The flow field created by swimming micro-organisms not only enables their locomotion but also leads to advective transport of nutrients. In this paper we address analytically and computationally the link between unsteady feeding and unsteady swimming on a model micro-organism, the spherical squirmer, actuating the fluid in a time-periodic manner. We start by performing asymptotic calculations at low Péclet number ($\mathit{Pe}$) on the advection–diffusion problem for the nutrients. We show that the mean rate of feeding as well as its fluctuations in time depend only on the swimming modes of the squirmer up to order ${\mathit{Pe}}^{3/ 2} $, even when no swimming occurs on average, while the influence of non-swimming modes comes in only at order ${\mathit{Pe}}^{2} $. We also show that generically we expect a phase delay between feeding and swimming of $1/ 8\mathrm{th} $ of a period. Numerical computations for illustrative strokes at finite $\mathit{Pe}$ confirm quantitatively our analytical results linking swimming and feeding. We finally derive, and use, an adjoint-based optimization algorithm to determine the optimal unsteady strokes maximizing feeding rate for a fixed energy budget. The overall optimal feeder is always the optimal steady swimmer. Within the set of time-periodic strokes, the optimal feeding strokes are found to be equivalent to those optimizing periodic swimming for all values of the Péclet number, and correspond to a regularization of the overall steady optimal.

Transfer of mechanical energy between solid spherical particles and a Newtonian carrier fluid has been explored in two-way coupled direct numerical simulations of turbulent channel flow. The inertial particles have been treated as individual point particles in a Lagrangian framework and their feedback on the fluid phase has been incorporated in the Navier–Stokes equations. At sufficiently large particle response times the Reynolds shear stress and the turbulence intensities in the spanwise and wall-normal directions were attenuated whereas the velocity fluctuations were augmented in the streamwise direction. The physical mechanisms involved in the particle–fluid interactions were analysed in detail, and it was observed that the fluid transferred energy to the particles in the core region of the channel whereas the fluid received kinetic energy from the particles in the wall region. A local imbalance in the work performed by the particles on the fluid and the work exerted by the fluid on the particles was observed. This imbalance gave rise to a particle-induced energy dissipation which represents a loss of mechanical energy from the fluid–particle suspension. An independent examination of the work associated with the different directional components of the Stokes force revealed that the dominating energy transfer was associated with the streamwise component. Both the mean and fluctuating parts of the Stokes force promoted streamwise fluctuations in the near-wall region. The kinetic energy associated with the cross-sectional velocity components was damped due to work done by the particles, and the energy was dissipated rather than recovered as particle kinetic energy. Componentwise scatter plots of the instantaneous velocity versus the instantaneous slip-velocity provided further insight into the energy transfer mechanisms, and the observed modulations of the flow field could thereby be explained.

Direct numerical simulations (DNS) are performed of a transient channel flow following a rapid increase of flow rate from an initially turbulent flow. It is shown that a low-Reynolds-number turbulent flow can undergo a process of transition that resembles the laminar–turbulent transition. In response to the rapid increase of flow rate, the flow does not progressively evolve from the initial turbulent structure to a new one, but undergoes a process involving three distinct phases (pre-transition, transition and fully turbulent) that are equivalent to the three regions of the boundary layer bypass transition, namely, the buffeted laminar flow, the intermittent flow and the fully turbulent flow regions. This transient channel flow represents an alternative bypass transition scenario to the free-stream-turbulence (FST) induced transition, whereby the initial flow serving as the disturbance is a low-Reynolds-number turbulent wall shear flow with pre-existing streaky structures. The flow nevertheless undergoes a ‘receptivity’ process during which the initial structures are modulated by a time-developing boundary layer, forming streaks of apparently specific favourable spacing (of about double the new boundary layer thickness) which are elongated streamwise during the pre-transitional period. The structures are stable and the flow is laminar-like initially; but later in the transitional phase, localized turbulent spots are generated which grow spatially, merge with each other and eventually occupy the entire wall surfaces when the flow becomes fully turbulent. It appears that the presence of the initial turbulent structures does not promote early transition when compared with boundary layer transition of similar FST intensity. New turbulent structures first appear at high wavenumbers extending into a lower-wavenumber spectrum later as turbulent spots grow and join together. In line with the transient energy growth theory, the maximum turbulent kinetic energy in the pre-transitional phase grows linearly but only in terms of ${u}^{\ensuremath{\prime} } $, whilst ${v}^{\ensuremath{\prime} } $ and ${w}^{\ensuremath{\prime} } $ remain essentially unchanged. The energy production and dissipation rates are very low at this stage despite the high level of ${u}^{\ensuremath{\prime} } $. The pressure–strain term remains unchanged at that time, but increases rapidly later during transition along with the generation of turbulent spots, hence providing an unambiguous measure for the onset of transition.

