Maintaining the stratification of the ocean requires deep mixing. Part of the energy that provides this mixing is transferred over topography from the surface tides into the internal tides. Numerous theoretical, numerical and experimental studies have aimed at quantifying the energy transfer of this conversion mechanism. Maas (J. Fluid Mech., this issue, vol. 684, 2011, pp. 5–24) constructs model topographies with localized response and no energy transfer into the propagating internal tide, a surprising result that raises questions about models of tidal conversion.

The consensus is that in a stratified sea a classical model of tidal flow over irregular but smooth topography necessarily leads to the generation of internal tides, regardless of the shape of the topography. This is referred to as tidal conversion. Here it is shown, however, that there exists a large class of topographies for which there is neither tidal conversion nor any scattering of incident internal waves. This result is obtained in a uniformly stratified, rigid-lid sea using a barotropic tide that, owing to its large horizontal scale, is supposed to be simply a mass-conserving, periodic back-and-forth flow. The baroclinic response at the tidal frequency is, upon non-dimensionalizing and stretching of coordinates, determined by a standard hyperbolic boundary value problem (BVP). We here solve this hyperbolic BVP by mapping a domain of complicated, yet a priori unknown shape, onto a uniform-depth channel for which the same hyperbolic problem is known to display neither conversion of the barotropic tide nor scattering of internal wave modes. The map achieving this is required to satisfy hyperbolic Cauchy–Riemann equations, defined as analogues of the Cauchy–Riemann equations that are used in solving elliptic problems. Mapping the rigid-lid surface in the original Cartesian frame onto a rigid-lid surface in the transformed frame, this map is solved in terms of one arbitrary function. Each particular function defines a new topographic shape that can be computed a posteriori. The map is unique provided the Jacobian of transformation does not vanish, which is guaranteed for subcritical bottom topography, whose slope is everywhere less than that of the characteristics. For topographies that can thus be mapped onto a channel, tidal conversion and scattering are absent. Examples discussed include the (classical) wedge, a (near-Gaussian) ridge, a continental slope and (near) sinusoidal topographies.

In this paper we present direct numerical simulations (DNS) of hypersonic turbulent boundary layers to study high-enthalpy effects. We study high- and low-enthalpy conditions, which are representative of those in hypersonic flight and ground-based facilities, respectively. We find that high-enthalpy boundary layers closely resemble those at low enthalpy. Many of the scaling relations for low-enthalpy flows, such as van-Driest transformation for the mean velocity, Morkovin’s scaling and the modified strong Reynolds analogy hold or can be generalized for high-enthalpy flows by removing the calorically perfect-gas assumption. We propose a generalized form of the modified Crocco relation, which relates the mean temperature and mean velocity across a wide range of conditions, including non-adiabatic cold walls and real gas effects. The DNS data predict Reynolds analogy factors in the range of those found in experimental data at low-enthalpy conditions. The gradient transport model approximately holds with turbulent Prandtl number and turbulent Schmidt number of order unity. Direct compressibility effects remain small and insignificant for all enthalpy cases. High-enthalpy effects have no sizable influence on turbulent kinetic energy (TKE) budgets or on the turbulence structure.

Boundary layer transition induced by the wake of a circular cylinder in the free stream has been investigated using the particle image velocimetry technique. Some differences between simulation and experimental studies have been reported in the literature, and these have motivated the present study. The appearance of spanwise vortices in the early stage is further confirmed here. A spanwise vortex appears to evolve into a /hairpin vortex; the flow statistics also confirm such vortices. With increasing Reynolds number, based on the cylinder diameter, and with decreasing cylinder height from the plate, the physical size of these hairpin-like structures is found to decrease. Some mean flow characteristics, including the streamwise growth of the disturbance energy, in a wake-induced transition resemble those in bypass transition induced by free stream turbulence. Streamwise velocity streaks that are eventually generated in the late stage often undergo sinuous-type oscillations. Similar to other transitional flows, an inclined shear layer in the wall-normal plane is often seen to oscillate and shed vortices. The normalized shedding frequency of these vortices, estimated from the spatial spacing and the convection velocity of these vortices, is found to be independent of the Reynolds number, similar to that in ribbon-induced transition. Although the nature of free stream disturbance in a wake-induced transition and that in a bypass transition are different, the late-stage features including the flow breakdown characteristics of these two transitions appear to be similar.

