We simulate numerically the full dynamics of Faraday waves in three dimensions for two incompressible and immiscible viscous fluids. The Navier–Stokes equations are solved using a finite-difference projection method coupled with a front-tracking method for the interface between the two fluids. The critical accelerations and wavenumbers, as well as the temporal behaviour at onset are compared with the results of the linear Floquet analysis of Kumar & Tuckerman (J. Fluid Mech., vol. 279, 1994, p. 49). The finite-amplitude results are compared with the experiments of Kityk et al (Phys. Rev. E, vol. 72, 2005, p. 036209). In particular, we reproduce the detailed spatio-temporal spectrum of both square and hexagonal patterns within experimental uncertainty. We present the first calculations of a three-dimensional velocity field arising from the Faraday instability for a hexagonal pattern as it varies over its oscillation period.

We formulate a new theoretical model for the swimming of a flexible body in a vortex street. We consider the class of periodic travelling-wave body motions, in the limit of small amplitude. We calculate the output power provided to the body by thrust forces, and the input power done against pressure forces, as functions of the aspect ratio and strength of the vortex street. We then formulate two optimization problems. In the first, we determine the body wave which provides maximum output power for fixed amplitude. We find a closed-form solution with a transition from power law to exponential decay of output power as the vortex street widens. In the second problem, we incorporate internal viscoelasticity to the swimming body and compute its contribution to the input power. We find the body wave which maximizes efficiency for a given output power. The body shape and resulting efficiency are found in closed form and simple approximate formulas are given. We find that efficiency scales as the inverse of the damping parameter. Finally, we compare our results with previous experiments and simulations. We find agreement in some aspects and disagreement in others. We give physical interpretations for agreements and disagreements in terms of the phase between the body wave and vortex street.

An incident shock wave/turbulent boundary layer interaction at Mach 2.1 is investigated using particle image velocimetry in combination with data processing using the proper orthogonal decomposition, to obtain an instantaneous and statistical description of the unsteady flow organization. The global structure of the interaction is observed to vary considerably in time. Although reversed flow is often measured instantaneously, on average no reversed flow is observed. On an instantaneous basis, the interaction exhibits a multi-layered structure, characterized by a relatively high-velocity outer region and low-velocity inner region. Discrete vortical structures are prevalent along their interface, which create an intermittent fluid exchange as they propagate downstream. A statistical analysis suggests that the instantaneous fullness of the incoming boundary layer velocity profile is (weakly) correlated with the size of the separation bubble and position of the reflected shock wave. The eigenmodes show an energetic association between velocity fluctuations within the incoming boundary layer, separated flow region and across the reflected shock wave, and portray subspace features that represent the phenomenology observed within the instantaneous realizations.

Turbulence measurements for a zero pressure gradient boundary layer over a two-dimensional roughness are presented and compared to previous results for a smooth wall and a three-dimensional roughness (Volino, Schultz & Flack, J. Fluid Mech., vol. 592, 2007, p. 263). The present experiments were made on transverse square bars in the fully rough flow regime. The turbulence structure was documented through the fluctuating velocity components, two-point correlations of the fluctuating velocity and swirl strength and linear stochastic estimation conditioned on the swirl and Reynolds shear stress. The two-dimensional bars lead to significant changes in the turbulence in the outer flow. Reynolds stresses, particularly and , increase, although the mean flow is not as significantly affected. Large-scale turbulent motions originating at the wall lead to increased spatial scales in the outer flow. The dominant feature of the outer flow, however, remains hairpin vortex packets which have similar inclination angles for all wall conditions. The differences between boundary layers over two-dimensional and three-dimensional roughness are attributable to the scales of the motion induced by each type of roughness. This study has shown three-dimensional roughness produces turbulence scales of the order of the roughness height k while the motions generated by two-dimensional roughness may be much larger due to the width of the roughness elements. It is also noted that there are fundamental differences in the response of internal and external flows to strong wall perturbations, with internal flows being less sensitive to roughness effects.

