We study the linear stability of gravitationally unstable, transient, diffusive boundary layers in porous media using non-modal stability theory. We first perform a classical optimization procedure, using an adjoint-based method, to obtain the perturbations at the initial time $t= {t}_{p} $ that have a maximum amplification at a final time $t= {t}_{f} $. We then investigate the sensitivity of the optimal perturbations to the initial time, ${t}_{p} $, and the final time, ${t}_{f} $, as well as different measures of perturbation amplification. Due to the transient nature of the base state, we demonstrate that there is an optimal initial perturbation time, ${ t}_{p}^{o} $. By rescaling the problem, we develop analytical relationships for the optimal initial time and wavenumber in terms of aquifer properties. We also demonstrate that the classical optimization procedure essentially recovers the dominant perturbation structures predicted by a quasi-steady modal analysis. Although the classical optimal perturbations are mathematically valid, we observe that due to physical constraints, they are unlikely to reflect analogous laboratory experiments. Therefore, we propose a modified optimization procedure (MOP) that constrains the optimization to physically admissible initial perturbation fields. We compare the results of the classical and modified optimization procedures with quasi-steady modal analyses and initial value problems commonly used in the literature. Finally, we validate the predictions of the modified optimization scheme by performing direct numerical simulations (DNS) that emulate the onset of convection in physical systems.

An oscillating continuously stratified flow past an isolated obstacle is investigated using scaling arguments and two-dimensional non-hydrostatic numerical experiments. A new dynamic scaling is introduced that incorporates the blocking of fluid with insufficient energy to overcome the background stratification and crest the obstacle. This clarifies the distinction between linear and nonlinear flow regimes near the crest of the obstacle. The flow is decomposed into propagating and non-propagating components. In the linear limit, the non-propagating component is related to the unstratified potential flow past the obstacle and the radiating component exhibits narrow wave beams that are tangent to the obstacle at critical points. When the flow is nonlinear, the near crest flow oscillates between states that include asymmetric, crest-controlled flows. Thin, fast, supercritical layers plunge in the lee, separate from the obstacle and undergo shear instability in the fluid interior. These flow features are localized to the neighbourhood of the crest where the flow transitions from subcriticality to supercriticality and are non-propagating. The nonlinear excitation of energetic non-propagating components reduces the efficiency of topographic radiation in comparison with linear dynamics.

Analysis based on the velocity potential free flow theory with the fully nonlinear boundary condition is made for the steady flow generated by a body gliding along a free surface. Employing the integral hodograph method, we derive analytical expressions for the complex velocity and for the derivative of the complex potential with the coordinate of a parameter plane. The boundary value problem is transformed into a system of two integro-differential equations for the velocity modulus on the free surface and for the slope of the wetted body surface in the parameter plane. The same slope and curvature of the free surface and the body surface at the intersection are adopted to determine the separation points of the flow and from the body. Numerical results are provided for a gliding flat plate and a circular cylinder. The pressure distribution along the body and the free surface shape are presented for a wide range of Froude numbers, within the limit for which the solution corresponding to non-breaking waves downstream can be obtained.

We note that the mirror-symmetric solution in plane Couette flow, found recently by Gibson, Halcrow & Cvitanović (J. Fluid Mech., vol. 611, 2009, pp. 107–130) and Itano & Generalis (Phys. Rev. Lett., vol. 102, 2009, p. 114501), belongs to the solution group classified as ‘ribbon’ in rotating-plane Couette flow (RPCF). It represents a subcritical (in terms of the system rotation) solution at zero rotation rate on the three-dimensional tertiary flow branch which bifurcates from the second streamwise-independent flow in RPCF. The way of its appearance is similar to that of the Nagata solution (J. Fluid Mech., vol. 217, 1990, pp. 519–527), which lies on the subcritical three-dimensional tertiary flow branch bifurcating from the first streamwise-independent flow in RPCF.

