Using Stokesian dynamics simulations, we examine the flow of a monodisperse, neutrally buoyant, homogeneous suspension of non-Brownian solid spheres in simple shear, starting from a large number of independent hard-sphere distributions and ensemble averaging the results. We construct a novel method for computing the gradient diffusivity via simulations on a homogeneous suspension and, although our results are only approximate due to the small number of particles used in the simulations, we present here the first values of this important parameter, both along and normal to the plane of shear, to be obtained directly either experimentally or numerically. We show furthermore that, although the system of equations describing the particle motions is deterministic, the particle displacements in the two directions normal to the bulk flow have Gaussian distributions with zero mean and a variance which eventually grows linearly in time thereby establishing that the system of particles is diffusive. For particle concentrations up to 45%, we compute the corresponding tracer diffusivities both from the slope of the mean-square particle displacement and by integrating the corresponding velocity autocorrelations and find good agreement between the two sets of results.

Free-surface flows past submerged obstacles in a channel are considered. The fluid is assumed to be inviscid and incompressible and the flow to be irrotational. In previous work involving a single obstacle (Dias & Vanden-Broeck 2002), new solutions called ‘generalized hydraulic falls’ were found. These solutions are characterized by a supercritical flow on one side of the obstacle and a train of waves on the other. However, in the case of a single submerged object, the generalized hydraulic falls are unphysical because the waves do not satisfy the radiation condition. In this paper new solutions for the flow past two obstacles of arbitrary shape are computed. These solutions are characterized by a train of waves ‘trapped’ between the obstacles. The generalized hydraulic falls are shown to describe locally the flow over one of the two obstacles when the distance between the two obstacles is large.

High-resolution shock-capturing finite-volume numerical methods are applied to investigate nonlinear geostrophic adjustment of rectilinear fronts and jets in the rotating shallow-water model. Numerical experiments for various jet/front configurations show that for localized initial conditions in the open domain an adjusted state is always attained. This is the case even when the initial potential vorticity (PV) is not positive-definite, the situation where no proof of existence of the adjusted state is available. Adjustment of the vortex, PV-bearing, part of the flow is rapid and is achieved within a couple of inertial periods. However, the PV-less low-energy quasi-inertial oscillations remain for a long time in the vicinity of the jet core. It is demonstrated that they represent a long-wave part of the initial perturbation and decay according to the standard dispersion law ${\sim}t^{-1/2}$. For geostrophic adjustment in a periodic domain, an exact periodic nonlinear wave solution is found to emerge spontaneously during the evolution of wave perturbations allowing us to conjecture that this solution is an attractor. In both cases of adjustment in open and periodic domains, it is shown that shock-formation is ubiquitous. It takes place immediately in the jet core and, thus, plays an important role in fully nonlinear adjustment. Although shocks dissipate energy effectively, the PV distribution is not changed owing to the passage of shocks in the case of strictly rectilinear flows.

The impact of turbulent fluctuations on the forces exerted by a fluid on a towed spherical particle is investigated by means of high-resolution direct numerical simulations. The measurements are carried out using a novel scheme to integrate the two-way coupling between the particle and the incompressible surrounding fluid flow maintained in a high-Reynolds-number turbulent regime. The main idea consists of combining a Fourier pseudo-spectral method for the fluid with an immersed-boundary technique to impose the no-slip boundary condition on the surface of the particle. This scheme is shown to converge as the power $3/ 2$ of the spatial resolution. This behaviour is explained by the ${L}_{2} $ convergence of the Fourier representation of a velocity field displaying discontinuities of its derivative. Benchmarking of the code is performed by measuring the drag and lift coefficients and the torque-free rotation rate of a spherical particle in various configurations of an upstream-laminar carrier flow. Such studies show a good agreement with experimental and numerical measurements from other groups. A study of the turbulent wake downstream of the sphere is also reported. The mean velocity deficit is shown to behave as the inverse of the distance from the particle, as predicted from classical similarity analysis. This law is reinterpreted in terms of the principle of ‘permanence of large eddies’ that relates infrared asymptotic self-similarity to the law of decay of energy in homogeneous turbulence. The developed method is then used to attack the problem of an upstream flow that is in a developed turbulent regime. It is shown that the average drag force increases as a function of the turbulent intensity and the particle Reynolds number. This increase is significantly larger than predicted by standard drag correlations based on laminar upstream flows. It is found that the relevant parameter is the ratio of the viscous boundary layer thickness to the dissipation scale of the ambient turbulent flow. The drag enhancement can be motivated by the modification of the mean velocity and pressure profile around the sphere by small-scale turbulent fluctuations. It is demonstrated that the variance of the drag force fluctuations can be modelled by means of standard drag correlations. Temporal correlations of the drag and lift forces are also presented.

