This paper considers the effect of continuous convection from small sources of buoyancy on the properties of the environment when the region of interest is bounded. The main assumptions are that the entrainment into the turbulent buoyant region is at a rate proportional to the local mean upward velocity, and that the buoyant elements spread out at the top of the region and become part of the non-turbulent environment at that level. Asymptotic solutions, valid at large times, are obtained for the cases of plumes from point and line sources and also periodically released thermals. These all have the properties that the environment is stably stratified, with the density profile fixed in shape, changing at a uniform rate in time at all levels, and everywhere descending (with ascending buoyant elements).

The analysis is carried out in detail for the point source in an environment of constant cross-section. Laboratory experiments have been conducted for this case, and these verify the major predictions of the theory. It is then shown how the method can be extended to include more realistic starting conditions for the convection, and a general shape of bounded environment. Finally, the model is applied quantitatively to a variety of problems in engineering, the atmosphere and the ocean, and the limitations on its use are discussed.

The two- and three-dimensional stability properties of the family of coherent shear-layer vortices discovered by Stuart are investigated. The stability problem is formulated as a non-separable eigenvalue problem in two independent variables, and solved numerically using spectral methods. It is found that there are two main classes of instabilities. The first class is subharmonic, and corresponds to pairing or localized pairing of vortex tubes; the pairing instability is most unstable in the two-dimensional limit, in which the perturbation has no spanwise variations. The second class repeats in the streamwise direction with the same periodicity as the basic flow. This mode is most unstable for spanwise wavelengths approximately 2/3 of the space between vortex centres, and can lead to the generation of streamwise vorticity and coherent ridges of upwelling. Comparison is made between the calculated instabilities and the observed pairing, helical pairing, and streak transitions. The theoretical and experimental results are found to be in reasonable agreement.

Theory and lattice-Boltzmann simulations are used to examine the effects of fluid inertia, at small Reynolds numbers, on flows in simple cubic, face-centred cubic and random arrays of spheres. The drag force on the spheres, and hence the permeability of the arrays, is determined at small but finite Reynolds numbers, at solid volume fractions up to the close-packed limits of the arrays. For small solid volume fraction, the simulations are compared to theory, showing that the first inertial contribution to the drag force, when scaled with the Stokes drag force on a single sphere in an unbounded fluid, is proportional to the square of the Reynolds number. The simulations show that this scaling persists at solid volume fractions up to the close-packed limits of the arrays, and that the first inertial contribution to the drag force relative to the Stokes-flow drag force decreases with increasing solid volume fraction. The temporal evolution of the spatially averaged velocity and the drag force is examined when the fluid is accelerated from rest by a constant average pressure gradient toward a steady Stokes flow. Theory for the short- and long-time behaviour is in good agreement with simulations, showing that the unsteady force is dominated by quasi-steady drag and added-mass forces. The short- and long-time added-mass coefficients are obtained from potential-flow and quasi-steady viscous-flow approximations, respectively.

The interaction of a laser-induced cavitation bubble with an elastic boundary and its dependence on the distance between bubble and boundary are investigated experimentally. The elastic boundary consists of a transparent polyacrylamide (PAA) gel with 80% water concentration with elastic modulus E = 0.25 MPa. At this E-value, the deformation and rebound of the boundary is very pronounced providing particularly interesting features of bubble dynamics. It is shown by means of high-speed photography with up to 5 million frames s−1 that bubble splitting, formation of liquid jets away from and towards the boundary, and jet-like ejection of the boundary material into the liquid are the main features of this interaction. The maximum liquid jet velocity measured was 960 m s−1. Such high-velocity jets penetrate the elastic boundary even through a water layer of 0.35 mm thickness. The jetting behaviour arises from the interaction between the counteracting forces induced by the rebound of the elastic boundary and the Bjerknes attraction force towards the boundary. General principles of the formation of annular and axial jets are discussed which allow the interpretation of the complex dynamics. The concept of the Kelvin impulse is examined with regard to bubble migration and jet formation. The results are discussed with respect to cavitation erosion, collateral damage in laser surgery, and cavitation-mediated enhancement of pulsed laser ablation of tissue.

