In flows where the ratio of inertia to gravity varies strongly, large variations in the fluid thickness appear and hydraulic jumps arise, as depicted by Rayleigh. We report a new family of hydraulic jumps, where the capillary effects dominate the gravitational acceleration. The Bond number – which measures the importance of gravitational body forces compared to surface tension – must be small in order to observe such objects using capillarity as a driving force. For water, the typical length should be smaller than 3 mm. Nevertheless, for such small scales, solid boundaries induce viscous stresses, which dominate inertia, and capillary jumps should not be described by the inertial shock wave theory that one would deduce from Bélanger or Rayleigh for hydraulic jumps. In order to get rid of viscous shears, we consider Plateau borders, which are the microchannels defined by the merging of three films inside liquid foams, and we show that capillary jumps propagate along these deformable conduits. We derive a simple model that predicts the velocity, geometry and shape of such fronts. A strong analogy with Rayleigh’s description is pointed out. In addition, we carried out experiments on a single Plateau border generated with soap films to observe and characterize these capillary jumps. Our theoretical predictions agree remarkably well with the experimental measurements.

We study unconfined homogeneous turbulence with a destabilizing background density gradient in the Boussinesq approximation. Starting from initial isotropic turbulence, the buoyancy force induces a transient phase toward a self-similar regime accompanied by a rapid growth of kinetic energy and Reynolds number, along with the development of anisotropic structures in the flow in the direction of gravity. We model this with a two-point statistical approach using an axisymmetric eddy-damped quasi-normal Markovian (EDQNM) closure that includes buoyancy production. The model is able to match direct numerical simulations (DNS) in a parametric study showing the effect of initial Froude number and mixing intensity on the development of the flow. We further improve the model by including the stratification timescale in the characteristic relaxation time for triple correlations in the closure. It permits the computation of the long-term evolution of unstably stratified turbulence at high Reynolds number. This agrees with recent theoretical predictions concerning the self-similar dynamics and brings new insight into the spectral energy distribution and anisotropy of the flow.

The structure and stability of Stewartson shear layers with different heights are investigated numerically via axisymmetric simulation and linear stability analysis, and a validation of the quasi-two-dimensional model is performed. The shear layers are generated in a rotating cylindrical tank with circular disks located at the lid and base imposing a differential rotation. The axisymmetric model captures both the thick and thin nested Stewartson layers, which are scaled by the Ekman number ($\mathit{E}\,$) as $\mathit{E}\,^{1/4}$ and $\mathit{E}\,^{1/3}$ respectively. In contrast, the quasi-two-dimensional model only captures the $\mathit{E}\,^{1/4}$ layer as the axial velocity required to invoke the $\mathit{E}\,^{1/3}$ layer is excluded. A direct comparison between the axisymmetric base flows and their linear stability in these two models is examined here for the first time. The base flows of the two models exhibit similar flow features at low Rossby numbers ($\mathit{Ro}$), with differences evident at larger $\mathit{Ro}$ where depth-dependent features are revealed by the axisymmetric model. Despite this, the quasi-two-dimensional model demonstrates excellent agreement with the axisymmetric model in terms of the shear-layer thickness and predicted stability. A study of various aspect ratios reveals that a Reynolds number based on the theoretical Ekman layer thickness is able to describe the transition of a base flow that is reflectively symmetric about the mid-plane to a symmetry-broken state. Additionally, the shear-layer thicknesses scale closely to the expected ${\it\delta}_{vel}\propto A\mathit{E}\,^{1/4}$ and ${\it\delta}_{vort}\propto A\mathit{E}\,^{1/3}$ for shear layers that are not affected by the confinement ($A\mathit{E}\,^{1/4}\lesssim 0.34$ in this system, the ratio of tank height to shear-layer radius). The linear stability analysis reveals that the ratio of Stewartson layer radius to thickness should be greater than $45$ for the stability of the flow to be independent of aspect ratio. Thus, for sufficiently small $A\mathit{E}\,^{1/4}$ and $A\mathit{E}\,^{1/3}$, the flow characteristics remain similar and the linear stability of the flow can be described universally when the azimuthal wavelength is scaled against $A$. The analysis also recovers an asymptotic scaling for the normalized azimuthal wavelength which suggests that ${\it\lambda}_{{\it\theta},c}^{\ast }\propto (|\mathit{Ro}|/\mathit{E}\,^{2})^{-1/5}$ for geometry-independent shear layers at marginal stability.