Careful reassessment of new and pre-existing data shows that recorded scatter in the hot-wire-measured near-wall peak in viscous-scaled streamwise turbulence intensity is due in large part to the simultaneous competing effects of the Reynolds number and viscous-scaled wire length l+. An empirical expression is given to account for these effects. These competing factors can explain much of the disparity in existing literature, in particular explaining how previous studies have incorrectly concluded that the inner-scaled near-wall peak is independent of the Reynolds number. We also investigate the appearance of the so-called outer peak in the broadband streamwise intensity, found by some researchers to occur within the log region of high-Reynolds-number boundary layers. We show that the ‘outer peak’ is consistent with the attenuation of small scales due to large l+. For turbulent boundary layers, in the absence of spatial resolution problems, there is no outer peak up to the Reynolds numbers investigated here (Reτ = 18830). Beyond these Reynolds numbers – and for internal geometries – the existence of such peaks remains open to debate. Fully mapped energy spectra, obtained with a range of l+, are used to demonstrate this phenomenon. We also establish the basis for a ‘maximum flow frequency’, a minimum time scale that the full experimental system must be capable of resolving, in order to ensure that the energetic scales are not attenuated. It is shown that where this criterion is not met (in this instance due to insufficient anemometer/probe response), an outer peak can be reproduced in the streamwise intensity even in the absence of spatial resolution problems. It is also shown that attenuation due to wire length can erode the region of the streamwise energy spectra in which we would normally expect to see kx−1 scaling. In doing so, we are able to rationalize much of the disparity in pre-existing literature over the kx−1 region of self-similarity. Not surprisingly, the attenuated spectra also indicate that Kolmogorov-scaled spectra are subject to substantial errors due to wire spatial resolution issues. These errors persist to wavelengths far beyond those which we might otherwise assume from simple isotropic assumptions of small-scale motions. The effects of hot-wire length-to-diameter ratio (l/d) are also briefly investigated. For the moderate wire Reynolds numbers investigated here, reducing l/d from 200 to 100 has a detrimental effect on measured turbulent fluctuations at a wide range of energetic scales, affecting both the broadband intensity and the energy spectra.

Building upon the recent experimentally verified modelling of turbulent plumes which are subject to decreases in their source strength (Scase et al., J. Fluid Mech., vol. 563, 2006b, p. 443), we consider the complementary case where the plume's source strength is increased. We consider the effect of increasing the source strength of an established plume and we also compare time-dependent plume model predictions for the behaviour of a starting plume to those of Turner (J. Fluid Mech., vol. 13, 1962, p. 356).

Unlike the decreasing source strength problems considered previously, the relevant solution to the time-dependent plume equations is not a simple similarity solution. However, scaling laws are demonstrated which are shown to be applicable across a large number of orders of magnitude of source strength increase. It is shown that an established plume that is subjected to an increase in its source strength supports a self-similar ‘pulse’ structure propagating upwards. For a point source plume, in pure plume balance, subjected to an increase in the source buoyancy flux F0, the rise height of this pulse in terms of time t scales as t3/4 while the vertical extent of the pulse scales as t1/4. The volume of the pulse is shown to scale as t9/4. For plumes in pure plume balance that emanate from a distributed source it is shown that the same scaling laws apply far from the source, demonstrating an analogous convergence to pure plume balance as that which is well known in steady plumes. These scaling law predictions are compared to implicit large eddy simulations of the buoyancy increase problem and are shown to be in good agreement.

We also compare the predictions of the time-dependent model to a starting plume in the limit where the source buoyancy flux is discontinuously increased from zero. The conventional model for a starting plume is well approximated by a rising turbulent, entraining, buoyant vortex ring which is fed from below by a ‘steady’ plume. However, the time-dependent plume equations have been defined for top-hat profiles assuming only horizontal entrainment. Therefore, this system cannot model either the internal dynamics of the starting plume's head or the extra entrainment of ambient fluid into the head due to the turbulent boundary of the vortex ring-like cap. We show that the lack of entrainment of ambient fluid through the head of the starting plume means that the time-dependent plume equations overestimate the rise height of a starting plume with time. However, by modifying the entrainment coefficient appropriately, we see that realistic predictions consistent with experiment can be attained.

