Owen Phillips grew up in Sydney, Australia, and following a distinguished record at a State high school and in the final NSW school examinations, he enrolled in the Faculty of Engineering at the University of Sydney in 1948. In the third year, he transferred to the Faculty of Science to do more advanced courses in Mathematics and Physics (with the idea of going back to Engineering after one year and qualifying for a Science degree on the way). Owen did so well, however, that he went on to do a fourth year in Mathematics and graduated with First Class Honours.

The transit of a two-dimensional elastic fluid-filled capsule through a channel with a side branch is investigated numerically. The mathematical formulation allows for a capsule carried in a pressure-driven flow of fluid of generally different viscosity to that inside the capsule. Far upstream and downstream in the main channel, and downstream in the side branch, the fluid velocity profiles are assumed to adopt those of unidirectional Poiseuille flow with prescribed flow rates. The capsule boundary is treated as a two-dimensional elastic membrane developing elastic tensions and bending moments according to simple constitutive laws. A boundary-integral formulation allows for the explicit computation of the fluid pressures upstream and downstream of the branching. The novelty of the approach is the inclusion of a notional boundary at the entrance to the side branch, which avoids the need to collocate the channel ends. The deformation experienced by the capsule in the region of the junction is found to depend strongly on the branch angle. The deformation is ameliorated by increasing the membrane stiffness or lowering the viscosity of the suspending fluid relative to the encapsulated fluid. When a capsule exits the branch region, a distance of many decades of capsule diameters is required before the capsule relaxes to an equilibrium shape. Capsule residence times in the vicinity of the branch region can be considerable, depending on the line of approach into the junction and the capsule deformability. The path selection of a cell at a branch junction can depend crucially on capsule deformability: capsules with different elastic properties may follow different routes out of the junction in otherwise identical flow conditions.

Two-layer fluids of finite depth under gravity are the simplest configuration capable of supporting internal wave motion. The extent to which such systems can be used to provide quantitative information on smoothly stratified fluids, in configurations relevant for geophysical applications, is analysed and a model of practical interest derived. The model is based on long-wave asymptotic expansions and on first-principle criteria for an optimal choice of effective two-layer parameters for the incompressible, smoothly stratified Euler equations. The accuracy of the model is extensively tested, via fully resolved numerical computations, on the class of travelling wave solutions supported by smooth stratification systems. It is found that, despite the severe restrictions posed by the discrete two-layer density assumption, solitary wave solutions corresponding to experimentally realizable parametric values can be accurately predicted, in both wave and fluid parcel markers, such as phase speed and density fields, respectively. Thanks to this analysis, explicit closed-form solutions are available for all relevant physical quantities. The agreement between the simple, optimized two-layer model and the parent smooth-stratification Euler system persists up to extreme cases, such as that of internal fronts, and even up to thicknesses of the pycnocline comparable to that of the effective layers.

We discuss how a solid textured with well-defined micropillars entrains a film when extracted out of a bath of wetting liquid. At low withdrawal velocity V, it is shown experimentally that the film exactly fills the gap between the pillars; its thickness hd is independent of V and corresponds to the pillar height hp. At larger velocity, hd slowly increases with V and tends towards the Landau–Levich–Derjaguin (LLD) thickness hLLD observed on a flat solid. We model the entrainment by adapting the LLD theory to a double layer consisting of liquid trapped inside the texture and covered by a free film. This model allows us to understand quantitatively our different observations and to predict the transition between hp and hLLD.

The effect of an incoming wake on the flow around and heat transfer from the stagnation region of a circular cylinder was studied using direct numerical simulations (DNSs). Four simulations were carried out at a Reynolds number (based on free-stream velocity and cylinder diameter D) of ReD = 13200: one two-dimensional (baseline) simulation and three three-dimensional simulations. The three-dimensional simulations comprised a baseline simulation with a uniform incoming velocity field, a simulation in which realistic wake data – generated in a separate precursor DNS – were introduced at the inflow plane and, finally, a simulation in which the turbulent fluctuations were removed from the incoming wake in order to study the effect of the mean velocity deficit on the heat transfer in the stagnation region. In the simulation with realistic wake data, the incoming wake still exhibited the characteristic meandering behaviour of a near-wake. When approaching the regions immediately above and below the stagnation line of the cylinder, the vortical structures from the wake were found to be significantly stretched by the strongly accelerating wall-parallel (circumferential) flow into elongated vortex tubes that became increasingly aligned with the direction of flow. As the elongated streamwise vortical structures impinge on the stagnation region, on one side they transport cool fluid towards the heated cylinder, while on the other side hot fluid is transported away from the cylinder towards the free stream, thereby increasing the heat transfer. The DNS results are compared with various semi-empirical correlations for predicting the augmentation of heat transfer due to free-stream turbulence.

