This paper presents the stochastic properties of orbital velocities of random water waves in intermediate water depth. Both the emergence effect and weak nonlinear effects are studied; the theoretical predictions are compared with measured kinematics and the deviations from linear theory are quantified.

This study includes new ideas in fluid dynamics. An analytic formula for probability distribution for velocities modified by the emergence effect as well as by nonlinearities of the wave motion in intermediate water depth is developed. This probability function gives us the first statistical moment, the second statistical moment for modified velocities in an analytical form, and by numerical integration the third statistical moment for modified velocities.

The theoretical formulae for the statistical moments for surface elevation and for velocities up to third order, with nonlinearities of the motion taken into account, for the case when the emergence effect can be neglected, i.e. below the surface layer, have been developed. This includes a generalized formula for free-surface elevation setdown and calculation of the asymmetry of the horizontal velocity, which is found to be negative in agreement with measurements of Anastasiou et al. (1982b).

From the first statistical moment of the modified horizontal velocity, the mean flux between any two levels in the wave flume may be calculated. When the integration is carried out from the bottom up to + ∞, it leads in approximation to the formula for total mean flux found by Phillips (1960). This agreement with Phillips’ formula encourages one to interpret the positive mean value of horizontal velocities as a ‘real current’. This interpretation also provides a new understanding of the fluid dynamic implications of results presented by Tung (1975).

Theoretical prediction of the measured kinematics has allowed a better estimation of the return flow in the wave flume, and in the vicinity of the mean water level currents in two different directions are noted. Firstly, the emergence effect gives rise to a current at the mean water level in the direction of the wave advance. Secondly, a flow in the opposite direction, interpreted as a return current in the wave flume, is noticed just below that level.

The aim of the paper is to examine the nonlinear dynamics of a truncated system modelling low-Prandtl-number thermal convection. The model describes situations where the primary flow is not a straight roll and the dynamics is dominated by the production of axial flow along the axis of bent rolls or of swirl along ring vortices. The physical mechanism for these processes is a spontaneous growth (i.e. bifurcation) of a vertical vorticity mode, breaking the two-dimensional symmetry of the system. A description of the model can be found in Massaguer & Mercader (1988) where the physics and the numerical results have been checked against laboratory experiments. The nonlinear dynamics of that model will be discussed in the more academic case of free boundaries, as it has been shown that for sufficiently small Prandtl numbers straight rolls cannot be expected to be the primary flow near the onset of convection (Busse & Bolton 1984). Two clearly differentiated time-dependent regimes have been found and they correspond to small and intermediate Péclet numbers. In the former regime there exists a transition to chaos with the whole scenario being dependent on a symmetry invariance common to a large number of confined flows. The route to chaos is made up of a sequence of homoclinic explosions giving rise to a cascade of period doublings, with the whole scenario being different from that of a Feigenbaum's cascade.

Planar flow past multiple successive blades and wakes is studied for nearly aligned configurations with normal non-symmetry inducing lift. The typical blade lies relatively near the centreline of the oncoming wake from the preceding blade. The central motion over a wide parameter range is in condensed periodic boundary layers and wakes with fixed displacement, buried within surrounding incident shear flow. This is accompanied, however, by streamwise jumps in the pressure, velocity and mass flux, across the leading edge of each blade, a new and surprising feature which is supported by the combination of incident shears and a solid surface and which is related to the normal flow through the multi-blade system. The leading-edge jumps are required in order to satisfy the equi-pressure condition at the trailing edge. Computational results include separating flows and show the lift and drag, and these are followed by a short-blade analysis which captures the main flow properties explicitly. The results agree qualitatively with experiments and direct simulations for rotor blade flows. The jump feature also extends for example to a single blade immersed in the relatively large wake of an upstream blade.

We consider steady compressible Euler flow corresponding to the compressible analogue of the well-known incompressible Hill's spherical vortex (HSV). We first derive appropriate compressible Euler equations for steady homentropic flow and show how these may be used to define a continuation of the HSV to finite Mach number M∞=U∞/C∞, where U∞, C∞ are the fluid velocity and speed of sound at infinity respectively. This is referred to as the compressible Hill's spherical vortex (CHSV). It corresponds to axisymmetric compressible Euler flow in which, within a vortical bubble, the azimuthal vorticity divided by the product of the density and the distance to the axis remains constant along streamlines, with irrotational flow outside the bubble. The equations are first solved numerically using a fourth-order finite-difference method, and then using a Rayleigh–Janzen expansion in powers of M2∞ to order M4∞. When M∞>0, the vortical bubble is no longer spherical and its detailed shape must be determined by matching conditions consisting of continuity of the fluid velocity at the bubble boundary. For subsonic compressible flow the bubble boundary takes an approximately prolate spheroidal shape with major axis aligned along the flow direction. There is good agreement between the perturbation solution and Richardson extrapolation of the finite difference solutions for the bubble boundary shape up to M∞ equal to 0.5. The numerical solutions indicate that the flow first becomes locally sonic near or at the bubble centre when M∞≈0.598 and a singularity appears to form at the sonic point. We were unable to find shock-free steady CHSVs containing regions of locally supersonic flow and their existence for the present continuation of the HSV remains an open question.

