Two-dimensional (2-D) quadrant analysis is generally used for investigating flow and sediment dynamics around a rigid structure in open channel flows. Given that particle distribution around rigid obstacles is not spatially uniform and changes in time, while vortices evolve to become three-dimensional (3-D) structures, 2-D quadrant analysis might be unsuitable to completely determine the sediment transport. Hence, 3-D quadrant and 3-D octant analyses should be considered, using the 3-D instantaneous velocity data and relative 3-D bursting process to define sediment transport surrounding the submerged square and circular cylinders. The turbulent kinetic energy (TKE), transition probabilities, occurrence probabilities, stress fraction and angles of inclination of 3-D bursting events are considered to quantify the coherent structures surrounding the cylinders and their interaction with bed particles. Experiments were conducted at the Hydraulics and Water Resources Engineering Laboratory, School of Infrastructure, Indian Institute of Technology, Bhubaneswar, Odisha, India, and velocity data were recorded at different cross-sections around the submerged cylinders using an acoustic Doppler velocimeter. Results show that the TKE is greater for internal ejection, external ejection, and internal sweep, external sweep in the upstream of the circular and square cylindrical structures. On the other part, the TKE is significant for internal ejection, external ejection, and internal sweep, external sweep in the downstream of the aligned square cylindrical structures, which justifies the highest scour depth that occurred upstream of the circular and square cylindrical structures and downstream of the aligned square cylindrical structure. The transition probability of the bursting events was determined using the Markov process from the measured velocity data to investigate the consecutive occurrence of bursting events. Further, the importance of sweeps and ejections on sediment erosion surrounding the cylinders within the scour hole at various stages of its development was investigated via 3-D quadrant analysis of the bursting occurrences. The outcomes show that external sweep and internal ejection events are active mechanisms for bed particle transport surrounding the cylinder. The maximum transition probability values are found around aligned cylindrical structures in comparison with the circular and square cylindrical structures in the transverse direction. This depicts the formation of a trailing vortex on both sides of the aligned square cylindrical object. The results reveal that the effect of inclination angles with respect to the water flow is greater for internal ejection and external sweep from upstream to downstream within the scour hole surrounding the cylindrical structures at various phases of development as horseshoe vortices and downflow develop upstream of the cylindrical structures while trailing vortices and wake vortices form at the top and downstream of the cylindrical structures. Internal and external ejection have a higher stress fraction than an internal and external sweep for square cylinders with alignment angles of 0°, 20° and circular cylinders over underdeveloped and developed scoured beds, respectively. With the higher percentage of fractional contributions for internal sweeps, the external sweep is predicted close to the cylindrical objects in comparison with the internal ejection and external ejection events because of the formation and warping of the horseshoe vortex close to the cylindrical objects, suggesting a significant probability of 3-D bursting occurrences with sediment movement near the cylindrical structures.

Steep wave breaking on a vertical cylinder (a typical foundation supporting offshore wind turbines) will induce slam loads. Many questions on the important violent wave loading and the associated secondary load cycle remain unanswered. We use laboratory experiments with unidirectional waves to investigate the fluid loading on vertical cylinders. We use a novel three-phase decomposition approach that allows us to separate different types of nonlinearity. Our findings reveal the existence of an additional quasi-impulsive loading component that is associated with the secondary load cycle and occurs in the backwards direction against that of the incoming waves. This quasi-impulsive force occurs at the end of the secondary load cycle and close to the passage of the downward zero-crossing point of the undisturbed wave. Wavelet analysis showed that the impulsive force exhibits superficially similar behaviour to a typical wave-slamming event but in the reverse direction. To monitor the scattered wave field and extract run-up on the cylinder, we installed a four-camera synchronised video system and found a strong temporal correlation between the arrival time of the Type-II scattered wave onto the cylinder and the occurrence of this quasi-impulsive force. The temporal characteristics of this quasi-impulsive force can be approximated by the Goda wave impact model, taking the collision of the Type-II scattered waves at the rear stagnation point as the impact source.

