Vortex formation in the near wake of shallow flow past a vertical cylinder can be substantially delayed by base bleed through a very narrow slot. The structure of the wake associated with this delay changes dramatically with the dimensionless momentum coefficient of the slot bleed. At very low values, where substantial vortex delay is attainable, the bleed flow is barely detectable. For progressively larger values, various forms of jets issue from the slot, and they undergo ordered, large-amplitude undulations, not necessarily synchronized with the formation of the large-scale vortices. When the cylinder is subjected to appropriate rotational perturbations, in the presence of small-magnitude base bleed, it is possible to transform the delayed vortex formation to a form characteristic of the naturally occurring vortices and, furthermore, to induce a large change of the phase, or timing, of the initially formed vortex, relative to the cylinder motion.

These features of the near-wake structure are assessed via a technique of high-image-density particle image velocimetry, which provides whole-field patterns of vorticity, Reynolds stress, amplitude distributions of spectral peaks, and streamline topology at and above the bed, for both the delayed and recovered states of the wake. Among the findings is that even small bleed can substantially alter the patterns of streamline topology and Reynolds stress at the bed, which has important consequences for the bed loading.

These alterations of the near-wake structure occur in conjunction with modifications of the shallow approach flow, which is incident upon the upstream face of the cylinder. The topology at the bed, which is altered in accord with attenuation of the well-defined vorticity concentration of the horseshoe (standoff) vortex, shows distinctive patterns involving new arrangements of critical points.

The two-layer model of a stably stratified medium has been used to investigate the stability of flows without inflection points on the profile of the velocity $V_x\,{=}\,u(y)$, which is monotonically increasing from zero at the bottom ($y\,{=}\,0$) to its maximum value $U_0$ (when $y\,{\to}\,\infty$). It is shown that in the case of flows of a general form (in which $u''(y)\,{<}\,0$ everywhere) an instability sets in for an arbitrarily small density difference; furthermore, perturbations of all scales build up simultaneously. With an enhancement of stratification, the real part $c_r$ of the phase velocity of unstable perturbations increases. The upper boundary of the instability domain is determined by the fact that at a certain stratification level (a particular one for perturbations of each scale), $c_r$ reaches $U_0$, the perturbation is no longer in phase resonance with the flow and turns into a neutral oscillation of the medium.

For flows of a special kind, having points of zero curvature (where $u''=0$) on the velocity profile (but having no inflection points as before), the influence of neutral modes, associated with these points, on the formation of the instability domain configuration is analysed, and an interpretation of this influence is given in terms of the resonance and non-resonance contributions to shear flow instability.

The paper considers the evolution of weakly nonlinear disturbances in linearly unstable stratified shear flows. We develop a generic Hamiltonian formulation for two-dimensional flows. The paper is focused on three-wave resonant interactions, which are always present in the stratified shear flows under consideration as the lowest-order nonlinear process. Two different types of shear flows are considered. The first one is the classical piecewise-linear model with constant density and vorticity in each layer. For such flows, linear instability is due to weak interaction of different modes. The second type is the Kelvin–Helmholtz model, consisting of two layers with different densities and velocities. Velocity shear is assumed to be weakly supercritical. We show that apart from the classical triplets consisting of stable waves, both flow types admit only triplets consisting of one weakly unstable and two neutrally stable waves, and we consider them in detail.

Universal evolution equations for three resonantly interacting wave packets are derived for both cases. For the first flow type, the generic equations coincide with the system derived earlier for a particular case of resonant interactions between unstable and neutral baroclinic waves in a quasi-geostrophic two-layer model. The evolution equations for the Kelvin–Helmholtz model are new, and are studied numerically and analytically in detail. In particular, we demonstrate that resonant interaction with neutral waves can stabilize the growth of the linearly unstable wave. This mechanism is essentially different from the well-known nonlinear stabilization mechanism due to cubic nonlinearity.

We present results for the average mass transfer to a spherical squirmer, a model micro-organism whose surface oscillates tangentially to itself. The surface motion drives a low-Reynolds-number flow which enables the squirmer either to swim relative to the fluid at infinity, at an average speed proportional to a streaming parameter, $W$, or to stir the fluid around it while remaining, on average, at rest (if $W\,{=}\,0$), as represented by a hovering parameter, $b$. We assume that the amplitude of the time-periodic surface distortions is scaled by a dimensionless small parameter $\epsilon$, and consider only high Péclet numbers $P$ – a measure of convection versus diffusion – by setting $P^{-1acute;\,{=}\,\epsilon^2 \gamma$, where $\gamma$ is a parameter of $O(1)$. It is shown that the average mass concentration distribution satisfies a steady convection–diffusion equation with an effective velocity field that is different from the actual mean velocity field. The model is used to calculate the mass transfer across the surface of the squirmer, measured by the mean Sherwood number $Sh$.