The effects of scale and aeration on violent breaking wave impacts with trapped and entrained air are investigated both analytically and numerically. By dimensional analysis we show that the impact pressures for Froude scaled conditions prior to the impact depend on the scale and aeration level. The Bagnold–Mitsuyasu scaling law for the compression of an air pocket by a piston of incompressible water is rederived and generalised to 3D air pockets of arbitrary shape. Numerical results for wall pressure, force and impulse are then presented for a flip-through impact, a low-aeration impact and a high-aeration impact, for nine scales and five levels of initial aeration. Two of these impact types trap a pocket of air at the wall. Among the findings of the paper is that for fixed initial aeration, impact pressures from the flip-through impact broadly follow Froude scaling. This is also the case for the two impact types with trapped air pockets for impact pressures below 318 kPa, while impact pressures above this value broadly follow the Bagnold–Mitsuyasu scaling law with full-scale pressures greater than those predicted by the Froude law. For all impact types, the effect of aeration is found to reduce the maximum impact pressure, maximum force and impulse. Good agreement with the asymptotic model of Peregrine & Thais (J. Fluid Mech., vol. 325, 1996, pp. 377–397) is found for the flip-through impact pressure and a fair agreement is found for the low- and high-aeration impacts. Based on the numerical results, a modified scaling curve that combines Froude scaling and the Bagnold–Mitsuyasu law is suggested. The practical implications of the findings are discussed and attention is drawn to the limitations of physical model tests.

This paper presents a numerical study on steady flow around two identical circular cylinders of various arrangements at a low subcritical Reynolds number ($\mathit{Re}=10^{3}$). The ratio of centre-to-centre pitch distance ($P$) to the diameter of the cylinder ($D$) ranges from 1.5 to 4, and the alignment angle $({\it\alpha})$ between the two cylinders and the direction of the cross-flow varies from 0 to 90°. The detailed flow information obtained from direct numerical simulation allows a comprehensive interpretation of the underlying physics responsible for some interesting flow features observed around two staggered cylinders. Four distinct vortex shedding regimes are identified and it is demonstrated that accurate classification of vortex shedding regimes around two staggered cylinders should consider the combination of the flow visualization with the analyses of lift forces and velocity signal in the wake. It is revealed that the change in pressure distribution, as a result of different vortex shedding mechanisms, leads to a variety of characteristics of hydrodynamic forces on both cylinders, including negative drag force, attractive and repulsive lift forces. Two distinct vortex shedding frequencies are identified and are attributed to the space differences based on the flow structures observed in the wake of the cylinders. It is also found that the three-dimensionality of flow in the gap and the shared wake region is significantly weakened in almost two of the classified flow regimes; however, compared with the flow around a single cylinder, active wake interaction at large ${\it\alpha}$ does not clearly increase the three-dimensionality.

One-dimensional planar premixed flames propagating in a uniform flow are susceptible to hydrodynamic instabilities known (generically) as Darrieus–Landau instabilities. Here, we extend that hydrodynamic linear stability analysis to include a lateral shear. This generalization is a situation of interest for laminar and turbulent flames when they travel into a region of shear (such as a jet or shear layer). It is shown that the problem can be formulated and solved analytically and a dispersion relation can be determined. The solution depends on a shear parameter in addition to the wavenumber, thermal expansion ratio, and Markstein lengths. The study of the dispersion relation shows that perturbations have two types of behaviour as wavenumber increases. First, for small shear, we recover the Darrieus–Landau results except for a region at small wavenumbers, large wavelengths, that is stable. Initially, increasing shear has a stabilizing effect. But, for sufficiently high shear, the flame becomes unstable again and its most unstable wavelength can be much smaller than the Markstein length of the zero-shear flame. Finally, the stabilizing effect of low shear can make flames with negative Markstein numbers stable within a band of wavenumbers.