We propose a cluster-based control strategy for feedback control of post-stall separated flows over an airfoil. The present approach partitions the flow trajectories (force measurements) into clusters, which correspond to characteristic coarse-grained phases in a low-dimensional feature space. A feedback control law (using blowing/suction actuation) is then sought for each cluster state through iterative evaluation and downhill simplex search to minimize power consumption in aerodynamic flight. The optimized control laws re-route the flow trajectories to the aerodynamically favourable regions in the feature space in a model-free manner. Utilizing a limited number of sensor measurements for both clustering and optimization, these feedback laws were determined in only $O(10)$ iterations. The objective of the present work is not necessarily to suppress flow separation but to minimize the desired cost function to achieve enhanced aerodynamic performance. The present approach is applied to the control of two- and three-dimensional separated flows over a NACA 0012 airfoil in large-eddy simulations at an angle of attack of $9^{\circ }$, Reynolds number $Re=23\,000$ and free-stream Mach number $M_{\infty }=0.3$. The optimized control laws avoid the intermittent occurrence of long-period shedding associated with high-drag clusters, thus lowering the mean drag. The present work aims to address some of the challenges associated with feedback control design for turbulent separated flows at moderate Reynolds number.

A pattern-recognition technique, applied to multi-point simultaneous velocity measurements obtained with 45° X-wire anemometer probes, is used to extract and characterize the underlying organized motions, i.e. coherent structures, within the near-wake region of a turbulent round jet discharged perpendicularly from a pipe into a crossflow. This flow has been found to be quite complex owing to its three-dimensional nature and the interactions between several flow regions. Analyses of the underlying coherent structures, which play an important role in the physics of the flow, are still rare and are mostly based on flow-visualization techniques. Using a pattern-recognition technique in conjunction with hot-wire measurements, we recently examined the wake regions of the pipe and jet at levels near the tip of the pipe, and found that Kármán-like vortex structures in the wake of the pipe are locked to similar structures in the jet-wake. In this paper we expand upon our previous work and characterize these structures throughout the wake of the jet up into the region of the bent-over jet – a region where they have not been identified previously. The complex geometry of these structures in the wake of the jet as well as their interaction with the bent-over jet are discussed. The results show that these structures split before they link to similar structures on the opposite side of the symmetry plane in the jet region. The results further suggest that the vorticity due to the structures in the wake of the jet contributes to the motion of the well-known counter-rotating vortex pair.

Nonlinear viscous droplet oscillations are analysed by solving the Navier-Stokes equation for an incompressible fluid. The method is based on mode expansions with modified solutions of the corresponding linear problem. A system of ordinary differential equations, including all nonlinear and viscous terms, is obtained by an extended application of the variational principle of Gauss to the underlying hydrodynamic equations. Results presented are in a very good agreement with experimental data up to oscillation amplitudes of 80% of the unperturbed droplet radius. Large-amplitude oscillations are also in a good agreement with the predictions of Lundgren & Mansour (boundary integral method) and Basaran (Galerkin-finite element method). The results show that viscosity has a large effect on mode coupling phenomena and that, in contradiction to the linear approach, the resonant mode interactions remain for asymptotically diminishing amplitudes of the fundamental mode.