We find asymptotic solutions for small and large $\gamma$ and numerical results for the whole range of values. While the large-$\gamma$ expansions are reproduced well by the numerical results, there is a discrepancy between the two at small $\gamma$. We believe this is due to very small recirculation regions, attached to the surface of the squirmer, which make boundary layer theory applicable only when $1/\gamma$ is immense.

For the parameters chosen in this study, results indicate that both hovering and streaming contribute to the mass transfer, although streaming has a greater effect. Also, energy dissipation considerations show that an optimum swimming mode exists, at least at small and large $\gamma$, for any given uptake rate. However, other factors have still to be taken into account, and the model realism improved, if we want to make predictions for real aquatic micro-organisms.

Consider Stokes flow in a cone of half-angle $\alpha$ filled with a viscous liquid. It is shown that in spherical polar coordinates there exist similarity solutions for the velocity field of the type $r^{\lambda} {\bm f}(\theta;\lambda)\exp{\rm i}m\phi $ where the eigenvalue $\lambda$ satisfies a transcendental equation. It follows, by extending an argument given by Moffatt (1964$a$), that if the eigenvalue $\lambda$ is complex there will exist, associated with the corresponding vector eigenfunction, an infinite sequence of eddies as $r\,{\rightarrow}\, 0$. Consequently, provided the principal eigenvalue is complex and the driving field is appropriate, such eddy sequences will exist. It is also shown that for each wavenumber $m$ there exists a critical angle $\alpha^*$ below which the principal eigenvalue is complex and above which it is real. For example, for $m\,{=}\,1$ the critical angle is about $74.45^{\circ}$. The full set of real and complex eigenfunctions, the inner eigenfunctions, can be used to compute the flow in a cone given data on the lid. There also exist outer eigenfunctions, those that decay for $r\,{\rightarrow}\, \infty$, and these can be generated from the inner ones. The two sets together can be used to calculate the flow in a conical container whose base and lid are spherical surfaces. Examples are given of flows in cones and in conical containers which illustrate how $\alpha$ and $r_0$, a length scale, affect the flow fields. The fields in conical containers exhibit toroidal corner vortices whose structure is different from those at a conical vertex; their growth and evolution to primary vortices is briefly examined.

The quasi-diagonal direct interaction approximation (QDIA) closure theory is formulated for the interaction of mean fields, Rossby waves and inhomogeneous turbulence over topography on a generalized $\beta$-plane. An additional small term, corresponding to the solid-body rotation vorticity on the sphere, is included in the barotropic equation and it is shown that this results in a one-to-one correspondence between the dynamical equations, Rossby wave dispersion relations, nonlinear stability criteria and canonical equilibrium theory on the generalized $\beta$-plane and on the sphere. The dynamics, kinetic energy spectra, mean field structures and mean streamfunction tendencies contributed by transient eddies are compared with the ensemble-averaged results from direct numerical simulations (DNS) at moderate resolution. A series of numerical experiments is performed to examine the generation of Rossby waves when eastward large-scale flows impinge on a conical mountain in the presence of moderate to strong two-dimensional turbulence. The ensemble predictability of northern hemisphere flows in 10-day forecasts is also examined on a generalized $\beta$-plane. In all cases, the QDIA closure is found to be in very good agreement with the statistics of DNS except in situations of strong turbulence and weak mean fields where ensemble-averaged DNS fails to predict mean field amplitudes correctly owing to sampling problems even with as many as 1800 ensemble members.

The general problem of propagation of three-dimensional disturbances in viscous supersonic flows is considered in the framework of characteristic analysis. Unlike previous results for linear disturbances we deduce a condition determining nonlinear characteristic surfaces which is exact and therefore allows both qualitative and quantitative studies of the speed of propagation as a function of various physical phenomena. These include negative and adverse pressure gradients, and effects of wall cooling and suction–blowing, which are studied in this work as an illustration of the general theory.

A numerical study of the three-dimensional fluid dynamics inside a model left ventricle during diastole is presented. The ventricle is modelled as a portion of a prolate spheroid with a moving wall, whose dynamics is externally forced to agree with a simplified waveform of the entering flow. The flow equations are written in the meridian body-fitted system of coordinates, and expanded in the azimuthal direction using the Fourier representation. The harmonics of the dependent variables are normalized in such a way that they automatically satisfy the high-order regularity conditions of the solution at the singular axis of the system of coordinates. The resulting equations are solved numerically using a mixed spectral–finite differences technique. The flow dynamics is analysed by varying the governing parameters, in order to understand the main fluid phenomena in an expanding ventricle, and to obtain some insight into the physiological pattern commonly detected. The flow is characterized by a well-defined structure of vorticity that is found to be the same for all values of the parameters, until, at low values of the Strouhal number, the flow develops weak turbulence.