The convective and absolute nature of instabilities in Rayleigh–Bénard–Poiseuille (RBP) mixed convection for viscoelastic fluids is examined numerically with a shooting method as well as analytically with a one-mode Galerkin expansion. The viscoelastic fluid is modelled by means of a general constitutive equation that encompasses the Maxwell model and the Oldroyd-B model. In comparison to Newtonian fluids, two more dimensionless parameters are introduced, namely the elasticity number ${\it\lambda}_{1}$ and the ratio ${\it\Gamma}$ between retardation and relaxation times. Temporal stability analysis of the basic state showed that the three-dimensional thermoconvective problem can be Squire-transformed. Therefore, one must distinguish mainly between two principal roll orientations: transverse rolls TRs (rolls with axes perpendicular to the Poiseuille flow direction) and longitudinal rolls LRs (rolls with axes parallel to the Poiseuille flow direction). The critical Rayleigh number for the appearance of LRs is found to be independent of the Reynolds number ($\mathit{Re}$). Depending on ${\it\lambda}_{1}$ and ${\it\Gamma}$, two different regimes can be distinguished. In the weakly elastic regime, the emerging LRs are stationary, while they are oscillatory in the strongly elastic regime. For TRs, it is found that in the weakly elastic regime, the stabilization effect of $\mathit{Re}$ is more important than in Newtonian fluids. Moreover, for sufficiently elastic fluids a jump is observed in the oscillation frequencies and wavenumbers for moderate $\mathit{Re}$. In the strongly elastic regime, the effect of the imposed throughflow is to promote the appearance of the upstream moving TRs for low values of $\mathit{Re}$, which are replaced by downstream moving TRs for higher values of $\mathit{Re}$. Moreover, the results proved that, contrary to the case where $\mathit{Re}=0$, the elasticity number ${\it\lambda}_{1}$ (the ratio ${\it\Gamma}$) has a strongly stabilizing (destabilizing) effect when the throughflow is added. The influence of the rheological parameters on the transition curves from convective to absolute instability in the Reynolds–Rayleigh number plane is also determined. We show that the viscoelastic character of the fluid hastens the transition to absolute instability and even may suppress the convective/absolute transition. Throughout this paper, similarities and differences with the corresponding problem for Newtonian fluids are highlighted.

We examine the dissolution of a vertical solid surface in the case where the heat and mass transfer is driven by turbulent compositional convection. A theoretical model of the turbulent dissolution of a vertical wall is developed, which builds on the scaling analysis presented by Kerr (J. Fluid Mech., vol. 280, 1994, pp. 287–302) for the turbulent dissolution of a horizontal floor or roof. The model has no free parameters and no dependence on height. The analysis is tested by comparing it with laboratory measurements of the ablation of a vertical ice wall in contact with salty water. The model is found to accurately predict the dissolution velocity for water temperatures up to approximately 5–$6\,^{\circ }\text{C}$, where there is a transition from turbulent dissolution to turbulent melting. We quantify the turbulent convective dissolution of vertical ice bodies in the polar oceans, and compare our results with some field observations.

The wake of a ship that sails at relatively large Froude numbers usually contains a number of components of different nature and with different heights, lengths, timings and propagation directions. We explore the possibilities of the spectrogram representation of one-point measurements of the ship wake to identify these components and to quantify their main properties. This representation, based on the short-time Fourier transform, facilitates a reliable decomposition of the wake into constituent components and makes it possible to quantify their variations in the time–space domain and the energy content of each component, from very low-frequency precursor waves up to high-frequency signals within the frequency range of typical wind-generated waves. A method for estimation of the ship speed and the distance of its sailing line from the measurement site is proposed, which only uses information available within the record of the ship wake surface elevation, but where it is assumed that the wake pattern does not deviate significantly from the classical Kelvin wake structure. The wake decomposition using the spectrogram method allows investigation of the energy content that can be attributed to each individual component of the wake. We demonstrate that the majority (60–80 %) of wake energy from strongly powered large ferries that sail at depth Froude numbers ${\sim}0.7$ is concentrated in components that are located near the edge of the wake wedge. Finally, we demonstrate that the spectrogram representation offers a convenient way to identify a specific signature of single types of ships.

