Evolving viscoplastic flows upon slopes are an important idealization of many flows in a variety of geophysical situations where yield stress is thought to play a role. For such models, asymptotic expansions suitable for slowly moving shallow fluid layers (lubrication theory) reduce the governing equations to a simpler problem in terms of the fluid thickness. We consider the version of the theory for fluids described by the Herschel–Bulkley constitutive law, and provide a variety of solutions to the reduced equation, both numerical and analytical. For extruded inclined domes, we derive the characteristic temporal behaviour of measures of the dome's dimensions.

Flow instabilities leading to the formation of streamwise vortices in a natural convection boundary layer over a heated inclined plate submerged in a water tank are manipulated using spanwise arrays of surface-mounted heating elements. The flow over the plate is driven by a two-ply surface heater comprised of a uniform, constant- heat flux heater and a mosaic of 32 × 12 individually controlled heating elements that are used as control actuators. Surface temperature distributions are measured using liquid crystal thermography and the fluid velocity in cross-stream planes is measured using particle image velocimetry (PIV). Time-invariant spanwise-periodic excitation over a range of spanwise wavelengths leads to the formation of arrays of counter-rotating streamwise vortex pairs and to substantial modification of the surface temperature and heat transfer. The increase in surface heat transfer is accompanied by increased entrainment of ambient fluid and, as a consequence, higher streamwise flowrate. Subsequent spanwise-periodic merging of groups of vortices farther downstream retards the streamwise increase of the surface heat transfer rate. Finally, the suppression of small-amplitude spanwise disturbances by linear cancellation is demonstrated.

Internal hydraulic jumps in two-layer flows are studied, with particular emphasis on their role in entrainment and mixing. For highly entraining internal jumps, a new closure is proposed for the jump conditions. The closure is based on two main assumptions: (i) most of the energy dissipated at the jump goes into turbulence, and (ii) the amount of turbulent energy that a stably stratified flow may contain without immediately mixing further is bounded by a measure of the stratification. As a consequence of this closure, surprising bounds emerge, for example on the amount of entrainment that may take place at the location of the jump. These bounds are probably almost achieved by highly entraining internal jumps, such as those likely to develop in dense oceanic over flows. The values obtained here are in good agreement with the existing observations of the spatial development of oceanic downslope currents, which play a crucial role in the formation of abyssal and intermediate waters in the global ocean.

Rayleigh–Bloch surface waves arise in many physical contexts including water waves and acoustics. They represent disturbances travelling along an infinite periodic structure. In the absence of any existence results, a number of authors have previously computed such modes for certain specific geometries. Here we prove that such waves can exist in the absence of any incident wave forcing for a wide class of structures.

This paper investigates the propagation of an air finger into a fluid-filled, axially uniform tube of elliptical or rectangular cross-section with transverse length scale a and aspect ratio α. Gravity is assumed to act parallel to the tube's axis. The problem is studied numerically by a finite-element-based direct solution of the free-surface Stokes equations.

In rectangular tubes, our results for the pressure drop across the bubble tip, Δp, are in good agreement with the asymptotic predictions of Wong et al. (1995b) at low values of the capillary number, Ca (ratio of viscous to surface-tension forces). At larger Ca, Wong et al.'s (1995b) predictions are found to underestimate Δp. In both elliptical and rectangular tubes, the ratio Δp(α)/Δp(α = 1) is approximately independent of Ca and thus equal to the ratio of the static meniscus curvatures.

In non-axisymmetric tubes, the air-liquid interface develops a noticeable asymmetry near the bubble tip at all values of the capillary number. The tip asymmetry decays with increasing distance from the bubble tip, but the decay rate becomes very small as Ca increases. For example, in a rectangular tube with α = 1.5, when Ca = 10, the maximum and minimum finger radii still differ by more than 10% at a distance 100a behind the finger tip. At large Ca the air finger ultimately becomes axisymmetric with radius r∞. In this regime, we find that r∞ in elliptical and rectangular tubes is related to r∞ in circular and square tubes, respectively, by a simple, empirical scaling law. The scaling has the physical interpretation that for rectangular and elliptical tubes of a given cross-sectional area, the propagation speed of an air finger, which is driven by the injection of air at a constant volumetric rate, is independent of the tube's aspect ratio.