Common simplified models for surface gravity waves result in parabolic-type equations. These equations mostly assume a negligible reflection from the bottom variations but account for both refraction and diffraction effects. A common deficiency of these equations is an inherent assumption of normally incident waves, which cause an increasing error, as the incident waves propagate from an increasing attack angle. In the nonlinear formulation of this type of equation, previous research added the assumption of small crossing angles between interacting waves, which also limits the applicability of these nonlinear models to narrow directional spectra. The current paper presents a parabolic approximation for oblique incident waves that overcomes these limitations. This is done by introducing a perturbation solution for the wave’s phase function, which in its lowest order corresponds to oblique incident waves on a bottom with no lateral changes. The resulting curved wave ray structure replaces the simplified straight one used in the derivation process of various former models. In the nonlinear model, the nonlinear interactions are calculated between the frequency modes while taking into account the different propagating directions. Numerical results of the oblique parabolic equation show great advantages compared to other formulations that use a straight wave ray structure and the small crossing angle assumption. The nonlinear model was used to investigate the nonlinear shoaling of two similar monochromatic waves approaching from different angles. This fundamental nonlinear interaction problem shows that there is an energy transfer to waves of the primary harmonic that approach at larger attack angles than the two incident waves. This is counter-intuitive. Unlike in the case of linear wave shoaling, where the waves reduce their attack angle in the refraction process, in the nonlinear shoaling process the attack angle can increase. Fundamental numerical simulations of the linear and nonlinear oblique parabolic models are shown to be in excellent agreement with the initial mild-slope equations. This confirms the benefits of applying these models to various shoaling scenarios for both linear and nonlinear waves.

Turbulent fluctuations induce the common phenomenon known as clustering in the spatial arrangement of small inertial particles transported by the fluid. Particles spread non-uniformly, and form clusters where their local concentration is much higher than in nearby rarefaction regions. The underlying physics has been exhaustively analysed in the so-called one-way coupling regime, i.e. negligible back-reaction of the particles on the fluid, where the mean flow anisotropy induces preferential orientation of the clusters. Turbulent transport in suspensions with significant mass in the disperse phase, i.e. particles back-reacting in the carrier phase (the two-way coupling regime), has instead been much less investigated and is still poorly understood. The issue is discussed here by addressing direct numerical simulations of particle-laden homogeneous shear flows in the two-way coupling regime. Consistent with previous findings, we observe an overall depletion of the turbulent fluctuations for particles with response time of the order of the Kolmogorov time scale. The depletion occurs in the energy-containing range, while augmentation is observed in the small-scale range down to the dissipative scales. Increasing the mass load results in substantial broadening of the energy cospectrum, thereby extending the range of scales driven by anisotropic production mechanisms. As discussed throughout the paper, this is due to the clusters which form the spatial support of the back-reaction field and give rise to a highly anisotropic forcing, active down to the smallest scales. A certain impact on two-phase flow turbulence modelling is expected from the above conclusions, since the frequently assumed small-scale isotropy is poorly recovered when the coupling between the phases becomes significant.

A model for the instantaneous wall-shear-stress distribution is presented for zero-pressure-gradient turbulent boundary layers. The model, based on empirical and theoretical considerations, is able to reconstruct a statistically representative fluctuating wall-shear-stress time-series, ${ \tau }_{w}^{\ensuremath{\prime} } (t)$, using only the low-frequency content of the streamwise velocity measured in the logarithmic region, away from the wall. Results, including spectra and second-order moments, show that the model is capable of successfully capturing Reynolds number trends as observed in other studies.