We investigate experimentally the formation of bedforms caused by the sustained flow of water and solid particles in a circular pipe ($\varnothing =30~\text{mm}$). The special feature of the tests carried out was the use of floating particles ($d=756~{\rm\mu}\text{m}$, ${\it\rho}_{s}=907~\text{kg}~\text{m}^{-3}$) whereas bedforms are usually studied with sedimental materials. A closed loop was used, so that the solid flux could be maintained for an infinite time. The finite size of the tube led to the saturation of the growth of the vortex ripples produced. For the set of parameters studied, the threshold of motion was obtained within a range of laminar to low turbulent flow. The saturated state was studied to characterise it for different flow rates and solid loads. The frequency, wavelength and propagating velocity of ripples were determined using different methodologies based on image analysis and pressure analysis. The frequency and propagating velocity show a clear linear dependence on the initial Shields number, while the wavelength seems to be constant in our experiments.

In a balanced oceanic front, the possible directions of the group velocity vector for internal waves depart from the classic Saint Andrew’s cross as a consequence of sloping isopycnals and the associated thermal wind shear. However, for waves oscillating at the Coriolis frequency $f$, one of these directions remains horizontal, while the other direction allows for vertical propagation of energy. This implies the existence of critical reflections from the ocean surface, after which wave energy, having propagated from below, cannot propagate back down. This is similar to the reflection of internal waves, propagating in a quiescent medium, from a bottom that runs parallel to the group velocity vector. We first illustrate this phenomenon with a series of linear Boussinesq numerical experiments on waves with various frequencies, ${\it\omega}$, exploring critical (${\it\omega}=f$), forward (${\it\omega}>f$), and backward (${\it\omega}<f$) reflections. We then conduct the nonlinear equivalents of these simulations. In agreement with the classical case, backward reflection inhibits triadic resonances and does not exhibit prominent nonlinear effects, while forward reflection shows strong generation of harmonics that radiate energy away from the surface. Surprisingly though, critical reflections are associated with oscillatory motions that extend down from the surface. These motions are not freely propagating waves but instead take the form of a cluster of non-resonant triads which decays with depth through friction.

We present an experimental investigation of entrainment and the dynamics near the turbulent/non-turbulent interface in a dense gravity current. The main goal of the study is to investigate changes in the interfacial physics due to the presence of stratification and to examine their impact on the entrainment rate. To this end, three-dimensional data sets of the density and the velocity fields are obtained through a combined scanning particle tracking velocimetry/laser-induced fluorescence approach for two different stratification levels with inflow Richardson numbers of $\mathit{Ri}_{0}=0.23$ and $\mathit{Ri}_{0}=0.46$, respectively, at a Reynolds number around $\mathit{Re}_{0}=3700$. An analysis conditioned on the instantaneous position of the turbulent/non-turbulent interface as defined by a threshold on enstrophy reveals an interfacial region that is in many aspects independent of the initial level of stratification. This is reflected most prominently in matching peaks of the gradient Richardson number $\mathit{Ri}_{g}\approx 0.1$ located approximately $10{\it\eta}$ from the position of the interface inside the turbulent region, where ${\it\eta}=({\it\nu}^{3}/{\it\epsilon})^{1/4}$ is the Kolmogorov scale, and ${\it\nu}$ and ${\it\epsilon}$ denote the kinematic viscosity and the rate of turbulent dissipation, respectively. A possible explanation for this finding is offered in terms of a cyclic evolution in the interaction of stratification and shear involving the buildup of density and velocity gradients through inviscid amplification and their subsequent depletion through molecular effects and pressure. In accordance with the close agreement of the interfacial properties for the two cases, no significant differences were found for the local entrainment velocity, $v_{n}$ (defined as the propagation velocity of an enstrophy isosurface relative to the fluid), at different initial stratification levels. Moreover, we find that the baroclinic torque does not contribute significantly to the local entrainment velocity. Comparing results for the surface area of the convoluted interface to estimates from fractal scaling theory, we identify differences in the interface geometry as the major factor in the reduction of the entrainment rate due to density stratification.

We investigate linear–quadratic dynamical systems with energy-preserving quadratic terms. These systems arise for instance as Galerkin systems of incompressible flows. A criterion is presented to ensure long-term boundedness of the system dynamics. If the criterion is violated, a globally stable attractor cannot exist for an effective nonlinearity. Thus, the criterion can be considered a minimum requirement for control-oriented Galerkin models of viscous fluid flows. The criterion is exemplified, for example, for Galerkin systems of two-dimensional cylinder wake flow models in the transient and the post-transient regime, for the Lorenz system and for wall-bounded shear flows. There are numerous potential applications of the criterion, for instance, system reduction and control of strongly nonlinear dynamical systems.