Clear differences in turbulence intensity profiles in smooth, transitional and fully rough zero-pressure-gradient boundary layers are demonstrated, using the diagnostic plot introduced by Alfredsson, Segalini & Örlü (Phys. Fluids, vol. 23, 2011, p. 041702) – ${u}^{\prime } / U$ versus $U/ {U}_{e} $, where ${u}^{\prime } $ and $U$ are the local (root mean square) fluctuating and mean velocities and ${U}_{e} $ is the free stream velocity. A wide range of published data are considered and all zero-pressure-gradient boundary layers yield outer flow ${u}^{\prime } / U$ values that are roughly linearly related to $U/ {U}_{e} $, just as for smooth walls, but with a significantly higher slope which is completely independent of the roughness morphology. The difference in slope is due largely to the influence of the roughness parameter ($ \mathrm{\Delta} {U}^{+ } $ in the usual notation) and all the data can be fitted empirically by using a modified form of the scaling, dependent only on $ \mathrm{\Delta} U/ {U}_{e} $. The turbulence intensity, at a location in the outer layer where $U/ {U}_{e} $ is fixed, rises monotonically with increasing $ \mathrm{\Delta} U/ {U}_{e} $ which, however, remains of $O(1)$ for all possible zero-pressure-gradient rough-wall boundary layers even at the highest Reynolds numbers. A measurement of intensity at a point in the outer region of the boundary layer can provide an indication of whether the surface is aerodynamically fully rough, without having to determine the surface stress or effective roughness height. Discussion of the implication for smooth/rough flow universality of differences in outer-layer mean velocity wake strength is included.

Sound induced by arrays of $10\times 4$ roughness elements in low-Mach-number turbulent boundary layers at ${\mathit{Re}}_{\theta } = 3065$ is studied with Lighthill’s theory and large-eddy simulation. Three roughness fetches consisting of hemispheres, cuboids and short cylinders are considered. The roughness elements of different shapes have the same height of $0. 124\delta $, the same element-to-element spacing of $0. 727\delta $ and the same flow blockage area. The acoustically compact roughness elements and their images in the wall radiate sound primarily as acoustic dipoles in the plane of wall. The dipole strength, orientation and spatial distribution show strong dependence on the roughness shape. Correlations between dipole sources associated with neighbouring elements are found to be small for these sparsely distributed roughness arrays. Correlations and coherence between roughness dipoles and surface pressure fluctuations are analysed, which reveals the importance of the impingement of upstream turbulence and surrounding vortical structures to dipole sound radiation, especially in the streamwise direction. For roughness shapes with sharp frontal edges, the edge-induced unsteady separation and reattachment also play important roles in sound generation. Large-scale turbulent structures in the boundary layer have a relatively low influence on roughness dipoles, except for the first row of elements.

The dynamics of a drop deforming, orienting and moving in a shear flow of a viscoelastic liquid near a wall is numerically investigated using a front-tracking finite-difference method and a semi-analytic theory. The viscoelasticity is modelled using the modified FENE-CR constitutive equation. In a Newtonian system, deformation in a drop breaks the reversal symmetry of the system resulting in a migration away from the wall. This study shows that the matrix elasticity reduces the migration velocity, the reduction scaling approximately linearly with viscoelasticity (product of the Deborah number De and the ratio of polymer viscosity to total viscosity $\beta $). Similar to a Newtonian system, for small Deborah numbers, the dynamics quickly reaches a quasi-steady state where deformation, inclination, as well as migration and slip velocities become independent of the initial drop–wall separation. They all approximately scale inversely with the square of the instantaneous separation except for deformation which scales inversely with the cube of separation. The deformation shows a non-monotonic variation with increasing viscoelasticity similar to the case of a drop in an unbounded shear and is found to influence little the change in migration. Two competing effects due to matrix viscoelasticity on drop migration are identified. The first stems from the reduced inclination angle of the drop with increasing viscoelasticity that tries to enhance migration velocity. However, it is overcome by the second effect inhibiting migration that results from the normal stress differences from the curved streamlines around the drop; they are more curved on the side away from the wall compared with those in the gap between the wall and the drop, an effect that is also present for a rigid particle. A perturbative theory of migration is developed for small ratio of the drop size to its separation from the wall that clearly shows the migration to be caused by the image stresslet field due to the drop in presence of the wall. The theory delineates the two competing viscoelastic effects, their relative magnitudes, and predicts migration that matches well with the simulation. Using the simulation results and the stresslet theory, we develop an algebraic expression for the quasi-steady migration velocity as a function of Ca, De and $\beta $. The transient dynamics of the migrating drop is seen to be governed by the finite time needed for development of the viscoelastic stresses. For larger capillary numbers, in both Newtonian and viscoelastic matrices, a viscous drop fails to reach a quasi-steady state independent of initial drop–wall separation. Matrix viscoelasticity tends to prevent drop breakup. Drops that break up in a Newtonian matrix are stabilized in a viscoelastic matrix if it is initially far away from the wall. Initial proximity to the wall enhances deformation and aids in drop breakup.