Steady flow of an incompressible homogeneous fluid over shallow topography in a rotating annulus is considered. The flow is driven by a differentially rotating lid. The recycling nature of the system means that prescribed upstream conditions are not available to close the problem. Consideration of the balance of transport across streamlines for the geostrophic flow leads to a general circulation condition; namely $\Gamma(\psi) = \frac{1}{2}\Gamma_T(\psi)$, where ϕ is the stream function for the geostrophic flow, Λ is the circulation around a streamline and ΛT is the circulation around the same path calculated using the prescribed upper surface velocity. Using this condition, stream functions for the linear viscous and nonlinear quasi-inviscid flow can be found. Solutions for these two limits, and linearized perturbation solutions for the transition regime between them, are presented for flow over ridges in the annulus.

A transonic-flow theory of thin, oblique wing of high aspect ratio is presented, which permits a delineation of the influence of wing sweep, the centre-line curvature, and other three-dimensional (3D) effects on the nonlinear mixed flow in the framework of small-disturbance theory. In the (parameter) domain of interest, the flow field far from the wing section pertains to a high subsonic, or linear sonic, outer flow, representable by a Prandtl–Glauert solution involving a swept, as well as curved, lifting line in the leading approximation.

Among the 3D effects is one arising from the compressibility correction to the velocity divergence, absent in classical works; this effect also leads to a correction in the outer flow in the form of an oblique line source. More important is the upwash corrections which includes the influence of both the near and far wakes, as well as the local curvature of the centre-line. For straight oblique wings, local similarities exist in the 3D flow structure, permitting the reduced equations to be solved once for all span stations. An analogy also exists between the oblique-wing problem and that of a 2D transonic flow which is weakly time-dependent; this provides an alternative method of solving numerically the inner airfoil problem.

Solutions to the reduced problem are demonstrated and compared with full-potential solutions for elliptic oblique wings involving high subcritical as well as slightly supercritical component flows.

When a fluid enters a rotating circular pipe, an angular momentum or swirl boundary layer appears at the wall and interacts with the axial momentum boundary layer. In the centre of the pipe, the fluid is free of swirl and is accelerated due to boundary layer growth. Below a critical flow number, defined as the ratio of average axial velocity to circumferential velocity of the pipe, there is flow separation, known in the turbomachinery context as part load recirculation. To describe this phenomenon analytically, we extended boundary layer theory to a swirl boundary layer interacting with the axial momentum boundary layer. The solution of the resulting generalized von Kármán momentum equation takes into account the influence of the Reynolds number and flow number. We show the impact of swirl on the axial boundary layer and conduct experiments in which we vary Reynolds number, flow number and surface roughness to validate the analytical results. The extended boundary layer theory predicts a critical flow number which is analytically derived and validated. Below this critical flow number, separation is expected.

We investigate theoretically and numerically the modulation of near-inertial waves by a larger-amplitude geostrophically balanced mean flow. Because the excited wave is initially trapped in the mixed layer, it projects onto a broad spectrum of vertical modes, each mode $n$ being characterized by a Burger number, $Bu_{n}$ , proportional to the square of the vertical scale of the mode. Using numerical simulations of the hydrostatic Boussinesq equations linearized about a prescribed balanced background flow, we show that the evolution of the wave field depends strongly on the spectrum of $Bu_{n}$ relative to the Rossby number of the balanced flow, $\unicode[STIX]{x1D716}$ , with smaller relative $Bu_{n}$ leading to smaller horizontal scales in the wave field, faster accumulation of wave amplitude in anticyclones and faster propagation of wave energy into the deep ocean. This varied behaviour of the wave may be understood by considering the dynamics in each mode separately; projecting the linearized hydrostatic Boussinesq equations onto modes yields a set of linear shallow water equations, with $Bu_{n}$ playing the role of the reduced gravity. The wave modes fall into two asymptotic regimes, defined by the scalings $Bu_{n}\sim O(1)$ for low modes and $Bu_{n}\sim O(\unicode[STIX]{x1D716})$ for high modes. An amplitude equation derived for the former regime shows that vertical propagation is weak for low modes. The high-mode regime is the basis of the Young & Ben Jelloul (J. Mar. Res., vol. 55, 1997, pp. 735–766) theory. This theory is here extended to $O(\unicode[STIX]{x1D716}^{2})$ , from which amplitude equations for the subregimes $Bu_{n}\sim O(\unicode[STIX]{x1D716}^{1/2})$ and $Bu_{n}\sim O(\unicode[STIX]{x1D716}^{2})$ are derived. The accuracy of each approximation is demonstrated by comparing numerical solutions of the respective amplitude equation to simulations of the linearized shallow water equations in the same regime. We emphasize that since inertial wave energy and shear are distributed across vertical modes, their overall modulation is due to the collective behaviour of the wave field in each regime. A unified treatment of these regimes is a novel feature of this work.

