  We study turbulent flows in planar channels with unstable thermal
stratification, using direct numerical simulations in a wide range of Reynolds
and Rayleigh numbers and reaching flow conditions which are representative of
asymptotic developed turbulence. The combined effect of forced and free
convection produces a peculiar pattern of quasi--streamwise rollers occupying
the full channel thickness with aspect--ratio considerably higher than unity;
it has been observed that they have an important redistributing effect on
temperature and momentum. The mean values and the variances of the flow
variables do not appear to follow Prandtl's scaling in the flow regime near
free convection, except for the temperature and vertical velocity fluctuations,
which are more affected by turbulent plumes. Nevertheless, we find that the
Monin--Obukhov theory still yields a useful representation of the main flow
features. In particular, the widely used Businger--Dyer relationships provide a
convenient way of accounting for the bulk effects of shear and buoyancy,
although individual profiles may vary widely from the alleged trends.
Significant deviations are found in DNS with respect to the commonly used
parametrization of the mean velocity in the light-wind regime, which may have
important practical impact in models of atmospheric dynamics. Finally, for
modelling purposes, we devise a set of empirical predictive formulas for the
heat flux and friction coefficients which can be used with about $10\%$ maximum
error in a wide range of flow parameters.
