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.

Keywords:

• Conjugate heat transfer
• effective boundary conditions
• multiscale homogenization
• natural convection
• ribbed surfaces

Disclosure statement

The authors report that there are no competing interests to declare.

Acknowledgment

The authors would like to thank Didier Lasseux (University of Bordeaux, France) and Francisco J. Valdés-Parada (Universidad Autónoma Metropolitana-Iztapalapa, Mexico) for several interesting discussions on the subject of upscaling.

Additional information

Funding

The financial support of the Italian Ministry of University and Research, program PRIN 2017, project 2017X7Z8S3 LUBRI-SMOOTH, is gratefully acknowledged.