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 = −zavs.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 zavs of the asymptotic virtual origin from the actual origin scales on the source diameter and is a function of the source parameter Γ ∝ Qˆ20Fˆ0/Mˆ5/20 which is a measure of the relative importance of the initial fluxes of buoyancy Fˆ0, momentum Mˆ0, and volume Qˆ0 in the plume. The virtual origin correction developed is valid for Γ > 1/2 and is therefore applicable to lazy plumes for which Γ > 1, pure plumes for which Γ = 1, and forced plumes in the range 1/2 < Γ < 1. The dimensionless correction z*avs decreases as Γ increases, and for Γ [Gt ] 1, z*avs → 0.853Γ−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.

We examine the dynamics of turbulent lazy plumes rising from horizontal area sources and from vertically distributed line sources into a quiescent environment of uniform density. First, we consider plumes with internal buoyancy flux gain and, secondly, plumes from horizontal area sources that have significant momentum flux deficits. We re-cast the conservation equations of Morton et al. (1956) for a constant entrainment coefficient $(\alpha)$ in terms of three dimensionless parameters: the plume radius $\beta$; a parameter $\Gamma$ characterizing the local balance of momentum, buoyancy and volume fluxes; and a parameter $\Lambda$ that characterizes the rate of internal buoyancy flux gain with height. For a plume with a linear internal buoyancy flux gain with height the flow is shown to be a constant-velocity lazy plume. For highly lazy area sources we derive exact solutions for the key plume parameters in terms of $\Gamma$ and an approximate solution for the variation of $\Gamma$ with height. We show that near the source there is a region of zero entrainment.

This paper describes the fluid mechanics within a box containing a vertical plane distributed source of buoyancy. A theoretical analysis is presented that models the development of plumes from such sources in an unconfined ambient of uniform density. Two extensions are considered. The first concerns a sealed box and the second involves the more general situation where the box is ventilated by openings at top and bottom. In the sealed box the stratification develops in much the same way as for a 'filling box' containing a single-point source of buoyancy on the floor. An initial front descends from the ceiling of the box and an asymptotic stratification eventually develops which is continuous in the vertical direction. In the case of the ventilated box it is found that a complex stratification develops where one or more horizontal intrusions are formed by detachment of the plume/boundary layer from the vertically distributed source where the buoyancy of the plume is less than, or equal to, that of the stratified ambient at a given height. Experimental results are presented to demonstrate the validity of the theory. The findings are relevant to both forced and naturally ventilated buildings containing non-adiabatic vertical surfaces.

Analytical solutions for the initial rise height $z_m$ of a turbulent fountain for the limits of both small and large source Froude number $\hbox{\it Fr}_0$ are presented. These solutions are based on a plume entrainment model. For large Froude number fountains, the established result $z_m/r_0{\,\sim\,}\hbox{\it Fr}_0$ is obtained ($r_0$ denoting the source radius). For intermediate Froude numbers, the relationship $z_m/r_0{\,\sim\,}\hbox{\it Fr}_0^2$ is found and the rise height is independent of the entrainment coefficient $\alpha$. For very small Froude numbers, the flow is hydraulically controlled at the source and $z_m/r_0{\,\sim\,}\hbox{\it Fr}_0^{2/3}$. Existing experimental and numerical results, as well as our own experimental results, are compared to our solutions and show good agreement. Comparison with experimental results also demonstrates that the appropriate entrainment coefficient for highly forced fountains is $\alpha_f{\,\approx\,}0.058$. This is significantly closer to the entrainment coefficient of a jet than of a plume.