Turbulent rotating convection is usually studied in a cylindrical geometry, as this is its most convenient experimental realization. In our previous work (Kunnen et al., J. Fluid Mech., vol. 688, 2011, pp. 422–442) we studied turbulent rotating convection in a cylinder with the emphasis on the boundary layers. A secondary circulation with a convoluted spatial structure has been observed in mean velocity plots. Here we present a linear boundary-layer analysis of this flow, which leads to a model of the circulation. The model consists of two independent parts: an internal recirculation within the sidewall boundary layer, and a bulk-driven domain-filling circulation. Both contributions exhibit the typical structure of the Stewartson boundary layer near the sidewall: a sandwich structure of two boundary layers of typical thicknesses ${E}^{1/ 4} $ and ${E}^{1/ 3} $, where $E$ is the Ekman number. Although the structure of the bulk-driven circulation may change considerably depending on the Ekman number, the boundary-layer recirculation is present at all Ekman numbers in the range $0. 72\times 1{0}^{- 5} \leq E\leq 5. 76\times 1{0}^{- 5} $ considered here.

Understanding the behaviour of particles entrained in a fluid flow upon changes in flow direction is crucial in problems where particle inertia is important, such as the erosion process in pipe bends. We present results on the impact of particles in a T-shaped channel in the laminar–turbulent transitional regime. The impacting event for a given system is described in terms of the Reynolds number and the particle Stokes number. Experimental results for the impact are compared with the trajectories predicted by theoretical particle-tracing models for a range of configurations to determine the role of the viscous boundary layer in retarding the particles and reducing the rate of collision with the substrate. In particular, a two-dimensional model based on a stagnation-point flow is used together with three-dimensional numerical simulations. We show how the simple two-dimensional model provides a tractable way of understanding the general collision behaviour, while more advanced three-dimensional simulations can be helpful in understanding the details of the flow.

We present the results of a combined experimental and theoretical investigation of droplets walking on a vertically vibrating fluid bath. Several walking states are reported, including pure resonant walkers that bounce with precisely half the driving frequency, limping states, wherein a short contact occurs between two longer ones, and irregular chaotic walking. It is possible for several states to arise for the same parameter combination, including high- and low-energy resonant walking states. The extent of the walking regime is shown to be crucially dependent on the stability of the bouncing states. In order to estimate the resistive forces acting on the drop during impact, we measure the tangential coefficient of restitution of drops impacting a quiescent bath. We then analyse the spatio-temporal evolution of the standing waves created by the drop impact and obtain approximations to their form in the small-drop and long-time limits. By combining theoretical descriptions of the horizontal and vertical drop dynamics and the associated wave field, we develop a theoretical model for the walking drops that allows us to rationalize the limited extent of the walking regimes. The critical requirement for walking is that the drop achieves resonance with its guiding wave field. We also rationalize the observed dependence of the walking speed on system parameters: while the walking speed is generally an increasing function of the driving acceleration, exceptions arise due to possible switching between different vertical bouncing modes. Special focus is given to elucidating the critical role of impact phase on the walking dynamics. The model predictions are shown to compare favourably with previous and new experimental data. Our results form the basis of the first rational hydrodynamic pilot-wave theory.