Direct and large eddy simulations are performed to study the internal waves generated by the oscillation of a barotropic tide over a model ridge of triangular shape. The objective is to go beyond linear theory and assess the role of nonlinear interactions including turbulence in situations with low tidal excursion number. The criticality parameter, defined as the ratio of the topographic slope to the characteristic slope of the tidal rays, is varied from subcritical to supercritical values. The barotropic tidal forcing is also systematically increased. Numerical results of the energy conversion are compared with linear theory and, in laminar flow at low forcing, they agree well in subcritical and supercritical cases but not at critical slope angle. In critical and supercritical cases with higher forcing, there are convective overturns, turbulence and significant reduction (as much as 25 %) of the radiated wave flux with respect to laminar flow results. Analysis of the baroclinic energy budget and spatial modal analysis are performed to understand the reduction. The near-bottom velocity is intensified at critical angle slope leading to a radiated internal wave beam as well as an upslope bore of cold water with a thermal front. In the critical case, the entire slope has turbulence while, in the supercritical case, turbulence originates near the top of the topography where the slope angle transitions through the critical value. The phase dependence of turbulence within a tidal cycle is examined and found to differ substantially between the ridge slope and the ridge top where the beams from the two sides cross.

This paper analyses subcritical transition to instability, also known as triggering in thermoacoustic systems, with an example of a Rijke tube model with an explicit time delay. Linear stability analysis of the thermoacoustic system is performed to identify parameter values at the onset of linear instability via a Hopf bifurcation. We then use the method of multiple scales to recast the model of a general thermoacoustic system near the Hopf point into the Stuart–Landau equation. From the Stuart–Landau equation, the relation between the nonlinearity in the model and the criticality of the ensuing bifurcation is derived. The specific example of a model for a horizontal Rijke tube is shown to lose stability through a subcritical Hopf bifurcation as a consequence of the nonlinearity in the model for the unsteady heat release rate. Analytical estimates are obtained for the triggering amplitudes close to the critical values of the bifurcation parameter corresponding to loss of linear stability. The unstable limit cycles born from the subcritical Hopf bifurcation undergo a fold bifurcation to become stable and create a region of bistability or hysteresis. Estimates are obtained for the region of bistability by locating the fold points from a fully nonlinear analysis using the method of harmonic balance. These analytical estimates help to identify parameter regions where triggering is possible. Results obtained from analytical methods compare reasonably well with results obtained from both experiments and numerical continuation.

We present an experimental approach used to measure both normal stress differences and the particle phase contribution to the normal stresses in suspensions of non-Brownian hard spheres. The methodology consists of measuring the radial profile of the normal stress along the velocity gradient direction in a torsional flow between two parallel discs. The values of the first and the second normal stress differences, ${N}_{1} $ and ${N}_{2} $, are deduced from the measurement of the slope and of the origin ordinate. The measurements are carried out for a wide range of particle volume fractions (between 0.2 and 0.5). As expected, ${N}_{2} $ is measured to be negative but ${N}_{1} $ is found to be positive. We discuss the validity of the method and present numerous tests that have been carried out in order to validate our results. The experimental setup also allows the pore pressure to be measured. Then, subtracting the pore pressure from the total stress, ${\mbrm{\Sigma} }_{\mathbf{22} } $, the contribution of the particles to the normal stress ${ \mbrm{\Sigma} }_{\mathbf{22} }^{\mathbi{p}} $ is obtained. Most of our results compare well with the different experimental and numerical data present in the literature. In particular, our results show that the magnitude of the particle stress tensor component and their dependence on the particle volume fraction used in the suspension model balance proposed by Morris & Boulay (J. Rheol., vol. 43, 1999, p. 1213) are suitable.

The spreading of an incompressible viscous liquid over an isotropic homogeneous unsaturated porous substrate is considered. It is shown that, unlike the dynamic wetting of an impermeable solid substrate, where the dynamic contact angle has to be specified as a boundary condition in terms of the wetting velocity and other flow characteristics, the ‘effective’ dynamic contact angle on an unsaturated porous substrate is completely determined by the requirement of existence of a solution, i.e. the absence of a non-integrable singularity in the spreading fluid’s pressure at the ‘effective’ contact line. The obtained velocity dependence of the ‘effective’ contact angle determines the critical point at which a transition to a different flow regime takes place, where the fluid above the substrate stops spreading whereas the wetting front inside it continues to propagate.