Experimental work to investigate plane Couette flow has been performed in the Reynolds number range of $750\,{\leq}\,\hbox{\it Re}\,(\,{=}\,hU_b/(2\nu))\,{\leq}\, 5000$ or $50\,{\leq}\, \hbox{\it Re}_*\,(\,{=}\,hu_*/\nu)\,{\leq}\, 253$, where $U_b$, $u_*$ and $h$ are moving wall speed, friction velocity and channel half-height, respectively. The low-Reynolds-number effect on the wall friction coefficient $C_f$, mean velocity profile and statistical turbulence quantities is discussed in relation to the turbulent Poiseuille flow properties. Since the shear stress is constant in Couette flow, the flow is free from the effect of shear stress gradient and the Reynolds number effect therefore can be seen explicitly, uncontaminated by this effect. A flow region diagram is given to show how the low-Reynolds-number effect penetrates into the wall region. The area of the buffer region is contracted by the low-Reynolds-number effect when $\hbox{\it Re}_*\leq 150$, so that the additive constant $B$ of the log law decrease as $\hbox{\it Re}_*$ decreases. Also, $C_f$ has a larger value than in Poiseuille flow in the low $\hbox{\it Re}_*$ range. The log-law area in Couette flow is 2–3 times as wide as that in Poiseuille flow. The defect law is $\hbox{\it Re}_*$-dependent and the non-dimensional velocity gradient at the core, $Rs=({\rm d}U_1/{\rm d}x_2)(h/u_*)$, increases from 3 to 4.2 as $\hbox{\it Re}_*$ increases from 50 to 253. The peak value of streamwise turbulence intensity $u_{1p}^+$ has a constant value of 2.88 but decreases sharply as $\hbox{\it Re}_*$ reduces below 150.

The large longitudinal vortices extending the entire height of the channel are shown to be sustained in Couette flow that is oscillating around their average position. This causes a slow fluctuation with large amplitude in the streamwise velocity component. These vortices make the Couette flow three-dimensional and the skin friction coefficient varies 20% sinuously in the spanwise direction, for example. Also, the zero-crossing time separation of streamwise velocity auto-correlation $R_{11}(\tau)$ becomes longer as $\tau=40h/U_b$, which is 3 times as long as that in Poiseuille flow.

The Benjamin–Feir instability is a modulational instability in which a uniform train of oscillatory waves of moderate amplitude loses energy to a small perturbation of other waves with nearly the same frequency and direction. The concept is well established in water waves, in plasmas and in optics. In each of these applications, the nonlinear Schrödinger equation is also well established as an approximate model based on the same assumptions as required for the derivation of the Benjamin–Feir theory: a narrow-banded spectrum of waves of moderate amplitude, propagating primarily in one direction in a dispersive medium with little or no dissipation. In this paper, we show that for waves with narrow bandwidth and moderate amplitude, any amount of dissipation (of a certain type) stabilizes the instability. We arrive at this stability result first by proving it rigorously for a damped version of the nonlinear Schrödinger equation, and then by confirming our theoretical predictions with laboratory experiments on waves of moderate amplitude in deep water. The Benjamin–Feir instability is often cited as the first step in a nonlinear process that spreads energy from an initially narrow bandwidth to a broader bandwidth. In this process, sidebands grow exponentially until nonlinear interactions eventually bound their growth. In the presence of damping, this process might still occur, but our work identifies another possibility: damping can stop the growth of perturbations before nonlinear interactions become important. In this case, if the perturbations are small enough initially, then they never grow large enough for nonlinear interactions to become important.

While many rheological studies are performed at a fixed concentration, most granular flows are constrained, not by concentration, but by an applied stress. The stress constraint sets the average concentration, but the material is free to vary that concentration slightly to match the applied stress with that generated internally. This study examines stress-controlled systems in light of recent findings that the elastic properties of the particles appear as constitutive parameters even in flowing situations. Stress-controlled flows are shown to behave very differently from flows at fixed concentration. In particular, if the stress is fixed and the shear rate is slowly increased, the flow exhibits the expected progression from elastic–quasi-static to elastic–inertial to inertial flow – a sequence opposite to that followed in fixed-concentration flows. Thus system-scale constraints can have a profound effect on granular rheology.

