The nonlinear shallow-water equations are often used to model flow over topography. In this paper we use these equations both analytically and numerically to study flow over two widely separated localised obstacles, and compare the outcome with the corresponding flow over a single localised obstacle. Initially we assume uniform flow with constant water depth, which is then perturbed by the obstacles. The upstream flow can be characterised as subcritical, supercritical and transcritical, respectively. We review the well-known theory for flow over a single localised obstacle, where in the transcritical regime the flow is characterised by a local hydraulic flow over the obstacle, contained between an elevation shock propagating upstream and a depression shock propagating downstream. Classical shock closure conditions are used to determine these shocks. Then we show that the same approach can be used to describe the flow over two widely spaced localised obstacles. The flow development can be characterised by two stages. The first stage is the generation of upstream elevation shock and downstream depression shock from each obstacle alone, isolated from the other obstacle. The second stage is the interaction of two shocks between the two obstacles, followed by an adjustment to a hydraulic flow over both obstacles, with criticality being controlled by the higher of the two obstacles, and by the second obstacle when they have equal heights. This hydraulic flow is terminated by an elevation shock propagating upstream of the first obstacle and a depression shock propagating downstream of the second obstacle. A weakly nonlinear model for sufficiently small obstacles is developed to describe this second stage. The theoretical results are compared with fully nonlinear simulations obtained using a well-balanced finite-volume method. The analytical results agree quite well with the nonlinear simulations for sufficiently small obstacles.

Assuming resonant nonlinear wave interactions to be the dominant physical mechanism of growing wind-driven seas we propose a concise relationship between instantaneous wave steepness and time or fetch of wave development expressed in dimensionless wave periods or lengths. This asymptotic physical law derived from the first principles of the theory of weak turbulence does not contain wind speed explicitly. The validity of this law is illustrated by results of numerical simulations, in situ measurements of growing wind seas and wind-wave tank observations. The impact of this new view of sea-wave physics is discussed in the context of conventional approaches to wave modelling and forecasting.

The high-speed impact of a droplet on a bulk fluid at high Weber number (We) is not well understood but is relevant to the production of marine aerosol by raindrop impact on the sea surface. These splashes produce a subsurface cavity and a crown which closes into a bubble canopy, but a floating layer of immiscible oil, such as a crude oil slick, alters the splash dynamics. The effects of oil layer fluid properties and thickness, droplet size and impact speed are examined by high-speed visualization. Oil layer rupture and crown behaviour are classified by dimensional scaling. The subsurface cavity volume for impact on thick layers is shown to depend on the Reynolds number (Re), although canopy formation at high Re introduces a competing We effect since rapid canopy closure is found to retard cavity expansion. Time-resolved kinematic measurements show that thin crude oil slicks similarly alter crown closure and cavity growth. The size and spatial distributions of airborne droplets are examined using high-speed holographic microscopy. The droplets have a bimodal distribution with peaks at 50 and $225~{\rm\mu}\text{m}$ and are clustered by size at different elevation angles. Small droplets ($50~{\rm\mu}\text{m}$) are ejected primarily at shallow angles, indicating production by splashing within the first $100~{\rm\mu}\text{s}$ and by breakup of microligaments. Larger droplets ($225~{\rm\mu}\text{m}$) are found at steeper elevation angles, indicating later production by capillary instability acting on large ligaments protruding upward from the crown. Intermittent droplet release while the ligaments grow and sweep upward is thought to contribute to the size-dependent spatial ordering. Greater numbers of small droplets are produced at high elevation angles when a crude oil layer is present, indicating satellite droplet formation from ligament breakup. A crude oil layer also increases the target fluid Ohnesorge number, leading to creation of an intact ejecta sheet, which then ruptures to form aerosolized oil droplets.