A method for predicting the outcome of vortex breakup in a rotating flow is introduced. The vortices dealt with here are subject to both centrifugal and barotropic instabilities. The prediction of the aftermath of the breakup relies on knowing how both centrifugal and barotropic instabilities would equilibrate separately. A theoretical model for non-linear equilibration in centrifugal instability is wedded to two-dimensional simulation of barotropic instability to predict the final vortices that emerge from the debris of the original vortex. This prediction method is tested against three-dimensional Navier–Stokes simulations. For vortices in which a rapid centrifugal instability triggers a slower barotropic instability, the method is successful both qualitatively and quantitatively. The skill of the prediction method decreases as the time scales of the two instabilities become comparable.

An analytical and computational study of Lagrangian trajectories for linear shear flow past a sphere or spheroid at low Reynolds numbers is presented. Using the exact solutions available for the fluid flow in this geometry, we discover and analyse blocking phenomena, local bifurcation structures and their influence on dynamical effects arising in the fluid particle paths. In particular, building on the work by Chwang & Wu, who established an intriguing blocking phenomenon in two-dimensional flows, whereby a cylinder placed in a linear shear prevents an unbounded region of upstream fluid from passing the body, we show that a similar blocking exists in three-dimensional flows. For the special case when the sphere is centred on the zero-velocity plane of the background shear, the separatrix streamline surfaces which bound the blocked region are computable in closed form by quadrature. This allows estimation of the cross-sectional area of the blocked flow showing how the area transitions from finite to infinite values, depending on the cross-section location relative to the body. When the sphere is off-centre, the quadrature appears to be unavailable due to the broken up-down mirror symmetry. In this case, computations provide evidence for the persistence of the blocking region. Furthermore, we document a complex bifurcation structure in the particle trajectories as the sphere centre is moved from the zero-velocity plane of the background flow. We compute analytically the emergence of different fixed points in the flow and characterize the global streamline topology associated with these fixed points, which includes the emergence of a three-dimensional bounded eddy. Similar results for the case of spheroids are considered in Appendix B. Additionally, the broken symmetry offered by a tilted spheroid geometry induces new three-dimensional effects on streamline deflection, which can be viewed as effective positive or negative suction in the horizontal direction orthogonal to the background flow, depending on the tilt orientation. We conclude this study with results on the case of a sphere embedded at a generic position in a rotating background flow, with its own prescribed rotation including fixed and freely rotating. Exact closed-form solutions for fluid particle trajectories are derived.

We analyse a simple ‘Stokesian squirmer’ model for the enhanced mixing due to swimming micro-organisms. The model is based on a calculation of Thiffeault & Childress (Phys. Lett. A, vol. 374, 2010, p. 3487), where fluid particle displacements due to inviscid swimmers are added to produce an effective diffusivity. Here we show that, for the viscous case, the swimmers cannot be assumed to swim an infinite distance, even though their total mass displacement is finite. Instead, the largest contributions to particle displacement, and hence to mixing, arise from random changes of direction of swimming and are dominated by the far-field stresslet term in our simple model. We validate the results by numerical simulation. We also calculate non-zero Reynolds number corrections to the effective diffusivity. Finally, we show that displacements due to randomly swimming squirmers exhibit probability distribution functions with exponential tails and a short-time superdiffusive regime, as found previously by several authors. In our case, the exponential tails are due to ‘sticking’ near the stagnation points on the squirmer's surface.