A depth-averaged model for turbulent open-channel flows with breaking roll waves on a sloping smooth bottom is derived under an assumption of independence between the wall turbulence and the roller turbulence. The model includes four variables – the water depth, the average velocity, and two variables called enstrophy, the shear enstrophy and the roller enstrophy – which take into account the deviations of the velocity with respect to its depth-averaged value due to shear effect and roller turbulence, respectively. The four equations of the model are the mass, momentum, energy and shear enstrophy balance equations, with the mathematical structure of the Euler equations of compressible fluids, with an additional transport equation and with source terms. The system is hyperbolic. The roller enstrophy is created by shocks. A non-zero value of the roller enstrophy indicates a breaking wave and a turbulent roller. The model is solved by a fast and well-known numerical scheme, with an explicit finite-volume method in one step. The model is used to simulate periodic and natural roll waves with a good agreement with existing experimental results. There is no parameter to calibrate in the model, which gives it a predictive character.

We consider a special type of hydraulic jumps (internal bores) which, in the vertically bounded system of two immiscible fluids with slightly different densities, conserve not only the mass and impulse but also the circulation and energy. This is possible only at specific combinations of the upstream and downstream states. Two such combinations are identified with arbitrary upstream and downstream interface heights. The first has a cross-symmetry between the interface height and shear on both sides of the jump. This symmetry, which is due to the invariance of the two-layer shallow-water system with swapping the interface height and shear, ensures the automatic conservation of the impulse and energy as well as the continuity of characteristic velocities across the jump. The speed at which such jumps propagate is defined uniquely by the conservation of the mass and circulation. The other possibility is a marginally stable shear flow that can have fully conservative jumps with discontinuous characteristic velocities. Both types of conservative jumps are shown to represent a long-wave approximation to the so-called solibores that appear as smooth permanent-shape solutions in a weakly non-hydrostatic model. A new analytical solution for solibores is obtained and found to agree very well with the previous DNS results for partial-depth lock release flow. The finding that certain large-amplitude hydraulic jumps can be fully conservative, while most are not such even in the inviscid approximation, points towards the wave dispersion as a primary mechanism behind the lossy nature of internal bores.

The tidal flow through a channel connecting two basins with different tidal regimes can be optimally controlled by means of a turbine fence or array to maximise the extracted mechanical power. The paper gives the optimal control strategy as a function of the blockage ratio $\sigma$, i.e. the ratio of the turbine cross-section to the cross-section of the local passage of a turbine. The results presented are a physically consistent generalisation of the results of Garrett & Cummins (Proc. R. Soc. Lond. A, vol. 461, 2060, pp, 2563–2572), valid only for $\sigma =1$ and turbine efficiency of one, now for arbitrary blockage ratio $0 < \sigma \leqslant 1$. Published research over the past decade on the same topic has taken the momentum equation and the turbine drag force as a starting point. The new approach presented here, in contrast, takes the energy equation as the starting point and uses the relative volume flow as the control variable. As the work shows, this new approach has three advantages. First, starting with the energy equation allows us to derive an optimal flow control problem resulting in an Euler–Lagrange equation using the physically consistent and experimentally validated actuator disk model for the free surface flow of Pelz et al. (J. Fluid Mech., vol. 889, 2020) in a direct and formal way. The optimal control problem is solved (a) numerically and (b) analytically. In the latter case, the turbine characteristics are approximated by a rational function in the relevant design and operating range. The analytical solution (b) validated against the numerical solution (a) is surprisingly concise and easy to apply in practice, as shown by use cases. Second, instead of the induction factor, we use the volume flow that is the same for all turbines in a cascade, i.e. a row of turbines in the direction of flow, which significantly reduces the complexity of the optimal control task of turbine arrays. Third, we obtain a well-founded energy estimate, whereas previous methods overestimate the energy yield due to inconsistent turbine disc models (for the consistency and valid parameter ranges of different models, also in comparison with experiments, see Pelz et al., J. Fluid Mech., vol. 889, 2020). The results can be used for the conceptual design of turbine arrays, but also for a sound physically realistic and consistent resource assessment of tidal power for a system consisting of two basins, a channel and a turbine fence with $0<\sigma \leqslant 1$ and operated in a complete tidal cycle.

