Graeme Hocking

The viscous Boussinesq equations are used to simulate the unsteady flow from an elevated source of a plume of heavier fluid into a lighter fluid. The solution is obtained with a spectral method. The source is located at some height above the bottom of a vertically confined layer and the flow is two dimensional. Results indicate that the small, initially circular bubble of heavier fluid expands until the top reaches some height above the source, after which it levels off and starts to flow downwards in a vertical plume before spreading horizontally. We discuss the results for different values of Reynolds number, flow rate, and density differential. It is found that there are several different plume types, the behaviours of which depend on the flow rate. At low flows, the plume forms an inverted mushroom shape, some parts of which may separate as it falls downward. At moderate flow rates, the inverted mushroom plume remains connected until it reaches the base, after which it spreads horizontally as a gravity current. Finally, at large flow rates, the central blob expands outward until it hits the bottom, after which it spreads horizontally. When the Reynolds number is small and viscosity is relatively high, the interface between the expanding region and the ambient fluid is very stable and exhibits few deviations and very little mixing, but as the Reynolds number increases, spirals begin to form around the edges, thus, enhancing the mixing.

The unsteady flow generated by a line source that is located at an arbitrary location beneath the free surface of a fluid of finite depth is considered when there is vertical barrier located nearby. The surface may have surface tension. The barrier is shown to have a significant effect on the wave height generated at the barrier and the outward travelling bore generated by the initiation of the flow. Simulations of free surface flows are very difficult due to the formation of curvature singularities on the surface. The method employed in this work does not appear to have these difficulties and the solutions can be computed almost up until the moment the wave breaks, including in some cases a significant spill at the front. A linearized solution and a fully nonlinear solution are presented and the results compared and discussed.

Rogue Waves are large waves that appear “out of nowhere” in the open ocean and in near-shore waters. The Mathematics in Industry Study Group organized by Professor David Mason at the University of Witwatersrand in January, 2024, considered these waves due to their frequency at locations off the South African coast. There are a number of quite sophisticated models of these waves, but in this work we try to reproduce something like a rogue wave with a very simple free surface model, and in particular consider a wave generated in a circular tank by wave activity at the outer edge. Linear and nonlinear equations are derived and integrated to obtain the wave activity due to different generation mechanisms. The results indicate that a relatively simple model can produce something that approaches the form of a rogue wave.

The human retina is supported by two distinct vascular systems: the highly vascular choroid located behind the retina, and the retinal vascular system, which is designed to minimally disrupt the light path. The avascular retinal layer, situated between these two layers, relies on the diffusion of metabolites through the tissue as it has no circulation. Diseases affecting the microvasculature, such as diabetes and hyper-tension can threaten oxygen supply to these layers, potentially leading to loss of sight. Accurately modeling retinal blood flow is crucial for understanding retinal oxygen supply and the complications arising from systemic vascular diseases. In this paper, we consider a model of just the choroid and avascular layer assuming axisymmetric flow and diffusion, identifying the dispersion pattern from the central region outward. This model captures several significant features of the exchange process and highlights the effects that must be considered in developing more sophisticated models and interpreting experimental results.

Six patents were secured by E. H. Lanier from 1930 to 1933 for aeroplane designs that were intended to be exceptionally stable. A feature of five of these was a flow-induced “vacuum chamber” which was thought to provide superior stability and increased lift compared to typical wing designs. Initially, this chamber was in the fuselage, but later designs placed it in the wing by replacing a section of the upper skin of the wing with a series of angled slats. We report upon an investigation of the Lanier wing design using inviscid aerodynamic theory and viscous numerical simulations. This took place at the 2005 Australia–New Zealand Mathematics-in-Industry Study Group. The evidence from this investigation does not support the claims but, rather, suggests that any improvement in lift and/or stability seen in the few prototypes that were built was, most probably, due to thicker airfoils than were typical at the time.

The effects of apparatus-induced dispersion on nonuniform, density-dependent flow in a cylindrical soil column were investigated using a finite-element model. To validate the model, the results with an analytical solution and laboratory column test data were analysed. The model simulations confirmed that flow nonuniformities induced by the apparatus are dissipated within the column when the distance to the apparatus outlet exceeds 3R/2, where R represents the radius of the cylindrical column. Furthermore, the simulations revealed that convergent flow in the vicinity of the outlet introduces additional hydrodynamic dispersion in the soil column apparatus. However, this effect is minimal in the region where the column height exceeds 3R/2. Additionally, it is found that an increase in the solution density gradient during the solute breakthrough period led to a decrease in flow velocity, which stabilized the flow and ultimately reduced dispersive mixing. Overall, this study provides insights into the behaviour of apparatus-induced dispersion in nonuniform, density-dependent flow within a cylindrical soil column, shedding light on the dynamics and mitigation of flow nonuniformities and dispersive mixing phenomena.

The unsteady flow generated when a line sink is located at an arbitrary location beneath the free surface of a fluid of finite depth is considered. Interest is in the evolution to a steady state flow, splashing flows and the drawdown of the surface. The location of the sink away from, but near to, a vertical barrier leads to some interesting new surface flows due to interactions and reflection, some of which have a profound impact on the nature of the critical drawdown flow parameters. Interesting new solutions with a delayed drawdown that are initiated by a reflected wave are presented. A linearized solution and a fully nonlinear solution are presented and the results discussed. Regions of steady, unsteady and supercritical flow are identified.

The two-dimensional, steady flow of an inviscid fluid induced by a line sink located near a vertical wall in a region of infinite depth is computed. The effects of surface tension are investigated. The solution in the limit of small Froude number is obtained analytically, and numerically for the nonlinear problem. The asymptotic solution is found to have a property that if the horizontal location of the sink, \(x_\mathrm{{s}} < 1\), there is only one stagnation point on the surface, at the wall. However, if the horizontal location \(x_{\mathrm{s}} > 1\), a second stagnation point forms on the free surface. Numerical solution for the nonlinear problem confirms these properties. The effect of moving the sink horizontally has also been considered. The maximum Froude numbers at which steady solutions exist are computed and compared with the previous work.

The flow of a jet of liquid emanating from an elevated, angled slot and impacting on a horizontal wall is considered. An exact solution for high flow speeds is computed, and a numerical technique is used to compute solutions for a range of slot widths, entry angles and separation heights as the flow rate decreases. It is shown that for a given height and slot width there is a minimum flow rate beneath which no steady solutions exist. The cause of this minimum flow is the formation of a stagnation point at the upper exit from the slot, but the height is also a factor in determining the flow rate at which this occurs. The proportion of outflow to the left and right along the horizontal surface for different flow rates and angles is computed.

A spectral method is detailed to simulate the viscous Boussinesq flow of a plume of heavy fluid emanating from a line source on the bottom of a channel. A small initially semi-circular “bubble” grows for small time, rising up to some height before it starts to flow outwards horizontally. We discuss the results for different values of Reynolds number, flow rate and density differential, considering the viscous and inertial effects.