C. P. Caulfield

The anatomy of the mixing transition in homogeneous and stratified free shear layers

We investigate the detailed nature of the ‘mixing transition’ through which turbulence may develop in both homogeneous and stratified free shear layers. Our focus is upon the fundamental role in transition, and in particular the associated ‘mixing’ (i.e. small-scale motions which lead to an irreversible increase in the total potential energy of the flow) that is played by streamwise vortex streaks, which develop once the primary and typically two-dimensional Kelvin–Helmholtz (KH) billow saturates at finite amplitude. Saturated KH billows are susceptible to a family of three-dimensional secondary instabilities. In homogeneous fluid, secondary stability analyses predict that the stream-wise vortex streaks originate through a ‘hyperbolic’ instability that is localized in the vorticity braids that develop between billow cores. In sufficiently strongly stratified fluid, the secondary instability mechanism is fundamentally different, and is associated with convective destabilization of the statically unstable sublayers that are created as the KH billows roll up. We test the validity of these theoretical predictions by performing a sequence of three-dimensional direct numerical simulations of shear layer evolution, with the flow Reynolds number (defined on the basis of shear layer half-depth and half the velocity difference) Re = 750, the Prandtl number of the fluid Pr = 1, and the minimum gradient Richardson number Ri (0) varying between 0 and 0.1. These simulations quantitatively verify the predictions of our stability analysis, both as to the spanwise wavelength and the spatial localization of the streamwise vortex streaks. We track the nonlinear amplification of these secondary coherent structures, and investigate the nature of the process which actually triggers mixing. Both in stratified and unstratified shear layers, the subsequent nonlinear amplification of the initially localized streamwise vortex streaks is driven by the vertical shear in the evolving mean flow. The two-dimensional flow associated with the primary KH billow plays an essentially catalytic role. Vortex stretching causes the streamwise vortices to extend beyond their initially localized regions, and leads eventually to a streamwise-aligned collision between the streamwise vortices that are initially associated with adjacent cores. It is through this collision of neighbouring streamwise vortex streaks that a final and violent finite-amplitude subcritical transition occurs in both stratified and unstratified shear layers, which drives the mixing process. In a stratified flow with appropriate initial characteristics, the irreversible small-scale mixing of the density which is triggered by this transition leads to the development of a third layer within the flow of relatively well-mixed fluid that is of an intermediate density, bounded by narrow regions of strong density gradient.

A laboratory study of explosive volcanic eruptions

This paper describes a series of laboratory experiments in which buoyant mixtures of methanol and ethylene glycol (MEG) are injected as a downward propagating jet into a tank of fresh water. As the MEG mixes with water, it becomes denser than the water. If the MEG mixes with sufficient water before its initial momentum is exhausted, the jet fluid may become dense and continue downward into the tank as a convecting plume. If the jet does not have sufficient initial momentum, then a collapsing fountain develops, and the material in the jet rises back toward the top of the tank and spreads laterally as a gravity current. Subsequent mixing can cause some of the material in the gravity current to become dense, separate from the current, and sink into the tank. Although the direction of gravity is reversed, these experiments simulate many of the important dynamical features of eruption columns which can develop during explosive volcanic eruptions. In the volcanic situation, a hot, dense, and dusty mixture of gas, ash, and clasts is erupted from a vent at high speed. If sufficient ambient air is entrained into the jet, then the mixture may become buoyant through heating and expansion of the air; it therefore continues rising high into the atmosphere. Otherwise, the material collapses back to the ground. These experiments allow one to investigate systematically the different styles of behavior of the erupted material as the eruption conditions change. Four different styles of behavior have been identified, with transitions from one style to the next as the initial momentum flux of the jet is decreased: (1) Plinian style convecting columns; (2) the periodic release of discrete convecting clouds originating close to the vent, from a collapsed fountain; (3) coignimbrite eruption columns centered some distance from the vent which are generated when a fraction of a pyroclastic flow becomes buoyant; and (4) pyroclastic flows in which the majority of the material remains relatively dense and therefore spreads laterally from the vent. Using a simple theoretical model of the laboratory experiments, an analytical expression describing the conditions necessary for collapse of the analogue laboratory columns has been derived. This is successfully compared with the laboratory experiments. The simple analysis predicts that column collapse may be induced by (1) increasing the density of the erupted material; (2) decreasing the maximum buoyancy the mixture can attain on mixing with ambient; (3) increasing the erupted mass flux for a given momentum flux; or (4) decreasing the initial momentum flux for a given mass flux, as may occur if the vent is eroded. These results are consistent with earlier studies (Wilson et al., 1980; Wilson and Walker, 1987; Bursik and Woods, 1991). Although the analogue experimental system is somewhat simplified, the observations of the periodic release of convecting thermals just after column collapse suggest a mechanism for the complex grading which is commonly found just below the collapse horizons in fall deposits, for example the Fogo A deposit (Walker and Croasdale, 1970) and the Vesuvius A.D. 79 deposit (Carey and Sigurdsson, 1987). In conjunction with partial column collapse, this periodicity also suggests a means by which flow deposits may become interspersed with fall deposits; this feature has been observed in a number of cases including the Taupo deposit (Wilson and Walker, 1985).

Time-dependent plumes and jets with decreasing source strengths

The classical bulk model for isolated jets and plumes due to Morton, Taylor & Turner ( Proc. R. Soc. Lond . A, vol. 234, 1956, p. 1) is generalized to allow for time-dependence in the various fluxes driving the flow. This new system models the spatio-temporal evolution of jets in a homogeneous ambient fluid and Boussinesq and non-Boussinesq plumes in stratified and unstratified ambient fluids. Separable time-dependent similarity solutions for plumes and jets are found in an unstratified ambient fluid, and proved to be linearly stable to perturbations propagating at the velocity of the ascending plume fluid. These similarity solutions are characterized by having time-independent plume or jet radii, with appreciably smaller spreading angles (

Three dimensionalization of the stratified mixing layer

\tan^{-1}(2\alpha/3)

The Relationship between Flux Coefficient and Entrainment Ratio in Density Currents

) than either constant-source-buoyancy-flux pure plumes (with spreading angle

Mixing efficiency in high-aspect-ratio Rayleigh-Taylor experiments

\tan^{-1}(6\alpha/5)

Turbulent gravitational convection from a point source in a non-uniformly stratified environment

) or constant-source-momentum-flux pure jets (with spreading angle

Plumes with non-monotonic mixing behaviour

\tan^{-1}(2\alpha)

Natural ventilation in interconnected chambers

), where

Blocked natural ventilation: the effect of a source mass flux

\alpha