The collision of a gravitationally driven horizontal current with a barrier following release from a confining lock is investigated using a shallow water model of the motion, together with a sophisticated boundary condition capturing the local interaction. The boundary condition permits several overtopping modes: supercritical, subcritical and blocked flow. The model is analysed both mathematically and numerically to reveal aspects of the unsteady motion and to compute the proportion of the fluid trapped upstream of the barrier. Several problems are treated. Firstly, the idealised problem of a uniform incident current is analysed to classify the unsteady dynamical regimes. Then, the extreme regimes of a very close or distant barrier are tackled, showing the progression of the interaction through the overtopping modes. Next, the trapped volume of fluid at late times is investigated numerically, demonstrating regimes in which the volume is determined purely by volumetric considerations, and others where transient inertial effects are significant. For a Boussinesq gravity current, even when the volume of the confined region behind the barrier is equal to the fluid volume, $30\,\%$ of the fluid escapes the domain, and a confined volume three times larger is required for the overtopped volume to be negligible. For a subaerial dam-break flow, the proportion that escapes is in excess of $60\,\%$ when the confined volume equals the fluid volume, and a barrier as tall as the initial release is required for negligible overtopping. Finally, we compare our predictions with experiments, showing a good agreement across a range of parameters.

A two-dimensional depth-averaged model is derived for open-channel flows in the smooth turbulent case. The derivation is consistently obtained with a method of matched asymptotic expansions in the outer and inner layers using a mixing length model of turbulence including the free-surface reduction of the eddy viscosity. The shear effects are taken into account by an extra tensor variable of the model called the enstrophy tensor. The friction coefficient is an explicit expression of the water depth. The three-dimensional velocity field and the friction velocity can be reconstructed from the values of the depth-averaged quantities. Numerical simulations show that the enstrophy can be used to evaluate the development of the turbulent boundary layer. In the case of subcritical unsteady flows, the reconstructed velocity can be described with a logarithmic law modified by Coles’ wake function with apparent von Kármán constant, integration constant and wake-strength parameter, which differ from their values in steady flows. In the viscous sublayer the steady-state relation between the velocity and the vertical coordinate, in the inner scaling, is not valid for unsteady flows. Large errors on the calculation of the von Kármán constant can be made if the validity of the steady-state relation is assumed for unsteady flows. The comparisons of the reconstructed velocity profiles in the case of one-dimensional unsteady open-channel flows and two-dimensional wide trapezoidal channels show a good agreement with experiments.

We show that in free surface flows, a uniform, streamwise current over small-amplitude wavy bottom topography generates cross-stream drift velocity. This drift mechanism, referred to as the current–bathymetry interaction-induced drift (CBIID), is specifically understood in the context of a simplified nearshore environment consisting of a uniform alongshore current, onshore-propagating surface waves and monochromatic wavy bottom making an oblique angle with the shoreline. The CBIID is found to originate from the steady, non-homogeneous solution of the governing system of equations. Similar to Stokes drift induced by surface waves, CBIID also generates a compensating Eulerian return flow to satisfy the no-flux lateral boundaries, e.g. the shoreline. The CBIID increases with an increase in particle's initial depth, bottom undulation amplitude and the strength of the alongshore current. Additionally, CBIID near the free (bottom) surface increases (decreases) with an increase in bottom undulation's wavelength. Maximum CBIID is obtained for long-wavelength bottom topography that makes an angle of approximately ${\rm \pi} /4$ with the shoreline. Unlike Stokes drift, particle excursions due to current–bathymetry interactions might not be small, and hence analytical expressions based on the small-excursion approximation could be inaccurate. We provide an alternative $z$-bounded approximation, which leads to highly accurate expressions for drift velocity and time period of particles especially located near the free surface. Realistic parametric analysis reveals that in some nearshore environments, CBIID's contribution to the net Lagrangian drift can be as important as Stokes drift, implying that CBIID can have major implications in cross-shelf tracer transport.

The propagation of dam-break waves on different rough beds was observed to be quasi-steady in the range $11.3 < x/h_{dam} < 18.8$, where $x$ is measured from the dam position. These quasi-steady propagation speeds converge with the steady ideal fluids model of Stoker (Water Waves, 1957, Interscience) when the tailwater depth $h_2$ becomes greater than ${\sim }0.5k_s$, in the range $0.001< k_s/h_{dam}< 0.2$, where $k_s$ is the roughness and $h_{dam}$ the depth behind the dam. Hence, this convergence encourages the use of Stoker's steady, ideal fluid solution to develop more general models, including friction effects due to bed roughness and/or viscosity. The new experimental data support a MacLaurin series for the celerity $c$, in analogy with the series in terms of $\sqrt [4]{{h_2}/{h_{dam}}}$, derived for Stoker's model, $h_2$ being the tailwater depth. Compared with the retarding effect of the tailwater, 1 mm of roughness is found to be equivalent to $\sim$13 mm of tailwater, and 1 $\mathrm {\mu }$m of viscous length (${\nu }/{\sqrt {gh_{dam}}}$, where ${\nu}$ is the kinematic viscosity and g the acceleration due to gravity) is equivalent to $\sim$1700 $\mathrm {\mu }$m of tailwater. While the MacLaurin series quantifies the similar effects of small roughness and small tailwater depths acting separately, the new data illustrate for the first time the complex interplay between tailwater and roughness on ‘wet beds’ with many details yet to be investigated. In particular, it was shown that a small amount of tailwater on a rough bed acts as a lubricant, so that $c(h_2, h_{dam}, k_s)$ is an increasing function of $h_2$ for $h_2 < 0.5k_s$.