It is known that the drag for flows driven by a pressure gradient in heated channels can be reduced below the level found in isothermal channels. This reduction occurs for spatially modulated heating and is associated with the formation of separation bubbles which isolate the main stream from direct contact with the solid wall. It is demonstrated that the use of a proper combination of spatially distributed and spatially uniform heating components results in an increase in the horizontal and vertical temperature gradients which lead to an intensification of convection which, in turn, significantly increases the drag reduction. An excessive increase of the uniform heating leads to breakup of the bubbles and the formation of complex secondary states, resulting in a deterioration of the system performance. This performance may, under certain conditions, still be better than that achieved using only spatially distributed heating. Detailed calculations have been carried out for the Prandtl number $\mathit{Pr}=0.71$ and demonstrate that this technique is effective for flows with a Reynolds number $\mathit{Re}<10$; faster flows wash away separation bubbles. The question of net gain remains to be settled as it depends on the method used to achieve the desired wall temperature and on the cost of the required energy. The presented results provide a basis for the design of passive flow control techniques utilizing heating patterns as controlling agents.

Immersed boundary-lattice Boltzmann simulations are used to examine the effects of particle rotation, at low particle Reynolds numbers, on flows in ordered and random arrays of mono-disperse spheres. The drag force, the Magnus lift force and the torque on the spheres, are determined at solid volume fractions up to the close-packed limits of the arrays. The rotational Reynolds number based on the angular velocity and the diameter of the spheres is used to characterize the rotational movement of spheres. The results show that the normalized Magnus lift force produced by particle rotation is approximately in direct proportion to the rotational Reynolds number, while the normalized drag force and torque acting on spheres are barely affected by this number. The Magnus lift force is negligible relative to the magnitude of the drag force when the rotational Reynolds number is low. However, it can be very significant, and even larger than the drag force, as the rotational Reynolds number increases up to $O(10^{2})$, especially for low solid volume fractions. Based on the simulation results, relations for the Magnus lift force and the torque for both ordered arrays and random arrays of rotating spheres at solid volume fractions from zero to close-packed limits are formulated. Further, the drag force relations in the literature are revised based on existing theories and the present simulation results for both arrays of spheres.

The flow past a sphere moving vertically at constant speeds in a salt-stratified fluid is investigated numerically at moderate Reynolds numbers $\mathit{Re}$. Time development of the flow shows that the violation of density conservation is the key process for the generation of the buoyant jet observed in the experiments. For example, if the sphere moves downward, isopycnal surfaces are simply deformed and dragged down by the sphere while the density is conserved along the flow. (The flow pattern is inverted if the sphere moves upward. Some explanations are given in the introduction.) Then, the flow will never become steady. As density diffusion becomes effective around the sphere surface and generates a horizontal hole in the isopycnal surface, fluid with non-conserved density is detached from the isopycnal surface and moves upward to generate a buoyant jet. These processes will constitute a steady state near the sphere. With lengths scaled by the sphere diameter and velocities by the downward sphere velocity, the duration of density conservation at the rear/upper stagnation point, or the maximum distance that the isopycnal surface is dragged downward, is proportional to the Froude number $\mathit{Fr}$, and estimated well by ${\rm\pi}\mathit{Fr}$ for $\mathit{Fr}\gtrsim 1$ and $\mathit{Re}\gtrsim 200$, corresponding to a constant potential energy.  The radius of a jet defined by the density and velocity distributions, which would have correlations with the density and velocity boundary layers on the sphere, is estimated well by $\sqrt{\mathit{Fr }/2\mathit{Re }\mathit{ Sc}}$ and $\sqrt{\mathit{Fr }/2\mathit{Re}}$ respectively for $\mathit{Fr}\lesssim 1$, where $\mathit{Sc}$ is the Schmidt number. Numerical results agree well with the previous experiments, and the origin of the conspicuous bell-shaped structure observed by the shadowgraph method is identified as an internal wave.

Inertial lift forces are exploited within inertial microfluidic devices to position, segregate and sort particles or droplets. However, the forces and their focusing positions can currently only be predicted by numerical simulations, making rational device design very difficult. Here we develop theory for the forces on particles in microchannel geometries. We use numerical experiments to dissect the dominant balances within the Navier–Stokes equations and derive an asymptotic model to predict the lateral force on the particle as a function of particle size. Our asymptotic model is valid for a wide array of particle sizes and Reynolds numbers, and allows us to predict how focusing position depends on particle size.