Two types of wall actuation in channel flow are considered: travelling waves of wall deformation (peristalsis) and travelling waves of blowing and suction. The flow response and its mechanisms are analysed using nonlinear and weakly nonlinear computations. We show that both actuations induce a flux in the channel in the absence of an imposed pressure gradient and can thus be characterized as pumping. In the context of flow control, pumping and drag reduction are strongly connected, and we seek to define them properly based on these two actuation examples. Movies showing the flow motion for the two types of actuation are available with the online version of this paper (journals.cambridge.org/FLM).

A computational fluid dynamics simulation of the flow in an overexpanded planar nozzle shows the existence of Mach-reflection hysteresis inside the nozzle. Previous simulations have dealt only with the flow outside the nozzle and thus concluded that the hysteresis phenomenon takes place outside the nozzle even when viscous effects are introduced. When including the geometry of the nozzle in the simulation it becomes evident that flow separation will occur before the transition from regular to Mach reflection for all relevant Mach numbers. The simulation reveals complex changes in the flow structure as the pressure ratio between the ambient and the jet is increased and decreased. The pressure along the nozzle wall downstream of the separation point is found to be less than the ambient pressure, and a modification of the Schilling curve fit is suggested for cases of extensive flow separation.

Several effects of nearly isotropic free-stream turbulence in transitionally rough turbulent boundary layers are studied using data obtained from laser Doppler anemometry measurements. The free-stream turbulence is generated with the use of an active grid, resulting in free-stream turbulence levels of up to 6.2%. The rough surface is characterized by a roughness parameter k+ ≈ 53, and measurements are performed at Reynolds numbers of up to Reθ = 11300. It is confirmed that the free-stream turbulence significantly alters the mean velocity deficit profiles in the outer region of the boundary layer. Consequently, the previously observed ability of the Zagarola & Smits (J. Fluid Mech., vol. 373, 1998, p. 33) velocity scale U∞δ*/δ to collapse results from both smooth and rough surface boundary layers, no longer applies in this boundary layer subjected to high free-stream turbulence. In inner variables, the wake region is significantly reduced with increasing free-stream turbulence, leading to decreased mean velocity gradient and production of Reynolds stress components. The effects of free-stream turbulence are clearly identifiable and significant augmentation of the streamwise Reynolds stress profiles throughout the entire boundary layer are observed, all the way down to the inner region. In contrast, the Reynolds wall-normal and shear stress profiles increase due to free-stream turbulence only in the outer part of the boundary layer due to the blocking effect of the wall. As a consequence, there is a significant portion of the boundary layer in which the addition of nearly isotropic turbulence in the free-stream, results in significant increases in anisotropy of the turbulence. To quantify which turbulence length scales contribute to this trend, second-order structure functions are examined at various distances from the wall. Results show that the anisotropy created by adding nearly isotropic turbulence in the free-stream resides mostly in the larger scales of the flow. Furthermore, by analysing the streamwise Reynolds stress equation, it can be predicted that it is the wall-normal gradient of 〈u2v〉 term that is responsible for the increase in 〈u2〉 profiles throughout the boundary layer (i.e. an efficient turbulent transport of turbulence away from the wall). Furthermore, a noticeable difference between the triple correlations for smooth and rough surfaces exists in the inner region, but no significant differences are seen due to free-stream turbulence. In addition, the boundary layer parameters δ*/δ95, H and cf are also evaluated from the experimental data. The flow parameters δ*/δ95 and H are found to increase due to roughness, but decrease due to free-stream turbulence, which has significance for flow control, particularly in delaying separation. Increases in cf due to high free-stream turbulence are also observed, associated with increased momentum flux towards the wall.