The effects of surface-active agents on drop deformation and breakup in extensional flows at low Reynolds numbers are described. In this free-boundary problem, determination of the interfacial velocity requires knowledge of the distribution of surfactant, which, in turn, requires knowledge of the interfacial velocity field. We account for this explicit coupling of the unknown drop shape and the evolving surfactant distribution. An analytical result valid for nearly spherical distortions is presented first. Finite drop deformation is studied numerically using the boundary-integral method in conjunction with the time-dependent convective–diffusion equation for surfactant transport. This procedure accurately follows interfacial tension variations, produced by non-uniform surfactant distribution, on the evolving interface. The numerical method allows for an arbitrary equation of state relating interfacial tension to the local concentration of surfactant, although calculations are presented only for the common linear equation of state. Also, only the case of insoluble surfactant is studied.

The analytical and numerical results indicate that at low capillary numbers the presence of surfactant causes larger deformation than would occur for a drop with a constant interfacial tension equal to the initial equilibrium value. The increased deformation occurs owing to surfactant being swept to the end of the drop where it acts to locally lower the interfacial tension, which therefore requires increased deformation to satisfy the normal stress balance. However, at larger capillary numbers and finite deformations, this convective effect competes with ‘dilution’ of the surfactant due to interfacial area increases. These two different effects of surface-active material are illustrated and discussed and their influence on the critical capillary number for breakup is presented.

The paper presents an analysis of laminar motion of a conducting liquid in a rectangular duct under a uniform transverse magnetic field. The effects of the duct having conducting walls are investigated. Exact solutions are obtained for two cases, (i) perfectly conducting walls perpendicular to the field and thin walls of arbitrary conductivity parallel to the field, and (ii) non-conducting walls parallel to the field and thin walls of arbitrary conductivity perpendicular to the field.

The boundary layers on the walls parallel to the field are studied in case (i) and it is found that at high Hartmann number (M), large positive and negative velocities of order MVc are induced, where Vc is the velocity of the core. It is suggested that contrary to previous assumptions the magnetic field may in some cases have a destabilizing effect on flow in ducts.

Fully developed incompressible turbulent pipe flow at bulk-velocity- and pipe-diameter-based Reynolds number ReD=44000 was simulated with second-order finite-difference methods on 630 million grid points. The corresponding Kármán number R+, based on pipe radius R, is 1142, and the computational domain length is 15R. The computed mean flow statistics agree well with Princeton Superpipe data at ReD=41727 and at ReD=74000. Second-order turbulence statistics show good agreement with experimental data at ReD=38000. Near the wall the gradient of with respect to ln(1−r)+ varies with radius except for a narrow region, 70 < (1−r)+ < 120, within which the gradient is approximately 0.149. The gradient of with respect to ln{(1−r)++a+} at the present relatively low Reynolds number of ReD=44000 is not consistent with the proposition that the mean axial velocity is logarithmic with respect to the sum of the wall distance (1−r)+ and an additive constant a+ within a mesolayer below 300 wall units. For the standard case of a+=0 within the narrow region from (1−r)+=50 to 90, the gradient of with respect to ln{(1−r)++a+} is approximately 2.35. Computational results at the lower Reynolds number ReD=5300 also agree well with existing data. The gradient of with respect to 1−r at ReD=44000 is approximately equal to that at ReD=5300 for the region of 1−r > 0.4. For 5300 < ReD < 44000, bulk-velocity-normalized mean velocity defect profiles from the present DNS and from previous experiments collapse within the same radial range of 1−r > 0.4. A rationale based on the curvature of mean velocity gradient profile is proposed to understand the perplexing existence of logarithmic mean velocity profile in very-low-Reynolds-number pipe flows. Beyond ReD=44000, axial turbulence intensity varies linearly with radius within the range of 0.15 < 1−r < 0.7. Flow visualizations and two-point correlations reveal large-scale structures with comparable near-wall azimuthal dimensions at ReD=44000 and 5300 when measured in wall units. When normalized in outer units, streamwise coherence and azimuthal dimension of the large-scale structures in the pipe core away from the wall are also comparable at these two Reynolds numbers.

Dissolution of a porous medium creates, under certain conditions, some highly conductive channels called wormholes. The mechanism of propagation is an unstable phenomenon depending on the microscopic properties at the pore scale and is controlled by the injection rate. The aim of this work is to test the ability of a Darcy-scale model to describe the different dissolution regimes and to characterize the influence of the flow parameters on the wormhole development. The numerical approach is validated by model experiments reflecting dissolution processes occurring during acid injection in limestone. Flow and transport macroscopic equations are written under the assumption of local mass non-equilibrium. The coupled system of equations is solved numerically in two dimensions using a finite volume method. Results are discussed in terms of wormhole propagation rate and pore volume injected.

