An analytical model was developed to predict the velocity evolution within a submerged canopy of finite width and used to explore the impact of plant flexibility and width on the velocity within the canopy. The analytical model was validated with laboratory experiments using canopies constructed from rigid cylinders and from individual model seagrass plants, consisting of six LDPE (low-density polyethylene) blades attached to a rigid sheath. Four canopy widths were considered, spanning 12.8 to 100 % of the channel width. As the canopy narrowed from the channel width (two-dimensional canopy) to finite width (12.8 % of the channel width), the velocity adjusted more rapidly at the leading edge and reached a lower fully developed in-canopy velocity. Predictions from the analytical model had good agreement with field and laboratory studies with real vegetation. Once validated, the model was applied to a range of field conditions to predict the within-canopy flow velocity and the adjustment length, which is the distance required for the flow to be fully developed.

Here we provide a unified theoretical description of two different physical situations in which liquid jets are expelled out of the bulk of a liquid as a consequence of the capillary collapse of a void. We demonstrate that the velocity field giving rise to the emergence of these jets can be calculated as the flow generated by a line of sinks with a length and an intensity that can be expressed in terms of the initial cavity radius and the wavelength and velocity of the capillary waves propagating along the cavity walls. The predicted jet speeds, which are expressed through algebraic equations, are in good quantitative agreement with those obtained from experiments and from the simulations of bubbles bursting on a free surface or after the implosion of the crater formed when a drop impacts a liquid pool.

The paper reports on three direct numerical simulations of sediment transport with different prototypical non-spherical particle shapes, ranging from prolate to oblate to a triaxial ellipsoidal shape, and a fourth simulation with spherical particles. All physical and numerical parameters of these simulations are identical, including bulk Reynolds number, equivalent particle diameter and sphericity of the non-spherical particles, so that the shape is the only difference. Numerous statistical quantities assessing the computed solutions are presented, such as mean particle velocity, sediment-bed porosity, equivalent roughness height, mean fluid velocity and Reynolds stresses. These allow a detailed analysis and reveal the considerable influence of the particle shape on the entire flow: fluid as well as particles. It is found that oblate spheroids yield the highest porosity among the four cases and spherical particles the lowest mean porosity. Prolate particles exhibit the strongest tendency to form spanwise clusters and generate the highest total bed friction. Triaxial ellipsoids representing natural medium-size sand particles yield results in between these cases, still exhibiting substantial differences with respect to spherical particles. The mean sediment transport rate is determined in all four cases showing that this quantity cannot be correlated to the mean shear stress of the fluid alone. This emphasises the importance of the particle shape in modelling the transport of natural sediment, and the results provide well-controlled data points for this task.

The generation of mean flows is a long-standing issue in rotating fluids. Motivated by planetary objects, we consider here a rapidly rotating fluid-filled spheroid, which is subject to weak perturbations of either the boundary (e.g. tides) or the rotation vector (e.g. in direction by precession, or in magnitude by longitudinal librations). Using boundary-layer theory, we determine the mean zonal flows generated by nonlinear interactions within the viscous Ekman layer. These flows are of interest because they survive in the relevant planetary regime of both vanishing forcings and viscous effects. We extend the theory to take into account (i) the combination of spatial and temporal perturbations, providing new mechanically driven zonal flows (e.g. driven by latitudinal librations), and (ii) the spheroidal geometry relevant for planetary bodies. Wherever possible, our analytical predictions are validated with direct numerical simulations. The theoretical solutions are in good quantitative agreement with the simulations, with expected discrepancies (zonal jets) in the presence of inertial waves generated at the critical latitudes (as for precession). Moreover, we find that the mean zonal flows can be strongly affected in spheroids. Guided by planetary applications, we also revisit the scaling laws for the geostrophic shear layers at the critical latitudes, and the influence of a solid inner core.

We investigate two-dimensional shallow granular flows on a rotating and gravitating elliptical body. This is motivated by regolith flow on small planetary bodies – also called minor planets – which is influenced by the rotation of the body, as well as its irregular topography and complex gravity field. Governing equations are obtained in an elliptic coordinate system attached to the body by extending the framework employed for terrestrial avalanches to incorporate effects of rotation, varying gravity and a curvilinear surface. Additionally, we introduce criteria to monitor grain shedding and to track flow initiation and cessation. We delineate different types of regolith motion that are governed by the rotation rate and surface roughness of the body. We find that grains migrate towards the minor and major axis of the body at low and high rotation rates, respectively. Grains are shed when the basal pressure vanishes, and shedding is encouraged by Coriolis effects during prograde flow. We observe the coexistence of regions of static and mobile regolith and their reorganization owing to the merging or division of flows. We also probe the formation and destruction of dunes – bulges arising from local grain accumulation – and find several aspects of their motion to be different from terrestrial situations. We then perform discrete element simulations that display a good match with theoretical predictions. Finally, we consider the evolution of a bi-disperse regolith. We find that big and small grains occupy, respectively, the top and bottom of the dunes formed on the surface, which is reminiscent of observations on asteroids like Itokawa.

