The evolution of the entrainment coefficient $K$ of the rotating fluid in a rotor–stator cavity with an inward throughflow and pre-rotation is studied according to the flow parameters. Measurements are obtained in water for a turbulent Batchelor type of flow with two separated boundary layers on the rotating and stationary disks by means of a laser Doppler anemometer, and the results are compared to those performed using pressure transducers. We show that the entrainment coefficient $K$ depends on a local flow rate coefficient $Cq_r$ according to a $5/7$ power law whose coefficients depend on the boundary conditions. A theoretical analysis confirms this behaviour of $K$.

Chaotic behaviour is found for sufficiently long triaxial ellipsoidal non-Brownian particles immersed in steady simple shear flow of a Newtonian fluid in an inertialess approximation. The result is first determined via numerical simulations. An analytic theory explaining the onset of chaotic rotation is then proposed. The chaotic rotation coexists with periodic and quasi-periodic motions. Quasi-periodic motions are depicted by regular closed loops and islands in the system Poincaré map, whereas chaotic rotations form a stochastic layer.

Particle-size segregation can have a significant feedback on the bulk motion of granular avalanches when the larger grains experience greater resistance to motion than the fine grains. When such segregation-mobility feedback effects occur the flow may form digitate lobate fingers or spontaneously self-channelize to form lateral levees that enhance run-out distance. This is particularly important in geophysical mass flows, such as pyroclastic currents, snow avalanches and debris flows, where run-out distance is of crucial importance in hazards assessment. A model for finger formation in a bidisperse granular avalanche is developed by coupling a depth-averaged description of the preferential transport of large particles towards the front with an established avalanche model. The coupling is achieved through a concentration-dependent friction coefficient, which results in a system of non-strictly hyperbolic equations. We compute numerical solutions to the flow of a bidisperse mixture of small mobile particles and larger more resistive grains down an inclined chute. The numerical results demonstrate that our model is able to describe the formation of a front rich in large particles, the instability of this front and the subsequent evolution of elongated fingers bounded by large-rich lateral levees, as observed in small-scale laboratory experiments. However, our numerical results are grid dependent, with the number of fingers increasing as the numerical resolution is increased. We investigate this pathology by examining the linear stability of a steady uniform flow, which shows that arbitrarily small wavelength perturbations grow exponentially quickly. Furthermore, we find that on a curve in parameter space the growth rate is unbounded above as the wavelength of perturbations is decreased and so the system of equations on this curve is ill-posed. This indicates that the model captures the physical mechanisms that drive the instability, but additional dissipation mechanisms, such as those considered in the realm of flow rheology, are required to set the length scale of the fingers that develop.

An experiment is described in which a turbulent boundary layer was generated along a sloping wall of a laboratory tank containing salt-stratified fluid. Initially, the pycnocline separating the upper fresh-water layer from the lower saline layer was relatively thin; it thickened in response to the mixing as the experiment proceeded. Two types of mean circulation developed as a result of the boundary mixing. In the boundary layer, counterflowing mean streams were observed that augmented the diffusion of salt up- and downslope. This augmented dispersion in turn tended to spread the salt beyond the depth range of the pycnocline in the ambient fluid, producing a mean convergence in the boundary layer and intrusions from the layer into the ambient fluid.

The evolving density structure in the ambient fluid was measured by conductivity-probe traverses, from which the net buoyancy flux (or salt flux) and volume flux in the boundary layer were determined. The level of zero volume flux was found to coincide closely with the level of maximum stability frequency N, so that the buoyancy transport across this level was entirely the result of turbulent dispersion in the boundary layer. At other levels, the convective transports up and down contribute significantly. A simple theory provides scaling in terms of the laboratory parameters; in terms of the inferred overall turbulent viscosity νe, the buoyancy transport resulting from boundary-layer dispersion was found to be\[F_{\rm B} = 0.60 \left\{\frac{\nu_{\rm e}N(z)}{\sin\theta}\right\}^{\frac{3}{2}}\cos^2\theta,\]and the intrusion velocity into the ambient fluid is\[v_1 = 0.42\frac{\nu_{\rm e}^{\frac{3}{2}}}{N^{\frac{1}{2}}h^2}\frac{\cos^2\theta}{(\sin\theta)^{\frac{3}{2}}},\]where θ is the angle of the sloping bed. These expressions must become invalid when θ becomes vanishingly small; their application to estuarine and continental slope flows is discussed briefly.