The problem of finite-amplitude thermal convection in a porous layer with finite conducting boundaries is investigated. The nonlinear problem of three-dimensional convection is solved by expanding the dependent variables in terms of powers of the amplitude of convection. The preferred mode of convection is determined by a stability analysis in which arbitrary infinitesimal disturbances are superimposed on the steady solutions. Square-flow-pattern convection is found to be preferred in a bounded region [Gcy ] in the (γb, γt)-space, where γb and γt are the ratios of the thermal conductivities of the lower and upper boundaries to that of the fluid. Two-dimensional rolls are found to be the preferred pattern outside [Gcy ]. The qualitative features of the convection problem appear to be essentially symmetric with respect to γb and γt. The dependence of the heat transported by convection on γb and γt is computed for the various solutions analysed in the paper.

The effect of rapid mean compression on compressible turbulence at a range of turbulent Mach numbers is investigated. Rapid distortion theory (RDT) and direct numerical simulation results for the case of axial (one-dimensional) compression are used to illustrate the existence of two distinct rapid compression regimes. These regimes – the nearly solenoidal and the ‘pressure-released’ – are defined by a single parameter involving the timescales of the mean distortion, the turbulence, and the speed of sound. A general RDT formulation is developed and is proposed as a means of improving turbulence models for compressible flows. In contrast to the well-documented observation that ‘compressibility’ (measured, for example, by the turbulent Mach number) is often associated with a decrease in the growth rate of turbulent kinetic energy, we find that under rapid distortion compressibility can produce an amplification of the kinetic energy growth rate. We also find that as the compressibility increases, the magnitude of the pressure–dilation correlation increases, in absolute terms, but its relative importance decreases compared to the magnitude of the kinetic energy production.

A coloured-smoke-visualization technique has been developed for the investigation of complex three-dimensional fluid flows. In particular, a coloured-smoke wire technique is used for the study of secondary flows in straight turbine cascades. Based on a large number of photographs and direct flow observation, the evolution of horseshoe and passage vortices through a high-turning turbine-blade passage is described.

The collapse of a spherical vapour cavity in the vicinity of a compliant boundary is examined numerically. The fluid is treated as a potential flow and a boundary-element method is used to solve Laplace's equation for the velocity potential. Full nonlinear boundary conditions are applied on the surface of the cavity. The compliant wall is modelled as a membrane with a spring foundation. At the interface between the fluid and the membrane, the pressure and vertical velocity in the flow are matched to the pressure and vertical velocity of the membrane using linearized conditions. The results of calculations are presented which show the effect of the parameters describing the flow (the initial cavity size and position, the fluid density and the pressure driving the collapse) and the parameters describing the compliant wall (the mass per unit area, membrane tension, spring constant and coating radius) on the interaction between the two. When the wall is rigid, the collapse of the cavity is characterized by the formation of a re-entrant jet that is directed toward the wall. However, if the properties of the compliant wall are chosen properly, the collapse can be made to occur spherically, as if the cavity were in an infinite fluid, or with the reentrant jet directed away from the wall, as if the cavity were adjacent to a free surface. This behaviour is in qualitative agreement with the experiments of Gibson & Blake (1982) and Shima, et al. (1989). Calculations of the transfer of energy between the flow and the coating are also presented.

Techniques of bifurcation theory are used to study the porous-medium analogue of the classical Rayleigh-Bénard problem: Lapwood convection in a two-dimensional saturated porous cavity heated from below. Two particular aspects of the problem are focused upon: (i) the existence of multiple steady solutions and (ii) the influence of aspect ratio.

Convection begins only when the applied temperature difference (say) exceeds a critical value defined by linear stability theory. The resulting convective flow pattern depends both on the magnitude of the temperature difference and on the aspect ratio of the cavity. A weakly nonlinear analysis reveals the roles played by so-called secondary bifurcations in determining the formation of further, anomalous patterns at fixed aspect ratio. In addition to giving rise to alternative stable flows for identical operating conditions, the secondary bifurcations are required for the modal exchanges which take place as the aspect ratio varies, a process which causes an abrupt change in preferred flow pattern at certain critical values of the aspect ratio.