We present the results of a combined experimental and theoretical investigation of millimetric droplets bouncing on a vertically vibrating fluid bath. We first characterize the system experimentally, deducing the dependence of the droplet dynamics on the system parameters, specifically the drop size, driving acceleration and driving frequency. As the driving acceleration is increased, depending on drop size, we observe the transition from coalescing to vibrating or bouncing states, then period-doubling events that may culminate in either walking drops or chaotic bouncing states. The drop’s vertical dynamics depends critically on the ratio of the forcing frequency to the drop’s natural oscillation frequency. For example, when the data describing the coalescence–bouncing threshold and period-doubling thresholds are described in terms of this ratio, they collapse onto a single curve. We observe and rationalize the coexistence of two non-coalescing states, bouncing and vibrating, for identical system parameters. In the former state, the contact time is prescribed by the drop dynamics; in the latter, by the driving frequency. The bouncing states are described by theoretical models of increasing complexity whose predictions are tested against experimental data. We first model the drop–bath interaction in terms of a linear spring, then develop a logarithmic spring model that better captures the drop dynamics over a wider range of parameter space. While the linear spring model provides a faster, less accurate option, the logarithmic spring model is found to be more accurate and consistent with all existing data.

Controlling vortex shedding using spanwise-varying passive or active actuation (namely three-dimensional control) has recently been reported as a very efficient method for regulating two-dimensional bluff-body wakes. However, the mechanism of how the designed controller regulates vortex shedding is not clearly understood. To understand this mechanism, we perform a linear stability analysis of two-dimensional wakes, the base flow of which is modified with a given spanwise waviness. Absolute and convective instabilities of the spanwise wavy base flows are investigated using Floquet theory. Two types of base-flow modification are considered: varicose and sinuous. Both of these modifications attenuate absolute instability of two-dimensional wakes. In particular, the varicose modification is found to be much more effective in the attenuation than the sinuous one, and its spanwise lengths resulting in maximum attenuation show good agreement with those in three-dimensional controls. The physical mechanism of the stabilization is found to be associated with the formation of streamwise vortices from tilting of two-dimensional Kármán vortices and the subsequent tilting of these streamwise vortices by the spanwise shear in the base flow. Finally, the sensitivity of absolute instability to spanwise wavy base-flow modification is investigated. It is shown that absolute instability of two-dimensional wakes is much less sensitive to spanwise wavy base-flow modification than to two-dimensional modification. This suggests that the high efficiency observed in several three-dimensional controls is not due to the sensitive response of the wake instability to the spanwise waviness in the base flow.

We present analytical predictions of the depth and onset slope of the steady surface flow of a cohesive granular material in an inclined channel. The rheology of Jop, Forterre & Pouliquen (Nature, vol. 441, 2006, pp. 727–730) is used, assuming co-axiality between the stress and strain-rate tensors, and a coefficient of friction dependent on the strain rate through the dimensionless inertial number $I$. This rheological law is augmented by a constant stress representing cohesion. Our analysis does not rely on a precise $\mu (I)$ functional, but only on its asymptotic power law in the limit of vanishing strain rates. Assuming a unidirectional flow, the Navier–Stokes equations can be solved explicitly to yield parametric equations of the iso-velocity lines in the plane perpendicular to the flow. Two types of channel walls are considered: rough and smooth, depicting walls whose friction coefficient is respectively larger or smaller than that of the flowing material. The steady flow starts above a critical onset angle and consists of a sheared zone confined between a surface plug flow and a deep dead zone. The details of the flow are discussed, depending on dimensionless parameters relating the static friction coefficient, cohesion strength of the material, incline angle, wall friction, and channel width. The depths of the flow at the centre of the channel and at the walls are calculated by a force balance on the flowing material. The critical angle for the onset of the flow is also calculated, and is found to be strongly dependent on the channel width, in agreement with experimental results on heap stability and in rotating drums. Our results predict the important conclusion that a cohesive material always starts to flow for an incline angle lower than 90° between smooth walls, whereas in a narrow enough channel with rough walls, it may not flow, even if the channel is inclined vertically.