An analytical solution is developed for studying transient water wave-induced responses inside an unsaturated poroelastic seabed of finite thickness. The soil skeleton and the pore fluid are compressible and the constitutive relationship of the soil skeleton is described by Hooke's law. Assuming that the horizontal length scale of wave motion is much larger than the seabed thickness, the leading-order analytical solutions for the seabed responses, including pore fluid pressure and soil skeleton motion, are obtained. The present solutions are suitable for general transient wave loading and for the shear modulus of the soil skeleton being of the same order of magnitude as the effective bulk modulus of elasticity of the pore fluid. The present theory is first validated by checking the solutions with the experimental data for the pore pressure induced by periodic-wave loading. The present analytical solutions are then used to investigate the seabed responses under transient waves, including linear periodic wave, a solitary wave and a bore. The effects of the wave-induced effective stresses on the bed failure potential are further analysed. The results show that the shear failure potential and its duration are highly dependent on the soil properties, such as saturation degree, shear modulus and permeability. Sensitivity analyses are presented.

A fully developed approach flow is necessary in open channel studies to maintain commonality among datasets obtained from different facilities. Two-component planar particle image velocimetry is used to study the characteristics of fully developed smooth open channel flow at a constant Reynolds number of 3.9 × 104 based on the maximum velocity and flow depth. The near-bed boundary layer is tripped to achieve a fully developed state and compared with the under- and over-tripped cases. The Reynolds stresses and higher-order moments are used as indicators to establish the fully developed state. Flow properties are explored by identifying uniform momentum zones (UMZs) using the probability density function of streamwise velocities. The instances are grouped based on the number of UMZs (NUMZ) and conditional averaging of flow variables of each group is used to evaluate the difference in flow properties between the developed and the developing flow. Large-scale ejections are found in the logarithmic layer when NUMZ is higher, whereas a lower number indicates the existence of large-scale sweeping motions. The distribution of the conditionally averaged ratio of the shear contribution from ejections and sweeps and velocity deficits shows a vertical variability in the fully developed state. The large-scale and pointwise quadrant events are used simultaneously to depict variability in inner flow properties between developing and fully developed flow which cannot be recognized in the mean flow characteristics. The sweep events have much higher shear generation in the outer flow in the fully developed state whereas the shear stress contribution from ejection is lower than that in developing flow.

The distinct turbulence dynamics and transport modulated by a common seagrass species were investigated experimentally using a flexible surrogate canopy in a refractive-index-matching environment that enabled full optical access. The surrogate seagrass replicated the dynamic behaviour and morphological properties of its natural counterpart. The flows studied were subcritical with Froude numbers $Fr<0.26$ and concerned five Reynolds numbers $Re\in [3.4\times 10^{4}, 1.1\times 10^{5}]$ and Cauchy numbers $Ca\in [120, 1200]$. Complementary rigid canopy experiments were also included to aid comparative insight. The flow was quantified in wall-normal planes in a developed region using high-frame-rate particle image velocimetry. Results show that the deflection and coordinated waving motion of the blades redistributed the Reynolds stresses above and below the canopy top. Critically, in-canopy turbulence associated with the seagrass lacked periodic stem wake vortex shedding present in the rigid canopy, yet the flexible canopy induced vortex shedding from the blade tips. Inspection of spatial and temporal characteristics of coherent flow structures using spectral proper orthogonal decomposition reveals that Kelvin–Helmholtz-type vortices are the dominant flow structures associated with the waving motion of the seagrass and that modulated the local flow exchange in both rigid and flexible canopies. A barrier-like effect produced by the blade deflections blocked large-scale turbulence transport, thereby reducing vortex penetration into the canopy. In addition, we uncovered a transition from sweep-dominated to ejection-dominated behaviour in the surrogate seagrass. We hypothesise that the vortices created during the upward blade motion period play a major role in the sweep-to-ejection-dominated transition. Conditionally averaged quadrant analysis on the downward and upward blade motion supports this contention.