As a complement to and an extension of the weakly nonlinear analysis, numerical methods are used to determine the bifurcation processes and to elucidate the modal exchange mechanisms in both weakly and strongly convective flows. The effect of container size is studied by continuation methods to predict the variation of the critical Rayleigh number of the bifurcation points for aspect ratios in the range 0.5 to 2.0. In this way a stability map is obtained which shows the alternative patterns expected for particular operating conditions. The Nusselt number is computed and it is found that the alternative stable modes transfer significantly different amounts of heat through the medium.

The study has provided new information on the existence and characteristics of, and interactions between, alternative steady modes of two-dimensional Lapwood convection. The results have important ramifications for the modelling and design of physical systems in which convective flow in a saturated porous medium is stimulated by an imposed unstable temperature gradient.

The method of characteristics is applied to the nonlinear equations describing two-phase flow in a fluidized bed. The method shows how a small disturbance changes with time and distance and can, eventually, produce a flow discontinuity similar to a shock wave in gases. The parameters entering the analysis are the amplitude of the initial disturbance, the wavelength of the original disturbance, the particulate pressure function, the particulate size, the uniform fluidization voidage, the uniform fluidization velocity, the fluid viscosity, the particulate density, and the fluid density. A parametric study shows that the following factors delay shock formation: a decrease in particulate size, an increase in bed density, an increase in fluid viscosity, and a decrease in particulate density. Experimental data on bubble formation in gas-particulate fluidized beds show that these same factors delay bubble formation. It is concluded, therefore, that the shock front and the bubble front are one and the same thing.

We present an improved upper bound on the energy dissipation rate in plane Couette flow. This is achieved through the numerical solution of the ‘background field’ variational problem formulated by Constantin and Doering with a one-dimensional unidirectional background field. The upper bound presented here both exhausts the bounding potential of the one-dimensional background field problem and also solves the provably equivalent problem formulated by Busse. The solution is calculated up to asymptotically large Reynolds number where we can estimate that the energy dissipation rate $\e \le 0.008553$ as $Re \rightarrow \infty$ (in units of $V^3/d$ where $V$ is the velocity difference across the plates separated by a distance $d$ and $Re = V d /\nu$, with $\nu$ kinematic viscosity). This represents a 21% improvement over the previous best value due to Nicodemus et al. A comparison is drawn between this numerical solution and the so-called multi-alpha asymptotic solutions discovered by Busse.

It is known from recent experiments that the disturbance due to a slender ship advancing in a shallow channel is essentially one-dimensional in the horizontal plane. In particular solitons can be radiated upstream in a transient manner. In this note we develop a theory for soliton radiation by slender bodies. It is shown that, when the ship speed is in the transcritical range, one-dimensional upstream influence can occur even when the channel width is nearly of the order of the ship length but much greater than the ship beam. The theory is also extended to one or more ships travelling in the same channel at near-critical speeds.

Several new families of nonlinear three-dimensional travelling wave solutions to the Navier–Stokes equation, also known as exact coherent states, are computed for Newtonian plane Poiseuille flow. The symmetries and streak/vortex structures are reported and their possible connections to critical layer dynamics are examined. While some of the solutions clearly display fluctuations that are localized around the critical layer (the surface on which the streamwise velocity matches the wave speed of the solution), for others this connection is not as clear. Dynamical trajectories along unstable directions of the solutions are computed. Over certain ranges of Reynolds number, two solution families are shown to lie on the basin boundary between laminar and turbulent flow. Direct comparison of nonlinear travelling wave solutions to turbulent flow in the same channel is presented. The state-space dynamics of the turbulent flow is organized around one of the newly identified travelling wave families, and in particular the lower-branch solutions of this family are closely approached during transient excursions away from the dominant behaviour. These observations provide a firm dynamical-systems foundation for prior observations that minimal channel turbulence displays time intervals of ‘active’ turbulence punctuated by brief periods of ‘hibernation’ (see, e.g., Xi & Graham, Phys. Rev. Lett., vol. 104, 2010, 218301). The hibernating intervals are approaches to lower-branch nonlinear travelling waves. Representing these solutions on a Prandtl–von Kármán plot illustrates how their bulk flow properties are related to those of Newtonian turbulence as well as the universal asymptotic state called maximum drag reduction (MDR) found in viscoelastic turbulent flow. In particular, the lower- and upper-branch solutions of the family around which the minimal channel dynamics is organized appear to approach the MDR asymptote and the classical Newtonian result respectively, in terms of both bulk velocity and mean velocity profile.

