  We study a Boussinesq system in a bounded domain with an outlet boundary
portion where fluid can leave or re-enter. On this boundary part, we consider a
do-nothing condition for the fluid flow, and a new artificial condition for the
heat transfer that couples nonlinearly the fluid velocity and temperature. The
latter can be further adjusted if convective or conductive phenomena are
dominant. We prove existence and, in some cases, uniqueness of weak solutions
to stationary and evolutionary problems by a fixed point strategy under
suitable assumptions on the data. A variety of numerical tests shows the
improved performance of the new artificial condition with respect to other
standard choices in the literature.