When two converging flows of unequal velocities and different flow directions come into contact, large-scale coherent structures are generated. For shallow conditions and small angles between the incoming streams, the mixing layer (ML) forming downstream of the confluence apex contains quasi-two-dimensional, Kelvin–Helmholtz (KH) vortices. Due to flow shallowness (e.g. stabilizing effect of bed friction), these vortices gradually lose their coherence at large distances from the ML origin. For large angles between the incoming streams, the spatial development of the shallow ML is more complex as strongly coherent, streamwise-oriented-vortical (SOV) cells form in the vicinity of the shallow ML and helical cells of secondary flow are generated due to curvature effects. The present paper uses three-dimensional eddy-resolving numerical simulations to study the effects of varying the angles between the two incoming channels and the downstream channel, the velocity ratio (VR) of the incoming flows and the flow depth on flow, turbulence structure and sediment entrainment mechanisms inside the ML and its surroundings. The simulations are performed for highly idealized conditions in which the ML develops in a wide channel (no interactions with the channel banks), over a flat bed and the flow depth is constant. Simulation results show that the SOV cells play an important role in the redistribution of the streamwise momentum. Some of the SOV cells are subject to bimodal oscillations in the lateral direction which induce strong interactions between the SOV cells and the ML vortices and sharply increase mixing. As for the case of a shallow ML developing between parallel streams, vortex pairing ceases some distance from the ML origin, the KH vortices start losing their coherence and the ML assumes an undulatory shape. The paper describes the effects of VR, angle between the incoming streams (α = 0° and 60°), planform geometry (symmetric vs. asymmetric confluences with α = 60°) and flow shallowness on the transverse shift of the ML centreline, ML width, dynamics of the KH vortices and on the formation, position and circulation of the SOV cells. In most of the α = 60° cases, the largest bed shear stresses are induced beneath the SOV cells rather than beneath the high-speed stream and the SOV cells play a major role in enhancing mixing between the two streams.

Direct numerical simulations are performed to investigate a stratified shear layer at high Reynolds number ($Re$) in a study where the Richardson number ($Ri$) is varied among cases. Unlike previous work on a two-layer configuration in which the shear layer resides between two layers with constant density, an unbounded fluid with uniform stratification is considered here. The evolution of the shear layer includes a primary Kelvin–Helmholtz shear instability followed by a wide range of secondary shear and convective instabilities, similar to the two-layer configuration. During transition to turbulence, the shear layers at low $Ri$ exhibit a period of thickness contraction (not observed at lower $Re$) when the momentum and buoyancy fluxes are counter-gradient. The behaviour in the turbulent regime is significantly different from the case with a two-layer density profile. The transition layers, which are zones with elevated shear and stratification that form at the shear-layer edges, are stronger and also able to support a significant internal wave flux. After the shear layer becomes turbulent, mixing in the transition layers is shown to be more efficient than that which develops in the centre of the shear layer. Overall, the cumulative mixing efficiency ($E^C$) is larger than the often assumed value of 1/6. Also, $E^C$ is found to be smaller than that in the two-layer configuration at moderate Ri. It is relatively less sensitive to background stratification, exhibiting little variation for $0.08 \leqslant Ri \leqslant 0.2$. The dependence of mixing efficiency on buoyancy Reynolds number during the turbulence phase is qualitatively similar to homogeneous sheared turbulence.

The paper addresses the fluid–structure interaction of submerged aquatic canopies, with particular focus on the complex interplay between coherent flow structures and the motion of vegetation elements. New insights into the underlying mechanisms are gained from a large eddy simulation of a submerged model canopy flow. The model canopy is made up of 800 highly flexible blades, each individually resolved by an immersed boundary method. The obtained high-resolution flow data reveal well-known coherent turbulent structures, including velocity streaks, Kelvin–Helmholtz (KH) vortices in the mixing layer as well as hairpin (HP) vortices in the outer flow region. The present results show that the interaction of these prototypical structures plays a key role creating unique turbulent features such as composite KH/HP vortices located between a high-speed and low-speed streak. Under the influence of these pronounced eddies, groups of blades respond by a strong local reconfiguration. Due to the convection of the coherent structures by the mean flow this causes an apparent wave-like motion of the canopy in streamwise direction, known as monami. A frequency analysis of this phenomenon shows that the vegetation responds almost passively, merely reflecting local flow conditions.