Consider the three-dimensional flow of a viscous Newtonian fluid upon an arbitrarily curved substrate when the fluid film is thin as occurs in many draining, coating and biological flows. We drive the lubrication model of the dynamics of the film expressed in terms of the film thickness. The comprehensive model accurately includes the effects of the curvature of the substrate, via a physical multiple-scale approach, and gravity and inertia, via more rigorous centre manifold techniques. This new approach theoretically supports the use of the model over a wide range of parameters and provides a sound basis for further development of lubrication models. Numerical simulations exhibit some generic features of the dynamics of such thin fluid films on substrates with complex curvature: we here simulate a film thinning at a corner, the flow around a torus, and draining of a film down a cylinder. The last is more accurate than other lubrication models. The model derived here describes well thin-film dynamics over a wide range of parameter regimes.

The internal gravity wave (IGW) field emitted by a stably stratified, initially turbulent, wake of a towed sphere in a linearly stratified fluid is studied using fully nonlinear numerical simulations. A wide range of Reynolds numbers, $\mathit{Re}= UD/ \nu \in [5\times 1{0}^{3} , 1{0}^{5} ] $ and internal Froude numbers, $\mathit{Fr}= 2U/ (ND)\in [4, 16, 64] $ ( $U$ , $D$ are characteristic body velocity and length scales, and $N$ is the buoyancy frequency) is examined. At the higher $\mathit{Re}$ examined, secondary Kelvin–Helmholtz instabilities and the resulting turbulent events, directly linked to a prolonged non-equilibrium (NEQ) regime in wake evolution, are responsible for IGW emission that persists up to $Nt\approx 100$ . In contrast, IGW emission at the lower $\mathit{Re}$ investigated does not continue beyond $Nt\approx 50$ for the three $\mathit{Fr}$ values considered. The horizontal wavelengths of the most energetic IGWs, obtained by continuous wavelet transforms, increase with $\mathit{Fr}$ and appear to be smaller at the higher $\mathit{Re}$ , especially at late times. The initial value of these wavelengths is set by the wake height at the beginning of the NEQ regime. At the lower $\mathit{Re}$ , consistent with a recently proposed model, the waves propagate over a narrow range of angles that minimize viscous decay along their path. At the higher $\mathit{Re}$ , wave motion is much less affected by viscosity, at least initially, and early-time wave propagation angles extend over a broader range of values which are linked to increased efficiency in momentum extraction from the turbulent wake source.

Numerical studies are described of the flows generated by a sphere moving vertically in a uniformly stratified fluid. It is found that the axisymmetric standing vortex usually found in homogeneous fluids at moderate Reynolds numbers (25 [les ] Re [les ] 200) is completely collapsed by stable stratification, generating a strong vertical jet. This is consistent with our experimental visualizations. For Re = 200 the complete collapse of the vortex occurs at Froude number F ≃ 19, and the critical Froude number decreases slowly as Re increases. The Froude number and the Reynolds number are here defined by F = W/Na and Re = 2Wa/v, with W being the descent velocity of the sphere, N the Brunt–Väisälä frequency, a the radius of the sphere and v the kinematic viscosity coefficient. The inviscid processes, including the generation of the vertical jet, have been investigated by Eames & Hunt (1997) in the context of weak stratification without buoyancy effects. They showed the existence of a singularity of vorticity and density gradient on the rear axis of the flow and also the impossibility of realizing a steady state. When there is no density diffusion, all the isopycnal surfaces which existed initially in front of the sphere accumulate very near the front surface because of density conservation and the fluid in those thin layers generates a rear jet when returning to its original position. In the present study, however, the fluid has diffusivity and the buoyancy effects also exist. The density diffusion prevents the extreme piling up of the isopycnal surfaces and allows the existence of a steady solution, preventing the generation of a singularity or a jet. On the other hand, the buoyancy effect works to increase the vertical velocity to the rear of the sphere by converting the potential energy to vertical kinetic energy, leading to the formation of a strong jet. We found that the collapse of the vortex and the generation of the jet occurs at much weaker stratifications than those necessary for the generation of strong lee waves, showing that jet formation is independent of the internal waves. At low Froude numbers (F [les ] 2) the lee wave patterns showed good agreement with the linear wave theory and the previous experiments by Mowbray & Rarity (1967). At very low Froude numbers (F [les ] 1) the drag on a sphere increases rapidly, partly due to the lee wave drag but mainly due to the large velocity of the jet. The jet causes a reduction of the pressure on the rear surface of the sphere, which leads to the increase of pressure drag. High velocity is induced also just outside the boundary layer of the sphere so that the frictional drag increases even more significantly than the pressure drag.

