We simulate high-velocity flow in a self-affine channel with a constant perpendicular opening by solving numerically the Navier–Stokes equations, and analyse the resulting flow qualitatively and quantitatively. At low velocity, i.e. vanishing inertia, the effective permeability is dominated by the narrowest constrictions measured perpendicular to the local flow direction and the flow field tends to fill the channel due to the diffusion generated by the viscous term in the Stokes equation. At high velocity (strong inertia), the high-velocity zones of the flow field resemble a narrow tube of essentially constant thickness in the direction of flow, since the transversal diffusion is weak compared to the longitudinal convection. The thickness of the flow tube decreases with Reynolds number. This narrowing in combination with mass balance results in an average velocity in the flow tube which increases faster with Reynolds number than the average velocity in the fracture. In the low-velocity zones, recirculation zones appear and the pressure is almost constant.

The flow tube consists of straight sections. This is due to inertia. The local curvature of the main stream reflects the flow-tube/channel-wall interaction. A boundary layer is formed where the curvature is large. This boundary layer is highly dissipative and governs the large pressure loss (inertial resistance) of the medium. Quantitatively, vanishing, weak and strong inertial flow regimes can be described by the Darcy, weak inertia and Forchheimer flow equations, respectively. We observe a cross-over flow regime from the weak to strong inertia, which extends over a relatively large range of Reynolds numbers.

The dynamic behaviour and stability of a two-dimensional immiscible droplet subject to shear or pressure-driven flow between parallel plates is studied under conditions of negligible inertial and gravitational forces. The droplet is attached to the lower plate and forms two contact lines that are either fixed or mobile. The boundary-integral method is used to numerically determine the flow along and dynamics of the free surface. For surfactant-free interfaces with fixed contact lines, the deformation of the interface is determined for a range of capillary numbers, droplet to displacing fluid viscosity ratios, droplet sizes and flow type. It is shown that as the capillary number or viscosity ratio or size of the droplet increases, the deformation of the interface increases and above critical values of the capillary number no steady shape exists. For small droplets, and at low capillary numbers, shear and pressure-driven flows are shown to yield similar steady droplet shapes. The effect of surfactants is studied assuming a fixed amount of surfactant that is subject to convective–diffusive transport along the interface and no transport to or from the bulk fluids. Increasing the surface Péclet number, the ratio of convective to diffusive transport, leads to an accumulation of surfactant at the downstream end of the droplet and creates Marangoni stresses that immobilize the interface and reduce deformation. The no-slip boundary condition is then relaxed and an integral form of the Navier-slip model is used to examine the effects of allowing the droplet to slip along the solid surface in a pressure-driven flow. For contact angles less than or equal to 90°, a stable droplet spreads along the wall until a steady shape is reached, when the droplet translates across the wall at a constant velocity. The critical capillary number is larger for these droplets compared to those with pinned contact lines. For contact angles greater than 90°, the wetted area between a stable droplet and the wall decreases until a steady shape is reached. The critical capillary number for these droplets is less than that for pinned droplets. Above the critical capillary number the droplet completely detaches for a contact angle of 120°, or part of it is pinched off leaving behind a smaller attached droplet for contact angles less than or equal to 90°.

The effect of Prandtl number on the dynamics of a convective turbulent flow is studied by numerical experiments. In particular, three series of experiments have been performed; in two of them the Rayleigh number spanned about two decades while the Prandtl number was set equal to 0.022 (mercury) and 0.7 (air). In the third series, in contrast, we fixed the Rayleigh number at 6×105 and the Prandtl number was varied from 0.0022 up to 15. The results have shown that, depending on the Prandtl number, there are two distinct flow regimes; in the first (Pr[lsim ]0.35) the flow is dominated by the large-scale recirculation cell that is the most important ‘engine’ for heat transfer. In the second regime, on the other hand, the large-scale flow plays a negligible role in the heat transfer which is mainly transported by the thermal plumes.

