We present a parametric experimental study of convective electrokinetic instability (EKI) in an isotropically etched, cross-shaped microchannel using quantitative epifluorescence imaging. The base state is a three-inlet, one-outlet electrokinetic focusing flow configuration where the centre sample stream and sheath flows have mismatched ionic conductivities. Electrokinetic flows with conductivity gradients become unstable when the electroviscous stretching and folding of conductivity interfaces grows faster than the dissipative effect of molecular diffusion. Scalar images, critical applied fields required for instability, and temporal and spatial scalar energy are presented for flows with a wide range of applied d.c. electric field and centre-to-sheath conductivity ratios. These parameters impose variations of the electric Rayleigh number across four orders of magnitude. We introduce a scaling for charge density in the bulk fluid as a function of local maximum conductivity gradients in the flow. This scaling shows that the flow becomes unstable at a critical electric Rayleigh number ($Ra_{e, \ell} \,{=}\, 205$) and applies to a wide range of applied field and centre-to-sheath conductivity ratios. This work is relevant to on-chip electrokinetic flows with conductivity gradients such as field amplified sample stacking, flow at the intersections of multi-dimensional assays, electrokinetic control and separation of sample streams with poorly specified chemistry, and low-Reynolds number micromixing.

The spatial instability problem in a slowly diverging rectangular duct is investigated. The mean flow for the present problem is three-dimensional and has been obtained asymptotically using lubrication theory. Using a WKBJ expansion for the disturbance quantities, the zeroth- and first-order equations are derived. The zeroth-order problem corresponds to a locally parallel flow approximation and the first-order problem yields the non-parallel-flow correction to the eigenvalues obtained from the former through the use of a solvability condition. The solution of these equations is discussed and the results used to determine the effect of the variation in duct geometry on the neutral curves.

Laminar flow of compressible Newtonian fluids in capillaries is analysed. A perturbation solution is obtained to the vorticity–stream function form of the hydrodynamics equations for weakly compressible flow, having a relatively simple form. In contrast with previous analyses of this flow, we find both a non-zero radial velocity and non-zero radial pressure gradient. Expressions for pressure drop and friction factor are also found that show significant deviations from the incompressible case (Hagen–Poiseuille equation) that arise from both fluid inertia and bulk viscosity.

This paper presents data on turbulent-spot propagation in the hypersonic boundary-layer flow over a blunted cylindrical body. Data are based on the measurement of time-dependent surface heat transfer rates using gauges positioned as arrays in either the axial or transverse directions. These are used to provide data on individual spots, including sectional profiles, characteristic spot planform geometries, propagation speeds, growth rates and some information on the development of an internal thermal cell structure and corresponding thermal streaks in the base or calm region of the spot.

The flow field around pairs of small particles moving and rotating in a shear flow close to a wall at low but finite Reynolds number (Re) is computed as a function of time by means of the lattice-Boltzmann technique. The total force and torque acting on each particle is computed at each time step and the position of the particles is updated. By considering the lift force and the disturbances induced by the particles, the trajectories of the pair of particles are explained as a function of the distances from the wall and the Reynolds number. It is shown that when particles are positioned in a particular form, they collide forming strings. In particular, we are interested in particle-bridge formation in shear flows, and two collided particles (a string) can be considered as a nucleus of a particle bridge.

A framework is developed to describe the two-point statistics of potential vorticity in rotating and stratified turbulence as described by the Boussinesq equations. The Kármán–Howarth equation for the dynamics of the two-point correlation function of potential vorticity reveals the possibility of inertial-range dynamics in certain regimes in the Rossby, Froude, Prandtl and Reynolds number parameters. For the case of large Rossby and Froude numbers, and for the case of quasi-geostrophic dynamics, a linear scaling law with 2/3 prefactor is derived for the third-order mixed correlation between potential vorticity and velocity, a result that is analogous to the Kolmogorov 4/5-law for the third-order velocity structure function in turbulence theory.

Valveless pumping can be achieved through the periodic compression of a pliant tube asymmetrically from its interfaces to different tubing or reservoirs. A mismatch of characteristic impedance between the flow channels is necessary for creating wave reflection sites. Previous experimental studies of the behaviour of such a pump were continued in order to demonstrate the wave mechanics necessary for the build-up of pressure and net flow. Specific measurements of the transient and resonant properties were used to relate the bulk responses to the pump mechanics. Ultrasound imaging through the tube wall allowed visualization of the wall motion concurrently with pressure and flow measurements. For analysis, a one-dimensional wave model was constructed which predicted many of the characteristics exhibited by the experiments.

