt number with the increase of N ribs from 0 to 80.The case N ribs ¼ 0 obviously represents a smooth isothermal surface, for which Nu % 63:00 as previously stated.At N ribs ¼ 80, the roughness elements are in direct contact and the inter-rib spaces vanish; the case of a smooth flat surface is, therefore, recovered again; however, the surface, which now consists of contiguous ribs, is either adiabatic (for j ¼ 0), isothermal (for j ! 1) or it resembles a smooth layer of low thermal conductivity (j ¼ 4) overlying the isothermal baseplate.For the case of adiabatic ribs, the heat transfer performance monotonically deteriorates with the increase of N ribs until the whole surface becomes thermally insulated Percentage deviations in the predictions are estimated relative to the reference results of the feature-resolving simulations.
(Nu ¼ 0 at N ribs ¼ 80).On the contrary, the behavior of the natural-convection heat transfer with the increase of N ribs is complex when the ribs are thermally conducting, even if the thermal conductivity is small (j ¼ 4).For this case, two opposing effects mainly take place when the number of ribs increases: (i) the flow moves mostly above the ribs' outer rims, where local enhancement  of the Nusselt number is expected (favorable effect); (ii) the inter-rib spaces become narrower, and the adverse effects of the hot dead recirculation zones between ribs on the local Nusselt number become therefore more pronounced.It is useful to interpret these phenomena using the illustrations in Figure 14, which highlight the local enhancement of the heat transfer performance over the outer edges of the ribs, the early loss of mainstream reattachment to the baseplate when shifting to a denser roughness pattern ( ¼ 1 50 ) and the consequent relative degradation of the Nusselt number through the inter-rib regions (compared to ¼ 1  10 ) along with the local thickening of the thermal boundary layer.The two opposing effects discussed above justify the nonmonotonic behavior of the average Nusselt number with the increase of the number of ribs for thermally conducting ribs, displayed previously in Figure 13.For the cases initially studied with the feature-resolving approach (revisit the points plotted in Figure 13), the homogenization-based model is further implemented and assessed.For each value of N ribs ¼ 1 , the rib size to pitch distance ratio is calculated as e l ¼ e L Â L l ¼ 1 80 Â N ribs : The value of e l is used to apply the compensation hypothesis given by Eq. ( 19) and, together with each of the three considered values of j, to evaluate the upscaled coefficients of the model; typical values of these coefficients are available in Figure 3. Predictions for the average Nusselt number agree well with the reference results of the full simulations, as shown in Figure 15, with deviations of the corrected model results always below 10% in magnitude.

Conclusions
The asymptotic homogenization technique was employed to upscale the steady conjugate heat transfer problem combining natural convection over a regularly micro-textured vertical surface and the thermal conduction through the roughness elements; the study of the effect of the ribs' thermal conductivity represents the novel contribution of the present work.The full domain was decomposed into two regions (microscopic/macroscopic), and two corresponding subproblems were formulated.An asymptotic analysis was conducted in the near-wall, microscopic layer to eventually derive effective boundary conditions (Eqs.(12a)-(12d)), second-order accurate in terms of a small parameter ¼ rib pitch ðlÞ plate height ðLÞ , for the velocity and temperature at a virtual plane interface of choice, where the two subproblems are coupled.A significant difference from the model developed by Ahmed et al. [16] is that the Robin-like temperature boundary condition includes a thermal-slip coefficient (k h ) which depends not only on the rib geometry but also on the rib-to-fluid thermal conductivity ratio, j: The ability of the proposed conditions to mimic the effects of the surface microstructure on the macroscale flow behavior and heat transfer performance was assessed by validating the model against corresponding full feature-resolving simulations.Several case studies were investigated, for the particular case of spanwise-elongated square ribs, varying the number of elements attached to the surface and the thermal conductivity ratio.A correction of the model predictions for the Nusselt number was proposed over an initial distance of the surface, as per Eq. ( 19), and was proved to be efficient, especially at not-so-small values of when the scale separation assumption is gradually lost.Besides validation, the analysis of the numerical results revealed the following: (i) the presence of ribs has a negative effect on the heat transfer performance of the surface, compared with the smooth plate case, at least as long as the flow remains steady; (ii) the lower the thermal conductivity of the ribs, the more the overall convective heat transfer rate deteriorates, despite the local relative enhancements realized across the inter-rib regions of the baseplate; (iii) considering elements of low thermal conductivity as adiabatic ribs in the analysis may be critical, as we found that the performance of wooden ribs (for instance) is intermediate between the isothermal and the adiabatic limiting cases; (iv) as the number of perfectly conducting ribs increases, the average Nusselt number, Nu, experiences a non-monotonic behavior, with an initial decrease and a subsequent increase up to the value of the isothermal flat surface, achieved when the ribs become contiguous; (v) as j decreases, until reaching the case of adiabatic ribs, a monotonically negative effect of increasing the ribs' density is found.
This work aims to reveal the numerical advantages of the multiscale homogenization approach as a rapid and accurate tool for the analysis of complex phenomena amenable to separation of scales.The proposed model represents a more general/versatile version of the earlier ones by Ahmed et al. [16] (valid exclusively for the case of perfectly conducting ribs) and by Introïni et al. [27] (where the buoyancy effect in the microscopic region was neglected).It is interesting to extend the present analytical procedure, along with numerical and experimental validation, to

Figure 1 .
Figure 1.Sketches showing: (left) a general description of the conjugate heat transfer problem under study, with indication of the base/fluid interface (I bf ), the base/rib interface (I br ) and the rib/fluid interface (I rf ); (right) the decomposed flow domain, with indication of a generic matching interface between the microscopic and the macroscopic regions.