For the low-Pr regime a model for the heat transfer is derived and the predictions are in qualitative and quantitative agreement with the results of the numerical simulations and of the experiments. All the hypotheses and the consequences of the model are directly checked and all the findings are consistent with the predictions and with experimental observations performed under similar conditions. Finally, in order to stress the effects of the large-scale flow some counter examples are shown in which the large-scale motion is artificially suppressed.

Large-eddy simulation is used to investigate fully developed turbulent flow in a neutral channel wherein the lower wall is sinusoidal. The numerical results are compared with experimental observations for wave slopes ranging from 0 to 0.628. Particular emphasis is placed on the separated flow induced by a large-amplitude wave. A detailed comparison with the data of Buckles, Hanratty & Adrian (1984) shows generally good agreement. Large-eddy simulation surface pressures are integrated to calculate form drag as a function of wave slope. Drag is found to increase quadratically with slope for small-amplitude waves, with a somewhat slower increase for larger amplitudes. However, comparison with experimental measurements is confounded by uncertainties with the values reported in the literature. An interesting feature characteristic of all wavy-surface simulations is an increase in transverse velocity fluctuations on the wave upslope. Although the precise mechanism responsible is not known, analysis shows it to be associated with temporally persistent vortex-like structures localized near the surface. The magnitude of the fluctuation increase appears to scale quadratically with slope for small-amplitude waves, in contrast to the streamwise fluctuations, which increase linearly.

The sound generated by vortex pairing in axisymmetric jets is determined by direct solution of the compressible Navier–Stokes equations on a computational grid that includes both the near field and a portion of the acoustic far field. At low Mach number, the far-field sound has distinct angles of extinction in the range of 60°–70° from the jet's downstream axis which can be understood by analogy to axisymmetric, compact quadrupoles. As the Mach number is increased, the far-field sound takes on a superdirective character with the dominant sound directed at shallow angles to the jet's downstream axis. The directly computed sound is compared to predictions obtained from Lighthill's equation and the Kirchhoff surface method. These predictions are in good agreement with the directly computed data. The Lighthill source terms have a large spatial distribution in the axial direction necessitating the introduction of a model to describe the source terms in the region downstream of the last vortex pairing. The axial non-compactness of the quadrupole sources must be adequately treated in the prediction method.

The barotropic shear layer in a rotating fluid is studied in a laboratory experiment. Through the rotation of circular sections in the base and lid of a circular tank relative to a background rotation of the entire system, a vertical layer of strong horizontal shear develops, the Stewartson layer. Above a critical shear, the shear layer breaks up through barotropic instability, which is an inertial instability. The flow then develops a string of vortices along the shear zone. It will be shown that the transition from an axisymmetric flow to regular vortices occurs through a Hopf bifurcation. Subsequent transitions to more complex flows, such as modulated vortices, chaos and highly irregular flow, will be presented briefly, while the main points of this paper are the primary instability, steady vortices and their nonlinear dynamics. Among the issues discussed is the sensitivity of the flow to the direction of the differential shear. The experimental data will be used to test the ability of boundary layer theory and quasi-geostrophic theory to predict the onset of instability and the range of unstable wavenumbers.

A model for turbulent skin friction, proposed by Orlandi & Jimenez, involving consideration of quasi-streamwise vortices in the cross-stream plane, is used to study the effect on the skin friction of oscillating the surface beneath the boundary layer in the spanwise direction. Using an exact solution of the Navier–Stokes equations, it is shown that the interaction between evolving, axially stretched, streamwise vortices and a modified Stokes layer on the oscillating surface beneath, leads to reduction in the skin friction, the Reynolds stress and the rate of production of kinetic energy, consistent with predictions based on experiments and direct numerical simulations.