The Euler-alpha and the vortex blob model are two different regularizations of incom- pressible ideal fluid flow. Here, a regularization is a smoothing operation which controls the fluid velocity in a stronger norm than $L^2$. The Euler-alpha model is the inviscid version of the Lagrangian averaged Navier–Stokes-alpha turbulence model. The vortex blob model was introduced to regularize vortex flows. This paper presents both models within one general framework, and compares the results when applied to planar and axisymmetric vortex filaments and sheets. By certain measures, the Euler-alpha model is closer to the unregularized flow than the vortex blob model. The differences that result in circular vortex filament motion, vortex sheet linear stability properties, and core dynamics of spiral vortex sheet roll-up are discussed.

The oscillatory convection for a real system of fluids under the joint action of buoyancy and thermocapillary effect is investigated. The nonlinear development of the oscillatory instability is studied. Two types of boundary condition are considered. In the case of periodic boundary conditions, regimes of either travelling waves or standing oscillations have been found, depending on the period of the flow. For rigid heat-insulated lateral walls, various types of symmetric and asymmetric standing waves are obtained. Transitions between the motions with different spatial structures are investigated. It is shown that in the case of rigid heat-insulated lateral walls the period of oscillations changes non-monotonically. The nonlinear oscillations exist in a finite interval of the Grashof number values, between the stability regions of a quiescent state and stationary convection.

The dynamics of buoyant water entering a rotating basin are studied using a series of laboratory experiments designed to elucidate the alongshore transport mechanisms in river plumes. Inflowing water, which is discharged perpendicular to the tank wall, is observed to form a growing anticyclonic bulge and a coastal current downstream of the bulge. Detailed simultaneous measurements of the velocity and buoyancy fields in the plume confirm that the bulge momentum is in a gradient–wind balance and the coastal current is geostrophic. The growth of the bulge and accumulation of fluid within it coincides with a reduction in coastal current transport to approximately 50% of the inflow discharge. The bulge is characterized by a depth scale, $h$, which is proportional to the geostrophic depth, $h_{g}$, and two time-dependent horizontal length scales, $y_{c}$, the displacement of the bulge centre from the wall, and $r_{b}$, the effective radius of the bulge. These two length scales are proportional to the inertial radius, $L_{i}$, and the local Rossby radius, $L_{b}$, respectively. When $r_{b}\gg y_{c}$, the bulge is held tightly to the wall, and a relatively large fraction of the inflow discharge is forced into the coastal current. For plumes with $y_{c}$ approaching $r_{b}$, the bulge is further from the wall, and the coastal current flux is reduced. Once ${y_{c}}/{r_{b}}\,{>}\,0.7$, the bulge separates from the wall causing flow into the coastal current to cease and the bulge to become unstable. In this state, the bulge periodically detaches from and re-attaches to the wall, resulting in pulsing transport in the coastal current. Scaling of the bulge growth based on $h_{g}$, $L_{i}$ and $L_{b}$ predicts that it will increase as $\hbox{\it Ro}^{1/4}$, where $\hbox{\it Ro}$ is the inflow Rossby number. The bulge growth, inferred from direct measurements of the coastal current transport, is proportional to $\hbox{\it Ro}^{0.32}$ and agrees with the predicted dependence within the experimental error.

Closed reduced equations analogous to the quasi-geostrophic equations are derived in the extratropics for small Rossby numbers and vertical scales that are comparable to or much larger than horizontal scales. On these scales, significant vertical motions are permitted and found to couple to balanced geostrophic dynamics. In the equatorial regions, similar reduced equations are derived for meridional scales much larger than the vertical and zonal scales. These equations are derived by a systematic exploration of different aspect ratios, and Froude and buoyancy numbers, and offer advantages similar to the standard quasi-geostrophic equations for studies of smaller-scale processes and/or of the equatorial regions.

A thin ‘rope’ of viscous fluid falling from a sufficient height onto a surface forms a series of regular coils. Here we investigate theoretically and experimentally a curious feature of this instability: the existence of multiple states with different frequencies at a fixed value of the fall height. Using a numerical model based on asymptotic ‘thin rope’ theory, we determine curves of coiling frequency $\Omega$vs. fall height $H$ as functions of the fluid viscosity $\nu$, the diameter $d$ of the injection hole, the volumetric injection rate $Q$, and the acceleration due to gravity $g$. In addition to the three coiling modes previously identified (viscous, gravitational and inertial), we find a new multivalued ‘inertio-gravitational’ mode that occurs at heights intermediate between gravitational and inertial coiling. In the limit when the rope is strongly stretched by gravity and $\Pi_1\,{\equiv}\, (\nu^5/g Q^3)^{1/5}\,{\gg}\, 1$, inertio-gravititational coiling occurs in the height range $O(\Pi_1^{-1/6})\,{\leq}\, H(g/\nu^2)^{1/3}\,{\leq}\, O(\Pi_1^{-5/48})$. The frequencies of the individual branches are proportional to $(g/H)^{1/2}$, and agree closely with the eigenfrequencies of a whirling liquid string with negligible resistance to bending and twisting. The number of coexisting branches scales as $\Pi_1^{5/32}$. The predictions of the numerical model are in excellent agreement with laboratory experiments performed by two independent groups using different apparatus and working fluids. The experiments further show that interbranch transitions in the inertio-gravitational regime occur via an intermediate state with a ‘figure of eight’ geometry that changes the sense of rotation of the coiling.