High-speed boundary-layer transition is extremely sensitive to the free-stream disturbances which are often uncertain. This uncertainty compromises predictions of models and simulations. To enhance the fidelity of simulations, we directly infuse them with available observations. Our methodology is general and can be adopted with any simulation tool, and is herein demonstrated using direct numerical simulations. An ensemble variational (EnVar) optimization is performed, whereby we determine the upstream flow that optimally reproduces the observations. The cost functional accounts for our relative confidence in the model and the observations, and judicious choice of the ensemble members improves convergence and reduces the prediction uncertainty. We demonstrate our observation-infused predictions for boundary-layer transition at Mach 4.5. Without prior knowledge of the free-stream condition, and using only observations of wall pressure at isolated locations from an independent computation (true flow), all of the relevant inflow disturbances are identified. We then evaluate the entire flow field, beyond the original limited wall observations, and interpret simulations consistently with data and vice versa. Our predicted flow compares favourably to the true ‘unknown’ state, and discrepancies are analysed in detail. We also examine the impact of weighting of observations. Improved convergence of the inverse problem and accuracy of the inflow amplitudes and phases are demonstrated, and are explained by aid of a simple example from two-dimensional unstable, chaotic convection.

We study the reactive displacement of two miscible fluids in channel flows and establish a quantitative link between fluid stretching and chemical reactivity. At the mixing interface, the two fluids react according to the instantaneous irreversible bimolecular reaction $A + B \rightarrow C$. We simulate the advection–diffusion–reaction problem using a random walk based reactive particle method that is free of numerical dispersion. The relative contributions of stretching and diffusion to mixing-limited reaction is controlled by changing the Péclet number, and the channel roughness is also systematically varied. We observe optimal ranges of fluid stretching that maximize reactivity, which are captured by a Lagrangian stretching measure based on an effective time period that honours the stretching history. We show that the optimality originates from the competition between the enhanced mixing by fluid stretching and the mass depletion of the reactants. We analytically derive the spatial distribution of reaction products using a lamellar formulation and successfully predict the optimal ranges of fluid stretching, which are consistent across different levels of channel roughness. These findings provide a mechanistic understanding of how the interplay between fluid stretching, diffusion and channel roughness controls mixing-limited reactions in rough channel flows, and show how reaction hot spots can be predicted from the concept of optimal fluid stretching.

The optimal steady wall heat flux is sought to delay transition to turbulence in a Mach 4.5 boundary layer, when the free stream disturbance is the nonlinearly most dangerous condition that causes the earliest breakdown to turbulence in the adiabatic flow. The transition-delay strategy is pursued using direct numerical simulations, and the problem is formulated as a constrained optimization where the objective is to minimize the viscous drag force and the control vector is the heat flux at the wall. The optimization is performed using an ensemble-variational approach. The algorithm identifies regions of the flow that are most sensitive to thermal treatment: most of the heating and cooling is applied upstream of the synchronization point of the slow and the fast modes. Attenuation of key instability waves by the optimal heat flux mitigates the nonlinear interactions that preceded breakdown to turbulence in the reference flow, and hence transition is appreciably delayed. Detailed analysis of the acoustic, entropic and solenoidal components of the momentum density vector reveals the stabilizing effects of the optimal heat flux. In addition, the influence of the thermal treatment is not limited to the pretransitional flow where it is concentrated, but rather it persists downstream even within the newly formed fully turbulent boundary layer.

We consider the inertia-free motion of an isotropic chemically active particle which is exposed to a weak uniform force field. This problem is characterised by two velocity scales, a ‘chemical’ scale associated with diffusio-osmosis and a ‘mechanical’ scale associated with the external force. The motion animated by the force deforms the originally spherically symmetric solute cloud surrounding the particle, thus resulting in a concomitant diffusio-osmotic flow which, in turn, modifies the particle speed. A weak-force linearisation furnishes a closed-form expression for the particle velocity as a function of the intrinsic Péclet number $\alpha$ associated with the chemical velocity scale. We find that the predicted velocity may become singular at $\alpha =4$, and that this happens under the same conditions on the surface parameters for which the associated unforced problem is known to exhibit, for $\alpha >4$, a symmetry-breaking instability giving rise to steady spontaneous motion (Michelin, Lauga & Bartolo, Phys. Fluids, vol. 25, 2013, 061701). Here, a local analysis in a distinguished region near $\alpha =4$, wherein the velocity scaling is amplified, yields a closed-form description of the imperfect bifurcation which bridges between a perturbed stationary state and a perturbed spontaneous motion. Remarkably, while the direction of spontaneous motion in the absence of an external force is random, in the perturbed case that motion is rendered steady solely in the directions parallel or antiparallel to the external force.