The spreading characteristics of jets from several asymmetric nozzles, and a set of rectangular orifices are compared, covering a jet Mach number range of 0.3–2.0. The effect of ‘tabs’ for a rectangular and a round nozzle is also included in the comparison. Compared to a round jet, the jets from the asymmetric nozzles spread only slightly more at subsonic conditions whereas at supersonic conditions, when ‘screech’ occurs, they spread much more. The dynamics of the azimuthal vortical structures of the jet, organized and intensified under the screeching condition, are thought to be responsible for the observed effect at supersonic conditions. Curiously, the jet from a ‘lobed’ nozzle spreads much less at supersonic condition compared to all other cases; this is due to the absence of screech with this nozzle. Screech stages inducing flapping, rather than varicose or helical, flow oscillation cause a more pronounced jet spreading. At subsonic conditions, only a slight increase in jet spreading with the asymmetric nozzles contrasts previous observations by others. The present results show that the spreading of most asymmetric jets is not much different from that of a round jet. This inference is further supported by data from the rectangular orifices. In fact, jets from the orifices with small aspect ratio (AR) exhibit virtually no increase in the spreading. A noticeable increase commences only when AR is larger than about 10. Thus, ‘shear layer perimeter stretching’, achieved with a larger AR for a given cross-sectional area of the orifice, by itself, proves to be a relatively inefficient mechanism for increasing jet spreading. In contrast, the presence of streamwise vortices or ‘natural excitation’ can cause a significant increase – effects that might explain the observations in the previous investigations. Thus far, the biggest increase in jet spreading is observed with the tabs. This is true in the subsonic regime, as well as in the supersonic regime, in spite of the fact that screech is eliminated by the tabs. The characteristic spreading of the tabbed jets is explained by the induced motion of the tab-generated streamwise vortex pairs. The tabs, however, incur thrust loss; the flow blockage and loss in thrust coefficient, vis-à-vis the spreading increase, are evaluated for various configurations.

We report here visualizations and quantitative measurements of scalar transport, under the influence of rotation, through permeable sediments with an overlying fluid layer. The experimental set-up considered here is a stationary cylinder containing a fluid-saturated porous medium up to its midheight, with supernatant water on top. A rotating lid generates, in the upper fluid region, a flow that partially percolates into the porous layer below. The velocity field in the fluid layer is obtained using particle image velocimetry (PIV). Further, dye transport from the sediment is studied using two different techniques. The first one is positron emission tomography (PET), a non-invasive method which allowed us to ‘see’ through the opaque solid matrix, and to obtain full three-dimensional pictures of dye transport through the sediment. The second one is digital photographic visualization from outside, and subsequent image processing in order to obtain the near-wall dye-washout depth. The experimental data suggest that the temporal evolution of washout depth for different sediments follows near-logarithmic behaviour. This finding is of importance for the a priori estimation of the transport of fluid and other solute substances in sandy aquatic sediments.

This paper describes laboratory experiments on the self-organizing character of geophysical turbulence. The experiments were carried out in a linearly stratified fluid, forced horizontally with sources and sinks around a horizontal ring. The flow was visualized with small particles illuminated by a horizontal light sheet, recorded with a camera and analysed with an advanced particle tracking system. Qualitative and quantitative data, such as flow patterns, velocity and vorticity fields, were obtained. In the experiments, the inverse energy cascade was clearly observed: the flow organized into a single quasi-steady, coherent vortex structure of the largest available scale. This vortex is maintained against diffusion of momentum by entrainment of vorticity from its exterior. In this process, patches of vorticity of the same sign as the core of this large structure intermittently cross the vorticity barrier around the vortex. Patches of opposite vorticity were observed to be effectively blocked by the barrier. The direction of rotation of the vortex was set by a slight bias in the experimental apparatus and could be changed by imposing a small initial circulation in the opposite sense, the magnitude of which suggested a measure for the bias. A detailed study of the effects of changing forcing parameters was carried out. The number of sinks (which play only a passive role) does not affect the flow, whereas the number of sources sets the lengthscale of the forcing and thereby determines the size of the vortices that are created close to the ring, as well as that of the large central vortex that emerges. However, after longer times of forcing, the vortex size also depends on the strength of the forcing. The velocities in the large vortex structure scale with the mean velocity from the sources, and with the square root of their number. Measurements were also taken of the decay of the vortex. After switching off the forcing it quickly becomes axisymmetric and a linear functional relationship is established between the vorticity and streamfunction. The spin-down time was observed to be much shorter than can be accounted for by vertical viscous diffusion alone: initially the short horizontal scale of the vorticity barrier causes a relatively fast decay, whereas at later times the size of the vortex as a whole is important.

