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 Γ ∝ Qˆ²₀Fˆ₀/Mˆ^5/2₀ 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.

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.

We examine the transient buoyancy-driven flow in a ventilated filling box that is subject to a continuous supply of buoyancy. A rectangular box is considered and the buoyancy input is represented as a turbulent plume, or as multiple non-interacting plumes, rising from the floor. Openings in the base and top of the box link the interior environment with a quiescent exterior environment of constant and uniform density. A theoretical model is developed to predict, as functions of time, the density stratification and the volume flow rate through the openings leading to the steady state. Comparisons are made with the results of small-scale analogue laboratory experiments in which saline solutions and fresh water are used to create density differences. Two characteristic timescales are identified: the filling box time ($T_f$), proportional to the time taken for fluid from a plume to fill a closed box; and the draining box time ($T_d$), proportional to the time taken for a ventilated box to drain of buoyant fluid. The timescale for the flow to reach the steady state depends on these two timescales, which are functions of the box height $H$ and cross-sectional area $S$, the ‘effective’ opening area $A^*$, and the strength, number and distribution of the buoyancy inputs. The steady-state flow is shown to be characterized by the ratio of these timescales ($\mu\,{=}\,T_d/T_f$) which is equivalent to the dimensionless vent area $A^*/H^2$. A feature of these flows is that for $\mu\,{>}\,\mu_c$ the depth of the buoyant upper layer may exceed, or ‘overshoot’, the steady layer depth during the initial transient. The value of $\mu_c$ is determined for both line and point-source plumes, and the sensitivity of the developing flow to the distribution of buoyancy input assessed.

The coalescence of two co-flowing axisymmetric turbulent plumes and the resulting single plume flow is modelled and compared to experiments. The point of coalescence is defined as the location at which only a single peak appears in the horizontal buoyancy profile, and a prediction is made for its height. The model takes into account the drawing together of the two plumes due to their respective entrainment fields. Experiments showed that the model tends to overestimate the coalescence height, though this discrepancy may be partly explained by the sensitivity of the prediction to the entrainment coefficient. A model is then developed to describe the resulting single plume and predict its virtual origin. This prediction and subsequent predictions of flow rate above the merge height compare very well with experimental results.