Stuart B. Dalziel

Gravity currents produced by lock exchange

The dynamics of gravity currents are believed to be strongly influenced by dissipation due to turbulence and mixing between the current and the surrounding ambient fluid. This paper describes new theory and experiments on gravity currents produced by lock exchange which suggest that dissipation is unimportant when the Reynolds number is sufficiently high. Although there is mixing, the amount of energy dissipated is small, reducing the current speed by a few percent from the energy-conserving value. Benjamin (J. Fluid Mech. vol. 31, 1968, p. 209) suggests that dissipation is an essential ingredient in gravity current dynamics. We show that dissipation is not important at high Reynolds number, and provide an alternative theory that predicts the current speed and depth based on energy-conserving flow that is in good agreement with experiments. We predict that in a deep ambient the front Froude number is 1, rather than the previously accepted value of √ 2. New experiments are reported for this case that support the new theoretical value. This paper provides an analysis of the motion of a gravity current produced by lock exchange. In a lock exchange experiment, fluids of different densities initially at rest are separated by a vertical barrier – the lock gate – in a tank. When the gate is removed, differences in the hydrostatic pressure cause the denser fluid to flow in one direction along the bottom boundary of the tank, while the lighter fluid flows in the opposite direction along the top boundary of the tank. Figure 1 shows the initial configurations for lock exchange flows: a full-depth release when the depths of heavy and light fluid on both sides of the gate are equal is shown in (a )a nd apartial-depth release when the dense fluid occupies only a fraction of the full depth is shown in (b). Figure 2 shows the flow resulting from a full-depth lock release experiment. In this case the densities on the two sides of the lock gate are very similar (the density ratio γ = ρ1/ρ2 < 1 is close to unity). A dense gravity current travels to the right along the lower boundary and a buoyant current travels to the left along the upper boundary. Visually the flows are very nearly symmetric, and the dense and light fronts travel at almost the same speeds (figure 2b). The currents occupy about half the channel depth in each case, although they may be shallower immediately behind the head where there is mixing. The speeds of the two currents are constant within experimental resolution. Previous similar observations led Benjamin (1968) to develop a theory for the propagation of a steadily advancing current. He considered one half of the flow shown in figure 2(a), say the dense current only. In a frame of reference moving with the current, the front

Self-similarity and internal structure of turbulence induced by Rayleigh–Taylor instability

This paper describes an experimental investigation of mixing due to Rayleigh{Taylor instability between two miscible fluids. Attention is focused on the gravitationally driven instability between a layer of salt water and a layer of fresh water with particular emphasis on the internal structure within the mixing zone. Three-dimensional numerical simulations of the same flow are used to give extra insight into the behaviour found in the experiments. The two layers are initially separated by a rigid barrier which is removed at the start of the experiment. The removal process injects vorticity into the flow and creates a small but signicant initial disturbance. A novel aspect of the numerical investigation is that the measured velocity eld for the start of the experiments has been used to initialize the simulations, achieving substantially improved agreement with experiment when compared with simulations using idealized initial conditions. It is shown that the spatial structure of these initial conditions is more important than their amplitude for the subsequent growth of the mixing region between the two layers. Simple measures of the growth of the instability are shown to be inappropriate due to the spatial structure of the initial conditions which continues to influence the flow throughout its evolution. As a result the mixing zone does not follow the classical quadratic time dependence predicted from similarity considerations. Direct comparison of external measures of the growth show the necessity to capture the gross features of the initial conditions while detailed measures of the internal structure show a rapid loss of memory of the ner details of the initial conditions. Image processing techniques are employed to provide a detailed study of the internal structure and statistics of the concentration eld. These measurements demonstrate that, at scales small compared with the conning geometry, the flow rapidly adopts self-similar turbulent behaviour with the influence of the barrier-induced perturbation conned to the larger length scales. Concentration power spectra and the fractal dimension of iso-concentration contours are found to be representative of fully developed turbulence and there is close agreement between the experiments and simulations. Other statistics of the mixing zone show a reasonable level of agreement, the discrepancies mainly being due to experimental noise and the nite resolution of the simulations.

Visualization and measurement of internal waves by 'synthetic schlieren'. Part 1. Vertically oscillating cylinder

We present measurements of the density and velocity fields produced when an oscillating circular cylinder excites internal gravity waves in a stratified fluid. These measurements are obtained using a novel, non-intrusive optical technique suitable for determining the density fluctuation field in temporally evolving flows which are nominally two-dimensional. Although using the same basic principles as conventional methods, the technique uses digital image processing in lieu of large and expensive parabolic mirrors, thus allowing more flexibility and providing high sensitivity: perturbations of the order of 1% of the ambient density gradient may be detected. From the density gradient field and its time derivative it is possible to construct the perturbation fields of density and horizontal and vertical velocity. Thus, in principle, momentum and energy fluxes can be determined. In this paper we examine the structure and amplitude of internal gravity waves generated by a cylinder oscillating vertically at different frequencies and amplitudes, paying particular attention to the role of viscosity in determining the evolution of the waves. In qualitative agreement with theory, it is found that wave motions characterized by a bimodal displacement distribution close to the source are attenuated by viscosity and eventually undergo a transition to a unimodal displacement distribution further from the source. Close quantitative agreement is found when comparing our results with the theoretical ones of Hurley & Keady (1997). This demonstrates that the new experimental technique is capable of making accurate measurements and also lends support to analytic theories. However, theory predicts that the wave beams are narrower than observed, and the amplitude is significantly under-predicted for low-frequency waves. The discrepancy occurs in part because the theory neglects the presence of the viscous boundary layers surrounding the cylinder, and because it does not take into account the effects of wave attenuation resulting from nonlinear wave–wave interactions between the upward and downward propagating waves near the source.