Gravity currents intruding into a uniformly stratified ambient are examined in a series of finite-volume full-depth lock-release laboratory experiments and in numerical simulations. Previous studies have focused on gravity currents which are denser than fluid at the bottom of the ambient or on symmetric cases in which the intrusion is the average of the ambient density. Here, we vary the density of the intrusion between these two extremes. After an initial adjustment, the intrusions and the internal waves they generate travel at a constant speed. For small departures from symmetry, the intrusion speed depends weakly upon density relative to the ambient fluid density. However, the internal wave speed approximately doubles as the waves change from having a mode-2 structure when generated by symmetric intrusions to having a mode-1 structure when generated by intrusions propagating near the bottom. In the latter circumstance, the interactions between the intrusion and internal waves reflected from the lock-end of the tank are sufficiently strong and so the intrusion stops propagating before reaching the end of the tank. These observations are corroborated by the analysis of two-dimensional numerical simulations of the experimental conditions. These reveal a significant transfer of available potential energy to the ambient in asymmetric circumstances.

The paper deals with the numerical investigation of the steady convective flow in a horizontal channel of rectangular cross-section subjected to a uniform longitudinal temperature gradient imposed at the walls. It is shown that at zero Prandtl number the solution of the problem corresponds to a plane-parallel flow along the channel axis. In this case, the fluid moves in the direction of the imposed temperature gradient in the upper part of the channel and in the opposite direction in the lower part. At non-zero values of the Prandtl number, such solution does not exist. At any small values of Pr all three components of the flow velocity differ from zero and in the channel cross-section four vortices develop. The direction of these vortices is such that the fluid moves from the centre to the periphery in the vertical direction and returns to the centre in the horizontal direction. The stability of these convective flows (uniform along the channel axis) with regard to small three-dimensional perturbations periodical in the direction of the channel axis is studied. It is shown that at low values of the Prandtl number the basic state loses its stability due to the steady hydrodynamic mode related to the development of vortices at the boundary of the counter flows. The growth of the Prandtl number results in the strong stabilization of this instability mode and, beyond a certain value of the Prandtl number depending on the cross-section aspect ratio, a new steady hydrodynamic instability mode becomes the most dangerous. This mode is characterized by the localization of the perturbations near the sidewalls of the channel. At still higher values of the Prandtl number, the spiral perturbations (rolls with axis parallel to the temperature gradient) become the most dangerous modes, at first the oscillatory spiral perturbations and then the Rayleigh-type steady spiral perturbations. The influence of the channel width on these different instabilities is also emphasized.

The stabilization of buoyant flows by a magnetic field is an important matter for crystal growth applications. It is studied here when the cavity is an infinite channel with rectangular cross-section typical of horizontal Bridgman configurations and when the magnetic field is applied in the vertical and transverse directions. The steady basic flow solution is first calculated: the usual counter flow structure is modified by the magnetic field and evolves towards jets in the cross-section corners when the magnetic field is vertical and towards a more uniform structure in the transverse direction when the magnetic field is transverse. The stability results show a very good stabilization of the convective flows for a vertical magnetic field with exponential increases of the thresholds for any width of the channel and for various Prandtl numbers Pr. The results for a transverse magnetic field are more surprising as a destabilizing effect corresponding to an initial decrease of the thresholds is obtained at Pr=0 and for small channel widths. A kinetic energy budget at the thresholds reveals that the main destabilizing factor is connected to the vertical shear of the longitudinal basic flow and that it is the modifications affecting this shear energy which are mainly responsible for the variation of the thresholds when a magnetic field is applied.