We investigate the mechanics of ice streams and glaciers flowing over a bed consisting of Coulomb-plastic subglacial sediment, or more generally, of channel flows with Coulomb or ‘solid’ friction laws at the boundary. Sliding is assumed to occur if shear stress at the glacier bed attains a prescribed, locally defined yield stress, while no sliding is assumed possible below that yield stress. Importantly, the location of regions of slip and no slip at the bed is not known a priori, but forms part of the solution. By analogy with friction problems in elasticity, we derive a weak formulation as a semi-coercive variational inequality, which admits a unique solution provided a solvability condition ensuring force balance is satisfied. The variational formulation is then exploited to calculate numerical solutions, and we investigate the effect of variations in subglacial water pressure, ice thickness and surface slope on the discharge of a valley glacier with a plastic bed. Significant differences are found between the behaviour of wide and narrow as well as steep and shallow-angled glaciers, and our results further indicate the need to develop models capable of accounting for longitudinal stresses.

The surface flow, pressure distributions, and vortex-shedding properties of the circular cylinder flow in the subcritical regime modulated by a self-excited vibrating rod were experimentally studied in a closed-circuit wind tunnel for Reynolds numbers between $2\,{\times}\,10^4$ and $1.8\,{\times}\,10^5$. The principle of galloping was employed to induce self-vibration of an elastic rod. The natural (uncontrolled) boundary layer showed significantly different flow patterns in two subranges of Reynolds number: $Re_D \,{<}\,0.55\,{\times}\,10^5$ and $Re_D \,{>}\, 0.55\,{\times}\,10^5$. In the low-Reynolds-number subrange, the laminar separation mode, which was similar to the result reported by previous investigators, was observed. While in the high-Reynolds-number subrange, the separation bubble mode, which was rarely discussed previously, was found. The term separation bubble mode was used to denote the existence of secondary recirculation bubbles induced by the reattachment of separated boundary layers. The surface flow modulated by the vibrating rod presented complex variations in these two subranges. Depending on the position of the control rod, the separation bubble mode or the turbulent separation mode might appear because of the increase of the turbulence kinetic energy. Modulating the surface flow patterns could significantly influence the surface pressure distributions on the main cylinder and the vortex shedding in the wake. Forcing the boundary layer at the positions upstream of the natural separation point would drastically lower the values of the minimum pressure coefficient and the base pressure coefficient, and therefore increase the lift. The drag coefficient, however, would not be apparently decreased. The frequency of the vortex shedding in the wake could ‘lock’ to the rod vibration frequency if the position of the vibrating rod was properly adjusted.

We consider the release from a point source of relatively heavy fluid into a porous saturated medium above an impermeable slope. We consider the case where the volume of the resulting gravity current increases with time like $t^\alpha$ and show that for $\alpha\,{<}\,3$ at short times the current spreads axisymmetrically, with radius $r\,{\sim}\,t^{(\alpha+1)/4}$, while at long times it spreads predominantly downslope. In particular, for long times the downslope position of the current scales like $t$ while the current extends a distance $t^{\alpha/3}$ across the slope. For $\alpha\,{>}\,3$, this situation is reversed with spreading occurring predominantly downslope for short times. The governing equations admit similarity solutions whose scaling behaviour we determine, with the full similarity form being evaluated by numerical computations of the governing partial differential equation. We find that the results of these analyses are in good quantitative agreement with a series of laboratory experiments. Finally, we discuss the implications of our work for the sequestration of carbon dioxide in aquifers with a sloping, impermeable cap.

The high-Reynolds-number axisymmetric wake behind a disk has been studied for $10 \leq x/D \leq 150$ ($36 \leq x/\theta \leq 552 $) using a rake of 15 hot wires. The disk had a diameter of 20 mm, and the Reynolds number based on the free-stream velocity was 26 400. Mean velocity profiles, root-mean-square profiles, and power spectra are presented. By using regression techniques to fit the velocity profiles it was possible to obtain accurate velocity centreline deficits and transverse length scales to even the last downstream position. Beyond the initial region which extends to $x/D=30$, the data are in excellent agreement with the high-Reynolds-number equilibrium similarity solution.