Numerical investigation of a sonic jet from a blunt body opposing a supersonic flow with a free stream Mach number was carried out using large-eddy simulation for two total pressure ratios of the jet to the free stream, i.e. and 1.633. Results have been validated carefully against experimental data. Various fundamental mechanisms dictating the flow phenomena, including shock/jet interaction, shock/shear-layer interaction, turbulent shear-layer evolution and coherent structures, have been studied systematically. Based on the analysis of the flow structures and features, two typical flow states, i.e. unstable and stable states corresponding to the two values of , are identified and the behaviours relevant to the flow states are discussed. Small-scale vortical structures mainly occur in the jet column, and large-scale vortices develop gradually in a recirculation region when the jet terminates through a Mach disk and reverses its orientation as a conical free shear layer. The turbulent fluctuations are enhanced by the rapid deviation of the shear layer and the interaction with shock waves. Moreover, the coherent structures of the flow motion are analysed using the proper orthogonal decomposition technique. It is found that the dominant mode in the cross-section plane exhibits an antisymmetric character for the unstable state and an axisymmetric one for the stable state, while statistical analysis of unsteady loads indicates that the side loads can be seen as a rotating vector uniformly distributed in the azimuthal direction. Further, we clarify a feedback mechanism whereby the unsteady motion is sustained by the upstream-propagating disturbance to the Mach disk through the recirculation subsonic region and downstream propagation in the conical shear layer. Feedback models are then proposed which can reasonably well predict the dominant frequencies of the two flow states. The results obtained in this study provide physical insight into the understanding of the mechanisms relevant to the opposing jet/supersonic flow interaction.

We consider near-inertial waves continuously excited by a localized source and their subsequent radiation and evolution on a two-dimensional -plane. Numerical simulations are used to quantify the wave propagation and the energy flux in a realistically stratified ocean basin. We focus on the dynamics near and poleward of the inertial latitude where the local value of the Coriolis parameter matches the forcing frequency , contrasting the behaviour of waves under the traditional approximation (TA), where only the component of the Earth’s rotation aligned with gravity is retained in the dynamics, with that obtained under the non-traditional approach (non-TA) in which the horizontal component of rotation is retained. Under the TA, assuming inviscid linear wave propagation in the WKB limit, all energy radiated from the source eventually propagates toward the equator, with the initially poleward propagation being internally reflected at the inertial latitude. Under the non-TA however, these waves propagate sub-inertially beyond their inertial latitude, exhibiting multiple reflections between internal turning points that lie poleward of the inertial latitude and the bottom. The numerical experiments complement and extend existing theory by relaxing the linearity and WKB approximations, and by illustrating the time development of the steadily forced flow and the spatial patterns of energy flux and flux divergence. The flux divergence of the flow at both the forcing frequency and its first harmonic reveal the spatial patterns of nonlinear energy transfer and highlight the importance of nonlinearity in the vicinity of near-critical bottom reflection at the inertial latitude of the forced waves.

In this numerical study we investigate the flow induced by metachronal coordination between beating cilia arranged in a densely packed layer by means of a continuum model. The continuum approach allows us to treat the problem as two-dimensional as well as stationary, in a reference frame moving with the speed of the metachronal wave. The model is used as a computationally efficient design tool to investigate cilia-induced transport of a Newtonian fluid in a plane channel. Contrary to prior continuum models, the present approach accounts for spatial variations in the porosity along the metachronal wave and thus ensures conservation of mass within the cilia layer. Using porous-media theory the governing volume-averaged Navier–Stokes (VANS) equations are derived and closure formulations are given explicitly for the model. This makes it possible to investigate cilia-induced flow with a continuum model in both the viscous regime and the inertial regime. The results show that metachronal coordination can act as a transport mechanism in both regimes. Porosity variations appear to be the key mechanism for correct prediction of the fluid transport in the viscous flow regime. The reason is that spatial variations in the porosity break the symmetry of the drag distribution along the metachronal wave. A new insight that has been gained is that the fluid transport reverses, thus switches from plectic to antiplectic metachronism, for the same cilia beat cycle when the wavespeed is increased such that inertial effects occur. Based on a parameter study, the net transport in the channel is described by a power-law relation of the amplitude, length and speed of the metachronal wave. It is found that the wavelength has the strongest effect on the viscosity-dominated fluid transport.

