Flow and transport through aquatic vegetation is characterized by a wide range of length scales: water depth (), plant height (), stem diameter (), the inverse of the plant frontal area per unit volume () and the scale(s) over which varies. Turbulence is generated both at the scale(s) of the mean vertical shear, set in part by , and at the scale(s) of the stem wakes, set by . While turbulence from each of these sources is dissipated through the energy cascade, some shear-scale turbulence bypasses the lower wavenumbers as shear-scale eddies do work against the form drag of the plant stems, converting shear-scale turbulence into wake-scale turbulence. We have developed a – model that accounts for all of these energy pathways. The model is calibrated against laboratory data from beds of rigid cylinders under emergent and submerged conditions and validated against an independent data set from submerged rigid cylinders and a laboratory data set from a canopy of live vegetation. The new model outperforms existing – models, none of which include the scale, both in the emergent rigid cylinder case, where existing – models break down entirely, and in the submerged rigid cylinder and live plant cases, where existing – models fail to predict the strong dependence of turbulent kinetic energy on . The new model is limited to canopies dense enough that dispersive fluxes are negligible.

The axisymmetric collapse of a cylindrical air cavity in water follows a universal power law with logarithmic corrections. Nonetheless, it has been suggested that the introduction of a small azimuthal disturbance induces a long-term memory effect, reflecting in oscillations which are no longer universal but remember the initial condition. In this work, we create non-axisymmetric air cavities by driving a metal disc through an initially quiescent water surface and observe their subsequent gravity-induced collapse. The cavities are characterized by azimuthal harmonic disturbances with a single mode number and amplitude . For small initial distortion amplitude (1 or 2 % of the mean disc radius), the cavity walls oscillate linearly during collapse, with nearly constant amplitude and increasing frequency. As the amplitude is increased, higher harmonics are triggered in the oscillations and we observe more complex pinch-off modes. For small-amplitude disturbances we compare our experimental results with the model for the amplitude of the oscillations by Schmidt et al. (Nature Phys., vol. 5, 2009, pp. 343–346) and the model for the collapse of an axisymmetric impact-created cavity previously proposed by Bergmann et al. (J. Fluid Mech., vol. 633, 2009b, pp. 381–409). By combining these two models we can reconstruct the three-dimensional shape of the cavity at any time before pinch-off.

The interaction of a shockwave with a gas bubble in a liquid medium is of interest in a variety of areas, e.g. shockwave lithotripsy, cavitation damage and the study of sonoluminescence. This study employs a high-resolution front-tracking framework to numerically investigate this phenomenon. The modelling paradigm is validated extensively and then used to explore the parametric space of interest. We provide a comprehensive qualitative analysis of the collapse process, which we categorize into three phases, based on the principal feature dominating each phase. This results in the characterization of numerous previously unidentified features important in the evolution of the process and in the emergence of peak temperatures and pressures. For example, we discover that the peak pressure does not occur as a result of the impact of the main transverse jet (also called the re-entrant jet) but later in the collapse. We perform fully three-dimensional simulations, showing that three-dimensional instabilities are limited to the small-scale details of collapse, and continue by comparing collapse of cylindrical and spherical bubbles. We detail a parametric investigation varying the shock strength from 100 MPa to 100 GPa. A counter-intuitive discovery is that the maximum gas density decreases with increasing shock strength.

Viscous laminar flow past a prolate spheroid has been investigated numerically at seven different Reynolds numbers; and . In contrast to all earlier investigations, the major axis of the spheroid was oriented perpendicular to the free stream flow. As expected, the flow field in the wake showed a strong resemblance of that observed behind a finite-length circular cylinder, yet had features observed in the axisymmetric wake behind a sphere. The following different flow regimes were observed in the present computational study: (i) steady laminar flow with massive flow separation and symmetry about the equatorial and the meridional planes at ; (ii) steady laminar flow with massive flow separation and symmetry about the equatorial and the meridional plane at , but the flow in the equatorial plane did no longer resemble the steady wake behind a circular cylinder; (iii) unsteady laminar flow with Strouhal number and symmetry about the equatorial plane at ; (iv) unsteady laminar flow with two distinct frequencies and without any planar symmetries at ; (v) transitional flow with a dominant shedding frequency and without any spatial symmetries at . For all but the two lowest hairpin vortices were alternately shed from the two sides of the spheroid and resulted in a ladder-like pattern of oppositely oriented vortex structures, in contrast with the single-sided shedding in the wake of a sphere. The contour of the very-near-wake mimicked the shape of the prolate spheroid. However, downstream the major axis of the wake became aligned with the minor axis of the spheroid. This implies that an axis switching occurred some downstream, i.e. the cross-section of the wake evolved such that the major and minor axes interchanged at a certain downstream location. This peculiar phenomenon has frequently been reported to arise for elliptical and rectangular jets, whereas observations of axis switching for asymmetric wakes are scarce.