For smaller Ca (Ca < Ca), the air finger is always non-axisymmetric and the persisting draining flows in the thin film regions far behind the bubble tip ultimately lead to dry regions on the tube wall. Ca increases with increasing α and for α > αˆ dry spots will develop on the tube walls at all values of Ca.

Thermal convection in a rotating spherical shell with free-slip boundaries can excite a dominant mean zonal flow in the form of differentially rotating cylinders concentric to the principal rotation axis. This process is studied numerically for Prandtl numbers of order 1, Ekman numbers in the range E = 3 × 10−4−10−5, and Rayleigh numbers up to 100× critical. Small-scale convection transfers kinetic energy into the mean zonal flow via Reynolds stresses. For low Ekman number and high Rayleigh number, the force balance is predominantly among the Coriolis, inertial and buoyancy forces, and viscosity plays a minor role. A modified Rayleigh number Ra* is introduced, which does not depend on viscosity or thermal diffusivity, and asymptotic scaling laws for the dependence of various properties on Ra* in the limit of negligible viscosity (E → 0) are estimated from the numerical results. The ratio of kinetic energy in the zonal flow to that in the non-zonal (convective) flow increases strongly with Ra* at low supercritical Rayleigh number, but drops at high values of Ra*. This is probably caused by the gradual loss of geostrophy of the convective columns and a corresponding decorrelation of Reynolds stresses. Applying the scaling laws to convection in the molecular hydrogen envelopes of the large gas planets predicts the observed magnitude of the zonal winds at their surfaces.

Results of an experimental and computational study of the flow in an annular region bounded by stationary inner and outer cylinders and driven by the rotation of the floor are presented. The top is a flat air/water interface, covered by an insoluble monolayer. We develop a technique to determine the surface shear viscosity from azimuthal velocity measurements at the interface which extends the range of surface shear viscosity that can be measured using a deep-channel viscometer in the usual Stokes flow regime by exploiting flow inertia. A Navier–Stokes-based model of bulk flow coupled to a Newtonian interface that has surface shear viscosity as the only interfacial property is developed. This is achieved by restricting the flow to regimes where the surface radial velocity vanishes. The use of inertia results in an improved signal-to-noise ratio of the azimuthal velocity measurements by an order of magnitude beyond that available in the Stokes flow limit. Measurements on vitamin K1 and stearic acid monolayers were performed, and their surface shear viscosities over a range of concentrations are determined and found to be in agreement with data in the literature.

The work presented extends previous research on linear controllers in temporal channel flow to spatially evolving boundary layer flow. The flows studied are those on an infinite swept wedge described by the Falkner–Skan–Cooke (FSC) velocity profiles, including the special case of the flow over a flat plate. These velocity profiles are used as the base flow in the Orr–Sommerfeld–Squire equations to compute the optimal feedback control through blowing and suction at the wall utilizing linear optimal control theory. The control is applied to a parallel FSC flow with unstable perturbations. Through an eigenvalue analysis and direct numerical simulations (DNS), it is shown that instabilities are stabilized by the controller in the parallel case. The localization of the convolution kernels for control is also shown for the FSC profiles.

Assuming that non-parallel effects are small a technique is developed to apply the same controllers to a DNS of a spatially evolving flow. The performance of these controllers is tested in a Blasius flow with both a Tollmien–Schlichting (TS) wave and an optimal spatial transiently growing perturbation. It is demonstrated that TS waves are stabilized and that transient growth is lowered by the controller. Then the control is also applied to a spatial FSC flow with unstable perturbations leading to saturated cross-flow vortices in the uncontrolled case. It is demonstrated that the linear controller successfully inhibits the growth of the cross-flow vortices to a saturated level and thereby delays the possibility of transition through secondary instabilities. It is also demonstrated that the controller works for relatively high levels of nonlinearity, and for stationary as well as time-varying perturbations.