Figure 2 .
Figure 2. A representative microscopic unit cell for the case of two-dimensional square ribs, with the geometric parameters indicated.

Figure 3 .
Figure 3. Dependence of the upscaled model coefficients on the aspect ratio e l :

1 dLFigure 4 .
Figure 4. General description of the computational domain and the boundary conditions used for (left) feature-resolving simulations and (right) homogenization-based simulations.

Figure 5 .
Figure 5. Fully resolved contours of H and U 1 next to the wall for four adiabatic ribs placed halfway through the plate, with plots of the typical values at the virtual interface (X 2 ¼ 0); ¼ 1 100 and e l ¼ 0:25:

Figure 6 .
Figure 6.Numerical results of the effective boundary conditions for U 1 and H, against the corresponding running-average values extracted from the full feature-resolving simulation.The fully resolved local values of the fields are also shown in the background.Same conditions and parameters as Figure 5.

Figure 7 .
Figure 7. Homogenization-based predictions for the profiles of U 1 and H across a section at X 1 ¼ 0:5, validated against the running-average results of the feature-resolving simulation.Adiabatic ribs are considered, with ¼ 1 100 and e l ¼ 0:25:

Figure 8 .
Figure 8. Homogenized model predictions for the Nusselt number, plotted together with the running-average results of the feature-resolving simulation, for validation.Adiabatic ribs are considered, with ¼ 1 100 and e l ¼ 0:25: The corresponding distribution of Nu along a smooth isothermal surface is also displayed.

Figure 9 .
Figure 9. Contours of the dimensionless temperature (H) and streamwise velocity (U 1 ) adjacent to the rough surface, displayed through two specific portions of the plate: (left) X 1 ¼ 0:1 to 0.25; (right) X 1 ¼ 0:85 to 1. Three distinct values of the rib-to-fluid thermal conductivity ratio are considered.Close-ups of the flow structures within two distant inter-rib regions, for the case of adiabatic ribs, are shown in the bottom frame.For all cases, it is ¼ 1 20 and e l ¼ 0:25:

Figure 10 .
Figure 10.Fully featured results of U 1 and H along a virtual plane at X 2 ¼ 0 (top) and patterns of the local Nusselt number (bottom), extracted from full simulations at three values of j, with ¼ 1 20 and e l ¼ 0:25: The yellow elements below the horizontal axis indicate the locations of the ribs.

Figure 11 .
Figure 11.Distributions of the Nusselt number along the plate for different values of j, with ¼ 1 20 and e l ¼ 0:25: Predictions of the homogenized model (filled circles) are validated against running-average results of full simulations (solid lines).The corresponding behavior along a smooth surface is also given (dashed black line).

Figure 12 .
Figure 12.Effect of the rib-to-fluid thermal conductivity ratio (j) on the average Nusselt number, based on the corrected homogenized-model results (filled circles) and the feature-resolving simulations (solid line).Percentage errors in the model results are plotted using dashed red line.The conditions under study are ¼ 1 20 , e l ¼ 0:25, Gr % 2:78 Â 10 8 , and Pr % 0:712:

Figure 13 .
Figure 13.Effect of the number of ribs on the average Nusselt number, at a fixed rib height to plate length ratio ( e L ¼ 1 80).The values plotted are based on feature-resolving simulations, with three values of the rib-to-fluid thermal conductivity ratio: j ¼ 0 (adiabatic ribs), j ¼ 4, and j ! 1 (perfectly conducting, isothermal ribs).Gr % 2:78 Â 10 8 and Pr % 0:712:

Figure 14 .
Figure 14.Contours of the fully featured temperature and streamwise velocity fields next to the ribbed surface, for two different values of : Here, the conjugate heat transfer problems are considered at j ¼ 4 with e L ¼ 1 80 : The contours are displayed, and the behaviors of the local Nusselt number are plotted, over the region from X 1 ¼ 0:4 to X 1 ¼ 0:6:

Figure 15 .
Figure 15.Effect of the number of ribs (N ribs ) on the average Nusselt number ( Nu), shown for: (top) isothermal ribs; (middle) a rib-to-fluid thermal conductivity ratio of 4; (bottom) adiabatic ribs.Results of the feature-resolving simulations (solid lines) are used as a reference to assess the accuracy of the corrected (dashed lines with filled circles) and the uncorrected (dotted lines with filled triangles) model predictions; percentage errors in the model results (corrected and uncorrected) are plotted in the right column.For all cases, it is e L ¼ 1 80 , Gr % 2:78 Â 10 8 and Pr % 0:712:

Table 1 .
Comparison between the corrected and the uncorrected homogenized-model predictions for the average Nusselt number at different values of j: