Turbulent entrainment in jets with arbitrary buoyancy

Abstract

Explosive volcanic jets present an unusual dynamic situation of reversing buoyancy. Their initially negative buoyancy with respect to ambient fluid first opposes the motion, but can change sign to drive a convective plume if a sufficient amount of entrainment occurs. The key unknown is the entrainment behaviour for the initial flow regime in which buoyancy acts against the momentum jet. To describe and constrain this regime, we present an experimental study of entrainment into turbulent jets of negative and reversing buoyancy. Using an original technique based on the influence of the injection radius on the threshold between buoyant convection and partial collapse, we show that entrainment is significantly reduced by negative buoyancy. We develop a new theoretical parameterization of entrainment as a function of the local (negative) Richardson number that (i) predicts the observed reduction of entrainment and (ii) introduces a similarity drift in the velocity and buoyancy profiles as a function of distance from source. This similarity drift allows us to reconcile the different estimates found in the literature for entrainment in plumes. † Values of α e are often quoted with respect to a Gaussian profile of axial velocity, but here we convert systematically to corresponding top-hat values.

Energy-consistent entrainment relations for jets and plumes

“…those of Delichatsios 1979;Yu 1990;Vul'fson & Borodin 2001;Scase et al 2006), whose properties and consistency can be checked with respect to an overarching framework by appealing to the mean energy equation (Craske & van Reeuwijk 2015b). In the context of entrainment, the use of a momentum-energy framework allows one to deduce that under certain conditions both unsteady jets and plumes remain straight-sided (Craske 2015;Craske & van Reeuwijk 2015b), and allowed Craske & van Reeuwijk (2015a) to obtain a decomposition of the entrainment coefficient for jets, similar to that of Kaminski et al (2005), but with the important distinction that (1) both turbulence and pressure contributions are included and (2) standard (top-hat) definitions for radius and velocity scales are used.…”

“…This can be attributed to differences in experimental set-ups, experimental error and uncertainty (Turner 1986;Kaminski, Tait & Carazzo 2005), and also source conditions (George 1989;Redford, Castro & Coleman 2012). However, it is evident that there is a systematic difference between the reported value of α for jets and plumes, which suggests a dependence of α on Γ .…”

“…Recognising that the assumption of self-similarity is often violated in the context of atmospheric convection, Telford (1966) suggests that the entrainment coefficient is proportional to the turbulence intensity, and solves an additional equation for the turbulence kinetic energy. Kaminski et al (2005) should be credited for reintroducing energy-based entrainment relations to the plume literature. Indeed, they built on the foundations laid by Fox (1970), and extended the analysis by relaxing the assumptions regarding the radial profile dependences.…”

“…This led to an entrainment relation valid for both the near field and the far field, which exposed the physics of entrainment in terms of leading-order quantities. While the main focus of Kaminski et al (2005) was on turbulent fountains, they argued that the ratio of the width of the velocity profile to the buoyancy (reduced gravity) profile, denoted λ, could help to reconcile the spread in α for pure plumes. In particular, they showed evidence that small variations of λ in the vertical direction had an effect on entrainment, a process they termed similarity drift.…”

A simple integral model of buoyancy‐generating plumes and its application to volcanic eruption columns

“…Previous laboratory experiments of buoyancy-generating plumes, such as Bhat and Narasimha [1996] and Kaminski et al [2005], were based on the theoretical framework for Morton's [1959] forced plume. Accordingly, they may have ignored some transitions of flow styles illustrated in Figure 1.…”