M. R. Flynn

Underwater breathing: the mechanics of plastron respiration

The rough, hairy surfaces of many insects and spiders serve to render them water-repellent; consequently, when submerged, many are able to survive by virtue of a thin air layer trapped along their exteriors. The diffusion of dissolved oxygen from the ambient water may allow this layer to function as a respiratory bubble or ‘plastron’, and so enable certain species to remain underwater indefinitely. Maintenance of the plastron requires that the curvature pressure balance the pressure difference between the plastron and ambient. Moreover, viable plastrons must be of sufficient area to accommodate the interfacial exchange of O 2 and CO 2 necessary to meet metabolic demands. By coupling the bubble mechanics, surface and gas-phase chemistry, we enumerate criteria for plastron viability and thereby deduce the range of environmental conditions and dive depths over which plastron breathers can survive. The influence of an external flow on plastron breathing is also examined. Dynamic pressure may become significant for respiration in fast-flowing, shallow and well-aerated streams. Moreover, flow effects are generally significant because they sharpen chemical gradients and so enhance mass transfer across the plastron interface. Modelling this process provides a rationale for the ventilation movements documented in the biology literature, whereby arthropods enhance plastron respiration by flapping their limbs or antennae. Biomimetic implications of our results are discussed.

Formation and stability of oxygen-rich bubbles that shape photosynthetic mats

Gas release in photic-zone microbialites can lead to preservable morphological biosignatures. Here, we investigate the formation and stability of oxygen-rich bubbles enmeshed by filamentous cyanobacteria. Sub-millimetric and millimetric bubbles can be stable for weeks and even months. During this time, lithifying organic-rich laminae surrounding the bubbles can preserve the shape of bubbles. Cm-scale unstable bubbles support the growth of centimetric tubular towers with distinctly laminated mineralized walls. In environments that enable high photosynthetic rates, only small stable bubbles will be enclosed by a dense microbial mesh, while in deep waters extensive microbial mesh will cover even larger photosynthetic bubbles, increasing their preservation potential. Stable photosynthetic bubbles may be preserved as sub-millimeter and millimeter-diameter features with nearly circular cross-sections in the crests of some Proterozoic conical stromatolites, while centrimetric tubes formed around unstable bubbles provide a model for the formation of tubular carbonate microbialites that are not markedly depleted in (13)C.

Low-mode Internal Tide Generation by Topography: An Experimental and Numerical Investigation

We analyse the low-mode structure of internal tides generated in laboratory experiments and numerical simulations by a two-dimensional ridge in a channel of finite depth. The height of the ridge is approximately half of the channel depth and the regimes considered span sub- to supercritical topography. For small tidal excursions, of the order of 1 % of the topographic width, our results agree well with linear theory. For larger tidal excursions, up to 15 % of the topographic width, we find that the scaled mode 1 conversion rate decreases by less than 15 %, in spite of nonlinear phenomena that break down the familiar wave-beam structure and generate harmonics and inter-harmonics. Modes two and three, however, are more strongly affected. For this topographic configuration, most of the linear baroclinic energy flux is associated with the mode 1 tide, so our experiments reveal that nonlinear behaviour does not significantly affect the barotropic to baroclinic energy conversion in this regime, which is relevant to large-scale ocean ridges. This may not be the case, however, for smaller scale ridges that generate a response dominated by higher modes.

Internal wave excitation by a vertically oscillating sphere

The properties of waves generated by a vertically oscillating sphere in a uniformly stratified fluid are examined both theoretically and experimentally. Existing predictions for the wave amplitude and phase structure are modified to account fo rt he effects of viscous attenuation. As with waves generated by an oscillating cylinder, the main effect of attenuation is to broaden the two peaks of the amplitude envelope on either flank of the wave beam so that far from the sphere the wave beam exhibits a single peak with a maximum along the centreline. The transition distance from bimodal to unimodal wave beam structure is shown to occur closer to the source than the corresponding distance calculated for the oscillating circular cylinder. For laboratory experiments, a recently developed ‘synthetic schlieren’ method is adapted so that quantitative measurements may be made of an axisymmetric wave field. This non-intrusive technique allows us to evaluate the amplitude of the waves everywhere in space and time. Experiments are performed to examine the amplitude of waves generated by small and large spheres oscillating with a range of amplitudes and frequencies. The wave amplitude is found to scale linearly with the oscillation amplitude A for A/a as large as 0.27, where a is the radius of the sphere. Generally good agreement between theory and experiment is found for the small sphere experiments. However, the theory overpredicts both the amplitude and the bimodal-to-unimodal transition distance for waves generated by the large sphere.