An analysis of flow visualization using small reflective flakes is introduced. This rational analysis is based on a stochastic treatment of Jeffery's (1922) solution for the motion of ellipsoidal particles in a viscous fluid, wherein thin flakes tend to align with stream surfaces. The predicted light fields are confirmed by examples of parallel flows, the flow over a rotating disk, and the spinup from rest in a cylindrical cavity. The Tollmien–Schlichting wave packet trailing a turbulent spot is taken as an example to discuss the suitability of the technique for visualizing small-amplitude waves. Attenuation of light through a suspension is described.

In treating unsteady particle motions in creeping flows, a quasi-steady approximation is often used, which assumes that the particle's motion is so slow that it is composed of a series of steady states. In each of these states, the fluid is in a steady Stokes flow and the total force and torque on the particle are zero. This paper examines the validity of the quasi-steady method. For simple cases of sedimenting spheres, previous work has shown that neglecting the unsteady forces causes a cumulative error in the trajectory of the spheres. Here we will study the unsteady motion of solid bodies in several more complex flows: the rotation of an ellipsoid in a simple shear flow, the sedimentation of two elliptic cylinders and four circular cylinders in a quiescent fluid and the motion of an elliptic cylinder in a Poiseuille flow in a two-dimensional channel. The motion of the fluid is obtained by direct numerical simulation and the motion of the particles is determined by solving their equations of motion with solid inertia taken into account. Solutions with the unsteady inertia of the fluid included or neglected are compared with the quasi-steady solutions. For some flows, the effects of the solid inertia and the unsteady inertia of the fluid are importanty quantitatively but not qualitatively. In other cases, the character of the particles' motion is changed. In particular, the unsteady effects tend to suppress the periodic oscillations generated by the quasi-steady approximation. Thus, the results of quasi-steady calculatioins are never uniformly valid and can be completely misleading. The conditions under which the unsteady effects at small Reynolds numbers are important are explored and the implications for modelling of suspension flows are addressed.

The conventional ‘bridge-feedback amplifier’ constant-temperature hot-wire anemometer is analysed to determine its static and dynamic response. The effects of moderate feedback amplifier gain, bridge imbalance, stray bridge reactance, amplifier offset voltage, lack of common mode rejection, amplifier frequency response and departure from constant transconducture are included. The root loci of the system are mapped out and the consequences of the analysis are discussed from the viewpoint of both the operator and the designer.

The results are reported of velocity measurements in oscillatory flow over rippled beds. Velocities were measured with a laser-doppler anemometer in both an oscillating tray rig and an oscillatory flow water channel. Both self-formed and artificial ripples were examined. In addition, some measurements were made with an apparently plane bed with intense sediment motion.

The experimental results were compared with the predictions of Kalkanis (1964), Kajiura (1968), Bakker (1973, 1974) and Sleath (1974). The closest agreement between theory and experiment was obtained with the method of Sleath.

Measurements of the Eulerian drift velocities showed drift towards the adjacent ripple crest in the immediate vicinity of the bed and away from the crest further out.