A strongly tilted, nearly axisymmetric vortex in dry air with asymmetric diabatic heating is analysed here by matched asymptotic expansions. The vortex is in gradient wind balance, with vortex Rossby numbers of order unity, and embedded in a quasi-geostrophic (QG) background wind with weak vertical shear. With wind speeds of , such vortices correspond to tropical storms or nascent hurricanes according to the Saffir–Simpson scale. For asymmetric heating, nonlinear coupling of the evolution equations for the vortex tilt, its core structure, and its influence on the QG background is found. The theory compares well with the established linear theory of precessing quasi-modes of atmospheric vortices, and it corroborates the relationship between vortex tilt and asymmetric potential temperature and vertical velocity patterns as found by Jones (Q. J. R. Meteorol. Soc., vol. 121, 1995, pp. 821–851) and Frank & Ritchie (Mon. Weath. Rev., vol. 127, 1999, pp. 2044–2061) in simulations of adiabatic tropical cyclones. A relation between the present theory and the local induction approximation for three-dimensional slender vortex filaments is established.

Breakup of an electrified viscous compound jet, surrounded by a dielectric gas, is investigated theoretically. The fluids are considered to be electrolytes and the core fluid viscosity is assumed to be much larger than that of the annular fluid. Axisymmetric configurations are considered with the three fluids bound by a cylindrical electrode that is held at a constant voltage potential. The model equations are investigated asymptotically in the long-wave limit, yielding two cases corresponding to a negligible surface charge with electrokinetic effects and a leaky dielectric model. A linear stability analysis for both cases is performed and the electrical effects are found to have a stabilizing effect, which is consistent with previous investigations of single electrified jet breakup at small wavenumbers. The one-dimensional equations are also solved numerically. The electric field is found to cause satellite formation in the core fluid, which does not occur in the purely hydrodynamic case, with the satellite size increasing with the strength of the electric field.

The first bifurcation and the instability mechanisms of shear-thinning and shear-thickening fluids flowing past a circular cylinder are studied using linear theory and numerical simulations. Structural sensitivity analysis based on the idea of a ‘wavemaker’ is performed to identify the core of the instability. The shear-dependent viscosity is modelled by the Carreau model where the rheological parameters, i.e. the power-index and the material time constant, are chosen in the range and . We show how shear-thinning/shear-thickening effects destabilize/stabilize the flow dramatically when scaling the problem with the reference zero-shear-rate viscosity. These variations are explained by modifications of the steady base flow due to the shear-dependent viscosity; the instability mechanisms are only slightly changed. The characteristics of the base flow, drag coefficient and size of recirculation bubble are presented to assess shear-thinning effects. We demonstrate that at critical conditions the local Reynolds number in the core of the instability is around 50 as for Newtonian fluids. The perturbation kinetic energy budget is also considered to examine the physical mechanism of the instability.

Mechanisms of magnetic field intensification by flows of an electrically conducting fluid in a rapidly rotating spherical shell are investigated using a numerical dynamo model with an Ekman number of . A strong dipolar solution with a magnetic energy 55 times larger than the kinetic energy of thermal convection is obtained. In a regime of small viscosity and inertia with the strong magnetic field, the convection structure consists of a few large-scale retrograde flows in the azimuthal direction and localized thin sheet-like plumes. A detailed term-by-term analysis of the magnetic field amplification processes shows that the magnetic field is amplified through stretching of magnetic lines, which occurs typically through four types of flow: the retrograde azimuthal flow near the outer boundary, the downwelling flow of the sheet plume, the prograde azimuthal flow near the rim of the tangent cylinder, and the cylindrical-radially alternating flows of the plume cluster. The current loop structure emerges as a result of stretching the magnetic lines along the magnetic field by the flow acceleration. The most remarkable effects of the generated magnetic field on the flow come from the strong azimuthal (toroidal) magnetic field. Similarities of the present model in the convection and magnetic field structures to previous studies at larger and even smaller Ekman numbers suggest universality of the dynamo mechanism in rotating spherical dynamos.