Electrokinetic effects at liquid/liquid interfaces have received considerably less attention than at solid/liquid interfaces. Because liquid/liquid interfaces are generally mobile, one might expect electrokinetic effects over a liquid/liquid interface to be faster than over an equivalent solid surface. The earliest predictions for the electrophoretic mobility of charged mercury drops – distinct approaches by Frumkin, along with Levich, and Booth – differed by , where is the radius of the drop and is the Debye length. Seeking to reconcile this rather striking discrepancy, Levine & O’Brien showed double-layer polarization to be the key ingredient. Without a physical mechanism by which electrokinetic effects are enhanced, however, it is difficult to know how general the enhancement is – whether it holds only for liquid metal surfaces, or more generally, for all liquid/liquid surfaces. By considering a series of systems in which a planar metal strip is coated with either a liquid metal or liquid dielectric, we show that the central physical mechanism behind the enhancement predicted by Frumkin is the presence of an unmatched electrical stress upon the electrolyte/liquid interface, which establishes a Marangoni stress on the droplet surface and drives it into motion. The source of the unbalanced electrokinetic stress on a liquid metal surface is clear – metals represent equipotential surfaces, so no field exists to drive an equal and opposite force on the surface charge. This might suggest that liquid metals represent a unique system, since dielectric liquids can support finite electric fields, which might be expected to exert an electrical stress on the surface charge that balances the electric stress. We demonstrate, however, that electrical and osmotic stresses on relaxed double layers internal to dielectric liquids precisely cancel, so that internal electrokinetic stresses generally vanish in closed, ideally polarizable liquids. The enhancement predicted by Frumkin for liquid mercury drops can thus be expected quite generally over ideally polarizable liquid drops. We then reconsider the electrophoretic mobility of spherical drops, and reconcile the approaches of Frumkin and Booth: Booth’s neglect of double-layer polarization leads to a standard electro-osmotic flow, without the enhancement, and Frumkin’s neglect of the detailed double-layer dynamics leads to the enhanced electrocapillary motion, but does not capture the (sub-dominant) electrophoretic motion. Finally, we show that, while the electrokinetic flow over electrodes coated with thin liquid films is faster than over solid/liquid interfaces, the Dukhin number, , which reflects the importance of surface conduction to bulk conduction, generally increases by a smaller amount [], where is the thickness of film and is the length of the electrode. This suggests that liquid/liquid interfaces may be utilized to enhance electrokinetic velocities in microfluidic devices, while delaying the onset of high- electrokinetic suppression.

We present a method of disentangling the anisotropies produced by the cliff structures in a turbulent velocity field. These cliff structures induce asymmetry in the velocity increments, which leads us to consider the plus and minus velocity structure functions (VSFs). We test the method in the system of turbulent Rayleigh–Bénard (RB) convection. It is found that in the RB system, the cliff structures in the velocity field are generated by thermal plumes. The plus velocity increments exclude cliff structures, while the minus ones include them. Our results show that the scaling exponents of the plus VSFs are in excellent agreement with those predicted for homogeneous and isotropic turbulence (HIT), whereas those of the minus VSFs exhibit significant deviations from HIT expectations in places where thermal plumes abound. These results demonstrate that plus and minus VSFs can be used to quantitatively study the effect of cliff structures in the velocity field and to effectively disentangle the associated anisotropies caused by these structures.