We examine the stability of two-dimensional marine ice sheets in steady state. The dynamics of marine ice sheets is described by a viscous thin-film model with two Stefan-type boundary conditions at the moving boundary or ‘grounding line’ that marks the transition from grounded to floating ice. One of these boundary conditions constrains ice thickness to be at a local critical value for flotation, which depends on depth to bedrock at the grounding line. The other condition sets ice flux as a function of ice thickness at the grounding line. Depending on the shape of the bedrock, multiple equilibria may be possible. Using a linear stability analysis, we confirm a long-standing heuristic argument that asserts that the stability of these equilibria is determined by a simple mass balance consideration. If an advance in the grounding line away from its steady-state position leads to a net mass gain, the steady state is unstable, and stable otherwise. This also confirms that grounding lines can only be stable in positions where bedrock slopes downwards sufficiently steeply.

Observations of the cavity produced by a droplet of rain falling on a plane water surface show that the cavity often takes the form of an inverted cone. In the present paper we discuss a simple analytic expression for the corresponding fluid flow.

The main source of rain noise has been found to occur when the flow entrains an air bubble at the vertex of the cone. In this paper we show that the magnitude of the sound produced by the fluid flow in the present model is consistent with observation.

The stability of circular Couette flow with radial heating across a vertically oriented annulus with inner cylinder rotating and outer cylinder stationary is investigated using linear stability theory. Infinite aspect ratio and constant fluid properties are assumed and critical stability boundaries are calculated for a conduction-regime base flow. Buoyancy is included through the Boussinesq approximation and stability is tested with respect to both toroidal and helical disturbances of uniform wavenumber. Symmetries of the linearized disturbance equations based on the sense of radial heating and the sense of cylinder rotation and their effect on the kinematics and morphology of instability waveforms are presented. The numerical investigation is primarily restricted to radius ratios 0.6 and 0.959 at Prandtl numbers 4.35, 15 and 100. The results follow the development of critical stability from Taylor cells at zero heating through a number of asymmetric modes to axisymmetric cellular convection at zero rotation. Increasing the Prandtl number profoundly destabilizes the flow in both wide and narrow gaps and the number of contending critical modes increases with increasing radius ratio. Specific calculations made to compare with the stability measurements of Snyder & Karlsson (1964) and Sorour & Coney (1979) exhibit good agreement considering the idealizations built into the linear stability analysis.

The discrete-vortex model is applied to simulate the separation bubble over a two- dimensional blunt flat plate with finite thickness and right-angled corners, which is aligned parallel to a uniform approaching stream. This flow situation is chosen because, unlike most previous applications of the model, the separation bubble is supposed to be strongly affected by a nearby solid surface. The major objective of this paper is to examine to what extent the discrete-vortex model is effective for such a flow. A simple procedure is employed to represent the effect of viscosity near the solid surface; in particular, the no-slip condition on the solid surface. A reduction in the circulation of elemental vortices is introduced as a function of their ages in order to represent the three-dimensional deformation of vortex filaments, An experiment was also performed for comparison purposes.

The calculation yielded reasonable predictions of the time-mean and r.m.s. values of the velocity and the surface-pressure fluctuations, together with correlations between their fluctuating components, over most of the separation bubble. The interrelation between instantaneous spatial variations of the surface-pressure and velocity fluctuations were also obtained. A comparison between the calculated and measured results suggests that, in the real flow, the three-dimensional deformation of vortex filaments will become more and more dominant as the reattachment point is approached.

Direct numerical simulations of subcritical rotating, stratified and magneto-hydrodynamic wall-bounded flows are performed in large computational domains, focusing on parameters where laminar and turbulent flow can stably coexist. In most cases, a regime of large-scale oblique laminar-turbulent patterns is identified at the onset of transition, as in the case of pure shear flows. The current study indicates that this oblique regime can be shifted up to large values of the Reynolds number by increasing the damping by the Coriolis, buoyancy or Lorentz force. We show evidence for this phenomenon in three distinct flow cases: plane Couette flow with spanwise cyclonic rotation, plane magnetohydrodynamic channel flow with a spanwise or wall-normal magnetic field, and open channel flow under stable stratification. Near-wall turbulence structures inside the turbulent patterns are invariably found to scale in terms of viscous wall units as in the fully turbulent case, while the patterns themselves remain large-scale with a trend towards shorter wavelength for increasing . Two distinct regimes are identified: at low Reynolds numbers the patterns extend from one wall to the other, while at large Reynolds number they are confined to the near-wall regions and the patterns on both channel sides are uncorrelated, the core of the flow being highly turbulent without any dominant large-scale structure.