Globally unstable wakes with co-flow at intermediate Reynolds numbers are studied, to quantify important spatial regions for the development and control of the global instability. One region of high structural sensitivity is found close to the inlet for all wakes, in agreement with previous findings for cylinder wakes. A second, elongated region of high structural sensitivity is seen downstream of the first one for unconfined wakes at . When base-flow modifications are considered, a spatially oscillating sensitivity pattern is found inside the downstream high-structural-sensitivity region. This implies that the same change in the base flow can either destabilize or stabilize the flow, depending on the exact position where it is applied. It is shown that the sensitivity pattern remains unchanged for different choices of streamwise boundary conditions and numerical resolution. The actual base-flow is modified in selected configurations, and the linear global modes recomputed. It is confirmed that the linear global eigenvalues move according to the predicted sensitivity pattern for small-amplitude base-flow modifications, for which the theory applies. We also look at the implications of a small control cylinder for the flow. Only the upstream high-sensitivity region proves to be robust in terms of control, but one should be careful not to disturb the flow in the downstream high-sensitivity region, in order to achieve control. The findings can have direct implications for the numerical resolution requirements for wakes at higher Reynolds numbers. Furthermore, they provide one more possible explanation for why confined wakes have a more narrow frequency spectrum than unconfined wakes.

We describe new experiments to examine the buoyancy-induced mixing which results from the injection of a small constant volume flux of fluid of density at the top of a long narrow vertical tank with square cross-section which is filled with fluid of density . The injected fluid vigorously mixes with the less dense fluid which initially occupies the tank, such that a dense mixed region of turbulent fluid propagates downwards during the initial mixing phase of the experiment. For an ideal source of constant buoyancy flux , we show that the height of the mixed region grows as and that the horizontally averaged reduced gravity at the top of tank increases as , where is the width of the tank. Once the mixed region reaches the bottom of the tank, the turbulent mixing continues in an intermediate mixing phase, and we demonstrate that the reduced gravity at each height increases approximately linearly with time. This suggests that the buoyancy flux is uniformly distributed over the full height of the tank. The overall density gradient between the top and bottom of the mixed region is hence time-independent for both the mixing phases before and after the mixed region has reached the bottom of the tank. Our results are consistent with previous models developed for the mixing of an unstable density gradient in a confined geometry, based on Prandtl’s mixing length theory, which suggest that the turbulent diffusion coefficient and the magnitude of the local turbulent flux are given by the nonlinear relations and , respectively. The constant relates the width of the tank to the characteristic mixing length of the turbulent eddies. Since the mixed region is characterized by a time-independent overall density gradient, we also tested the predictions based on a linear model in which the turbulent diffusion coefficient is approximated by a constant . We solve the corresponding nonlinear and linear turbulent diffusion equations for both mixing phases, and show a good agreement with experimental profiles measured by a dye attenuation technique, in particular for the solutions based on the nonlinear model.

The present work investigates the influence of bubble clustering on the propagation of shock waves in bubbly liquids. A continuum model is developed to describe the macroscopic response of a bubbly liquid with a cluster structure, using a two-step homogenization technique. The proposed methodology allows us to simulate shock wave propagation over long distances with a small computation time and to study the effect of bubble clustering on the shock structure. It is shown that the typical length of the shock profile is related to the global response of the clusters instead of the single-bubble dynamics, as in homogeneous bubbly flows. The accuracy of the proposed modelling is assessed through comparisons with axisymmetric simulations, in which clusters are directly specified, with given positions and sizes, and with experimental data.

We analyse particle electrophoresis in the thin-double-layer limit for asymptotically large applied electric fields. Specifically, we consider fields scaling as , being the dimensionless Debye thickness. The dominant advection associated with the intense flow mandates a uniform salt concentration in the electro-neutral bulk. The large tangential fields in the diffuse part of the double layer give rise to a novel ‘surface conduction’ mechanism at moderate zeta potentials, where the Dukhin number is vanishingly small. The ensuing electric current emerging from the double layer modifies the bulk electric field; the comparable transverse salt flux, on the other hand, is incompatible with the nil diffusive fluxes at the homogeneous bulk. This contradiction is resolved by identifying the emergence of a diffusive boundary layer of thickness, resembling thermal boundary layers at large-Reynolds-number flows. The modified electric field within the bulk gives rise to an irrotational flow, resembling those in moderate-field electrophoresis. At leading order, the particle electrophoretic velocity is provided by Smoluchowski’s formula, describing linear variation with applied field.