The linear spatiotemporal stability properties of axisymmetric laminar capillary jets with fully developed initial velocity profiles are studied for large values of both the Reynolds number, , and the Froude number, , where is the injector radius, the volume flow rate, the kinematic viscosity and the gravitational acceleration. The downstream development of the basic flow and its stability are addressed with an approximate formulation that takes advantage of the jet slenderness. The base flow is seen to depend on two parameters, namely a Stokes number, , and a Weber number, , where is the surface tension coefficient, while its linear stability depends also on the Reynolds number. When non-parallel terms are retained in the local stability problem, the analysis predicts a critical value of the Weber number, , below which a pocket of local absolute instability exists within the near field of the jet. The function is computed for the buoyancy-free jet, showing marked differences with the results previously obtained with uniform velocity profiles. It is seen that, in accounting for gravity effects, it is more convenient to express the parametric dependence of the critical Weber number with use made of the Morton and Bond numbers, and , as replacements for and . This alternative formulation is advantageous to describe jets of a given liquid for a known value of , in that the resulting Morton number becomes constant, thereby leaving as the only relevant parameter. The computed function for a water jet under Earth gravity is shown to be consistent with the experimental results of Clanet and Lasheras for the transition from jetting to dripping of water jets discharging into air from long injection needles, which cannot be properly described with a uniform velocity profile assumed at the jet exit.

The effects of transverse rotation on the dynamics of a droplet released into a uniform free stream are numerically investigated. The range of the dimensionless rotation rate is limited to , to avoid any possibility of the droplet breaking up. Droplet dynamics and deformations undergo distinct changes when the dimensionless rotational rate reaches a critical value. The critical rotational rate is sensitive to the change in the density ratio, but less dependent on the viscosity ratio and interfacial tension. Below , the droplet drag coefficients are reduced marginally as the effect of the rotation is quickly suppressed by the free stream. Above , the drag coefficients decrease initially as the rotation effect dominates at earlier times, resulting in a global minimum. The drag coefficients increase monotonically at later times, when the rotation effects decrease and the free-stream effects become dominant. The only exception is with the increase in the viscosity ratio and the surface tension, which either inhibits droplet deformation or restores the droplet to a more spherical shape in the late stages of droplet evolution. The droplet also experiences lift due to the effects of the transverse rotation. It is observed that the lift coefficients are less dependent on the droplet frontal area as the lift is generated by the velocity difference between the upper and lower interface. In general, the lift coefficients increase with at earlier times and decrease at later times as the difference in the velocity between the upper and lower interface decreases. In some extreme cases, the lift coefficients even become negative.

Experiments have been conducted in a large oscillatory flow tunnel to investigate the effects of acceleration skewness on oscillatory boundary layer flow over fixed beds. As well as enabling experimental investigation of the effects of acceleration skewness, the new experiments add substantially to the relatively few existing detailed experimental datasets for oscillatory boundary layer flow conditions that correspond to full-scale sea wave conditions. Two types of bed roughness and a range of high-Reynolds-number, , oscillatory flow conditions, varying from sinusoidal to highly acceleration-skewed, are considered. Results show the structure of the intra-wave velocity profile, the time-averaged residual flow and boundary layer thickness for varying degrees of acceleration skewness, . Turbulence intensity measurements from particle image velocimetry (PIV) and laser Doppler anemometry (LDA) show very good agreement. Turbulence intensity and Reynolds stress increase as the flow accelerates after flow reversal, are maximum at around maximum free-stream velocity and decay as the flow decelerates. The intra-wave turbulence depends strongly on but period-averaged turbulent quantities are largely independent of . There is generally good agreement between bed shear stress estimates obtained using the log-law and using the momentum integral equation, and flow acceleration skewness leads to high bed shear stress asymmetry between flow half-cycles. Turbulent Reynolds stress is much less than the shear stress obtained from the momentum integral; analysis of the stress contributors shows that significant phase-averaged vertical velocities exist near the bed throughout the flow cycle, which lead to an additional shear stress, ; near the bed this stress is at least as large as the turbulent Reynolds stress.

The high-Reynolds-number stability of unsteady pipe flow to axisymmetric disturbances is studied using asymptotic analysis. It is shown that as the disturbance amplitude is increased, nonlinear effects first become significant within the critical layer, which moves away from the pipe wall as a result. It is found that the flow stabilizes once the basic profile has become sufficiently fully developed. By tracing the nonlinear neutral curve back to earlier times, it is found that in addition to the wall mode, which arises from a classical upper branch linear stability analysis, there also exists a nonlinear neutral centre mode, governed primarily by inviscid dynamics. The centre mode problem is solved numerically and the results show the existence of a concentrated region of vorticity centred on or close to the pipe axis and propagating downstream at almost the maximum fluid velocity. The connection between this structure and the puffs and slugs of vorticity observed in experiments is discussed.