An axisymmetric particle sedimenting in an otherwise quiescent Newtonian fluid, in the Stokes regime, retains its initial orientation. For the special case of a spheroidal geometry, we examine analytically the effects of weak inertia and viscoelasticity in driving the particle towards an eventual steady orientation independent of initial conditions. The generalized reciprocal theorem, together with a novel vector spheroidal harmonics formalism, is used to find closed-form analytical expressions for the $O(\mathit{Re})$ inertial torque and the $O(\mathit{De})$ viscoelastic torque acting on a sedimenting spheroid of an arbitrary aspect ratio. Here, $\mathit{Re}=UL/{\it\nu}$ is the Reynolds number, with $U$ being the sedimentation velocity, $L$ the semi-major axis and ${\it\nu}$ the fluid kinematic viscosity, and is a measure of the inertial forces acting at the particle scale. The Deborah number, $\mathit{De}=({\it\lambda}U)/L$, is a dimensionless measure of the fluid viscoelasticity, with ${\it\lambda}$ being the intrinsic relaxation time of the underlying microstructure. The analysis is valid in the limit $\mathit{Re},\mathit{De}\ll 1$, and the effects of viscoelasticity are therefore modelled using the constitutive equation of a second-order fluid. The inertial torque always acts to turn the spheroid broadside-on, while the final orientation due to the viscoelastic torque depends on the ratio of the magnitude of the first ($N_{1}$) to the second normal stress difference ($N_{2}$), and the sign (tensile or compressive) of $N_{1}$. For the usual case of near-equilibrium complex fluids – a positive and dominant $N_{1}$ ($N_{1}>0$, $N_{2}<0$ and $|N_{1}/N_{2}|>1$) – both prolate and oblate spheroids adopt a longside-on orientation. The viscoelastic torque is found to be remarkably sensitive to variations in ${\it\kappa}$ in the slender-fibre limit (${\it\kappa}\gg 1$), where ${\it\kappa}=L/b$ is the aspect ratio, $b$ being the radius of the spheroid (semi-minor axis). The angular dependence of the inertial and viscoelastic torques turn out to be identical, and one may then characterize the long-time orientation of the sedimenting spheroid based solely on a critical value ($\mathit{El}_{c}$) of the elasticity number, $\mathit{El}=\mathit{De}/\mathit{Re}$. For $\mathit{El}<\mathit{El}_{c}~({>}\mathit{El}_{c})$, inertia (viscoelasticity) prevails with the spheroid settling broadside-on (longside-on). The analysis shows that $\mathit{El}_{c}\sim O[(1/\text{ln}\,{\it\kappa})]$ for ${\it\kappa}\gg 1$, and the viscoelastic torque thus dominates for a slender rigid fibre. For a slender fibre alone, we also briefly analyse the effects of elasticity on fibre orientation outside the second-order fluid regime.