Intrusive gravity currents in two-layer fluids

We investigate the dynamics of a gravity current that propagates along the interface of a two-layer fluid. The results of the well-studied symmetric case are reproduced in which the upper- and lower-layer depth of the ambient are equal and the density of the intrusion is the average density of the ambient. In addition, we present the first detailed examination of asymmetric circumstances in which the density of the intrusion differs from the mean density of the ambient and in which the upper- and lower-layer fluid depths are unequal. The general equations derived by J. Y. Holyer & H. E. Huppert (J. Fluid Mech. vol. 100, 1980, pp. 739-767,), which predict the speed and vertical extent of the gravity current head, are re-expressed in a simpler form that employs the Boussinesq approximation. Approximate analytic solutions are determined using perturbation theory. The predictions are compared with the results of laboratory experiments. We find excellent agreement if the density of the gravity current is the average of the upper- and lower-layer densities weighted by the respective depths of the two layers. However, exact theory significantly underpredicts the gravity current speeds if the current density differs from this weighted-mean average. The discrepancy is attributed to the generation of waves that lead and trail the gravity current head. Empirical support for this assertion is provided through an examination of the observed wave characteristics.

Intrusive gravity currents and internal gravity wave generation in stratified fluid

The excitation of internal gravity waves by fluid intrusions that propagate along the interface between a uniform upper layer and a uniformly stratified lower layer is examined by way of laboratory experiments. Intrusions are generated using a simple lock-release apparatus. Experiments are conducted in which the density gradient of the uniformly stratified layer, the density jump across the interface and the density difference between the lock fluid and the uniform upper layer are varied. In all cases, the fluid intrusions travelled at a constant speed. The forcing imparted by the generated internal gravity waves did not deform the intrusion head or significantly retard the intrusions rate of forward advance. For a limited range of density parameters, good agreement was obtained between the experimental data and the two-layer analytical theory of J. Y. Holyer & H. E. Huppert ( J. Fluid Mech. vol. 100 (1980), pp. 739–767) which provides estimates for the intrusion speed and depths of penetration into the upper and lower layers. Internal gravity wave excitation is due to the initial collapse of the lock fluid and the forcing imparted by the head of the intrusion. Waves are visualized and their amplitudes measured using ‘synthetic schlieren’. The vertical flux of horizontal momentum due to internal gravity wave excitation is related to measurable properties of the fluid intrusion. This analysis suggests that outflows produced by tall convective storms that travel along the tropopause may excite non-hydrostatic internal gravity waves in the stratosphere whose momentum flux, at least during the transient generation time, is comparable to that of waves generated by topographic forcing.

Natural ventilation in interconnected chambers

Ventilation of adjacent, connected chambers, forced in one chamber by an isolated point source of buoyancy is investigated. There are floor- and ceiling-level external openings in the forced and unforced chambers, respectively, while the partition between the chambers has both a floor- and ceiling-level opening. The flow evolves on the time scale over which the volume flux associated with the plume at the ceiling would fill both chambers. The steady state in the forced chamber is analogous to the single chamber flow described by Linden, Lane-Serff & Smeed (J. Fluid Mech., vol. 212, 1990, p. 309), with a well-mixed buoyant upper layer which is deeper than in the single chamber flow due to the extra pressure drop at the upper interior opening. The steady state in the unforced chamber inevitably exhibits vertical stratification, and depends on the transient flow, all the opening areas, and the relative plan area of the two chambers, as is verified by laboratory experiments. When the upper interior opening is relatively large, the buoyant layer in the unforced chamber is deeper than the buoyant layer in the forced chamber, which contradicts model predictions based on the assumption that the layers are always well-mixed.

