A two-component laser-Doppler velocimeter, with high spatial and temporal resolution, was used to study how the introduction of a drag-reducing surfactant to water changes the fully-developed velocity field in an enclosed rectangular channel. Measurements were made for four different Reynolds numbers, Re = 13300; 19100; 32000, and 49100 (based on the bulk viscosity, the half-height of the channel, and the viscosity of water). For a fixed volumetric flow the pressure drop was reduced by 62 to 76% when compared to a Newtonian flow with an equal wall viscosity. Measurements were made of the mean streamwise velocity, the root mean square of two components of the fluctuating velocity, the Reynolds shear stress and the spectral density function of the fluctuating velocity in the streamwise direction. The Reynolds shear stress is found to be zero over the whole channel and the spectra of the streamwise velocity fluctuations show a sharp cutoff at a critical frequency, fc. The ratio of the cutoff frequency to the root mean square of the streamwise velocity fluctuations is found to be approximately equal to 1 mm−1. The observation of a zero Reynolds shear stress indicates the existence of additional mean shear stresses (or mean transfers of momentum) that are not seen with a Newtonian fluid. Furthermore, the presence of a random fluctuating velocity field suggests a production of turbulence by a mechanism other than that usually found for a fully developed flow. Possible explanations for this behaviour are presented.

Thermo-solutal chimney-plume flows from a solidifying dendritic mushy region have been promoted in thin-walled glass tubes of internal radii from 0.3 mm–2.0 mm. Flow rates, liquid compositions and temperatures were measured as functions of the depth of immersion of capillary tubes in the advancing mushy region. The results demonstrate competition between buoyancy pressures and the restrictions of liquid recirculation within the dendritic array and have been analysed to provide permeability data for the mushy region at high liquid fractions. These data have been used to make some assessments of channel/plume dimensions for naturally occurring, unconstrained flows in the same system.

Spin-up of a turbulent flow in a cylindrical tank caused by a rotating bluff body has been investigated using flow visualization, fluid velocity measurements, and hydrodynamic torque measurements. During the spin-up process three distinct temporal regimes exist. These regimes are: (i) a build-up regime where the torque and the tangential velocity fluctuations in the close proximity of the body remain constant; (ii) a decay regime where these quantities decay with power-law relations; and (iii) a mean flow steady state where these values remain relatively constant. Experiments were conducted in two tanks differing in volume by a factor of 80 and with a large range of bluff body sizes. A non-dimensional time scale, τ, based upon turbulent diffusion is determined and the tangential velocity fluctuations and torque coefficient start to decay at a fixed value of τ. Likewise, steady state is attained at a larger fixed value of τ. This time scaling is physically based upon the time required for momentum to be transferred over the entire tank volume due to turbulent diffusion, and is general for any body size, tank size, rotation rate, and acceleration rate.

Linear and nonlinear dilational and sinuous capillary waves on thin inviscid infinite and semi-infinite planar liquid sheets in a void are analysed in a unified manner by means of a method that reduces the two-dimensional unsteady problem to a one-dimensional unsteady problem. For nonlinear dilational waves on infinite sheets, the accuracy of the numerical solutions is verified by comparing with an analytical solution. The nonlinear dilational wave maintains a reciprocal relationship between wavelength and wave speed modified from the linear theory prediction by a dependence of the product of wavelength and wave speed on the wave amplitude. For the general dilational case, nonlinear numerical simulations show that the sheet is unstable to superimposed subharmonic disturbances on the infinite sheet. Agreement for both sinuous and dilational waves is demonstrated for the infinite case between nonlinear simulations using the reduced one-dimensional approach, and nonlinear two-dimensional simulations using a discrete-vortex method. For semi-infinite dilational and sinuous distorting sheets that are periodically forced at the nozzle exit, linear and nonlinear analyses predict the appearance of two constant-amplitude waves of nearly equal wavelengths, resulting in a sheet disturbance characterized by a long-wavelength envelope of a short-wavelength oscillation. For semi-infinite sheets with sinuous waves, qualitative agreement between the dimensionally reduced analysis and experimental results is found. For example, a half-wave thinning and a sawtooth wave shape is found for the nonlinear sinuous mode. For the semi-infinite dilational case, a critical frequency-dependent Weber number is found below which one component of the disturbances decays with downstream distance. For the semi-infinite sinuous case, a critical Weber number equal to 2 is found; below this value, only one characteristic is emitted in the positive time direction from the nozzle exit.

Linear Rossby waves in a two-layer ocean with a corrugated bottom relief (the isobaths are straight parallel lines) are investigated. The case of a rough bottom relief (the wave scale L is much greater than the bottom relief scale Lb) is studied analytically by the method of multiple scales. A special numerical technique is developed to investigate the waves over a periodic bottom relief for arbitrary relationships between L and Lb.