Coherent axisymmetric structures in a turbulent jet are modelled as linear instability modes of the mean velocity profile, regarded as the profile of a, fictitious laminar inviscid flow. The usual multiple-scales expansion method is used in conjunction with a family of profiles consistent with similarity laws for the initial mixing region and approximating the profiles measured by Crow & Champagne (1971), Moore (1977) and other investigators, to deal with the effects of flow divergence. The downstream growth and approach to peak amplitude of axisymmetric wave modes with prescribed real frequency is calculated numerically, and comparisons are made with various sets of experimental data. Excellent agreement is found with the wavelength measurements of Crow & Champagne. Quantities such as the amplitude gain which depend on cumulative effects are less well predicted, though the agreement is still quite tolerable in view of the facts that this simple linear model of slowly diverging flow is being applied far outside its range of strict validity and that many of the published measurements are significantly contaminated by nonlinear effects. The predictions show that substantial variations are to be expected in such quantities as the phase speed and growth rate, according to the flow signal (velocity, pressure, etc.) measured, and that these variations depend not only on the axial measurement location but also on the cross-stream position. Trends of this kind help to explain differences in, for example, the preferred Strouhal number found by investigators using hot wires or pressure probes on the centre-line, in the mixing layer or in the near field.

We present the first application of an artificial neural network trained through a deep reinforcement learning agent to perform active flow control. It is shown that, in a two-dimensional simulation of the Kármán vortex street at moderate Reynolds number ($Re=100$), our artificial neural network is able to learn an active control strategy from experimenting with the mass flow rates of two jets on the sides of a cylinder. By interacting with the unsteady wake, the artificial neural network successfully stabilizes the vortex alley and reduces drag by approximately 8 %. This is performed while using small mass flow rates for the actuation, of the order of 0.5 % of the mass flow rate intersecting the cylinder cross-section once a new pseudo-periodic shedding regime is found. This opens the way to a new class of methods for performing active flow control.

This is a study of the interactions of solitary waves climbing up a circular island. A series of large-scale laboratory experiments with waves of different incident height-to-depth ratios and different crest lengths is described. Detailed two-dimensional run-up height measurements and time histories of surface elevations around the island are presented. A numerical model based on the two-dimensional shallow-water wave equations including runup calculations was developed. Numerical model predictions agreed very well with the laboratory data and the model was used to study wave trapping and the effect of slope. Under certain conditions, enhanced runup and wave trapping on the lee side of the island were observed, suggesting a possible explanation for the devastation reported by field surveys in Babi Island off Flores, Indonesia, and in Okushiri Island, Japan.

We have used the technique of laser-Doppler velocimetry to study the transition to turbulence in a fluid contained between concentric cylinders with the inner cylinder rotating. The experiment was designed to test recent proposals for the number and types of dynamical regimes exhibited by a flow before it becomes turbulent. For different Reynolds numbers the radial component of the local velocity was recorded as a function of time in a computer, and the records were then Fourier-transformed to obtain velocity power spectra. The first two instabilities in the flow, to time-independent Taylor vortex flow and then to time-dependent wavy vortex flow, are well known, but the present experiment provides the first quantitative information on the subsequent regimes that precede turbulent flow. Beyond the onset of wavy vortex flow the velocity spectra contain a single sharp frequency component and its harmonics; the flow is strictly periodic. As the Reynolds number is increased, a previously unobserved second sharp frequency component appears at R/Rc = 10·1, where Rc is the critical Reynolds number for the Taylor instability. The two frequencies appear to be irrationally related; hence this is a quasi-periodic flow. A chaotic element appears in the flow at R/Rc ≃ 12, where a weak broadband component is observed in addition to the sharp components; this flow can be described as weakly turbulent. As R is increased further, the component that appeared at R/Rc= 10·1 disappears at R/Rc = 19·3, and the remaining sharp component disappears at R/Rc = 21·9, leaving a spectrum with only the broad component and a background continuum. The observance of only two discrete frequencies and then chaotic flow is contrary to Landau's picture of an infinite sequence of instabilities, each adding a new frequency to the motion. However, recent studies of nonlinear models with a few degrees of freedom show a behaviour similar in most respects to that observed.