Extensive numerical simulations of fetch-limited growth of wind-driven waves are analysed within two approaches: a ‘traditional’ wind-speed scaling first proposed by Kitaigorodskii (Bull. Acad. Sci. USSR, Geophys. Ser., Engl. Transl., vol. N1, 1962, p. 105) in the early 1960s and an alternative weakly turbulent scaling developed recently by Badulin et al. (J. Fluid Mech.591, 2007, 339–378). The latter one uses spectral fluxes of wave energy, momentum and action as physical scales of the problem and allows for advanced qualitative and quantitative analysis of wind-wave growth and features of air–sea interaction. In contrast, the traditional approach is shown to be descriptive rather than proactive. Numerical simulations are conducted on the basis of the Hasselmann kinetic equation for deep-water waves in a wide range of wind speeds from 5 to 30 m s −1 and for the ideal case of fetch-limited growth: permanent wind blowing perpendicularly to a straight coastline. Two different wave input functions, Sin, and two methods for calculating the nonlinear transfer term Snl (Gaussian quadrature method, or GQM, a quasi-exact method based on the use of Gaussian quadratures, and the discrete interaction approximation, or DIA) are used in the simulations. Comparison of the corresponding results firstly shows the relevance of the analysis of wind-wave growth in terms of the proposed weakly turbulent scaling, and secondly, allows us to highlight some critical points in the modelling of wind-generated waves. Three stages of wind-wave development corresponding to qualitatively different balance of the source terms, Sin, Sdiss and Snl, are identified: initial growth, growing sea and fully developed sea. Validity of the asymptotic weakly turbulent approach for the stage of growing wind sea is determined by the dominance of nonlinear transfers, which results in a rigid link between spectral fluxes and wave energy. This stage of self-similar growth is investigated in detail and presented as a consequence of three sub-stages of qualitatively different coupling of air flow and growing wind waves. The key self-similarity parameter of the asymptotic theory is estimated to be αss = 0.68 ± 0.1.

Further prospects of wind-wave modelling in the context of the presented weakly turbulent scaling are discussed.

An analytical solution is presented for steady inviscid separated flows modelled by hollow vortices, that is, by closed vortex sheets bounding a region with fluid at rest. Steady flows past arbitrary obstacles protruding from an infinite wall are considered. The solution is similar to that of the vortex patch model; it depends on two free parameters that define the size of the hollow vortex and the location of the separation point. When a sharp edge constrains the separation point (Kutta condition), the solution depends on a single parameter. As with the vortex patch model, families of growing vortices exist, which represent the continuation of desingularized point vortices. Numerical results are presented for the flows past a semicircular bump, a Ringleb snow cornice and a normal flat plate. The differences from the previous results found in the literature are analysed and discussed with the present solutions for the flow past a normal flat plate.

Global-mode analysis is applied to a cold, M = 2.5 laminar jet. Global modes of the non-parallel jet capture directly both near-field dynamics and far-field acoustics which, in this case, are coupled by Mach wave radiation. In addition to type (a) modes corresponding to Kelvin–Helmholtz instability, it is found that the jet also supports upstream-propagating type (b) modes which could not be resolved by previous analyses of the parabolized stability equations. The locally neutrally propagating part of a type (a) mode consists of the growth and decay of an aerodynamic wavepacket attached to the jet, coupled with a beam of acoustic radiation at a low angle to the jet downstream axis. Type (b) modes are shown to be related to the subsonic family of modes predicted by Tam & Hu (1989). Finally, significant transient growth is recovered by superposing damped, but non-normal, global modes, leading to a novel interpretation of jet noise production. The mechanism of optimal transient growth is identified with a propagating aerodynamic wavepacket which emits an acoustic wavepacket to the far field at an axial location consistent with the peaks of the locally neutrally propagating parts of type (a) modes.

Poiseuille and thermal transpiration flows of a highly rarefied gas are investigated on the basis of the linearized Boltzmann equation, with a special interest in the over-concentration of molecules on velocities parallel to the walls. An iterative approximation procedure with an explicit error estimate is presented, by which the structure of the over-concentration is clarified. A numerical computation on the basis of the procedure is performed for a hard-sphere molecular gas to construct a database that promptly gives the induced net mass flow for an arbitrary value of large Knudsen numbers. An asymptotic formula of the net mass flow is also presented for molecular models belonging to Grad's hard potential. Finally, the resemblance of the profiles between the heat flow of the Poiseuille flow and the flow velocity of the thermal transpiration is pointed out. The reason is also given.

This paper is concerned with an analysis of the near-tip region of a fluid-driven fracture propagating in a permeable saturated rock. The analysis is carried out by considering the stationary problem of a semi-infinite fracture moving at constant speed V. Two basic dissipative processes are taken into account: fracturing of the rock and viscous flow in the fracture, and two fluid balance mechanisms are considered – leak-off and storage of the fracturing fluid in the fracture. It is shown that the solution is characterized by a multiscale singular behaviour at the tip, and that the nature of the dominant singularity depends both on the relative importance of the dissipative processes and on the scale of reference. This solution provides a framework to understand the interaction of representative physical processes near the fracture tip, as well as to track the changing nature of the dominant tip process(es) with the tip velocity and its impact on the global fracture response. Furthermore, it gives a universal scaling of the near-tip processes on the scale of the entire fracture and sets the foundation for developing efficient numerical algorithms relying on accurate modelling of the tip region.

