Buoyancy induced flows due to energy sources in cold quiescent pure and saline water

From the simple density state equation of Gebhart and Mollendorf (1977), a relatively compact similarity formulation has resulted for vertical boundary layer flows, as developed by Gebhart and Mollendorf (1978). This treatment accommodates the occurrence of density extrema in the flow field. Calculations have been given there, and elsewhere, for many different vertical flow conditions, as well as for some horizontal ones. However, similarity does not in general result for non-isothermal assigned or resulting midplane conditions, when an unstratified ambient medium is not at the temperature at which the density extremum occurs. However, this is a common circumstance in applications. Here this limitation is removed in a perturbation analysis of three vertical flows: that arising from a vertical surface with uniform heat flux, a plane plume and the flow arising from a concentrated horizontal line source imbedded in a vertical adiabatic surface, i.e., a wall plume. Comparison of perturbation and similarity solutions is also made for the isothermal condition, to corroborate the perturbation formulation. The new formulation also avoids some calculation difficulties which attend buoyancy force reversal across the boundary region. Additional similarity solutions are also given in this range for the isothermal surface condition.