There are three types of modes in the two-layer case: barotropic, topographic, and baroclinic. The structure and frequencies of the modes depend substantially on the ratio Δ = (Δh/h2)/(L/a) measuring the relative strength of the topography and β-effect. Here Δh/h2 is the typical relative height of topographic inhomogeneity and a is the Earth's radius. The topographic and barotropic mode frequencies depend weakly on the stratification for small and large Δ and increase monotonically with increasing Δ. Both these modes become close to pure topographic modes for Δ>1.

The dependence of the baroclinic mode on Δ is more non-trivial. The frequency of this mode is of the order of f0L2i/aL (Li is the internal Rossby scale) irrespective of the magnitude of Δ. At the same time the spatial structure of the mode depends strongly on Δ. With increasing Δ the relative magnitude of motion in the lower layer decreases. For Δ>1 the motion in the mode is confined mainly to the upper layer and is very weak in the lower one. A similar concentration of mesoscale motion in an upper layer over an abrupt bottom topography has been observed in the real ocean many times.

Another important physical effect is the so-called ‘screening’. It implies that for Lb<Li the small-scale component of the wave with scale Lb is confined to the lower layer, whereas in the upper layer the scale of the motion L is always greater than or of the order of, Li. In other words, the stratification prevents the ingress of motion with scale smaller than the internal Rossby scale into the main thermocline.

Linear Rossby waves in a continuously stratified ocean over a corrugated rough-bottomed topography are investigated by asymptotic methods. The main results are obtained for the case of constant buoyancy frequency. In this case there exist three types of modes: a topographic mode, a barotropic mode, and a countable set of baroclinic modes. The properties of these modes depend on the type of mode, the relative height δ of the bottom bumps, the wave scale L, the topography scale Lb and the Rossby scale Li. For small δ the barotropic and baroclinic modes are transformed into the ‘usual’ Rossby modes in an ocean of constant depth and the topographic mode degenerates. With increasing δ the frequencies of the barotropic and topographic modes increase monotonically and these modes become close to a purely topographic mode for sufficiently large δ. As for the baroclinic modes, their frequencies do not exceed O(βL) for any δ. For large δ the so-called ‘displacement’ effect occurs when the mode velocity becomes small in a near-bottom layer and the baroclinic mode does not ‘feel’ the actual rough bottom relief. At the same time, for some special values of the parameters a sort of resonance arises under which the large- and small-scale components of the baroclinic mode intensify strongly near the bottom.

As in the two-layer model, a so-called ‘screening’ effect takes place here. It implies that for Lb<Li the small-scale component of the mode is confined to a near-bottom boundary layer (Lb/Li)H thick, whereas in the region above the layer the scale L of motion is always larger than or of the order of Li.

The influence of a non-monotonic viscosity–concentration relationship on miscible displacements in porous media is studied for radial source flows and the quarter five-spot configuration. Based on linear stability results, a parametric study is presented that demonstrates the dependence of the dispersion relations on both the Péclet number and the parameters of the viscosity profile. The stability analysis suggests that any displacement can become unstable provided only that the Péclet number is sufficiently high. In contrast to rectilinear flows, for a given end-point viscosity ratio an increase of the maximum viscosity generally has a destabilizing effect on the flow. The physical mechanisms behind this behaviour are examined by inspecting the eigensolutions to the linear stability problem. Nonlinear simulations of quarter five-spot displacements, which for small times correspond to radial source flows, confirm the linear stability results. Surprisingly, displacements characterized by the largest instability growth rates, and consequently by vigorous viscous fingering, lead to the highest breakthrough recoveries, which can even exceed that of a unit mobility ratio flow. It can be concluded that, for non-monotonic viscosity profiles, the interaction of viscous fingers with the base-flow vorticity can result in improved recovery rates.

We present a numerical study of steady convection in a two-dimensional mushy layer during solidification of a binary mixture at a constant speed V. The mushy layer is modelled as a reactive porous medium whose permeability is a function of the local solid fraction. The flow in the liquid region above the mushy layer is modelled using the Stokes equations (i.e. the Prandtl number is taken to be infinite). The calculations follow the development of buoyancy-driven convection as the flow amplitude is increased to the level where the solid fraction is driven to zero at some point within the mushy region. We show that this event cannot occur before the local buoyancy-driven volume flux exceeds the solidification rate V. Further increases in the flow amplitude lead to the formation of a region with negative solid fraction, indicating the need to switch from the Darcy approximation to the Stokes flow approximation. These regions ultimately become what are known as chimneys. We exhibit solutions which give the detailed structure of the temperature, solute, flow and solid fraction fields within the mushy layer. A key finding of the numerics is that these fledgling chimneys emerge from the interior of the mushy layer, rather than eating their way down from the top of the layer, as the amplitude of the steady convection is increased. We discuss some qualitative features of the resulting liquid inclusions and, in the light of these, reassess the interfacial conditions between mushy and liquid regions.