Streamline patterns and their bifurcations in two-dimensional Navier–Stokes flow of an incompressible fluid near a non-simple degenerate critical point close to a stationary wall are investigated from the topological point of view by considering a Taylor expansion of the velocity field. Using a five-order normal form approach we obtain a much simplified system of differential equations for the streamlines. Careful analysis of the simplified system gives possible bifurcations for non-simple degeneracies of codimension three. Three heteroclinic connections from three on-wall separation points merge at an in-flow saddle point to produce two separation bubbles with opposite rotations which occur only near a non-simple degenerate critical point. The theory is applied to the patterns and bifurcations found numerically in the studies of Stokes flow in a double-lid-driven rectangular cavity.

A plane wall-jet flow is numerically investigated and compared to experiments. The measured base flow is matched to a boundary-layer solution developing from a coupled Blasius boundary layer and Blasius shear layer. Linear stability analysis is performed, revealing high instability of two-dimensional eigenmodes and non-modal streaks. The nonlinear stage of laminar-flow breakdown is studied with three-dimensional direct numerical simulations and experimentally visualized. In the direct numerical simulation, an investigation of the nonlinear interaction between two-dimensional waves and streaks is made. The role of subharmonic waves and pairing of vortex rollers is also investigated. It is demonstrated that the streaks play an important role in the breakdown process, where their growth is transformed from algebraic to exponential as they become part of the secondary instability of the two-dimensional waves. In the presence of streaks, pairing is suppressed and breakdown to turbulence is enhanced.

Experiments are reported on the sustained release of saline and particle-laden fluid into a long, but relatively narrow, flume, filled with fresh water. The dense fluid rapidly spread across the flume and flowed away from the source: the motion was then essentially two-dimensional. In the absence of a background flow in the flume, the motion was symmetric, away from the source. However, in the presence of a background flow the upstream speed of propagation was slowed and the downstream speed was increased. Measurements of this motion are reported and, when the excess density was due to the presence of suspended sediment, the distribution of the deposited particles was also determined. Alongside this experimental programme, new theoretical models of the motion were developed. These were based upon multi-layered depth-averaged shallow-water equations, in which the interfacial drag and mixing processes were explicitly modelled. While the early stages of the motion are independent of these interfacial phenomena to leading order, they play an increasingly important dynamical role as the the flow is slowed, or even arrested. In addition a new integral model is proposed. This does not resolve the interior dynamics of the flow, but may be readily integrated and obviates the need for more lengthy numerical calculations. It is shown that the predictions from both the shallow-layer and integral models are in close agreement with the experimental observations.

The finite wavelength instability of viscosity-stratified three-layer flow down an inclined wall is examined for small but finite Reynolds numbers. It has previously been demonstrated using linear theory that three-layer zero-Reynolds-number instabilities can have growth rates that are orders of magnitude larger than those that arise in two-layer structures. Although the layer configurations yielding large growth instabilities have been well characterized, the physical origin of the three-layer inertialess instability remains unclear. Using analytic, numerical and experimental techniques, we investigate the origin and evolution of these instabilities. Results from an energy equation derived from linear theory reveal that interfacial shear and Reynolds stresses contribute to the energy growth of the instability at finite Reynolds numbers, and that this remains true in the limit of zero Reynolds number. This is thus a rare example that demonstrates how the Reynolds stress can play an important role in flow instability, even when the Reynolds number is vanishingly small. Numerical solutions of the Navier–Stokes equations are used to simulate the nonlinear evolution of the interfacial deformation, and for small amplitudes the predicted wave shapes are in excellent agreement with those obtained from linear theory. Further comparisons between simulated interfacial deformations and linear theory reveal that the linear evolution equations are surprisingly accurate even when the interfaces are highly deformed and nonlinear effects are important. Experimental results obtained using aqueous gelatin systems exhibit large wave growth and are in agreement with both the theoretical predictions of small-amplitude behaviour and the nonlinear simulations of the large-amplitude behaviour. Quantitative agreement is confounded owing to water diffusion driven by differences in gelatin concentration between the layers in experiments. However, the qualitative agreement is sufficient to confirm that the correct mechanism for the experimental instability has been determined.

We combine elements of poroelasticity and of fluid mechanics to construct a mathematical model of the human brain and ventricular system. The model is used to study hydrocephalus, a pathological condition in which the normal flow of the cerebrospinal fluid is disturbed, causing the brain to become deformed. Our model extends recent work in this area by including flow through the aqueduct, by incorporating boundary conditions that we believe accurately represent the anatomy of the brain and by including time dependence. This enables us to construct a quantitative model of the onset, development and treatment of this condition. We formulate and solve the governing equations and boundary conditions for this model and give results that are relevant to clinical observations.