A dynamical instability is observed in experimental studies on micro-channels of rectangular cross-section with smallest dimension 100 and $160~\mathrm{\mu} \mathrm{m} $ in which one of the walls is made of soft gel. There is a spontaneous transition from an ordered, laminar flow to a chaotic and highly mixed flow state when the Reynolds number increases beyond a critical value. The critical Reynolds number, which decreases as the elasticity modulus of the soft wall is reduced, is as low as 200 for the softest wall used here (in contrast to 1200 for a rigid-walled channel). The instability onset is observed by the breakup of a dye-stream introduced in the centre of the micro-channel, as well as the onset of wall oscillations due to laser scattering from fluorescent beads embedded in the wall of the channel. The mixing time across a channel of width 1.5 mm, measured by dye-stream and outlet conductance experiments, is smaller by a factor of 105 than that for a laminar flow. The increased mixing rate comes at very little cost, because the pressure drop (energy requirement to drive the flow) increases continuously and modestly at transition. The deformed shape is reconstructed numerically, and computational fluid dynamics (CFD) simulations are carried out to obtain the pressure gradient and the velocity fields for different flow rates. The pressure difference across the channel predicted by simulations is in agreement with the experiments (within experimental errors) for flow rates where the dye stream is laminar, but the experimental pressure difference is higher than the simulation prediction after dye-stream breakup. A linear stability analysis is carried out using the parallel-flow approximation, in which the wall is modelled as a neo-Hookean elastic solid, and the simulation results for the mean velocity and pressure gradient from the CFD simulations are used as inputs. The stability analysis accurately predicts the Reynolds number (based on flow rate) at which an instability is observed in the dye stream, and it also predicts that the instability first takes place at the downstream converging section of the channel, and not at the upstream diverging section. The stability analysis also indicates that the destabilization is due to the modification of the flow and the local pressure gradient due to the wall deformation; if we assume a parabolic velocity profile with the pressure gradient given by the plane Poiseuille law, the flow is always found to be stable.

This study refers to the article of Chicón, Castellanos & Martion (J. Fluid Mech., vol. 344, 1997, pp. 43–66), who presented a numerical study of electroconvection in a layer of dielectric liquid induced by unipolar injection. An important characteristic of the numerical strategy proposed by Chicón et al. lies in the fact that the Navier–Stokes equations are never solved to obtain the velocity field, which is subsequently needed in the charge density transport equation. Instead, the velocity field is explicitly provided by an expression obtained with some assumptions about the flow structure and related to the electric field (the imposed velocity field approach; IVF). The validity of the above simplification is examined through a direct comparison of the solutions obtained by solving the Navier–Stokes equations (the Navier–Stokes computation approach; NSC). It is clearly demonstrated that, even in the strong injection regime ($C= 10$), the results look very similar for a given range of the mobility parameter $M$; however, in the weak injection regime ($C= 0. 1$), significant discrepancies are observed. The rich flow structures obtained with the NSC approach invalidate the use of the IVF approach in the weak injection regime.

Three-dimensional exact, finite-amplitude solutions are presented for the problem of channel flow subject to a system rotation about a spanwise axis. The solutions are of travelling wave form, and may bifurcate as tertiary flows from the two-dimensional streamwise-independent secondary flow, or as secondary flows directly from the basic flow. For the tertiary flows, we consider solutions of spanwise superharmonic and subharmonic type. We distinguish flows on the basis of symmetry, originating eigenmode and major solution branch, and thus identify 15 distinct flows: 5 superharmonic tertiary, 5 subharmonic tertiary and 5 secondary flows. The tertiary flows all feature a single layer of vortical structures in the spanwise–wall-normal plane, the secondary flows feature single-, double-, triple- or quadruple-layer flow structures in this plane. All flows feature low-speed streamwise-orientated streaks in the streamwise velocity component and/or pulses of low-speed streamwise velocity. The streaks may be sinusoidal or varicose. Sinusoidal streaks are flanked by staggered streamwise vortices, varicose streaks and pulses are flanked by aligned vortices. A comparison with previous simulation and experimental studies finds that the simplest three-dimensional flows observed previously correspond to superharmonic tertiary flows bifurcating from the upper branch of the secondary flow. The mean absolute vorticity of the present flows is also considered. A flattening of the profile of this vorticity is observed in the central region of the channel for two-dimensional secondary and many of the three-dimensional flows, with two-step profiles also observed. This phenomenon is attributed to mixing of the vorticity across zones of the channel in which streamwise vortex structures exist, and is demonstrated by a two-dimensional model. The phenomenon appears to be distinct to that observed in fully turbulent rotating channel flows.