Nonlinear dynamics of the concentric, two-phase flow of two immiscible fluids in a circular tube of variable cross-section is studied for parameter values where the steady core–annular flow (CAF) is linearly unstable. The simulations are based on a pseudo-spectral numerical method. They are carried out assuming axial symmetry, that the total flow rate remains constant and that all dependent variables are periodic in the axial direction, which includes the minimum necessary number of repeated units so that the obtained solution is independent of this number. The time integration originates with the numerically computed steady CAF or the steady CAF seeded with either the most unstable mode or random small disturbances. Only a limited number of the most interesting cases are presented. For the most part, the values of the majority of the dimensionless parameters are such that oil flows in the centre of the tube driven by an applied pressure gradient against gravity, whereas water is flowing in the annulus. It is shown that, whereas the steady (unstable) solution may indicate that the heavier water flows countercurrently with respect to the oil, the time periodic (observable) solution may indicate the same, albeit at a much smaller core flow rate or that concurrent flow occurs. This is due to the water being trapped between the large-amplitude interfacial waves that are generated and being convected by the oil. It is also shown that increasing the inverse Weber number increases the wave amplitude to the point that the flow of the core fluid may become discontinuous with a mechanism that depends on the viscosity ratio between the two fluids. Increasing the amplitude of the sinusoidal variation of the tube leads to a combination of travelling and standing waves, which interact to produce a time periodic solution with a long period associated with the time it takes the travelling wave to travel through the computational domain and a second much shorter period that is related to their interaction time. Qualitative agreement has been obtained upon comparing our numerical simulations with limited experimental reports, even though the experimental conditions were not identical to those in our model.

A new analytical model is developed for the equilibrium range of the spectrum of wind-forced ocean surface gravity waves. We first show that the existing model of Phillips (1985) does not satisfy overall momentum conservation at high winds. This constraint is satisfied by applying recent understanding of the wind forcing of waves. Waves exert a drag on the air flow so that they support a fraction of the applied wind stress, which thus leaves a smaller turbulent stress near the surface to force growth of shorter wavelength waves. Formulation of the momentum budget accounting for this sheltering constrains the overall conservation of momentum and leads to a local turbulent stress that reduces as the wavenumber increases. This local turbulent stress then forces wind-induced wave growth. Following Phillips (1985), the wind sea is taken to be a superposition of linear waves, and equilibrium is maintained by a balance between the three sources and sinks of wave action.

These assumptions lead to analytical formulae for the local turbulent stress and the degree of saturation, B(k), of waves in the equilibrium range. We identify a sheltering wavenumber, ks, over which the local turbulent stress is significantly reduced by longer waves. At low wavenumbers or at low winds, when k [Lt ] ks, the sheltering is weak and B(k) has a similar form to the model of Phillips (1985). At higher wavenumbers or at higher winds, ks, B(k) makes a transition to being proportional to k0. The additional constraint of conservation of momentum also yields a formula for the coefficient that appears in the solution for B(k). The spectra for mature seas are calculated from the model and are shown to agree with field observations. In particular, our model predicts more realistic spectral levels toward the high wavenumber limit compared to the previous model of Phillips (1985).

We suggest that the model may explain the overshoot phenomena observed in the spectral energy levels as the fetch increases.

It is recognized that flow in the melt can have a profound influence on the dynamics of a solidifying interface and hence on the quality of the solidified material. To better understand the effect of fluid flow on the interface morphological stability and on the cellular and dendritic growth, directional solidification experiments were carried out in a horizontally placed Hele-Shaw cell with and without externally imposed parallel shear flow. The specimen material used was SCN–1.0 Wt% acetone. The experiment shows that the transient parallel flow has a stabilizing effect on the planar interface by damping the existing initial perturbations. The left–right symmetry of crystal cells was broken by the parallel flow, with cells tilting toward the incoming flow direction. The tilting angle increased with the velocity ratio. The secondary dendrites were found to either not appear or appear much later on the downstream side of the crystal cells. The wavelengths of the initial perturbations and of the cellular interface were insensitive to the imposed flow.

A linear stability analysis as well as nonlinear simulations are performed in order to analyse the coupling between the directional solidification of a binary alloy and the flow in its melt. An incompressible, potential flow model is assumed, whose validity is tested through comparisons with the accompanying experiments of Zhang & Maxworthy (2002) in a Hele-Shaw cell. The linear stability analysis predicts that a uniform flow parallel to the interface reduces the growth rates of directional solidification instabilities. In addition, the dominant wavelength is shifted to larger values by the flow, and a small propagation velocity in the downstream direction is observed. These findings are confirmed by the nonlinear simulations as well. While the overall stabilization is confirmed by the experiments, the predicted values of the dominant wavenumber and its growth rate are too high by factors of two and four, respectively. These differences are attributed to the existence of a velocity boundary layer in the melt, which strongly affects the lateral solute transport.