Five isothermal round jets at Mach number and Reynolds number originating from a pipe nozzle are computed by large-eddy simulations to investigate the effects of initial turbulence on flow development and noise generation. In the pipe, the boundary layers are untripped in the first case and tripped numerically in the four others in order to obtain, at the exit, mean velocity profiles similar to a Blasius laminar profile of momentum thickness equal to 1.8 % of the jet radius, yielding Reynolds number , and peak turbulence levels around 0, 3 %, 6 %, 9 % or 12 % of the jet velocity . As the initial turbulence intensity increases, the shear layers develop more slowly with much lower root-mean-square (r.m.s.) fluctuating velocities, and the jet potential cores are longer. Velocity disturbances downstream of the nozzle exit also exhibit different structural characteristics. For low , they are dominated by the first azimuthal modes , 1 and 2, and show significant skewness and intermittency. The growth of linear instability waves and a first stage of vortex pairings occur in the shear layers for . For higher , three-dimensional features and high azimuthal modes prevail, in particular close to the nozzle exit where the wavenumbers naturally found in turbulent wall-bounded flows clearly appear. Concerning the sound fields, strong broadband components mainly associated with mode are noticed around the pairing frequency for the untripped jet. With rising , however, they become weaker, and the noise levels decrease asymptotically down to those measured for jets at , which are likely to be initially turbulent and to emit negligible vortex-pairing noise. These results correspond well to experimental observations, made separately for either mixing layers, jet flow or sound fields.

A model for steady flow in a ventilated space containing a heat source is developed, taking account of the main heat transfers at the upper and lower boundaries. The space has an opening at low level, allowing cool ambient air to enter the space, and an opening near the ceiling, allowing warm air to leave the space. The flow is driven by the temperature contrast between the air inside and outside the space (natural ventilation). Conductive heat transfer through the ceiling and radiant heat transfer from the ceiling to the floor are incorporated into the model, to investigate how these heat transports affect the flow and temperature distribution within the space. In the steady state, a layer of warm air occupies the upper part of the space, with the lower part of the space filled with cooler air (although this is warmer than the ambient air when the radiant transfer from ceiling to floor is included). Suitable scales are derived for the heat transfers, so that their relative importance can be characterized. Explicit relationships are found between the height of the interface, the opening area and the relative size of the heat transfers. Increasing heat conduction leads to a lowering of the interface height, while the inclusion of the radiant transfer tends to increase the interface height. Both of these effects are relatively small, but the effect on the temperatures of the layers is significant. Conductive heat transfer through the upper boundary leads to a significant lowering of the temperature in the space as a proportion of the injected heat flux is taken out of the space by conduction rather than advection. Radiative transfer from the ceiling to floor results in the lower layer becoming warmer than the ambient air. The results of the model are compared with full-scale laboratory results and a more complex unsteady model, and are shown to give results that are much more accurate than models which ignore the heat transfers.

We investigate the stability of the pressure-driven, low-Reynolds-number flow of Brownian suspensions with spherical particles in microchannels. We find two general families of stable/unstable modes: (i) degenerate modes with symmetric and antisymmetric patterns; (ii) single modes that are either symmetric or antisymmetric. The concentration profiles of degenerate modes have strong peaks near the channel walls, while single modes diminish there. Once excited, both families would be detectable through high-speed imaging. We find that unstable modes occur in concentrated suspensions whose velocity profiles are sufficiently flattened near the channel centreline. The patterns of growing unstable modes suggest that they are triggered due to Brownian migration of particles between the central bulk that moves with an almost constant velocity, and a highly-sheared low-velocity region near the wall. Modes are amplified because shear-induced diffusion cannot efficiently disperse particles from the cavities of the perturbed velocity field.

Local dissipation-scale distributions and high-order statistics of the energy dissipation rate are examined in turbulent channel flow using very high-resolution direct numerical simulations at Reynolds numbers , and . For sufficiently large , the dissipation-scale distributions and energy dissipation moments in the channel bulk flow agree with those in homogeneous isotropic turbulence, including only a weak Reynolds-number dependence of both the finest and largest scales. Systematic, but -independent, variations in the distributions and moments arise as the wall is approached for . In the range , there are substantial differences in the moments between the lowest and the two larger values of . This is most likely caused by coherent vortices from the near-wall region, which fill the whole channel for low .