We present a model for thixotropic gravity currents flowing down an inclined plane that combines lubrication theory for shallow flow with a rheological constitutive law describing the degree of microscopic structure. The model is solved numerically for a finite volume of fluid in both two and three dimensions. The results illustrate the importance of the degree of initial ageing and the spatio-temporal variations of the microstructure during flow. The fluid does not flow unless the plane is inclined beyond a critical angle that depends on the ageing time. Above that critical angle and for relatively long ageing times, the fluid dramatically avalanches downslope, with the current becoming characterized by a structured horseshoe-shaped remnant of fluid at the back and a raised nose at the advancing front. The flow is prone to a weak interfacial instability that occurs along the border between structured and de-structured fluid. Experiments with bentonite clay show broadly similar phenomenological behaviour to that predicted by the model. Differences between the experiments and the model are discussed.

The three-dimensional transition of the flow behind a circular cylinder with a near-wake wire disturbance has been investigated experimentally. The asymmetric placement of a wire in the near-wake region of the cylinder causes an unnatural mode of shedding to occur, namely mode C. We performed flow visualization and particle image velocimetry (PIV) experiments to investigate the influence of the wire on various properties of the flow, such as the dynamics of the streamwise secondary vortices. Experiments were performed at the Reynolds number range of Re = 165–300. From these experiments, it can be concluded that mode C structures are formed as secondary streamwise vortices around the primary von Kármán vortices. The spanwise wavelength of those mode C structures is determined to be approximately two cylinder diameters. The presence of the wire also triggered the occurrence of period doubling in the wake. Each new set of mode C structures is out of phase with the previous set, i.e. doubling the shedding period. This period-doubling phenomenon is due to a feedback mechanism between the consecutively shed upper vortices.

Analytic results are derived for the apparent slip length, the change in drag and the optimum air layer thickness of laminar channel and pipe flow over an idealised superhydrophobic surface, i.e. a gas layer of constant thickness retained on a wall. For a simple Couette flow the gas layer always has a drag reducing effect, and the apparent slip length is positive, assuming that there is a favourable viscosity contrast between liquid and gas. In pressure-driven pipe and channel flow blockage limits the drag reduction caused by the lubricating effects of the gas layer; thus an optimum gas layer thickness can be derived. The values for the change in drag and the apparent slip length are strongly affected by the assumptions made for the flow in the gas phase. The standard assumptions of a constant shear rate in the gas layer or an equal pressure gradient in the gas layer and liquid layer give considerably higher values for the drag reduction and the apparent slip length than an alternative assumption of a vanishing mass flow rate in the gas layer. Similarly, a minimum viscosity contrast of four must be exceeded to achieve drag reduction under the zero mass flow rate assumption whereas the drag can be reduced for a viscosity contrast greater than unity under the conventional assumptions. Thus, traditional formulae from lubrication theory lead to an overestimation of the optimum slip length and drag reduction when applied to superhydrophobic surfaces, where the gas is trapped.

We present a modelling study of locomotion over a layer of viscoplastic fluid motivated by the self-propulsion of marine and terrestrial gastropods. Our model comprises a layer of viscoplastic mucus lying beneath a fluid-filled foot that is laced internally by muscular fibres under tension and overlain by the main body of the locomotor, which is assumed to be rigid. The mucus is described using lubrication theory and the Bingham constitutive law, and the foot using a continuum approximation for the action of the muscle fibres. The model is first used to study the retrograde strategy of locomotion employed by marine gastropods, wherein the muscle fibres create a backwards-travelling wave of predominantly normal displacements along the surface of the foot. Once such a retrograde forcing pattern is switched on, the system is shown to converge towards a steady state of locomotion in a frame moving with the wave. The steady speed of locomotion decreases with the yield stress, until it vanishes altogether above a critical yield stress. Despite the absence of locomotion above this threshold, waves still propagate along the foot, peristaltically pumping mucus in the direction of the wave. The model is next used to study the prograde strategy employed by terrestrial gastropods, wherein the muscle fibres create a forwards-travelling wave of predominantly tangential displacements of the foot surface. In this case, a finite yield stress is shown to be necessary for locomotion, with the speed of locomotion initially increasing with the yield stress. Beyond a critical yield stress, localized rigid plugs form across the depth of the mucus layer, adhering parts of the foot to the base. These stop any transport of mucus, but foot motions elsewhere still drive locomotion. As the yield stress is increased further, the rigid plugs widen horizontally, increasing the viscous drag and eventually reducing the speed of locomotion, which is therefore maximized for an intermediate value of the yield stress.