In the present paper turbulent mixing in the Stokes–Ekman bottom boundary layer is investigated analytically and by wall-resolving large-eddy simulations (LES). The analytical solution shows that when the Rossby number oscillation and rotation interact with each other, this gives rise to a thickening of the boundary layer compared with the purely oscillating or the purely rotating case. The solution also shows the presence of elliptical patterns developing on the horizontal planes that shrink when approaching the low latitudes. In the turbulent regime, a LES was applied for an east–west tidal current, considered at three different latitudes in the Northern Hemisphere, namely, the polar case, the mid-latitude case () and the quasi-equatorial case (). The Reynolds number of the simulation, based on the viscous penetration depth and the frequency of the purely oscillatory flow, was set equal to . The analysis suggests that rotation has two main effects on the flow field: in the polar case, rotation tends to delay the cyclic re-transition to turbulence and to narrow the turbulent phases of the cycle. Also, rotation suppresses vertical fluctuations of velocity and redistributes energy from the streamwise direction to the spanwise direction. It is noteworthy that the high-latitude effect makes the turbulent field substantially different from the reference Stokes boundary layer case, whereas the low-latitude effects appear to be of secondary importance, owing to the weakness of the rotation rate. Consequently, the study shows that the Stokes boundary layer may be representative of the oceanic bottom boundary layer in the low-latitude cases ( cases in our simulations). Conversely, it cannot be considered as archetypal of the oceanic boundary layer at high latitudes ( case of our study), where the vertical background vorticity profoundly modifies the turbulent field.

A gravity-driven film flow on a slippery inclined plane is considered within the framework of long-wave and boundary layer approximations. Two coupled depth-averaged equations are derived in terms of the local flow rate and the film thickness . Linear stability analysis of the averaged equations shows good agreement with the Orr–Sommerfeld analysis. The effect of a slip at the wall on the primary instability has been found to be non-trivial. Close to the instability onset, the effect is destabilising whereas it becomes stabilising at larger values of the Reynolds number. Nonlinear travelling waves are amplified by the presence of the slip. Comparisons to direct numerical simulations show a remarkable agreement for all tested values of parameters. The averaged equations capture satisfactorily the speed, shape and velocity distribution in the waves. The Navier slip condition is observed to significantly enhance the backflow phenomenon in the capillary region of the solitary waves with a possible effect on heat and mass transfer.

The dynamics of two deformable drops placed in tandem and subjected to a sudden acceleration by a gaseous flow is investigated. A finite-volume scheme coupled with the method of moving mesh interface tracking is employed. The unsteady interactions between the droplet pair are studied by varying the minimum initial separation distance () from to with being the radius of the initial spherical droplets. The influence of the interactions on the droplet dynamics is examined by comparing with the case of a single isolated droplet at three initial Weber numbers of 40, 4 and 0.4. The computations show that for small initial separation distances the dynamics of the downstream droplet is significantly affected by the presence of the upstream droplet. A mushroom shape is formed by the droplet pair at the two largest Weber numbers, while the two drops experience small deformation and shape oscillations at . The drag coefficient of the downstream droplet is dramatically reduced, especially for the two largest Weber numbers with smaller initial separation distances due to the sheltering effects, while the drag force of the upstream drop is slightly decreased. For the cases with smaller , a thin film is formed between the two drops at the later stages, and this leads to a sudden increase in the drag of the trailing drop, but a sharp reduction in the drag of the leading drop.