The paper is concerned with the interaction between the boundary layer on a smooth body surface and the outer inviscid compressible flow in the vicinity of a sonic point. First, a family of local self-similar solutions of the Kármán–Guderley equation describing the inviscid flow behaviour immediately outside the interaction region is analysed; one of them was found to be suitable for describing the boundary-layer separation. In this solution the pressure has a singularity at the sonic point with the pressure gradient on the body surface being inversely proportional to the cubic root dpw/dx ∼ (−x)−1/3 of the distance (−x) from the sonic point. This pressure gradient causes the boundary layer to interact with the inviscid part of the flow. It is interesting that the skin friction in the boundary layer upstream of the interaction region shows a characteristic logarithmic decay which determines an unusual behaviour of the flow inside the interaction region. This region has a conventional triple-deck structure. To study the interactive flow one has to solve simultaneously the Prandtl boundary-layer equations in the lower deck which occupies a thin viscous sublayer near the body surface and the Kármán–Guderley equations for the upper deck situated in the inviscid flow outside the boundary layer. In this paper a numerical solution of the interaction problem is constructed for the case when the separation region is entirely contained within the viscous sublayer and the inviscid part of the flow remains marginally supersonic. The solution proves to be non-unique, revealing a hysteresis character of the flow in the interaction region.

The modal system describing nonlinear sloshing with inviscid flows in a rectangular rigid tank is revised to match both shallow fluid and secondary (internal) resonance asymptotics. The main goal is to examine nonlinear resonant waves for intermediate depth/breadth ratio 0.1 [lsim ] h/l [lsim ] 0.24 forced by surge/pitch excitation with frequency in the vicinity of the lowest natural frequency. The revised modal equations take full account of nonlinearities up to fourth-order polynomial terms in generalized coordinates and h/l and may be treated as a modal Boussinesq-type theory. The system is truncated with a high number of modes and shows good agreement with experimental data by Rognebakke (1998) for transient motions, where previous finite depth modal theories failed. However, difficulties may occur when experiments show significant energy dissipation associated with run-up at the walls and wave breaking. After reviewing published results on damping rates for lower and higher modes, the linear damping terms due to the linear laminar boundary layer near the tank's surface and viscosity in the fluid bulk are incorporated. This improves the simulation of transient motions. The steady-state response agrees well with experiments by Chester & Bones (1968) for shallow water, and Abramson et al. (1974), Olsen & Johnsen (1975) for intermediate fluid depths. When h/l [lsim ] 0.05, convergence problems associated with increasing the dimension of the modal system are reported.

The objective of the present work is to investigate experimentally the secondary instability of the one-dimensional voidage waves occurring in two-dimensional liquid- fluidized beds and to examine the physical origin of bubbles, i.e. regions devoid of particles, which arise in fluidization. In the case of moderate-density glass particles, we observe the formation of transient buoyant blobs clearly resulting from the destabilization of the one-dimensional wavy structure. With metallic beads of the same size but larger density, the same destabilization occurs but it leads to the formation of real bubbles. Comparison with previous analytical and numerical studies is attempted. Whereas the linear and weakly nonlinear analytical models are not appropriate, the direct nonlinear simulations provide a qualitative agreement with the observed destabilization mechanism.

The nonlinear long waves generated by a disturbance moving at subcritical, critical and supercritical speed in unbounded shallow water are investigated. The problem is formulated by a new modified generalized Boussinesq equation and solved numerically by an implicit finite-difference algorithm. Three-dimensional upstream solitary waves with significant amplitude are generated with a periodicity by a pressure distribution or slender strut advancing on the free surface. The crestlines of these solitons are almost perfect parabolas with decreasing curvature with respect to time. Behind the disturbance, a complicated, divergent Kelvin-like wave pattern is formed. It is found that, unlike the wave breaking phenomena in a narrow channel at Fh [ges ] 1.2, the three- dimensional upstream solitons form several parabolic water humps and are blocked ahead of the disturbance at supercritical speed in an unbounded domain for large time.