Virtual origin correction for lazy turbulent plumes

The location of the asymptotic virtual origin of positively buoyant turbulent plumes with a deficit of initial momentum flux when compared with equivalent pure plumes is investigated. These lazy plumes are generated by continuous steady releases of momentum, buoyancy and volume into a quiescent uniform environment from horizontal sources (at z = 0) of finite area, and are shown to be equivalent to the far-field flow above point source pure plumes, of buoyancy only, rising from the asymptotic virtual source located below the actual source at z = -z(avs). An analytical expression for the location of the asymptotic virtual source relative to the actual source of the lazy plume is developed. The plume conservation equations are solved for the volume flow rate, and the position of the asymptotic virtual origin is deduced from the scaling for the volume flow rate at large distances from the source. The displacement z(avs) of the asymptotic virtual origin from the actual origin scales on the source diameter and is a function of the source parameter Gamma proportional to (Q) over cap (2)(0)(F) over cap (0)/(M) over cap (5/2)(0) which is a measure of the relative importance of the initial fluxes of buoyancy (F) over cap (0), momentum (M) over cap (0), and volume (Q) over cap (0) in the plume. The virtual origin correction developed is valid for Gamma > 1/2 and is therefore applicable to lazy plumes for which Gamma > 1, pure plumes for which Gamma = 1, and forced plumes in the range 1/2 < Gamma < 1. The dimensionless correction z(avs)* decreases as Gamma increases, and for r much greater than 1, z(avs)* --> 0.853 Gamma (-1/5). Comparisons made between the predicted location of the asymptotic virtual origin and the location inferred from measurements of lazy saline plumes in the laboratory show close agreement.