Decay of rotating turbulence: some particle tracking experiments

Recent development of measurement techniques based on particle image velocimetry (PIV) are enabling more detailed measurements to be made over extended regions of a flow than have been previously possible. These techniques are of particular value for turbulent flows where the structures present within such flows are incompletely understood and are not readily accessible to traditional measurement techniques. Unfortunately the considerable processing time and specialised equipment required with most PIV techniques limits their applicability when ensemble statistics are required for an evolving turbulent flow. This paper reports on the development and application of an efficient, fully automated particle tracking system. The system was developed as part of a study of the decay of turbulence in a rotating environment. Ensemble descriptions of the temporally evolving flow were required over an extended measurement domain. For each set of parameters particles were tracked with a sampling frequency of 12.5Hz over 60 seconds for 25 realisations. Typically 350 particles were identified and tracked at each time step. Processing speeds in the region ten to fifteen sample images per minute were achieved using a PC/AT compatible computer. The results of the experiments were found to be in broad agreement with previous investigations. However it was found that the method of generating the initial turbulent flow had a profound affect on the subsequent evolution due to the forcing of a strong, large scale systematic flow.

Rayleigh-Taylor instability: experiments with image analysis

Rayleigh-Taylor instability is investigated in the laboratory using a simple apparatus of novel design to set up the unstable initial conditions. Visualisation techniques give a qualitative view of the development of the instability. Quantitative measurements are obtained through digital image analysis. Of primary interest is the evolution of the velocity field. Measurements are made over a two dimensional slice of the flow using an efficient, high-resolution particle tracking technique. This technique is described and its strengths and limitations are discussed in comparison with traditional measurement techniques. The visualisations and velocity measurements are compared with the experi- mental and numerical results of previous workers.

Dispersion mechanisms in a street canyon

In this article, we investigate experimentally and analytically the dispersion mechanisms of a passive tracer in a two-dimensional model of a street canyon. The principal concern is the concentration transfer between the street and the external flow. In contrast to previous studies, the mass fluxes are not only inferred from mean concentration measurements but also directly measured thanks to a Particle Tracking Velocimetry technique. Visualizations of the evolution of the concentration field show the role of the shear layer at the top of the street canyon. Analytical transfer and dispersion models are derived, demonstrating the importance of external turbulence properties on the transfer. Those models are in excellent agreement with the measurements. The results presented in this article strongly suggest that the transfer in a street canyon does depend on the structure of the incoming turbulence, i.e. on the local stability conditions and on the upwind buildings.

Two-layer hydraulics : a functional approach

A new approach for investigating two-layer hydraulic exchange flows in channels is introduced. The approach is based on the functional formalism of Gill (1977) and applied to the flow through a contraction in width and to flow over a simple sill. The sill geometry is an extension of that looked at by earlier workers, in particular Farmer & Armi (1986) who used a Froude-number-plane approach. In the present paper a simple relationship between the composite Froude number and the hydraulic functional is derived, though the functional approach may also be applied to channels where a Froude number is not readily defined. The ability to trace roots of this functional from one reservoir to the other is a prerequisite for the flow to be realizable. Two hydraulic transitions are required for the flow to be fully controlled and the exchange flow rate to be maximal. If only one hydraulic transition is present, the flow is governed by the conditions in one of the reservoirs and the exchange flow rate is found to be submaximal. The flow along a channel is found to be very sensitive to small departures from symmetry about a horizontal plane. The response of the interface to the introduction of a net (barotropic) flow is found to be a discontinuous function of the strength of the forcing for some range of sill heights.

Structure formation in homogeneous freely decaying rotating turbulence

One of the most striking features of rotating turbulence is the inevitable appearance of large-scale columnar structures. Whilst these structures are frequently observed, the processes by which they are created are still poorly understood. In this paper we consider the emergence of these structures from freely decaying, rotating turbulence with Ro ∼ 1. Our study follows the conjecture by Davidson, Staplehurst & Dalziel ( J. Fluid Mech. , vol. 557, 2006, p. 135) that the structure formation may be due to linear inertial wave propagation, which was shown to be consistent with the growth of columnar eddies in inhomogeneous turbulence. Here we extend that work and consider the case of homogeneous turbulence. We describe laboratory experiments where homogeneous turbulence is created in a rotating tank. The turbulence is generated with Ro ∼ 1, and as the energy decays, the formation of columnar vortices is observed. The axial growth of these columnar structures is then measured using two-point correlations and in all cases the results are consistent with structure formation via linear inertial wave propagation. In particular, we obtain a self-similar collapse of the two-point correlations when the axial coordinate is normalized by Ω tb , where b is a measure of the integral scale in the horizontal plane and Ω is the rotation rate. Although our results do not exclude the possibility of significant nonlinear dynamics, they are consistent with the conjecture of Davidson et al . (2006) that linear dynamics play a strong guiding hand in structure formation.

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 (

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

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