The effect of free-surface drift layers on the maximum height that a steady wave can attain without breaking is explored through experiments and numerical simulations. In the experiments, the waves are generated by towing a two-dimensional fully submerged hydrofoil at constant depth, speed and angle of attack. The drift layer is generated by towing a plastic sheet on the water surface ahead of the hydrofoil. It is found that the presence of this drift layer (free-surface wake) dramatically reduces the maximum non-breaking wave height and that this wave height correlates well with the surface drift velocity. In the simulations, the inviscid two-dimensional fully nonlinear Euler equations are solved numerically. Initially symmetric wave profiles are superimposed on a parallel drift layer whose mean flow characteristics match those in the experiments. It is found that for large enough initial wave amplitudes a bulge forms at the crest on the forward face of the wave and the vorticity fluctuations just under the surface in this region grow dramatically in time. This behaviour is taken as a criterion to indicate impending wave breaking. The maximum non-breaking wave elevations obtained in this way are in good agreement with the experimental findings.

We consider the critical Weber number (Wec≡ ρV20D/σ) at which the transition from dripping to jetting occurs when a Newtonian liquid of density ρ and surface tension σ is injected with a velocity V0 through a tube of diameter D downward into stagnant air, under gravity g. We extend Taylor's (1959) model for the recession speed of a free edge, and obtain in the inviscid limit an exact solution which includes gravity and inertia effects. This solution provides a criterion for the transition which is shown to occur at a critical Weber number

formula here

where Bo and Boo are the Bond numbers (Bo≡[ρgD2/(2σ)]1/2), respectively based on the inside and outside diameter of the tube, and K is a constant equal to 0.37 for the case of water injected in air. This critical Weber number is shown to be in good agreement with existing experimental values as well as with new measurements performed over a wide range of Bond numbers.

Motivated by the study of blood flow in the coronary arteries, this paper examines the flow of an incompressible Newtonian fluid in a tube of time-dependent curvature. The flow is driven by an oscillatory pressure gradient with the same dimensionless frequency, α, as the curvature variation. The dimensionless governing parameters of the flow are α, the curvature ratio δ0, a secondary streaming Reynolds number Rs and a parameter Rt representing the time-dependence of curvature. We consider the parameter regime δ0<Rt<1 (Rs and α remain O(1) initially) in which the effect of introducing time-dependent curvature is to perturb the flow driven by an oscillatory pressure gradient in a fixed curved tube. Flows driven by low- and high-frequency pressure gradients are then considered. At low frequency (δ0<Rt<α<1) the flow is determined by using a sequence of power series expansions (Rs=O(1)). At high frequency (δ0<Rt<1/α2<1) the solution is obtained using matched asymptotic expansions for the region near the wall (Stokes layer) and the region away from the wall in the interior of the pipe. The behaviour of the flow in the interior is then determined at both small and intermediate values of Rs. For both the low and high frequency cases, we find the principal corrections introduced by the time-varying curvature to the primary and secondary flows, and hence to the wall shear stress. The physiological application to flow in the coronary arteries is discussed.

The polyelectrolyte layer coating mammalian cells, known as the glycocalyx, is important in communicating flow information to the cell. In this paper, the layer is modelled as a semi-infinite, doubly periodic array of parallel charged cylinders. The electric potential and ion distributions surrounding such an array are found using the Poisson–Boltzmann equation and an iterative domain decomposition technique. Similar methods are used to calculate Stokes flows, driven either by a shear at infinity or by an electric field, parallel or transverse to the cylinders. The resulting electric streaming currents due to flow over endothelial cells, and the electrophoretic mobilities of red blood cells are deduced as functions of polymer concentration and electrolyte molarity. It is shown that only the top portion of the layer is important in these effects.