Non-isothermal liquid evaporation in micro-pore structures is studied experimentally and numerically using the lattice Boltzmann method. A hybrid thermal entropic multiple-relaxation-time multiphase lattice Boltzmann model (T-EMRT-MP LBM) is implemented and validated with experiments of droplet evaporation on a heated hydrophobic substrate. Then liquid evaporation is investigated in two specific pore structures, i.e. spiral-shaped and gradient-shaped micro-pillar cavities, referred to as SMS and GMS, respectively. In SMS, the liquid receding front follows the spiral pattern; while in GMS, the receding front moves layer by layer from the pillar rows with large pitch to the rows with small one. Both simulations agree well with experiments. Moreover, evaporative cooling effects in liquid and vapour are observed and explained with simulation results. Quantitatively, in both SMS and GMS, the change of liquid mass with time coincides with experimental measurements. The evaporation rate generally decreases slightly with time mainly because of the reduction of liquid–vapour interface. Isolated liquid films in SMS increase the evaporation rate temporarily resulting in local peaks in evaporation rate. Reynolds and capillary numbers show that the liquid internal flow is laminar and that the capillary forces are dominant resulting in menisci pinned to the pillars. Similar Péclet number is found in simulations and experiments, indicating a diffusive type of heat, liquid and vapour transport. Our numerical and experimental studies indicate a method for controlling liquid evaporation paths in micro-pore structures and maintaining high evaporation rate by specific geometry designs.

The near-field breakup and interaction of a hollow-cone liquid sheet with coannular swirling air flow have been examined using high-speed diagnostics. Time-resolved PIV (particle image velocimetry; $3500~\text{frames}~\text{s}^{-1}$) is employed to capture the spatio-temporal behaviour of the swirling air flow field. The combined liquid–gas phase interaction is visualized with the help of high-speed ($20\,000~\text{frames}~\text{s}^{-1}$) shadowgraphy. In this study, the transition from weak to strong spray–swirl interaction is explained based on the momentum ratio. Proper orthogonal decomposition (POD) is implemented on instantaneous PIV and shadowgraphy images to extract the energetic spatial eigenmodes and characteristic modal frequencies. The POD results suggest the dominance of the KH (Kelvin–Helmholtz) instability mechanism (pure axial shear, axial plus azimuthal shear) in swirl–spray interaction. In addition, linear stability analysis also shows the destabilization of the liquid–air interface caused by KH waves ($\unicode[STIX]{x1D706}_{p}$), which arises from the formation of a vorticity layer of thickness $\unicode[STIX]{x1D6FF}_{g}$ near the liquid–air interface. The frequency values obtained from the primary KH wavelength ($\unicode[STIX]{x1D706}_{p}$) exhibit good agreement with the POD modal frequencies. Scaling laws are proposed to elucidate the relationships between the global length scales (breakup length, spray spread) and the primary wavelength. The breakup length scale and liquid sheet oscillations are meticulously analysed in the time domain to reveal the breakup dynamics of the liquid sheet. Furthermore, the large-scale coherent structures of the swirl flow exhibit different sheet breakup phenomena in the spatial domain. For instance, flapping breakup is induced by the central toroidal recalculation zone in the swirling flow field. Finally, the ligament formation mechanism and its diameter, i.e. the size of first-generation droplets, are measured with phase Doppler interferometry. The measured sizes scale reasonably with KH waves.

When a millimetre-sized liquid drop approaches a deep liquid pool, both the interface of the drop and the pool deform before the drop touches the pool. The build-up of air pressure prior to coalescence is responsible for this deformation. Due to this deformation, air can be entrained at the bottom of the drop during the impact. We quantify the amount of entrained air numerically, using the boundary integral method for potential flow for the drop and the pool, coupled to viscous lubrication theory for the air film that has to be squeezed out during impact. We compare our results with various experimental data and find excellent agreement for the amount of air that is entrapped during impact onto a pool. Next, the impact of a rigid sphere onto a pool is numerically investigated and the air that is entrapped in this case also matches with available experimental data. In both cases of drop and sphere impact onto a pool the numerical air bubble volume $V_{b}$ is found to be in agreement with the theoretical scaling $V_{b}/V_{drop/sphere}\sim \mathit{St}^{-4/3}$, where $\mathit{St}$ is the Stokes number. This is the same scaling as has been found for drop impact onto a solid surface in previous research. This implies a universal mechanism for air entrainment for these different impact scenarios, which has been suggested in recent experimental work, but is now further elucidated with numerical results.