New intermediate models for the rotating shallow water (RSW) equations are derived by considering the nonlinear interactions between subsets of the eigenmodes for the linearized equations. It is well-known that the two-dimensional quasi-geostrophic (QG) equation results when the nonlinear interactions are restricted to include only the vortical eigenmodes. Continuing past QG in a non-perturbative manner, the new models result by including subsets of interactions which include inertial-gravity wave (IG) modes. The such simplest model adds nonlinear interactions between one IG mode and two vortical modes. In sharp contrast to QG, the latter model behaves similar to the full RSW equations for decay from balanced initial conditions as well as unbalanced random initial conditions with divergence-free velocity. Quantitative agreement is observed for statistics that measure structure size, intermittency and cyclone/anticyclone asymmetry. In particular, dominance of anticyclones is observed for Rossby numbers Ro in the range 0.1 < Ro < 1 (away from the QG parameter regime Ro → 0). A hierarchy of models is explored to determine the effects of wave-vortical and wave–wave interactions on statistics such as the skewness of vorticity in decaying turbulence. Possible advantages over previously derived intermediate models include (i) the non-perturbative nature of the new models (not restricting them a priori to any particular parameter regime) and (ii) insight into the physical and mathematical consequences of vortical–wave interactions.

Compositional gravity current flows produced by the instantaneous release of a finite-volume, heavier lock fluid in a rectangular horizontal plane channel are investigated using large eddy simulation. The first part of the paper focuses on the evolution of Boussinesq lock-exchange gravity currents with a large initial volume of the release during the slumping phase in which the front of the gravity current propagates with constant speed. High-resolution simulations are conducted for Grashof numbers = 3150 (LGR simulation) and = 126000 (HGR simulation). The Grashof number is defined with the channel depth h and the buoyancy velocity ub = (g′ is the reduced gravity). In the HGR simulation the flow is turbulent in the regions behind the two fronts. Compared to the LGR simulation, the interfacial billows lose their coherence much more rapidly (over less than 2.5h behind the front), which results in a much faster decay of the large-scale content and turbulence intensity in the trailing regions of the flow. A slightly tilted, stably stratified interface layer develops away from the two fronts. The concentration profiles across this layer can be approximated by a hyperbolic tangent function. In the HGR simulation the energy budget shows that for t > 18h/ub the flow reaches a regime in which the total dissipation rate and the rates of change of the total potential and kinetic energies are constant in time. The second part of the paper focuses on the study of the transition of Boussinesq gravity currents with a small initial volume of the release to the buoyancy–inertia self-similar phase. When the existence of the back wall is communicated to the front, the front speed starts to decrease, and the current transitions to the buoyancy–inertia phase. Three high-resolution simulations are performed at Grashof numbers between = 3 × 104 and = 9 × 104. Additionally, a calculation at a much higher Grashof number ( = 106) is performed to understand the behaviour of a bottom-propagating current closer to the inviscid limit. The three-dimensional simulations correctly predict a front speed decrease proportional to t−α (the time t is measured from the release time) over the buoyancy–inertia phase, with the constant α approaching the theoretical value of 1/3 as the current approaches the inviscid limit. At Grashof numbers for which > 3 × 104, the intensity of the turbulence in the near-wall region behind the front is large enough to induce the formation of a region containing streaks of low and high streamwise velocities. The streaks are present well into the buoyancy–inertia phase before the speed of the front decays below values at which the streaks can be sustained. The formation of the velocity streaks induces a streaky distribution of the bed friction velocity in the region immediately behind the front. This distribution becomes finer as the Grashof number increases. For simulations in which the only difference was the value of the Grashof number ( = 4.7 × 104 versus = 106), analysis of the non-dimensional bed friction velocity distributions shows that the capacity of the gravity current to entrain sediment from the bed increases with the Grashof number. Past the later stages of the transition to the buoyancy–inertia phase, the temporal variations of the potential energy, the kinetic energy and the integral of the total dissipation rate are logarithmic.