At sufficient mass loading and in the presence of a mean body force (e.g. gravity), an initially random distribution of particles may organize into dense clusters as a result of momentum coupling with the carrier phase. In statistically stationary flows, fluctuations in particle concentration can generate and sustain fluid-phase turbulence, which we refer to as cluster-induced turbulence (CIT). This work aims to explore such flows in order to better understand the fundamental modelling aspects related to multiphase turbulence, including the mechanisms responsible for generating volume-fraction fluctuations, how energy is transferred between the phases, and how the cluster size distribution scales with various flow parameters. To this end, a complete description of the two-phase flow is presented in terms of the exact Reynolds-average (RA) equations, and the relevant unclosed terms that are retained in the context of homogeneous gravity-driven flows are investigated numerically. An Eulerian–Lagrangian computational strategy is used to simulate fully developed CIT for a range of Reynolds numbers, where the production of fluid-phase kinetic energy results entirely from momentum coupling with finite-size inertial particles. The adaptive filtering technique recently introduced in our previous work (Capecelatro et al., J. Fluid Mech., vol. 747, 2014, R2) is used to evaluate the Lagrangian data as Eulerian fields that are consistent with the terms appearing in the RA equations. Results from gravity-driven CIT show that momentum coupling between the two phases leads to significant differences from the behaviour observed in very dilute systems with one-way coupling. In particular, entrainment of the fluid phase by clusters results in an increased mean particle velocity that generates a drag production term for fluid-phase turbulent kinetic energy that is highly anisotropic. Moreover, owing to the compressibility of the particle phase, the uncorrelated components of the particle-phase velocity statistics are highly non-Gaussian, as opposed to systems with one-way coupling, where, in the homogeneous limit, all of the velocity statistics are nearly Gaussian. We also observe that the particle pressure tensor is highly anisotropic, and thus additional transport equations for the separate contributions to the pressure tensor (as opposed to a single transport equation for the granular temperature) are necessary in formulating a predictive multiphase turbulence model.

When a drop impacts on a solid surface, its rapid deceleration is cushioned by a thin layer of air, which leads to the entrapment of a bubble under its centre. For large impact velocities the lubrication pressure in this air layer becomes large enough to compress the air. Herein we use high-speed interferometry, with 200 ns time-resolution, to directly observe the thickness evolution of the air layer during the entire bubble entrapment process. The initial disc radius and thickness shows excellent agreement with available theoretical models, based on adiabatic compression. For the largest impact velocities the air is compressed by as much as a factor of 14. Immediately following the contact, the air disc shows rapid vertical expansion. The radial speed of the surface minima just before contact, can reach 50 times the impact velocity of the drop.

In comparison to the flow in a rigid channel, there is a multifold reduction in the transition Reynolds number for the flow in a microchannel when one of the walls is made sufficiently soft, due to a dynamical instability induced by the fluid–wall coupling, as shown by Verma & Kumaran (J. Fluid Mech., vol. 727, 2013, pp. 407–455). The flow after transition is characterised using particle image velocimetry in the $x{-}y$ plane, where $x$ is the streamwise direction and $y$ is the cross-stream coordinate along the small dimension of the channel of height 0.2–0.3 mm. The flow after transition is characterised by a mean velocity profile that is flatter at the centre and steeper at the walls in comparison to that for a laminar flow. The root mean square of the streamwise fluctuating velocity shows a characteristic sharp increase away from the wall and a maximum close to the wall, as observed in turbulent flows in rigid-walled channels. However, the profile is asymmetric, with a significantly higher maximum close to the soft wall in comparison to that close to the hard wall, and the Reynolds stress is found to be non-zero at the soft wall, indicating that there is a stress exerted by fluid velocity fluctuations on the wall. The maximum of the root mean square of the velocity fluctuations and the Reynolds stress (divided by the fluid density) in the soft-walled microchannel for Reynolds numbers in the range 250–400, when scaled by suitable powers of the maximum velocity, are comparable to those in a rigid channel at Reynolds numbers in the range 5000–20 000. The near-wall velocity profile shows no evidence of a viscous sublayer for $(yv_{\ast }/{\it\nu})$ as low as two, but there is a logarithmic layer for $(yv_{\ast }/{\it\nu})$ up to approximately 30, where the von Karman constants are very different from those for a rigid-walled channel. Here, $v_{\ast }$ is the friction velocity, ${\it\nu}$ is the kinematic viscosity and $y$ is the distance from the soft surface. The surface of the soft wall in contact with the fluid is marked with dye spots to monitor the deformation and motion along the fluid–wall interface. Low-frequency oscillations in the displacement of the surface are observed after transition in both the streamwise and spanwise directions, indicating that the velocity fluctuations are dynamically coupled to motion in the solid.

