  Heat can generally transfer via thermal conduction, thermal radiation, and
thermal convection. All the existing theories of transformation thermotics and
optics can treat thermal conduction and thermal radiation, respectively.
Unfortunately, thermal convection has never been touched in transformation
theories due to the lack of a suitable theory, thus limiting applications
associated with heat transfer through fluids (liquid or gas). Here, we develop,
for the first time, a general theory of transformation thermal convection by
considering the convection-diffusion equation, the Navier-Stokes equation, and
the Darcy law. By introducing porous media, we get a set of coupled equations
keeping their forms under coordinate transformation. As model applications, the
theory helps to show the effects of cloaking, concentrating, and camouflage.
Our finite element simulations confirm the theoretical findings. This work
offers a general transformation theory for thermal convection, thus revealing
some novel behaviors of thermal convection; it not only provides new hints on
how to control heat transfer by combining thermal conduction, thermal
radiation, and thermal convection, but also benefits the study of mass
diffusion and other related fields that contain a set of equations and need to
transform velocities at the same time.
