The transport is largely governed by the mixed convection parameter, also known as Richardson number. Several relationships have been obtained, as reviewed in various papers and books. No general expression has been found to satisfy different cases.

The Ri number provides a parameter (value) to discern whether there is free, forced or mixed convection. But as you point out, there is no generalized expression involving all three cases in only one expression. Your actual research sounds very interesting!

interesting research area