The high-Reynolds-number axisymmetric wake behind a disk has been studied from $x/D\,{=}\,30$ to $x/D\,{=}\,150$ using the proper orthogonal decomposition (POD) applied to measurements of the streamwise fluctuating velocity. It was found that the energetic structure of the axisymmetric wake can very efficiently be described in terms of POD modes. The first radial (or lowest-order) POD mode has 56 % of the energy. Two major features dominate the eigenspectra, manifested as two major peaks. The first peak is an azimuthal mode-1 peak at a frequency corresponding to the Strouhal number of the near wake. The second is an azimuthal mode-2 peak at near-zero frequency. The mode-1 peak dies out faster than the mode-2 peak, so that the far wake is dominated by the latter.

This evolution from azimuthal mode-1 dominance in the near wake to mode-2 dominance in the far wake corresponds closely to the approach to equilibrium similarity. Once azimuthal mode-2 becomes equally important as azimuthal mode-1 (after $x/D\,{=}\,30$ or $x/\theta\,{=}\,110$), the ratio of turbulence intensity to centreline velocity deficit is constant, the mean deficit and turbulence intensity collapse in similarity variables, and the wake grows as $x^{1/3}$.

The interactions of vortices shed from two coaxially joined cylinders with a diameter ratio of 0.51 have been studied in a water channel, at Reynolds numbers, based on the large cylinder diameter, between 63 and 1100. Spectral and wavelet analyses of streamwise velocity measurements have identified the formation of three distinct spanwise vortex-shedding cells away from the far ends of the two cylinders, including a low-frequency cell, which appeared in a cyclic manner near the step and behind the large cylinder. Analysis of flow-visualization images has identified two distinct types of vortex connection at cell boundaries: some vortices in a cell of lower frequency connected across the boundary to counterparts in the cell of higher frequency, while others formed a half-loop to connect to a vortex shed from the opposite side of the cylinder in the same cell. Additional cross-boundary vortex connections balanced the total vortex strengths. The interactions of two types of streamwise vortices, identified as edge vortices and junction vortices, with spanwise vortices have also been documented.

Time-averaged two-point measurements of the fluctuating wall pressure and instantaneous measurements of the full fluctuating wall pressure field were performed in incompressible turbulent impinging jets with nozzle-to-plate spacings of 2 and 4 diameters, Reynolds numbers of 23 300, and Mach numbers of 0.03. An azimuthal Fourier decomposition of the fluctuating wall pressure revealed that the fluctuations in the stagnation region were dominated by azimuthal mode 1 to 3, and the contributions from these modes were larger for the jet with $H/D=4$. Azimuthal modes 0 and 1 made significant contributions to the pressure fluctuations in the wall jet region of both jets, indicating that the large-scale ring structures formed in the jet shear-layer play a prominent role in this region. The contributions from modes 0 and 1 in the wall jet were smaller for the jet with $H/D=4$ indicating that the ring structures play a less prominent role in the wall jet region as the nozzle-to-plate distance increased. A wavelet analysis of the transient fluctuations indicated that azimuthal mode 1 had dominant high-frequency and low-frequency components, while azimuthal mode 0 had only the higher-frequency oscillations. The higher-frequency components of azimuthal modes 0 and 1 occurred in both the stagnation and the wall jet regions and are attributed to the asymmetric evolution of the ring structures in the jet. The low-frequency oscillations were primarily associated with azimuthal mode 1 and were evident only in the stagnation region. These oscillations became more prominent as the nozzle-to-plate distance was increased.

The evolution of a finite-amplitude three-dimensional localized disturbance, having an initial dipole Gaussian vorticity distribution, embedded in an external unbounded irrotational plane stagnation flow (${{\bm U}}\,{=}\,(Ay,Ax,0)$) is investigated. Using the fluid impulse integral as a characteristic of such a disturbance, the viscous vorticity equation is integrated analytically. Accordingly, the fluid impulse associated with such disturbances decays and grows exponentially along the principal axes $x\,{=}\,y$ and $x\,{=}\,{-}y$, respectively. Numerical simulations, carried out for both linear and nonlinear disturbances at a Reynolds number of 40, confirm the above predictions. The simu- lations have also been compared with the solution of the linear viscous vorticity disturbance equation. While the solution predicts the vorticity distribution for the linear case, it fails to predict the essential characteristics of a nonlinear disturbance associated with its self-induced movement. Finally, it is shown that the fluid impulse and the disturbance kinetic energy follow the same trend, i.e. when the fluid impulse increases with time so does the kinetic energy and vice versa. The correspondence between them suggests the use of the fluid impulse to predict the stability of a localized disturbance.