We calculate the dynamics of an uncharged colloidal sphere embedded in a quenched polyelectrolyte hydrogel to (i) an oscillatory (optical and magnetic) force, as adopted in classical micro-rheology, and (ii) an oscillatory electric field, as adopted in electrical micro-rheology and electro-acoustics. The hydrogel is modelled as a linearly elastic porous medium with the charge fixed to the skeleton and saturated with a Newtonian electrolyte; and the colloidal inclusion is modelled as a rigid, impenetrable sphere. The dynamic micro-rheological susceptibility, defined as the ratio of the particle displacement to the strength of an applied oscillatory force, depends on the fixed-charge density and ionic strength and is bounded by the limits for incompressible and uncharged, compressible skeletons. Nevertheless, the influences of fixed charge and ionic strength vanish at frequencies above the reciprocal draining time, where the polymer and the electrolyte hydrodynamically couple as a single incompressible phase. Generally, the effects of fixed charge and ionic strength are small compared with, for example, the influences of polymer slip at the particle surface. The electrical susceptibility, defined as the ratio of the particle displacement to the strength of an applied oscillatory electric field, is directly influenced by charge at all frequencies, irrespective of skeleton compressibility. At low frequencies, polymer charge modulates the driving (electro-osmotic) and restoring (electrostatically enhanced elastic) forces, whereas charge has no influence on the restoring force at high frequencies where dilational strain is suppressed by hydrodynamic coupling with the electrolyte. In striking contrast to charged inclusions in uncharged hydrogels (Wang & Hill, J. Fluid Mech., vol. 640, 2009, pp. 357–400), the electrical susceptibility at high frequencies is independent of electrolyte concentration. Rather, the dynamics primarily reflect the elastic modulus, charge and hydrodynamic permeability, with a relatively weak dependence on particle size. Interestingly, the dynamic mobility in the zero-momentum reference frame, which is central to the electro-acoustic response, is qualitatively different from the dynamic mobility in the skeleton-fixed reference frame. Finally, we propose a phenomenological harmonic-oscillator model to address – in an approximate manner – the dynamics of charged particles in charged hydrogels. This shows that particle dynamics at low frequencies are dominated by particle charge, whereas high-frequency dynamics are dominated by hydrogel charge.

Experiments are conducted to study the planing and skipping of a rectangular paddle on the surface of a shallow stream. The paddle is allowed to move freely up and down by attaching it to a pivoted arm. A steady planing state, in which the lift force from the water balances the weight on the paddle, is found to be stable for small stream velocities but to become unstable above a certain threshold velocity which depends upon the weight and the angle of attack. Above this threshold, the paddle oscillates in the water and can take off into a continual bouncing, or skipping, motion, with a well-defined amplitude and frequency. The transition is sometimes bistable so that both a steady planing state and a regular skipping state are possible for the same experimental parameters. Shallow-water theory is used to construct simple models that explain the qualitative features of the planing and skipping states in the experiments. It is found that a simple parameterisation of the lift force on the paddle proportional to the depth of entry is not sufficient to explain the observations, and it is concluded that the rise of water ahead of the paddle, in particular the way this varies over time, is responsible for causing the planing state to become unstable and for enabling a continual skipping state.

A study is presented of the generation of internal tides by barotropic tidal flow over topography in the shape of a double ridge. An iterative map is constructed to expedite the search for the closed ray paths that form wave attractors in this geometry. The map connects the positions along a ray path of consecutive reflections from the surface, which is double-valued owing to the presence of both left- and right-going waves, but which can be made into a genuine one-dimensional map using a checkerboarding algorithm. Calculations are then presented for the steady-state scattering of internal tides from the barotropic tide above the double ridges. The calculations exploit a Green function technique that distributes sources along the topography to generate the scattering, and discretizes in space to calculate the source density via a standard matrix inversion. When attractors are present, the numerical procedure appears to fail, displaying no convergence with the number of grid points used in the spatial discretizations, indicating a failure of the Green function solution. With the addition of dissipation into the problem, these difficulties are avoided, leading to convergent numerical solutions. The paper concludes with a comparison between theory and a laboratory experiment.