We report on the influence of rotation about a vertical axis on heat transport by turbulent Rayleigh–Bénard convection in a cylindrical vessel with an aspect ratio ( is the diameter and the height of the sample) and compare the results with those for larger . The working fluid was water at where the Prandtl number  is 4.38. For rotation rates , corresponding to inverse Rossby numbers between zero and twenty, we measured the Nusselt number for six Rayleigh numbers in the range . For small rotation rates and at constant , the reduced Nusselt number initially increased slightly with increasing , but at it suddenly became constant or decreased slightly depending on . At a second sharp transition occurred in to a state where increased with increasing . We know from direct numerical simulation that the transition at corresponds to the onset of Ekman vortex formation reported before for at and for at (Weiss et al., Phys. Rev. Lett., vol. 105, 2010, 224501). The -dependence of can be explained as a finite-size effect that can be described phenomenologically by a Ginzburg–Landau model; this model is discussed in detail in the present paper. We do not know the origin of the transition at . Above , increased with increasing up to . We discuss the -dependence of in this range in terms of the average Ekman vortex density as predicted by the model. At even larger there is a decrease of that can be attributed to two possible effects. First, the Ekman pumping might become less efficient when the Ekman layer is significantly smaller than the thermal boundary layer, and second, for rather large , the Taylor–Proudman effect in combination with boundary conditions suppresses fluid flow in the vertical direction.

We study both theoretically and numerically surface quasi-geostrophic turbulence regularized by the usual molecular viscosity, with an emphasis on a number of classical predictions. It is found that the system’s number of degrees of freedom , which is defined in terms of local Lyapunov exponents, scales as , where is the Reynolds number expressible in terms of the viscosity, energy dissipation rate and system’s integral scale. For general power-law energy spectra , a comparison of with the number of dynamically active Fourier modes, i.e. the modes within the energy inertial range, yields . This comparison further renders the scaling for the exponential dissipation rate at the dissipation wavenumber. These results have been predicted on the basis of Kolmogorov’s theory. Our approach thus recovers these classical predictions and is an analytic alternative to the traditional phenomenological method. The implications of the present findings are discussed in conjunction with related results in the literature. Support for the analytic results is provided through a series of direct numerical simulations.

Direct numerical simulation of open channel flow over a geometrically rough wall has been performed at a bulk Reynolds number of . The wall consisted of a layer of spheres in a square arrangement. Two cases have been considered. In the first case the spheres are small (with diameter equivalent to wall units) and the limit of the hydraulically smooth flow regime is approached. In the second case the spheres are more than three times larger ( wall units) and the flow is in the transitionally rough flow regime. Special emphasis is given to the characterisation of the force and torque acting on a particle due to the turbulent flow. It is found that in both cases the mean drag, lift and spanwise torque are to a large extent produced at the top region of the particle surface. The intensity of the particle force fluctuations is significantly larger in the large-sphere case, while the trend differs for the fluctuations of the individual components of the torque. A simplified model is used to show that the torque fluctuations might be explained by the spheres acting as a filter with respect to the size of the flow scales which can effectively generate torque fluctuations. Fluctuations of both force and torque are found to exhibit strongly non-Gaussian probability density functions with particularly long tails, an effect which is more pronounced in the small-sphere case. Some implications of the present results for sediment erosion are briefly discussed.

An idealised model is presented to study the formation of sorted bed forms generated by a wind-driven along-shore current. The study employs a linear stability analysis to describe the time development of perturbations of both bottom composition and bed elevation, superimposed on a flat bed composed of a sediment mixture homogeneously distributed in space. The model considers both bed and suspended loads and takes into account the averaged influence of waves on the flow field and the transport of sediment. The results show that the positive coupling between waves, along-shore current and the erodible heterogeneous bed leads to the amplification of two modes, which exhibit distinct characteristics. A first mode is found to be dominant when moderate hydrodynamic conditions are considered and is primarily amplified by the convergence of sediment transport induced by the changes in the bed elevation. This mode has wavelengths of the order of hundred metres and has coarse (fine) sediments in its troughs (crests). By increasing the height of the waves and/or the strength of the steady current, the second mode can become dominant. This mode is characterised by shorter wavelengths and results from the interaction between the convergence of sediment transport related to changes in the bottom composition and that induced by perturbations of the bed elevation. These bed features can have an up-current or a down-current shift between the centre of the coarse-grained bands and the trough of the bottom wave. Typical growth times of the amplified features are of the order of hundreds of days and the migration rates, in the direction of the along-shore current, range between 0.1 and 10 m per day. A qualitative comparison of the model results with field observations indicates that the generation of two distinct modes provides a possible explanation for the broad range of characteristics of the natural bed features.