Using a slip-length-based level-set approach with adaptive mesh refinement, we have simulated axisymmetric droplet spreading for a dimensionless slip length down to $O(1{0}^{\ensuremath{-} 4} )$. The main purpose is to validate, and where necessary improve, the asymptotic analysis of Cox (J. Fluid Mech., vol. 357, 1998, pp. 249–278) for rapid droplet spreading/dewetting, in terms of the detailed interface shape in various regions close to the moving contact line and the relation between the apparent angle and the capillary number based on the instantaneous contact-line speed, $\mathit{Ca}$. Before presenting results for inertial spreading, simulation results are compared in detail with the theory of Hocking & Rivers (J. Fluid Mech., vol. 121, 1982, pp. 425–442) for slow spreading, showing that these agree very well (and in detail) for such small slip-length values, although limitations in the theoretically predicted interface shape are identified; a simple extension of the theory to viscous exterior fluids is also proposed and shown to yield similar excellent agreement. For rapid droplet spreading, it is found that, in principle, the theory of Cox can predict accurately the interface shapes in the intermediate viscous sublayer, although the inviscid sublayer can only be well presented when capillary-type waves are outside the contact-line region. However, $O(1)$ parameters taken to be unity by Cox must be specified and terms be corrected to ${\mathit{Ca}}^{+ 1} $ in order to achieve good agreement between the theory and the simulation, both of which are undertaken here. We also find that the apparent angle from numerical simulation, obtained by extrapolating the interface shape from the macro region to the contact line, agrees reasonably well with the modified theory of Cox. A simplified version of the inertial theory is proposed in the limit of negligible viscosity of the external fluid. Building on these results, weinvestigate the flow structure near the contact line, the shear stress and pressure along the wall, and the use of the analysis for droplet impact and rapid dewetting. Finally, we compare the modified theory of Cox with a recent experiment for rapid droplet spreading, the results of which suggest a spreading-velocity-dependent dynamic contact angle in the experiments. The paper is closed with a discussion of the outlook regarding the potential of using the present results in large-scale simulations wherein the contact-line region is not resolved down to the slip length, especially for inertial spreading.

We report measurements of the properties of turbulent thermal convection of a fluid with a Prandtl number $\mathit{Pr}= 4. 38$ in a cylindrical cell with an aspect ratio $\Gamma = 0. 50$. The rotational symmetry was broken by a small tilt of the sample axis relative to gravity. Measurements of the heat transport (as expressed by the Nusselt number Nu), as well as properties of the large-scale circulation (LSC) obtained from temperature measurements along the sidewall, are presented. In contradistinction to similar experiments using containers of aspect ratio $\Gamma = 1. 00$ (Ahlers et al., J. Fluid Mech., vol. 557, 2006b, pp. 347–367) and $\Gamma = 0. 50$ (Chillà et al., Eur. Phys. J. B, vol. 40, 2004, pp. 223–227; Sun, Xi & Xia, Phys. Rev. Lett., vol. 95, 2005, p. 074502; Roche et al., New J. Phys., vol. 12, 2010, p. 085014), we see a very small increase of the heat transport for tilt angles up to about 0.1 rad. Based on measurements of properties of the LSC we explain this increase by a stabilization of the single-roll state (SRS) of the LSC and a destabilization of the double-roll state (DRS) (it is known from previous work that the SRS has a slightly larger heat transport than the DRS). Quantitative measurements of the strength and the orientation of the LSC show that its azimuthal diffusion is suppressed with increasing tilt whereas the torsional oscillation becomes more pronounced and its frequency increases.

As a generalization of the mass–flux based classical stream tube, the concept of momentum and energy transport tubes is discussed as a flow visualization tool. These transport tubes have the property that no fluxes of momentum or energy exist over their respective tube mantles. As an example application using data from large eddy simulation, such tubes are visualized for the mean-flow structure of turbulent flow in large wind farms, in fully developed wind-turbine-array boundary layers. The three-dimensional organization of energy transport tubes changes considerably when turbine spacings are varied, enabling the visualization of the path taken by the kinetic energy flux that is ultimately available at any given turbine within the array.