Computations are presented of the upward force on a two-dimensional wedge descending towards a plane surface due to the Stokes flow of an intervening viscous fluid. The predictions are compared with those of lubrication theory and an approximate analytical solution; all three predict a logarithmic divergence of the force with the minimum separation. An object falling vertically under gravity will therefore make contact with an underlying plane surface in finite time if roughened by asperities with sharp corners (with smooth surfaces, contact is made only after infinite time). Contact is still made in finite time if the roughened object also moves horizontally or rotates as it falls.

A numerical analysis of flow around a circular cylinder oscillating in-line with a steady flow is carried out over a range of driving frequencies $(f_{d})$ at relatively low amplitudes $(A)$ and a constant Reynolds number of 175 (based on the free-stream velocity). The vortex shedding is investigated, especially when the shedding frequency $(f_{s})$ synchronises with the driving frequency. A series of modes of synchronisation are presented, which are referred to as the $p/q$ modes, where $p$ and $q$ are natural numbers. When a $p/q$ mode occurs, $f_{s}$ is detuned to $(p/q)f_{d}$, representing the shedding of $p$ pairs of vortices over $q$ cycles of cylinder oscillation. The $p/q$ modes are further characterised by the periodicity of the transverse force over every $q$ cycles of oscillation and a spatial–temporal symmetry possessed by the global wake. The synchronisation modes $(p/q)$ with relatively small natural numbers are less sensitive to the change of external control parameters than those with large natural numbers, while the latter is featured with a narrow space of occurrence. Although the mode of synchronisation can be almost any rational ratio (as shown for $p$ and $q$ smaller than 10), the probability of occurrence of synchronisation modes with $q$ being an even number is much higher than $q$ being an odd number, which is believed to be influenced by the natural even distribution of vortices in the wake of a stationary cylinder.

We consider the linearized form of the regularized 13-moment equations (R13) to model the slow, steady gas dynamics surrounding a rigid, heat-conducting sphere when a uniform temperature gradient is imposed far from the sphere and the gas is in a state of rarefaction. Under these conditions, the phenomenon of thermophoresis, characterized by forces on the solid surfaces, occurs. The R13 equations, derived from the Boltzmann equation using the moment method, provide closure to the mass, momentum and energy conservation laws in the form of constitutive, transport equations for the stress and heat flux that extend the Navier–Stokes–Fourier model to include non-equilibrium effects. We obtain analytical solutions for the field variables that characterize the gas dynamics and a closed-form expression for the thermophoretic force on the sphere. We also consider the slow, streaming flow of gas past a sphere using the same model resulting in a drag force on the body. The thermophoretic velocity of the sphere is then determined from the balance between thermophoretic force and drag. The thermophoretic force is compared with predictions from other theories, including Grad’s 13-moment equations (G13), variants of the Boltzmann equation commonly used in kinetic theory, and with recently published experimental data. The new results from R13 agree well with results from kinetic theory up to a Knudsen number (based on the sphere’s radius) of approximately 0.1 for the values of solid-to-gas heat conductivity ratios considered. However, in this range of Knudsen numbers, where for a very high thermal conductivity of the solid the experiments show reversed thermophoretic forces, the R13 solution, which does result in a reversal of the force, as well as the other theories predict significantly smaller forces than the experimental values. For Knudsen numbers between 0.1 and 1 approximately, the R13 model of thermophoretic force qualitatively shows the trend exhibited by the measurements and, among the various models considered, results in the least discrepancy.

In the dense-inertial regime of granular flow, the stresses scale inertially, but the flow is dominated by clusters of particles. This paper describes observations of cluster development in this regime. Clusters were seen to form for both elastic and inelastic reasons: elastic when the shear rate pushes the particles together faster than the contacts can elastically disperse them, and inelastic as large energy dissipation leads to cluster formation. Furthermore, large particle surface friction leads to cluster formation both for structural reasons, because it generates stronger clusters, and for energetic reasons, as friction dissipates energy. However, the most intriguing result of this work is that clusters appear to have little effect on the rheology of the dense inertial regime, which suggests that one can model the dense inertial regime with entirely collisional hard sphere models, and not have to worry about the complexities of modelling clusters. But at the same time it presents a physical puzzle, as one would normally expect the rheology to be strongly dependent on microstructural features such as clusters, particularly as they present an elastic pathway for internal momentum transport. There is no completely satisfying explanation for why the clusters can be ignored, but two possibilities suggest themselves. Because the clusters are short-lived, it is possible that they do not survive long enough to make a significant contribution to the momentum transport. And it is also possible for the granular temperature that governs transport between clusters to act as a rate-limiting bottleneck that is in overall control of the momentum transport rate.