The global temporal stability of three-dimensional compressible flow about a yawed parabolic body of infinite span is investigated using an iterative eigenvalue technique in combination with direct numerical simulations. The computed global spectrum provides a comprehensive picture of the temporal perturbation dynamics of the flow, and a wide and rich variety of modes has been uncovered for the investigated parameter choices: stable and unstable boundary-layer modes, different types of stable and unstable acoustic modes, and stable wavepacket modes have been found. A parameter study varying the spanwise perturbation wavenumber and the sweep Reynolds number reproduced a preferred spanwise length scale and a critical Reynolds number for a boundary-layer or acoustic instability. Convex leading-edge curvature has been found to have a strongly stabilizing effect on boundary-layer modes but only a weakly stabilizing effect on acoustic modes. Furthermore, for certain parameter choices, the acoustic modes have been found to dominate the boundary-layer modes.

Direct numerical simulation (DNS) of a spatially developing turbulent boundary layer (TBL) over a wall roughened with regularly arrayed cubes was performed to investigate the effects of three-dimensional (3-D) surface elements on the properties of the TBL. The cubes were staggered in the downstream direction and periodically arranged in the streamwise and spanwise directions with pitches of px/k = 8 and pz/k = 2, where px and pz are the streamwise and spanwise spacings of the cubes and k is the roughness height. The Reynolds number based on the momentum thickness was varied in the range Reθ = 300−1300, and the roughness height was k = 1.5θin, where θin is the momentum thickness at the inlet, which corresponds to k/δ = 0.052–0.174 from the inlet to the outlet; δ is the boundary layer thickness. The characteristics of the TBL over the 3-D cube-roughened wall were compared with the results from a DNS of the TBL over a two-dimensional (2-D) rod-roughened wall. The introduction of cube roughness affected the turbulent Reynolds stresses not only in the roughness sublayer but also in the outer layer. The present instantaneous flow field and linear stochastic estimations of the conditional averaging showed that the streaky structures in the near-wall region and the low-momentum regions and hairpin packets in the outer layer are dominant features in the TBLs over the 2-D and 3-D rough walls and that these features are significantly affected by the surface roughness throughout the entire boundary layer. In the outer layer, however, it was shown that the large-scale structures over the 2-D and 3-D roughened walls have similar characteristics, which indicates that the dimensional difference between the surfaces with 2-D and 3-D roughness has a negligible effect on the turbulence statistics and coherent structures of the TBLs.

Aerodynamic interference between two cylinders involves most of the generic flow features associated with multiple structures, thus providing an excellent model for gaining physical insight into the wake of multiple cylindrical structures. This work aims to provide an experimental systematic study of the flow behind two side-by-side square cylinders. The square cylinder is a representative model for bluff bodies with sharp corners, characterized by a fixed flow separation point, which are distinct from those of continuous curvature with oscillating separation points, typically represented by the circular cylinder. Experiments were performed at a Reynolds number Re of 4.7 × 104 and a cylinder centre-to-centre spacing ratio T/d (d is the cylinder height) of 1.02–6.00. The flow was measured using different techniques, including hot wires, load cell, particle imaging velocimetry and laser-induced fluorescence flow visualization. Four distinct flow regimes and their corresponding T/d ranges are identified for the first time on the basis of the flow structure and the Strouhal number. Physical aspects in each regime, such as interference between shear layers, gap flow deflection and changeover, multiple flow modes, entrainment, recirculation bubble, vortex interactions and formation lengths, are investigated in detail and are connected to the characteristics of the time-averaged and fluctuating fluid forces. The flow displays a marked difference in many facets from that behind two side-by-side circular cylinders, which is linked to their distinct flow separation natures. A crucial role played by the gap flow and its passage geometry in contributing to the observed difference is also unveiled.

In this paper, we generalize the concept of low-Mach-number approximation to multi-phase flows and apply it to the two-phase flow model of Papalexandris (J. Fluid Mech., vol. 517, 2004, p. 103) for granular materials. In our approach, the governing system of equations is first non-dimensionalized with values that correspond to a reference thermodynamic state of the phase with the smaller speed of sound. By doing so, the Mach number based on this reference state emerges as a perturbation parameter of the equations in hand. Subsequently, we expand each variable in power series of this parameter and apply singular perturbation techniques to derive the low-Mach-number equations. As expected, the resulting equations are considerably simpler than the unperturbed compressible equations. Our methodology is quite general and can be directly applied for the systematic reduction of continuum models for granular materials and for many different types of multi-phase flows. The structure of the low-Mach-number equations for two special cases of particular interest, namely, constant-density flows and the equilibrium limit is also discussed and analysed. The paper concludes with some proposals for experimental validation of the equations.