Transient energy growth of disturbances to co-rotating pairs of vortices with axial core flows is investigated in an analysis where vortex core expansion and vortex merging are included by adopting a time-evolving base flow. The dynamics of pairs are compared with those of individual vortices in order to highlight the effect of vortex interaction. Three typical vortex pair cases are studied, with the pairs comprised respectively of individually inviscidly unstable vortices at the streamwise wavenumber that maximizes the individual instabilities, viscously unstable vortices also at the streamwise wavenumber maximizing the individual instabilities and asymptotically stable vortices at streamwise wavenumber zero. For the inviscidly unstable case, the optimal perturbation takes the form of a superposition of two individual helical unstable modes and the optimal energy growth is similar to that predicted for an individual inviscid unstable vortex, while where the individual vortices are viscously unstable, the optimal disturbances within each core have similar spatial distributions to the individually stable case. For both of these cases, time horizons considered are much lower than those required for the merger of the undisturbed vortices. However, for the asymptotically stable case, large linear transient energy growth of optimal perturbations occurs for time horizons corresponding to vortex merging. Linear transient disturbance energy growth exhibited by pairs in this stable case is two to three orders of magnitude larger than that for a corresponding individual vortex. The superposition of the perturbation and the base flow shows that the perturbation has a displacement effect on the vortices in the base flow. Direct numerical simulations of stable pairs seeded by optimal initial perturbations have been carried out and acceleration/delay of vortex merging associated with a dual vortex meandering and vortex breakup related to axially periodic acceleration and delay of vortex merging are observed. For axially invariant cases, the sign of perturbation has an effect, as well as magnitude; the sign dependence relates to whether or not the perturbation adds to or subtracts from the swirl of the base flow. For a two-dimensional perturbation that adds to the swirl of the base flow, seeding with the linear optimal disturbance at a relative energy level induces the pair to move towards each other and approximately halves the time required for merger. Direct numerical simulation shows that the optimal three-dimensional perturbation can induce the vortex system to break up before merging occurs, since the two-dimensional nature of vortex merging is broken by the development of axially periodic perturbations.

This paper deals with the mass transport induced by sea waves propagating over a sloping beach and fully reflected from a wall. It is shown that for moderate slopes the classical recirculation cell structure holds for small Reynolds numbers only. When the Reynolds number is large, the cells interact among themselves giving rise to the merging of the negative cells and the confinement of the positive ones near the bottom. Under such circumstances the fluid moves onshore near the bottom and offshore near the free surface. The seaward decrease of the vorticity produced at the bottom appears to be the reason for the merging phenomenon.

The flow in the far field of an isolated static momentum source is considered, taking into account the entrainment of fluid by the turbulent jet which develops far downstream irrespective of the type of device. The result is a simple analytical model for the irrotational region, which depends only on the thrust applied. This equation is implied by Stewart (J. Fluid Mech., vol. 1, 1956, pp. 593–606) for a jet. For a rotor, the model is radically different from the classical one derived from an actuator disk without turbulence or mixing in the wake, which led to a sink flow in the far field. The velocities decay like rather than , where is the radius, and are everywhere directed in the direction opposite to the thrust, rather than pointing towards the origin. The momentum source drives a co-flow which converges towards the turbulent region, thus supplying the entrained fluid. This flow pattern supports the assumption that the fluid surrounding the turbulent region is irrotational, better than the sink-flow model would. The model depends only on one empirical constant, a measure of the entrainment in a fully developed jet, for which a range of values is determined from the experimental literature. If the rotor is climbing, the sink flow is recovered; however, the limit of that equation as the climb velocity tends to zero, leading to hover, is singular. For both jets and rotors, this model used in a boundary condition should eliminate extraneous parameters and reduce the computational cost of numerical simulations, and may guide the design of chambers used for experiments, following Ricou & Spalding (J. Fluid Mech., vol. 11, 1960, pp. 21–32).

A mathematical model is developed to study the behaviour of an oscillating wave energy converter in a channel. During recent laboratory tests in a wave tank, peaks in the hydrodynamic actions on the converter occurred at certain frequencies of the incident waves. This resonant mechanism is known to be generated by the transverse sloshing modes of the channel. Here the influence of the channel sloshing modes on the performance of the device is further investigated. Within the framework of a linear inviscid potential-flow theory, application of Green’s theorem yields a hypersingular integral equation for the velocity potential in the fluid domain. The solution is found in terms of a fast-converging series of Chebyshev polynomials of the second kind. The physical behaviour of the system is then analysed, showing sensitivity of the resonant sloshing modes to the geometry of the device, which concurs in increasing the maximum efficiency. Analytical results are validated with available numerical and experimental data.