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We study turbulent flows in planar channels with unstable thermalstratification, using direct numerical simulations in a wide range of Reynoldsand Rayleigh numbers and reaching flow conditions which are representative ofasymptotic developed turbulence. The combined effect of forced and freeconvection produces a peculiar pattern of quasi--streamwise rollers occupyingthe full channel thickness with aspect--ratio considerably higher than unity;it has been observed that they have an important redistributing effect ontemperature and momentum. The mean values and the variances of the flowvariables do not appear to follow Prandtl's scaling in the flow regime nearfree convection, except for the temperature and vertical velocity fluctuations,which are more affected by turbulent plumes. Nevertheless, we find that theMonin--Obukhov theory still yields a useful representation of the main flowfeatures. In particular, the widely used Businger--Dyer relationships provide aconvenient 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 usedparametrization of the mean velocity in the light-wind regime, which may haveimportant practical impact in models of atmospheric dynamics. Finally, formodelling purposes, we devise a set of empirical predictive formulas for theheat flux and friction coefficients which can be used with about $10\%$ maximumerror in a wide range of flow parameters.

Mixed convection in turbulent channels with unstable stratification