We discuss energetic restrictions on the entrainment coefficient α for axisymmetric jets and plumes. The resulting entrainment relation includes contributions from the mean flow, turbulence and pressure, fundamentally linking α to the production of turbulence kinetic energy, the plume Richardson number Ri and the profile coefficients associated with the shape of the buoyancy and velocity profiles. This entrainment relation generalises the work by Kaminski et al. (J. Fluid Mech., vol. 526, 2005, pp. 361-376) and Fox (J. Geophys. Res., vol. 75, 1970, pp. 6818-6835). The energetic viewpoint provides a unified framework with which to analyse the classical entrainment models implied by the plume theories of Morton et al. , vol. 81, 1954, pp. 144-157). Data for pure jets and plumes in unstratified environments indicate that to first order the physics is captured by the Priestley and Ball entrainment model, implying that (1) the profile coefficient associated with the production of turbulence kinetic energy has approximately the same value for pure plumes and jets, (2) the value of α for a pure plume is roughly a factor of 5/3 larger than for a jet and (3) the enhanced entrainment coefficient in plumes is primarily associated with the behaviour of the mean flow and not with buoyancy-enhanced turbulence. Theoretical suggestions are made on how entrainment can be systematically studied by creating constant-Ri flows in a numerical simulation or laboratory experiment.

We present a direct numerical simulation (DNS) data set for a statistically axisymmetric turbulent jet, plume, and forced plume in a domain of size 40r0×40r0×60r0, where r0 is the source diameter. The data set supports the validity of the Priestley-Ball entrainment model in unstratified environments (excluding the region near the source) [Priestley and Ball, Q. J. R. Meteor. Soc. 81, 144 (1955)], which is corroborated further by the Wang-Law and Ezzamel et al. experimental data sets [Wang and Law, J. Fluid Mech. 459, 397 (2002); Ezzamel et al., J. Fluid Mech. 765, 576 (2015)], the latter being corrected for a small but influential coflow that affected the statistics. We show that the second-order turbulence statistics in the core region of the jet and the plume are practically indistinguishable from each other, although there are significant differences near the plume edge. The DNS data indicate that the turbulent Prandtl number is about 0.7 for both jets and plumes. For plumes, this value is a result of the difference in the ratio of the radial turbulent transport of radial momentum and buoyancy. For jets, however, the value originates from a different spread of the buoyancy and velocity profiles, in spite of the fact that the ratio of radial turbulent transport terms is approximately unity. The DNS data do not show any evidence of similarity drift associated with gradual variations in the ratio of buoyancy profile to velocity profile widths

We study the physics of unsteady turbulent jets using direct numerical simulation (DNS) by introducing an instantaneous step change (both up and down) in the source momentum flux. Our focus is on the propagation speed and rate of spread of the resulting front. We show that accurate prediction of the propagation speed requires information about the energy flux in addition to the momentum flux in the jet. Our observations suggest that the evolution of a front in a jet is a self-similar process that accords with the classical dispersive scaling z ∼ √ t. In the analysis of the problem we demonstrate that the use of a momentum-energy framework of the kind used by Priestley & Ball (Q. J. R. Meteorol. Soc., vol. 81, 1955, pp. 144-157) has several advantages over the classical mass-momentum formulation. In this regard we generalise the approach of Kaminski et al. (J. Fluid Mech., vol. 526, 2005, pp. 361-376) to unsteady problems, neglecting only viscous effects and relatively small boundary terms in the governing equations. Our results show that dispersion originating from the radial dependence of longitudinal velocity plays a fundamental role in longitudinal transport. Indeed, one is able to find dispersion in the steady state, although it has received little attention because its effects can then be absorbed into the entrainment coefficient. Specifically, we identify two types of dispersion. Type I dispersion exists in a steady state and determines the rate at which energy is transported relative to the rate at which momentum is transported. In unsteady jets type I dispersion is responsible for the separation of characteristic curves and thus the hyperbolic, rather than parabolic, nature of the governing equations, in the absence of longitudinal mixing. Type II dispersion is equivalent to Taylor dispersion and results in the longitudinal mixing of the front. This mixing is achieved by a deformation of the self-similar profiles that one finds in steady jets. Using a comparison with the local eddy viscosity, and by examining dimensionless fluxes in the vicinity of the front, we show that type II dispersion provides a dominant source of longitudinal mixing.

In this paper we develop an integral model for an unsteady turbulent jet that incorporates longitudinal dispersion of two distinct types. The model accounts for the difference in the rate at which momentum and energy are advected (type I dispersion) and for the local deformation of velocity profiles that occurs in the vicinity of a sudden change in the momentum flux (type II dispersion). We adapt the description of dispersion in pipe flow by Taylor (Proc. R. Soc. Lond. A, vol. 219, 1953, pp. 186-203) to develop a dispersion closure for the longitudinal transportation of energy in unsteady jets. We compare our model's predictions to results from direct numerical simulation and find a good agreement. The model described in this paper is robust and can be solved numerically using a simple central differencing scheme. Using the assumption that the longitudinal velocity profile in a jet has an approximately Gaussian form, we show that unsteady jets remain approximately straight-sided when their source area is fixed. Straight-sidedness provides an algebraic means of reducing the order of the governing equations and leads to a simple advection-dispersion relation. The physical process responsible for straight-sidedness is type I dispersion, which, in addition to determining the local response of the area of the jet, determines the growth rate of source perturbations. In this regard the Gaussian profile has the special feature of ensuring straight-sidedness and being insensitive to source perturbations. Profiles that are more peaked than the Gaussian profile attenuate perturbations and, following an increase (decrease) in the source momentum flux, lead to a local decrease (increase) in the area of the jet. Conversely, profiles that are flatter than the Gaussian amplify perturbations and lead to a local increase (decrease) in the area of the jet.

Abstract