The behaviour of converging spherical shock waves is considered using Whitham's theory of geometrical shock dynamics. An analysis of converging shocks whose initial shape takes the form of regular polyhedra is presented. The analysis of this problem is motivated by the earlier work on converging cylindrical shocks discussed in Schwendeman & Whitham (1987). In that paper, exact solutions were reported for converging polygonal shocks in which the initial shape re-forms repeatedly as the shock contracts. For the polyhedral case, the analysis is performed both analytically and numerically for an equivalent problem involving shock propagation in a converging channel with triangular cross-section. It is found that a repeating sequence of shock surfaces composed of nearly planar pieces develops, although the initial planar surface does not re-form, and that the increase in strength of the shock at each iterate in the sequence follows the same behaviour as for a converging spherical shock independent of the convergence angle of the channel. In this sense, the shocks are stable and the result is analogous to that found in the two-dimensional case. A numerical study of converging spherical shocks subject to smooth initial perturbations in strength shows a strong tendency to form surfaces composed of nearly planar pieces suggesting that the stability result is fairly general.

A two-time-scale perturbation expansion is used to derive a cross-section-averaged convection–dispersion equation for the particle distribution in the flow of a concentrated suspension of neutrally buoyant, non-colloidal particles through a straight, circular tube. Since the cross-streamline motion of particles is governed by shear-induced migration, the Taylor-dispersion coefficient ${\mathscr{D}}_{eff} $ scales as ${U}^{\prime } {R}^{3} / {a}^{2} $, ${U}^{\prime } $, $R$ and $a$ being the characteristic velocity scale, the tube radius and the particle radius, respectively. Here ${\mathscr{D}}_{eff} $ is found to decrease monotonically with an increase in the particle concentration. The linear dependence of ${\mathscr{D}}_{eff} $ on ${U}^{\prime } $ implies that changes in the cross-section averaged axial concentration profile are dependent only on the total axial strain experienced by the suspension. This stipulates that the spatial evolution of a fluctuation in the concentration of particles in the flowing suspension, or the width of the mixing zone between two regions of different concentrations in the tube will be independent of the suspension velocity in the tube. A second interesting feature in particulate dispersion is that the effective velocity of the particulate phase is concentration-dependent, which, by itself (i.e. without considering Taylor dispersion), can produce either sharpening or relaxation of concentration gradients. In particular, shocks with positive concentration gradients along the flow direction can asymptotically evolve into time-independent distributions in an appropriately chosen frame of reference, and concentration pulses relax asymmetrically. These trends are contrasted with those expected from the classical problem of Taylor dispersion of a passive tracer in the same geometry. The results in this paper are especially relevant for suspension flows through microfluidic geometries, where the induction lengths for shear-induced migration are short.

The shear rheology of an emulsion of viscous drops in the presence of finite inertia is investigated using direct numerical simulation. In the absence of inertia, emulsions display a non-Newtonian rheology with positive first and negative second normal stress differences. However, recently it was discovered that a small amount of drop-level inertia alters their signs – the first normal stress difference becomes negative and the second one becomes positive, each in a small range of capillary numbers (Li & Sarkar, J. Rheol., vol. 49, 2005, pp. 1377–1394). Sign reversal was shown numerically and analytically, but only in the limit of a dilute emulsion where drop–drop interactions were neglected. Here, we compute the rheology of a density- and viscosity-matched emulsion, accounting for the interactions in the volume fraction range of 5 %–27 % and Reynolds number range of 0.1–10. The computed rheological properties (effective shear viscosity and first and second normal stress differences) in the Stokes limit match well with previous theoretical (Choi–Schowalter in the dilute limit) and simulated results (for concentrated systems) using the boundary element method. The two distinct components of the rheology arising from the interfacial stresses at the drop surface and the perturbative Reynolds stresses are investigated as functions of the drop Reynolds number, capillary number and volume fraction. The sign change is caused by the increasing drop inclination in the presence of inertia, which in turn directly affects the interfacial stresses. Increase of the volume fraction or capillary number increases the critical Reynolds number for sign reversals due to enhanced alignment of the drops with the flow directions. The effect of increasing the volume fraction on the rheology is explained by relating it to interactions and specifically to the contact pair-distribution function computed from the simulation. The excess stresses are seen to show an approximately linear behaviour with the Reynolds number in the range of 0.1–5, while with the capillary number and volume fraction, the variation is weakly quadratic.