Laboratory observations and numerical experiments have shown that a variety of compound vortices can emerge in two-dimensional flow due to the instability of isolated circular vortices. The simple geometrical features of these compound vortices suggest that their description may take a simple form if an appropriately chosen set of functions is used. We employ a set which is complete on the infinite plane for vorticity distributions with finite total enstrophy. Through projection of the vorticity equation (Galerkin method) and subsequent truncation we derive a dynamical system which is used to model the observed behaviour in as simple as possible a fashion. It is found that at relatively low-order truncations the observed behaviour is qualitatively captured by the dynamical system. We determine what the necessary ingredients are for saturation of instabilities at finite amplitude in terms of wave–wave interactions and feedback between various azimuthal components of the vorticity field.

The statistical features of turbulence can be studied either through spectral quantities, such as the kinetic energy spectrum, or through structure functions, which are statistical moments of the difference between velocities at two points separated by a variable distance. In this paper structure function relations for two-dimensional turbulence are derived and compared with calculations based on wind data from 5754 airplane flights, reported in the MOZAIC data set. For the third-order structure function two relations are derived, showing that this function is generally positive in the two-dimensional case, contrary to the three-dimensional case. In the energy inertial range the third-order structure function grows linearly with separation distance and in the enstrophy inertial range it grows cubically with separation distance. A Fourier analysis shows that the linear growth is a reflection of a constant negative spectral energy flux, and the cubic growth is a reflection of a constant positive spectral enstrophy flux. Various relations between second-order structure functions and spectral quantities are also derived. The measured second-order structure functions can be divided into two different types of terms, one of the form r2/3, giving a k−5/3-range and another, including a logarithmic dependence, giving a k−3-range in the energy spectrum. The structure functions agree better with the two-dimensional isotropic relation for larger separations than for smaller separations. The flatness factor is found to grow very fast for separations of the order of some kilometres. The third-order structure function is accurately measured in the interval [30, 300] km and is found to be positive. The average enstrophy flux is measured as Πω≈1.8×10−13 s−3 and the constant in the k−3-law is measured as [Kscr ]≈0.19. It is argued that the k−3-range can be explained by two-dimensional turbulence and can be interpreted as an enstrophy inertial range, while the k−5/3-range can probably not be explained by two-dimensional turbulence and should not be interpreted as a two-dimensional energy inertial range.

We present the results of an experimental investigation of the generation of coherent vortical structures by buoyant line plumes in rotating fluids. Both uniform and stratified ambients are considered. By combining the scalings describing turbulent plumes and geostrophically balanced vortices, we develop a simple model which predicts the scale of the coherent vortical structures in excellent accord with laboratory experiments.

We examine the motion induced by a constant buoyancy flux per unit length B, released for a finite time ts, from a source of length L into a fluid rotating with angular speed Ω = f/2. When the plume discharges into a uniformly stratified environment characterized by a constant Brunt–Väisälä frequency, N>f, the fluid rises to its level of neutral buoyancy unaffected by the system rotation before intruding as a gravity current. Rotation has a strong impact on the subsequent dynamics: shear develops across the spreading neutral cloud which eventually goes unstable, breaking into a chain of anticyclonic lenticular vortices. The number of vortices n emerging from the instability of the neutral cloud, n = (0.65±0.1)Lf1/2/ (t1/2sB1/3), is independent of the ambient stratification, which serves only to prescribe the intrusion height and aspect ratio of the resulting vortex structures. The experiments indicate that the Prandtl ratio characterizing the geostrophic vortices is given by P = Nh/(fR) = 0.47±0.12; where h and R are, respectively, the half-height and radius of the vortices. The lenticular vortices may merge soon after formation, but are generally stable and persist until they are spun-down by viscous effects.

When the fluid is homogeneous, the plume fluid rises until it impinges on a free surface. The nature of the flow depends critically on the relative magnitudes of the layer depth H and the rotational lengthscale Lf = B1/3/f. For H>10Lf, the ascent phase of the plume is influenced by the system rotation and the line plume breaks into a series of unstable anticylonic columns of characteristic radius (5.3±1.0)B1/3/f which typically interact and lose their coherence before surfacing. When H<10Lf, the system rotation does not influence the plume ascent, but does control the spreading of the gravity current at the free surface. In a manner analogous to that observed in the stratified ambient, shear develops across the surface current, which eventually becomes unstable and generates a series of anticyclonic surface eddies with characteristic radius (1.6±0.2)B1/3t1/3s /f2/3. These surface eddies are significantly more stable than their columnar counterparts, but less so than the lenticular eddies arising in the uniformly stratified ambient.