An experimental study of turbulent wave–current boundary layer flows is performed using a state-of-the-art oscillating water tunnel (OWT) for flow generation and a particle image velocimetry system for velocity measurements. The current velocity profiles in the presence of sinusoidal waves indicate a two-log-profile structure suggested by the widely-used Grant–Madsen model. However, for weak currents in the presence of nonlinear waves, the two-log-profile structure is contaminated or even totally obliterated by the boundary layer streaming which is produced by the asymmetry of turbulence in successive half-periods of nonlinear waves. To interpret experimental results, a semi-analytical model which adopts a rigorous way to account for a time-varying turbulent eddy viscosity is developed. The model can accurately predict turbulence asymmetry streaming, which leads to successful predictions of the mean velocity embedded in nonlinear-wave tests and the current velocity profiles in the presence of either sinusoidal or nonlinear waves. Since the Longuet-Higgins-type streaming due to wave propagation is absent in OWT flows and not included in the semi-analytical model, future work is necessary to extend this study for applications in the coastal environment.

We experimentally investigate the thrust and propulsive efficiency of a NACA 0012 airfoil undergoing oscillating pitching motion at a Reynolds number of $1.7\times 10^{4}$. While previous studies have computed thrust and power indirectly through measurements of momentum deficit in the object’s wake, we use a pair of force transducers to measure fluid forces directly. Our results help solidify a variety of experimental, theoretical and computational answers to this classical problem. We examine trends in propulsive performance with flapping frequency, amplitude and Reynolds number. We also examine the measured unsteady forces on the airfoil and compare them with linear theory dating from the first half of the 20th century. While linear theory significantly overpredicts the mean thrust on the foil, its prediction for the amplitude and phase of the time-varying component is surprisingly accurate. We conclude with evidence that the thrust force produced by the pitching airfoil is largely insensitive to most wake vortex arrangements.

Large eddy simulation (LES) is used to investigate the evolution of Boussinesq gravity currents propagating through a channel of height $H$ containing a staggered array of identical cylinders of square cross-section and edge length $D$. The cylinders are positioned with their axes horizontal and perpendicular to the (streamwise) direction along which the lock-exchange flow develops. The effects of the volume fraction of solids, ${\it\phi}$, the Reynolds number and geometrical parameters describing the array of obstacles on the structure of the lock-exchange flow, total drag force acting on the gravity current, front velocity and global energy budget are analysed. Simulation results show that the currents rapidly transition to a state in which the extra resistance provided by the cylinders strongly retards the motion and dominates the dissipative processes. A shallow layer model is also formulated and similarity solutions for the motion are found in the regime where the driving buoyancy forces are balanced by the drag arising from the interaction with the cylinders. The numerical simulations and this shallow layer model show that low-Reynolds-number currents transition to a drag-dominated regime in which the resistance is linearly proportional to the flow speed and, consequently, the front velocity, $U_{f}$, is proportional to $t^{-1/2}$, where $t$ is the time measured starting at the gate release time. By contrast, high-Reynolds-number currents, for which the cylinder Reynolds number is sufficiently high that the drag coefficient for most of the cylinders can be considered constant, transition first to a quadratic drag-dominated regime in which the front speed determined from the simulations is given by $U_{f}\sim t^{-0.25}$, before undergoing a subsequent transition to the aforementioned linear drag regime in which $U_{f}\sim t^{-1/2}$. Meanwhile, away from the front, the depth-averaged gravity current velocity is proportional to $t^{-1/3}$, a result that is in agreement with the shallow water model. It is suggested that the difference between these two is due to mixing processes, which are shown to be significant in the numerical simulations, especially close to the front of the motion. Direct estimation of the drag coefficient $C_{D}$ from the numerical simulations shows that the combined drag parameter for the porous medium, ${\it\Gamma}_{D}=C_{D}{\it\phi}(H/D)/(1-{\it\phi})$, is the key dimensionless grouping of variables that determines the speed of propagation of the current within arrays with different $C_{D},{\it\phi}$ and $D/H$.