On Transformation Form-Invariance in Thermal Convection

Abstract

Over the past two decades, effective control of physical fields, such as light fields or acoustics fields, has greatly benefited from transforming media. One of these rapidly growing research areas is transformation thermotics, especially embodied in the thermal conductive and radiative modes. On the other hand, transformation media in thermal convection has seldom been studied due to the complicated governing equations involving both fluid motion and heat transfer terms. The difficulty lies in the robustness of form invariance in the Navier–Stokes equations or their simplified forms under coordinate transformations, which determines whether the transformation operations can be executed on thermal convection to simultaneously regulate the flow and thermal fields. In this work, we show that thermal convection in two-dimensional Hele–Shaw cells keeps form-invariance, while its counterpart in general creeping flows or general laminar flows does not. This conclusion is numerically verified by checking the performances of invisible devices made of transformation media in convective environments. We further exploit multilayered structures constituted of isotropic homogeneous natural materials to realize the anisotropic inhomogeneous properties required for transformation media. Our results clarify the long-term confusion about the validation of the transformation method in thermal convection and provide a rigorous foundation and classical paradigm on inspiring various fascinating metadevices in both thermal and flow fields.