Intrusive gravity currents

(Received 5 May 2006 and in revised form 1 August 2006) The speed of a fluid intrusion propagating along a sharp density interface is predicted using conservation of mass, momentum and energy. For the special case in which the intrusion density equals the depth-weighted mean density of the upper and lower layers, the theory of Holyer & Huppert (J. Fluid Mech., vol. 100, 1980, p. 739) predicts that the intrusion occupies one-half the total depth, its speed is one-half the interfacial long-wave speed and the interface ahead of the intrusion remains undisturbed. For all other intrusion densities, the interface is deflected vertically by a long wave that travels ahead of the intrusion and thereby changes the local upstream conditions. In these cases, the conservation equations must be matched to an exact solution of the two-layer shallow water equations, which describe the spatial evolution of the nonlinear wave. We obtain predictions for the intrusion speed that match closely with experiments and numerical simulations, and with a global energy balance analysis by Cheong, Keunen & Linden (J. Fluid Mech., vol. 552, 2006, p. 1). Since the latter does not explicitly include the energetics of the upstream wave, it is inferred that the energy carried by the wave is a small fraction of the intrusion energy. However, the new more detailed model also shows that the kinematic influence of the upstream wave in changing the level of the interface is a critical component of the flow that has previously been ignored.

Plastron induced drag reduction and increased slip on a superhydrophobic sphere

On low contact angle hysteresis superhydrophobic surfaces, droplets of water roll easily. It is intuitively appealing, but less obvious, that when such material is immersed in water, the liquid will flow more easily across its surface. In recent experiments it has been demonstrated that superhydrophobic surfaces with the same high contact angle and low contact angle hysteresis may not, in fact, have the same drag reducing properties. A key performance parameter is whether the surface is able to retain a layer of air (i.e. a plastron) when fully immersed. In this report, we consider an analytical model of Stokes flow (i.e. low Reynolds number, Re, creeping flow) across a surface retaining a continuous layer of air. The system is based on a compound droplet model consisting of a solid sphere encased in a sheathing layer of air and is the extreme limit of a solid sphere with a superhydrophobic surface. We demonstrate that an optimum thickness of air exists at which the drag on this compound object is minimized and that the level of drag reduction can approach 20 to 30%. Physically, drag reduction is caused by the ability of the external flow to transfer momentum across the water–air interface generating an internal circulation of air within the plastron. We also show that the drag experienced by the plastron-retaining sphere can be viewed as equivalent to the drag on a non-plastron retaining sphere, but with the no-slip boundary condition replaced by a slip boundary condition. If the plastron layer becomes too thin, or the liquid-gas interface is rigidified, circulation is no longer possible and drag increases to the value expected for a solid object in direct contact with water. We discuss the implications of this physical understanding in terms of its general applicability to the intelligent design of drag reducing superhydrophobic surfaces at low Re. We emphasize that the length scales and connectivity of surface topography generating superhydrophobicity are also likely to determine whether a plastron is of a suitable size to reduce drag.

The role of diffusion on the interface thickness in a ventilated filling box.

We examine the role of diffusivity, whether molecular or turbulent, on the steady-state stratification in a ventilated filling box. The buoyancy-driven displacement ventilation model of Linden et al. (J. Fluid Mech., vol. 212, 1990, p. 309) predicts the formation of a two-layer stratification when a single plume is introduced into an enclosure with vents at the top and bottom. The model assumes that diffusion plays no role in the development of the ambient buoyancy stratification: diffusion is a slow process and the entrainment of ambient fluid into the plume from the diffuse interface will act to thin the interface resulting in a near discontinuity of density between the upper and lower layers. This prediction has been corroborated by small-scale salt bath experiments; however, full-scale measurements in ventilated rooms and complementary numerical simulations suggest an interface that is not sharp but rather smeared out over a finite thickness. For a given plume buoyancy flux, as the cross-sectional area of the enclosure increases the volume of fluid that must be entrained by the plume to maintain a sharp interface also increases. Therefore the balance between the diffusive thickening of the interface and plume-driven thinning favours a thicker interface. Conversely, the interface thickness decreases with increasing source buoyancy flux, although the dependence is relatively weak. Our analysis presents two models for predicting the interface thickness as a function of the enclosure height, base area, composite vent area, plume buoyancy flux and buoyancy diffusivity. Model results are compared with interface thickness measurements based on previously reported data. Positive qualitative and quantitative agreement is observed.