Turbulent entrainment properties of thermohaline internal hydraulic jumps in quiescent ambient water are investigated. Underflow and overflow jumps are considered. The study is mainly concerned with thermohaline buoyancy effects on flow development. The thermohaline buoyancy is determined from a seawater equation of state, which is nonlinear in temperature and linear in salinity, and is accurate throughout the range of temperature and salinity of interest (T = 0 to 40 °C and S = 0 to 40 ppt). The results indicate that thermohaline buoyancy produces smaller less diluted underflow jumps and larger more diluted overflow jumps. These nonlinear buoyancy effects are particularly significant when buoyancy arising from temperature and salinity act in opposite directions. Hydraulic controls by vertical and/or horizontal constrictions downstream reveal, using a matching technique, unique stationary miscible jump solutions.

This paper presents some results coming from a linear stability analysis of turbulent depth-averaged open-channel flows (OCFs) with secondary currents. The aim was to identify plausible mechanisms underpinning the formation of large-scale turbulence structures, which are commonly referred to as large-scale motions (LSMs) and very-large-scale motions (VLSMs). Results indicate that the investigated flows are subjected to a sinuous instability whose longitudinal wavelength compares very well with that pertaining to LSMs. In contrast, no unstable modes at wavelengths comparable to those associated with VLSMs could be found. This suggests that VLSMs in OCFs are triggered by nonlinear mechanisms to which the present analysis is obviously blind. We demonstrate that the existence of the sinuous instability requires two necessary conditions: (i) the circulation of the secondary currents $\omega$ must be greater than a critical value $\omega _c$; (ii) the presence of a dynamically responding free surface (i.e. when the free surface is modelled as a frictionless flat surface, no instabilities are detected). The present paper draws some ideas from the work by Cossu, Hwang and co-workers on other wall flows (i.e. turbulent boundary layers, pipe, channel and Couette flows) and somewhat supports their idea that LSMs and VLSMs might be governed by an outer-layer cycle also in OCFs. However, the presence of steady secondary flows makes the procedure adopted herein much simpler than that used by these authors.

The paper deals with the steady axially symmetric flow of a viscous liquid layer over a rotating disk. The liquid is fed near the axis of rotation and spreads due to inertia and the centrifugal force. The viscous shallow-water approach gives a system of ordinary differential equations governing the flow. We consider inertia, gravity, centrifugal and Coriolis forces and estimate the effect of surface tension. We found four qualitatively different flow regimes. Transition through these regimes shows the continuous evolution of the flow structure from a hydraulic jump on a static disk to a monotonic thickness decrease on a fast rotating one. We show that, in the absence of surface tension, the intensity of the jump gradually vanishes at a finite distance from the axis of rotation while the angular velocity increases. The surface tension decreases the jump radius and destroys the steady solution for a certain range of parameters.

Highly turbulent free-surface flows are characterised by complex and rapidly varying air–water surface features, leading to enhanced surface roughness, breakup and disintegration processes. Such a strong free-surface turbulence has an impact on a number of environmental flows, and a deeper understanding of its physical nature is fundamental. Unsteady breaking bores are of particular interest because of their recirculating motion, with large air entrainment and splashes, resulting in highly fluctuating and rapidly varying free-surface flows. Herein, new methodologies and innovative approaches are used in support of a deeper understanding of the physical processes within a breaking roller, inclusive of a comprehensive assessment of its free-surface dynamics. Because of the unsteadiness of the flow, multiple repetitions were necessary and all results were based upon an ensemble statistical analysis. Ultra-high-speed videos recorded from both top and side views allowed for a detailed characterisation of the roller's free surface, providing a description and classification of the most recurring air–water surface features. A quantification of their main properties in terms of geometry, duration and frequency of appearance revealed an evolution of these features during their lifespans. In parallel, the use of optical flow techniques provided a characterisation of the surface velocity fields, yielding information on the free-surface kinematic properties and revealing a strong link between air–water surface features, energy dissipation and time/length scales.