We investigate solutions of the two-dimensional Navier–Stokes equation in a $\lrm{\pi} \ensuremath{\times} \lrm{\pi} $ square box with stress-free boundary conditions. The flow is steadily forced by the addition of a source $\sin nx\sin ny$ to the vorticity equation; attention is restricted to even $n$ so that the forcing has zero integral. Numerical solutions with $n= 2$ and $6$ show that at high Reynolds numbers the solution is a domain-scale vortex condensate with a strong projection on the gravest mode, $\sin x\sin y$. The sign of the vortex condensate is selected by a symmetry-breaking instability. We show that the amplitude of the vortex condensate has a finite limit as $\nu \ensuremath{\rightarrow} 0$. Using a quasilinear approximation we make an analytic prediction of the amplitude of the condensate and show that the amplitude is determined by viscous selection of a particular solution from a family of solutions to the forced two-dimensional Euler equation. This theory indicates that the condensate amplitude will depend sensitively on the form of the dissipation, even in the undamped limit. This prediction is verified by considering the addition of a drag term to the Navier–Stokes equation and comparing the quasilinear theory with numerical solution.

Resuspension and saltation of nearly neutrally buoyant polystyrene beads $({d}_{p} = 583\pm 14. 4~\lrm{\ensuremath{\mu}} \mathrm{m} , {\rho }_{p} = 1050~\mathrm{kg} ~{\mathrm{m} }^{\ensuremath{-} 3} )$ in a turbulent boundary layer were studied using time-resolved particle image velocimetry and particle tracking velocimetry in a horizontal water channel facility $({\mathit{Re}}_{h} = 7353)$. The time difference between frames was $ \mrm{\Delta} {t}^{+ } = 0. 297$, comparable to the particle Stokes number, ${ \tau }_{p}^{+ } = 0. 267$. Near-wall coherent structures were visualized using spatial distributions of vorticity and swirling strength in combination with those of the instantaneous ${u}_{1} {u}_{2} $ correlations and ${u}_{1} $. Two case studies, the first on resuspension and the second on saltation, showed that in all cases lift-off coincided with the passage of a vortex core, creating an ejection-sweep cycle (‘burst’) responsible for lift-off. In all cases beads left the wall when immersed in near-wall ejections and exposed to positive shear. As a consequence, a high shear-induced lift force coincided with bead lift-off, while the Magnus force due to bead rotation and translation-induced lift were negligible. The wall-normal component of the drag force mostly opposed lift-off, causing the bead’s deceleration. The difference between resuspension and saltation was governed by the type of coherent flow structures encountered by the beads when lifted out of the viscous sublayer. Resuspension was observed when beads were carried upwards by the combined action of a strong, spatially coherent upstream fast moving $({u}_{1} \gt 0)$ flow structure and a downstream ejection. On the other hand, saltation was accompanied by similar but weaker and spatially less coherent near-wall turbulence structures.

We study the thickness ${h}_{d} $ of the liquid film left on a wet surface after scraping it with an elastic wiper (length $L$, rigidity $B$) moved at a velocity $V$. The scraper is clamped vertically at a given distance above the substrate, and ${h}_{d} $ is maximal when the tip of the scraper is just tangent to the surface. We show experimentally and theoretically that this maximum thickness is ${h}_{\mathit{max}} \simeq 0. 33L \mathop{ (\eta V{L}^{2} / B)}\nolimits ^{3/ 4} , $ where $\eta $ is the liquid viscosity. The deposition law is found to be sensitive to the shape of the wiper: the film thickness can also be tuned by using wipers with a permanent curvature, and varying this curvature.