The relevance of the study to the formation of coherent vortical structures by leads in the polar ocean and hydrothermal venting is discussed.

A parametric study of multiple steady states, their stability, onset of oscillatory instability, and some supercritical unsteady regimes of convective flow of a Boussinesq fluid in laterally heated rectangular cavities is presented. Cavities with four no-slip boundaries, isothermal vertical and perfectly insulated horizontal boundaries are considered. Four distinct branches of steady-state flows are found for this configuration. A complete study of stability of each branch is performed for the aspect ratio A (length/height) of the cavity varying continuously from 1 to 11 and for two fixed values of the Prandtl number: Pr = 0 and Pr = 0.015. The results are represented as stability diagrams showing the critical parameters (critical Grashof number and the frequency at the onset of the oscillatory instability) corresponding to transitions from steady to oscillatory states, appearance of multi-roll states, merging of multiple states and backwards transitions from multi-roll to single-roll states. For better comparison with the existing experimental data, an additional stability study for varying Prandtl number (0.015 [les ] Pr [les ] 0.03) and fixed value of the aspect ratio A = 4 was carried out. It was shown that the dependence of the critical Grashof number on the aspect ratio and the Prandtl number is very complicated and a very detailed parametric study is required to reproduce it correctly. Comparison with the available experimental data for A = 4 shows that the results of a two-dimensional stability analysis are in good agreement with the experimental results if the width ratio (width/height) of the experimental container is sufficiently large. The study is carried out numerically with the use of two independent numerical approaches based on the global Galerkin and finite-volume methods.

The scattering of plane surface water waves by both surface-piercing and submerged axisymmetric bed formations has been investigated by a number of authors. We extend previous contributions in two ways: we employ the recently derived modified mild-slope equation to approximate the fluid motion and we consider arbitrary axisymmetric topography, subject only to the restriction imposed by the mild-slope approximation. We thereby derive a robust method for calculating the scattered wave field for the small seabed slopes typical of most real situations and make a contribution to the phenomena of near-resonance and near-trapping over submerged axisymmetric shoals, which have previously been detected for only one idealized bedform.

The sedimentation of a small dense sphere through a suspension of neutrally buoyant fibres is investigated via a numerical simulation technique that includes both fibre–fibre contact forces and long-range hydrodynamic interactions. In situations where the diameter of the sphere is smaller than the length of the fibres, calculations that exclude the effect of contacts between fibres severely underestimate the drag force on the sphere measured in experiments. By including fibre–fibre contacts in our simulations we are to able to account for this discrepancy, and also the strong dependence of the drag on the initial orientation of the fibres. At low and moderate values of nL3, where n is the number of fibres per unit volume and L the fibre length, hydrodynamic interactions are found to be important in moderating the effect of contacts between fibres.

An asymptotic solution is presented for the limit when the sphere diameter is much smaller than both the fibre length and inter-fibre spacing, but large compared to the fibre thickness. This is found to be in good agreement with the simulations.

Results of calculations on sedimentation through a monolayer of fibres are also presented, as a model of a semi-concentrated suspension. Collisions between fibres are much more frequent, due to the geometric confinement.

This work brings new insight to the question of heat transfer in near–critical fluids under Earth gravity conditions. The interplay between buoyant convection and thermoacoustic heat transfer (piston effect) is investigated in a two-dimensional non-insulated cavity containing a local heat source, to reproduce the conditions used in recent experiments. The results were obtained by means of a finite-volume numerical code solving the Navier–Stokes equations written for a low-heat-diffusing near-critical van der Waals fluid. They show that hydrodynamics greatly affects thermoacoustics in the vicinity of the upper thermostated wall, leading to a rather singular heat transfer mechanism. Heat losses through this wall govern a cooling piston effect. Thus, the thermal plume rising from the heat source triggers a strong enhancement of the cooling piston effect when it strikes the middle of the top boundary. During the spreading of the thermal plume, the cooling piston effect drives a rapid thermal quasi-equilibrium in the bulk fluid since it is further enhanced so as to balance the heating piston effect generated by the heat source. Then, homogeneous fluid heating is cancelled and the bulk temperature stops increasing. Moreover, diffusive and convective heat transfers into the bulk are very weak in such a low-heat-diffusing fluid. Thus, even though a steady state is not obtained owing to the strong and seemingly continuous instabilities present in the flow, the bulk temperature is expected to remain quasi-constant. Comparisons performed with a supercritical fluid at initial conditions further from the critical point show that this thermalization process is peculiar to near-critical fluids. Even enhanced by the thermal plume, the cooling piston effect does not balance the heating piston effect. Thus, overall piston-effect heating lasts much longer, while convection and diffusion progressively affect the thermal field much more significantly. Ultimately, a classical two-roll convective-diffusive structure is obtained in a perfect gas, without thermoacoustic heat transfer playing any role.