In this paper, we study inertial and gravity wave transmissions near radiative–convective boundaries on the f-plane. Two configurations have been considered: waves propagate from the convective layer to the radiative stratified stable layer, or the other way around. It has been found that waves prefer to survive at low latitudes when the stable layer is strongly stratified ($N^2/(2\varOmega )^2>1$). When the stable layer is weakly stratified ($N^2/(2\varOmega )^2 < 1$), however, waves can survive at any latitude if the meridional wavenumber is large. Then we have discussed transmission ratios for two buoyancy frequency structures: the uniform stratification and the continuously varying stratification. For the uniform stratification, we have found that the transmission is efficient when the rotation is rapid, or when the wave is near the critical colatitude. For the continuously varying stratification, we have discussed the transmission ratio when the square of buoyancy frequency is an algebraic function $N^2\propto z^{\nu } (\nu >0)$. We have found that the transmission can be efficient when the rotation is rapid, or when the wave is near the critical colatitude, or when the thickness of the stratification layer is far greater than the horizontal wavelength. The transmission ratio does not depend on the configurations (radiative layer sits above convective layer, or vice versa; wave propagates outward or inward), but only on characteristics of the wave (frequency and wavenumber) and the fluid (degree of stratification).

In inertial-range turbulence, structure functions can diagnose transfer or dissipation rates of energy and enstrophy, which are difficult to calculate directly in flows with complex geometry or sparse sampling. However, existing relations between third-order structure functions and these rates only apply under isotropic conditions. We propose new relations to diagnose energy and enstrophy dissipation rates in anisotropic two-dimensional (2-D) turbulence. These relations use second-order advective structure functions that depend on spatial increments of vorticity, velocity, and their advection. Numerical simulations of forced-dissipative anisotropic 2-D turbulence are used to compare new and existing relations against model-diagnosed dissipation rates of energy and enstrophy. These simulations permit a dual cascade where forcing is applied at an intermediate scale, energy is dissipated at large scales, and enstrophy is dissipated at small scales. New relations to estimate energy and enstrophy dissipation rates show improvement over existing methods through increased accuracy, insensitivity to sampling direction, and lower temporal and spatial variability. These benefits of advective structure functions are present under weakly anisotropic conditions, and increase with the flow anisotropy as third-order structure functions become increasingly inappropriate. Several of the structure functions also show promise for diagnosing the forcing scale of 2-D turbulence. Velocity-based advective structure functions show particular promise as they can diagnose both enstrophy and energy cascade rates, and are robust to changes in the effective resolution of local derivatives. Some existing and future datasets that are amenable to advective structure function analysis are discussed.

Soft corals, such as the bipinnate sea plume Antillogorgia bipinnata, are colony building animals that feed by catching food particles brought by currents. Because of their flexible skeleton, they bend and sway back and forth with the wave swell. In addition to this low-frequency sway of the whole colony, branches of A. bipinnata vibrate at high frequency with small amplitude and transverse to the flow as the wave flow speed peaks. In this paper, we investigate the origin of these yet unexplained vibrations and consider their effect on soft corals. Estimation of dynamical variables along with finite element implementation of the wake-oscillator model favour vortex-induced vibrations (VIVs) as the most probable origin of the observed rapid dynamics. To assess the impact of the dynamics on filter feeding, we simulated particles advected by the flow around a circular cylinder and calculated the capture rate with an in-house monolithic fluid–structure interaction (FSI) finite element solver and Python code. We observe that vibrating cylinders can capture up to 40 % more particles than fixed cylinders at frequency lock-in. Therefore, VIVs plausibly offer soft corals a better food capture.