We investigate the instability of a single helical vortex filament of small pitch with respect to displacement perturbations whose wavelength is large compared to the vortex core size. We first revisit previous theoretical analyses concerning infinite Rankine vortices, and consider in addition the more realistic case of vortices with Gausssian vorticity distributions and axial core flow. We show that the various instability modes are related to the local pairing of successive helix turns through mutual induction, and that the growth rate curve can be qualitatively and quantitatively predicted from the classical pairing of an array of point vortices. We then present results from an experimental study of a helical vortex filament generated in a water channel by a single-bladed rotor under carefully controlled conditions. Various modes of displacement perturbations could be triggered by suitable modulation of the blade rotation. Dye visualisations and particle image velocimetry allowed a detailed characterisation of the vortex geometry and the determination of the growth rate of the long-wave instability modes, showing good agreement with theoretical predictions for the experimental base flow. The long-term (downstream) development of the pairing instability leads to a grouping and swapping of helix loops. Despite the resulting complicated three-dimensional structure, the vortex filaments surprisingly remain mostly intact in our observation interval. The characteristic distance of evolution of the helical wake behind the rotor decreases with increasing initial amplitude of the perturbations; this can be predicted from the linear stability theory.

We study the freely decaying weak turbulence of capillary waves by direct numerical solution of the primitive Euler equations. By introducing a small amount of wave dissipation, measured by the viscosity magnitude ${\it\gamma}_{0}$, we are able to recover phenomena observed in experiments that are not described by weak-turbulence theory (WTT), including the exponential modal decay and time variation of the width and power-law spectral slope ${\it\alpha}$ of the inertial range. In contrast to WTT, this problem also involves non-constant inter-modal energy transfer across the inertial range, which imposes a difficulty in quantifying and measuring the energy flux $P$ associated with a certain power-law spectrum. We propose an effective and novel way to evaluate $P$ in such cases by physically considering the unsteady effects of the spectrum and variation of the inter-modal energy transfer. Our results show the fundamental difference between the energy flux $P$ and the total energy dissipation rate ${\it\Gamma}$, which is due to significant energy dissipation within the inertial range. This settles the previous debate on the measurement of $P$ which assumes the equivalence of the two. Based on our numerical data, we obtain a general form of the time-evolving inertial-range spectrum, where the parameters involved are functions of ${\it\gamma}_{0}$ only. The value of the spectral slope ${\it\alpha}$ at each time moment in the decay, however, is found to be uniquely related to the spectral magnitude at that time and irrespective of ${\it\gamma}_{0}$, in the range we consider. This physically reveals the dominant effect of nonlinear wave interaction in forming the power-law spectrum within the inertial range. The evolutions of the inertial-range energy are shown to be predicted by analytical integration of the evolving spectra for different values of ${\it\gamma}_{0}$.

We analyse the low-Reynolds-number flow generated by a cylinder (of radius $a$) in a stream (of velocity $U_{\infty }$) which has a uniform through-surface blowing component (of velocity $U_{b}$). The flow is characterized in terms of the Reynolds number $Re$ ($=2aU_{\infty }/{\it\nu}$, where ${\it\nu}$ is the kinematic viscosity of the fluid) and the dimensionless blow velocity ${\it\Lambda}$ ($=U_{b}/U_{\infty }$). We seek the leading-order symmetric solution of the vorticity field which satisfies the near- and far-field boundary conditions. The drag coefficient is then determined from the vorticity field. For the no-blow case Lamb’s (Phil. Mag., vol. 21, 1911, pp. 112–121) expression is retrieved for $Re\rightarrow 0$. For the strong-sucking case, the asymptotic limit, $C_{D}\approx -2{\rm\pi}{\it\Lambda}$, is confirmed. The blowing solution is valid for ${\it\Lambda}<4/Re$, after which the flow is unsymmetrical about ${\it\theta}={\rm\pi}/2$. The analytical results are compared with full numerical solutions for the drag coefficient $C_{D}$ and the fraction of drag due to viscous stresses. The predictions show good agreement for $Re=0.1$ and ${\it\Lambda}=-5,0,5$.