The majority of existing single-unit devices for extracting power from sea waves relies on resonance at the peak frequency of the incident wave spectrum. Such designs usually call for structural dimensions not too small compared to a typical wavelength and yield high efficiency only within a limited frequency band. A recent innovation in Norway departs from this norm by gathering many small buoys in a compact array. Each buoy is too small to be resonated in typical sea conditions. In this article a theoretical study is performed to evaluate this new design. Within the framework of linearization, we consider a periodic array of small buoys with similarly small separation compared to the typical wavelength. The method of homogenization (multiple scales) is used to derive the equations governing the macroscale behaviour of the entire array. These equations are then applied to energy extraction by an infinite strip of buoys, and by a circular array. In the latter case, advantages are found when compared to a single buoy of equal volume.

The flow of a partial-depth lock-exchange gravity current past an isolated bottom-mounted obstacle is studied by means of two-dimensional direct numerical simulations and steady shallow-water theory. The simulations indicate that the flux of the current downstream of the obstacle is approximately constant in space and time. This information is employed to extend the shallow-water models of Rottman et al. (J. Hazard. Mater., vol. 11, 1985, pp. 325–340) and Lane-Serff, Beal & Hadfield (J. Fluid Mech., vol. 292, 1995, pp. 39–53), in order to predict the height and front speed of the downstream current as functions of the upstream Froude number and the ratio of obstacle to current height. The model predictions are found to agree closely with the simulation results. In addition, the shallow-water model provides an estimate for the maximum drag that lies within 10% of the simulation results for obstacles much larger than the boundary-layer thickness.

The prevailing view of the dynamics of flapping flags is that the onset of motion is caused by temporal instability of the initial planar state. This view is re-examined by considering the linearized two-dimensional motion of a flag immersed in a high-Reynolds-number flow and taking account of forcing by a ‘street’ of vortices shed periodically from its cylindrical pole. The zone of nominal instability is determined by analysis of the self-induced motion in the absence of shed vorticity, including the balance between flag inertia, bending rigidity, varying tension and fluid loading. Forced motion is then investigated by separating the flag deflection into ‘vortex-induced’ and ‘self’ components. The former is related directly to the motion that would be generated by the shed vortices if the flag were absent. This component serves as an inhomogeneous forcing term in the equation satisfied by the ‘self’ motion. It is found that forced flapping is possible whenever the Reynolds number based on the pole diameter ReD ≳ 100, such that a wake of distinct vortex structures is established behind the pole. Such conditions typically prevail at mean flow velocities significantly lower than the critical threshold values predicted by the linear theory. It is therefore argued that analyses of the onset of flag motion that are based on ideal, homogeneous flag theory are incomplete and that consideration of the pole-induced fluid flow is essential at all relevant wind speeds.

Previous laboratory measurements on drag of tandem rigid bodies moving in viscous incompressible fluids found that a following body experienced less drag than a leading one. Very recently a laboratory experiment (Ristroph & Zhang, Phys. Rev. Lett., vol. 101, 2008) with deformable bodies (rubble threads) revealed just the opposite – the leading body had less drag than the following one. The Reynolds numbers in the experiment were around 104. To find out how this qualitatively different phenomenon may depend on the Reynolds number, a series of numerical simulations are designed and performed on the interaction of a pair of tandem flexible flags separated by a dimensionless vertical distance (0 ≤ D ≤ 5.5) in a flowing viscous incompressible fluid at lower Reynolds numbers (40 ≤ Re ≤ 220) using the immersed boundary (IB) method. The dimensionless bending rigidity b and dimensionless flag mass density used in our work are as follows: 8.6 × 10−5 ≤ b ≤ 1.8 × 10−3, 0.8 ≤ ≤ 1.0. We obtain an interesting result within these ranges of dimensionless parameters: when Re is large enough so that the flapping of the two flags is self-sustained, the leading flag has less drag than the following one; when Re is small enough so that the flags maintain two nearly static line segments aligned with the mainstream flow, the following flag has less drag than the leading one. The transitional range of Re separating the two differing phenomena depends on the value of b. With Re in this range, both the flapping and static states are observed depending on the separation distance D. When D is small enough, the flags are in the static state and the following flag has less drag; when D is large enough the flags are in the constant flapping state and the leading flag has less drag. The critical value of D depends on b.