We show that shock polars for an ideal non-polytropic gas (thermally but not calorically perfect) have a unique velocity angle maximum, the critical shock, assuming a convex equation of state (positive fundamental derivative) and other standard conditions. We also show that the critical shock is always transonic. These properties are explained by a brief informal mass flux argument, which is then extended into a precise calculation. In the process we show that temperature, pressure, energy, enthalpy, normal mass flux and entropy are increasing along the forward Hugoniot curves, and hence along the polar from vanishing to normal shock; speed is decreasing along the entire polar, mass flux and, importantly, Mach number are decreasing on subsonic parts of the polar. If the equation of state is not convex, counterexamples can be given with multiple critical shocks, permitting more than two shocks that attain the same velocity angle, in particular, more than one shock of weak type, which would cause theoretical problems and practical risks of misprediction. For dissociating diatomic gases, numerical experiments suggest that positive fundamental derivative and uniqueness of critical shocks hold at all realistic pressures, although both fail at very low purely theoretical pressures.

The QSQH theory is extended to all three velocity components taking into account the fluctuations of the direction of the large-scale component of the wall friction. This effect is found to be significant. It explains the large sensitivity of the fluctuations of longitudinal and spanwise velocities to variations in the Reynolds number in comparison with the sensitivity of the mean velocity, the Reynolds stress and the wall-normal velocity fluctuations. The analysis shows that the variation of the longitudinal velocity fluctuations with the Reynolds number is dominated by the variation of the amplitude and wall-normal-scale modulation of the universal mean velocity profile by the outer, large-scale, Reynolds-number-dependent motions. The variation of spanwise velocity fluctuations is dominated by the fluctuations of the direction of the large-scale component of the wall friction. The Reynolds number dependence of the other second moments is not dominated by these mechanisms because the mean wall-normal velocity and the mean spanwise velocity are zero. Explicit relationships between the differences in the second moments of velocity in any two high-Reynolds-number near-wall flows were derived. The comparisons gave a satisfactory agreement for the root-mean square of the wall-parallel velocity components in the range of the distances from the wall where modulation by large-scale motions dominates. Relationships between the differences of the constants of the logarithmic law, the shape of the mean velocity profile and the differences of the second moments of velocity caused by the differences in large-scale motions were derived and estimated quantitatively.

Formation of a microlayer in nucleate boiling results in substantial changes to the heat-transfer dynamics of the process, and establishing a criterion to distinguish between the microlayer and contact-line regimes encountered is necessary to obtain a quantitative description of the nucleate-boiling phenomenon. In this work, we describe microlayer formation as a dewetting transition in the presence of phase change. We derive a specific criterion for modelling the transition based on a synthesis of existing theoretical, experimental and numerical data. As a result, a new formulation of the transition criterion is developed, and applied to (i) reference data from a dewetting experiment of a volatile liquid, and (ii) results derived from a high-resolution direct numerical simulation of nucleate boiling. In both cases, very good agreement is found for Jakob numbers $\lesssim$75, a range which covers many important boiling situations.

Spontaneous motion of liquid droplets can occur on hydrophobic, micro-structured, solid surfaces comprising a structural gradient. In this study, we examine such motion experimentally and explain our observations by invoking variable droplet–surface interactions (both actuation and resistance forces) arising from the structural gradient. The oscillatory motion of the droplet constitutes an integral aspect of the behaviour and this is incorporated into the overall modelling. The theoretical model features a truncated spheroid for the drop shape (flattened in the region of solid contact) coupled with the oscillatory and alternate leading and trailing motion of the contact line. Results from the model and experiments provide good qualitative and quantitative agreement. The component of the vertical oscillation is found to help overcome wetting hysteresis and actuate the motion, this being a key element for the completeness of the model.

Particles of differing sizes are notoriously prone to segregation in shear driven flows under the action of gravity. This has important implications in many industrial processes, where particle-size segregation can lead to flow problems and reduced product quality, as well as longer product development and start-up times. Particle-size segregation also readily occurs in many hazardous geophysical mass flows (such as snow avalanches, debris flows and volcanic pyroclastic flows) and can lead to the formation of destructive bouldery flow fronts and significantly longer runouts. Although general theories exist to model particle-size segregation, the detailed functional dependence of the segregation flux on the shear rate, gravity, pressure, particle concentration, grain size and grain-size ratio is still not known. This paper describes refractive-index matched oscillatory shear-cell experiments that shed light on the segregation velocity in the two extreme cases of (i) a single large intruder rising up through a matrix of smaller grains, and (ii) a single small intruder percolating down through a matrix of large particles. Despite the sometimes markedly different time scales for segregation in these two situations, a unifying scaling law has been found that is able to collapse all the experimental data over a wide range of shear rates and grain-size ratios in the range $[1.17,4.17]$. The resulting functional form is easily generalizable to intermediate concentrations and can quantitatively capture laboratory experiments and numerical simulations with a $50\ {:}\ 50$ mix of large and small grains.