The paper describes the energy fluxes simultaneously occurring in the space of scales and in the physical space of wall-turbulent flows. The unexpected behaviour of the energy fluxes consists of spiral-like paths in the combined physical/scale space where the controversial reverse energy cascade plays a central role. Two dynamical processes are identified as driving mechanisms for the fluxes, one in the near-wall region and a second one further away from the wall. The former, stronger, one is related to the dynamics involved in the near-wall turbulence regeneration cycle. The second suggests an outer self-sustaining mechanism which is asymptotically expected to take place in the eventual log layer and could explain the debated mixed inner/outer scaling of the near-wall statistics. The observed behaviour may have strong repercussions on both theoretical and modelling approaches to wall turbulence, as anticipated by a simple equation which is shown able to capture most of the rich dynamics of the shear-dominated region of the flow.

A concept which explains unstable modes by resonance between two neutral waves is applied to instability in shear flows with non-uniform potential vorticity distribution. In basic flows where shear instability occurs, a pair of neutral waves which should resonate to form an unstable mode does not exist mathematically unless the basic potential vorticity is uniform, owing to the shear of the basic flow. However, if a small window region is made where the potential vorticity of the basic flow is uniform, regular neutral modes can exist. The neutral wave located at the centre of this window region, which corresponds to a mode where the jump in its first derivative at one end of the window region cancels that at the other end, can be considered to be the hidden neutral wave. It is equivalent to a continuous mode which does not have a jump in its first derivative at the critical point even though it can have a singularity of $(y\ensuremath{-} {y}_{c} )\log \vert y\ensuremath{-} {y}_{c} \vert $, in the limit of an infinitely small window region. Thus, a method for retrieving neutral waves hidden in the shear of the basic flow is proposed here. By sweeping the basic flow range with this uniform potential vorticity window, all the features of the hidden neutral wave appear. By applying this method, shear instability in a flow with a tanh-shaped basic velocity profile is well explained via the resonance between neutral waves.

Research into high-Reynolds-number turbulent boundary layers in recent years has brought about a renewed interest in the larger-scale structures. It is now known that these structures emerge more prominently in the outer region not only due to increased Reynolds number (Metzger & Klewicki, Phys. Fluids, vol. 13(3), 2001, pp. 692–701; Hutchins & Marusic, J. Fluid Mech., vol. 579, 2007, pp. 1–28), but also when a boundary layer is exposed to an adverse pressure gradient (Bradshaw, J. Fluid Mech., vol. 29, 1967, pp. 625–645; Lee & Sung, J. Fluid Mech., vol. 639, 2009, pp. 101–131). The latter case has not received as much attention in the literature. As such, this work investigates the modification of the large-scale features of boundary layers subjected to zero, adverse and favourable pressure gradients. It is first shown that the mean velocities, turbulence intensities and turbulence production are significantly different in the outer region across the three cases. Spectral and scale decomposition analyses confirm that the large scales are more energized throughout the entire adverse pressure gradient boundary layer, especially in the outer region. Although more energetic, there is a similar spectral distribution of energy in the wake region, implying the geometrical structure of the outer layer remains universal in all cases. Comparisons are also made of the amplitude modulation of small scales by the large-scale motions for the three pressure gradient cases. The wall-normal location of the zero-crossing of small-scale amplitude modulation is found to increase with increasing pressure gradient, yet this location continues to coincide with the large-scale energetic peak wall-normal location (as has been observed in zero pressure gradient boundary layers). The amplitude modulation effect is found to increase as pressure gradient is increased from favourable to adverse.

When the linear stability analysis is applied to the time-averaged flow past a circular cylinder after the primary instability of the wake, a nearly marginally stable global mode is predicted with a frequency in time equal to that of the saturated vortex shedding. This behaviour has recently been shown to hold up to Reynolds number $\mathit{Re}= 600$ by direct numerical simulations. In the present work we verify that the global stability analysis provides reasonable estimation also when applied to experimental velocity fields measured in the wake past a porous circular cylinder at $\mathit{Re}\simeq 3. 5\ensuremath{\times} 1{0}^{3} $. Different intensities of continuous suction and blowing through the entire surface of the cylinder are considered. The global direct and adjoint stability modes, derived from the experimental data, are used to sort the random instantaneous snapshots of the velocity field in phase. The proposed method is remarkable, sorting the snapshots in phase with respect to the vortex shedding, allowing phase-averaged velocity fields to be extracted from the experimental database. The phase-averaged flow fields are analysed in order to study the effect of the transpiration on the kinematical characteristics of the large-scale wake vortices.