We report the normal stresses in a non-Brownian suspension in plane Couette flow determined from Stokesian Dynamics simulations. The presence of normal stresses that are linear in the shear rate in a viscometric flow indicates a non-Newtonian character of the suspension, which is otherwise Newtonian. While in itself of interest, this phenomenon is also important because it is believed that normal stresses determine the migration of particles in flows with inhomogeneous shear fields. We simulate plane Couette flow by placing a layer of clear fluid adjacent to one wall in the master cell, which is then replicated periodically. From a combination of the traceless hydrodynamic stresslet on the suspended particles, the stresslet due to (non-hydrodynamic) inter-particle forces, and the total normal force on the walls, we determine the hydrodynamic and inter-particle force contributions to the isotropic ‘particle pressure’ and the first normal stress difference. We determine the stresses for a range of the particle concentration and the Couette gap. The particle pressure and the first normal stress difference exhibit a monotonic increase with the mean particle volume fraction ϕ. The ratio of normal to shear stresses on the walls also increases with ϕ, substantiating the result of Nott & Brady (1994) that this condition is required for stability to concentration fluctuations. We also study the microstructure by extracting the pair distribution function from our simulations; our results are in agreement with previous studies showing anisotropy in the pair distribution, which is the cause of normal stresses.

Multi-point velocity measurements have been performed in turbulent pipe flow at ReD = 1.5 × 105 and combined with cross-spectral and proper orthogonal decomposition analysis to elucidate information on the structure of the large- and very-large-scale motions in the outer layer of wall-bounded flows. The results indicate that in the outer layer the large-scale motions (LSM) may be composed of detached eddies with a wide range of azimuthal scales, whereas in the logarithmic layer they are attached. The very-large-scale motions (VLSM) have large radial scales, are concentrated around a single azimuthal mode and make a smaller angle with the wall compared to the LSM. The results support a hypothesis that only the detached LSM in the outer layer align to form the VLSM.

Superhydrophobic surfaces can retain gas pockets within their microscale textures when submerged in water. This property reduces direct contact between water and solid surfaces and presents opportunities for improving hydrodynamic performance by decreasing viscous drag. In most realistic applications, however, the flow regime is turbulent and retaining the gas pockets is a challenge. In order to overcome this challenge, it is crucial to develop an understanding of physical mechanisms that can lead to the failure of superhydrophobic surfaces in retaining gas pockets when the overlying liquid flow is turbulent. We present a study of the onset of failure in gas retention by analysing direct numerical simulations (DNS) of turbulent flows over superhydrophobic surfaces coupled with the deformation of air–water interfaces that hold the gas pockets. The superhydrophobic surfaces are modelled as periodic textures with patterned slip and no-slip boundary conditions on the overlying water flow. The liquid–gas interface is modelled via a linearized Young–Laplace equation, which is solved coupled with the overlying turbulent flow. A wide range of texture sizes and interfacial Weber numbers are considered in this study. Our analysis identifies flow-induced upstream-travelling capillary waves that are coherent in the spanwise direction as one mechanism for failure in retention of gas pockets. To confirm physical understanding of these waves, a semianalytical inviscid linear analysis is developed; the wave speeds obtained from the space–time correlations in the DNS data were found to match with the predictions of the semianalytical model. The magnitude of the pressure fluctuations due to these waves was found to increase with decreasing surface tension, and increase with a much stronger dependence on the slip velocity, when all numbers are reported in wall units. Based on a fitted scaling, a threshold criterion for the failure of superhydrophobic surfaces is developed that is based on estimates of the onset condition required for the motion of contact lines. The second contribution of this work is the development of boundary maps that identify stable and unstable zones in a parameter space consisting of working parameter and design parameters including texture size and material contact angle. We provide a brief description of previously identified failure modes of superhydrophobic surfaces, namely the stagnation pressure and shear-driven drainage mechanisms. In an overlay map, the stable and unstable zones due to each mechanism are presented. For various input conditions, we provide scaling laws that identify the most critical mechanism limiting the stability of gas retention by superhydrophobic surfaces.