Universal solutions for Boussinesq and non-Boussinesq plumes

Abstract

Closed-form solutions describing the behaviour of buoyant axisymmetric turbulent rising plumes and fountains, emitted vertically from area sources in unconfined quiescent environments of uniform density, are proposed in a form that is universally applicable to Boussinesq and non-Boussinesq plumes. This paper, thereby, generalizes the results obtained separately for steady Boussinesq and non-Boussinesq plumes, including asymptotic virtual source corrections. The flux balance parameter Γ = Γ( z ), a local Richardson number, is instrumental in describing the behaviour of steady plumes and the initial rise behaviour of fountains with height z . Non-dimensional graphs (cf. the ‘scale diagrams’ of Morton & Middleton, J. Fluid Mech ., vol. 58, 1973, pp. 165–176) are plotted, showing certain characteristic heights for different source conditions, characterized by one single source flux balance parameter, giving a unique representation of the behaviour of Boussinesq fountains and both Boussinesq and non-Boussinesq plumes. Finally, a length scale has been identified that characterizes the height over which non-Boussinesq effects are important for lazy plumes rising from area sources.