The common assumption that the energy of waves on a non-uniform current U is propagated with a velocity (U + c) where cg is the group-velocity, and that no further interaction takes place, is shown in this paper to be incorrect. In fact the current does additional work on the waves at a rate γijSij where γij is the symmetric rate-of-strain tensor associated with the current, and Sij is the radiation stress tensor introduced earlier (Longuet-Higgins & Stewart 1960).

In the present paper we first obtain an asymptotic solution for the combined velocity potential in the simple case (1) when the non-uniform current U is in the direction of wave propagation and the horizontal variation of U is compensated by a vertical upwelling from below. The change in wave amplitude is shown to be such as would be found by inclusion of the radiation stress term.

In a second example (2) the current on the x-axis is assumed to be as in (1), but the horizontal variation in U is compensated by a small horizontal inflow from the sides. It is found that in that case the wave amplitude is also affected by the horizontal advection of wave energy from the sides.

From cases (1) and (2) the general law of interaction between short waves and non-uniform currents is inferred. This is then applied to a third example (3) when waves encounter a current with vertical axis of shear, at an oblique angle. The change in wave amplitude is shown to differ somewhat from the previously accepted value.

The conclusion that non-linear interactions affect the amplification of the waves has some bearing on the theoretical efficiency of hydraulic and pneumatic breakwaters.

Direct numerical simulations (DNS) and optimal control theory are used in a predictive control setting to determine controls that effectively reduce the turbulent kinetic energy and drag of a turbulent flow in a plane channel at Reτ = 100 and Reτ = 180. Wall transpiration (unsteady blowing/suction) with zero net mass flux is used as the control. The algorithm used for the control optimization is based solely on the control objective and the nonlinear partial differential equation governing the flow, with no ad hoc assumptions other than the finite prediction horizon, T, over which the control is optimized.

Flow relaminarization, accompanied by a drag reduction of over 50%, is obtained in some of the control cases with the predictive control approach in direct numerical simulations of subcritical turbulent channel flows. Such performance far exceeds what has been obtained to date in similar flows (using this type of actuation) via adaptive strategies such as neural networks, intuition-based strategies such as opposition control, and the so-called ‘suboptimal’ strategies, which involve optimizations over a vanishingly small prediction horizon T+ → 0. To achieve flow relaminarization in the predictive control approach, it is shown that it is necessary to optimize the controls over a sufficiently long prediction horizon T+ [gsim ] 25. Implications of this result are discussed.

The predictive control algorithm requires full flow field information and is computationally expensive, involving iterative direct numerical simulations. It is, therefore, impossible to implement this algorithm directly in a practical setting. However, these calculations allow us to quantify the best possible system performance given a certain class of flow actuation and to qualify how optimized controls correlate with the near-wall coherent structures believed to dominate the process of turbulence production in wall-bounded flows. Further, various approaches have been proposed to distil practical feedback schemes from the predictive control approach without the suboptimal approximation, which is shown in the present work to restrict severely the effectiveness of the resulting control algorithm. The present work thus represents a further step towards the determination of optimally effective yet implementable control strategies for the mitigation or enhancement of the consequential effects of turbulence.

The dynamics of gravity currents are believed to be strongly influenced by dissipation due to turbulence and mixing between the current and the surrounding ambient fluid. This paper describes new theory and experiments on gravity currents produced by lock exchange which suggest that dissipation is unimportant when the Reynolds number is sufficiently high. Although there is mixing, the amount of energy dissipated is small, reducing the current speed by a few percent from the energy-conserving value. Benjamin (J. Fluid Mech. vol. 31, 1968, p. 209) suggests that dissipation is an essential ingredient in gravity current dynamics. We show that dissipation is not important at high Reynolds number, and provide an alternative theory that predicts the current speed and depth based on energy-conserving flow that is in good agreement with experiments. We predict that in a deep ambient the front Froude number is 1, rather than the previously accepted value of $\sqrt 2$. New experiments are reported for this case that support the new theoretical value.

The proper orthogonal decomposition technique (Lumley's decomposition) is applied to the turbulent flow in a channel, to extract coherent structures by decomposing the velocity field into characteristic eddies with random coefficients. In the homogeneous spatial directions a generalization of the shot-noise expansion is used to determine the characteristic eddies. In this expansion the Fourier coefficients of the characteristic eddy cannot be obtained from second-order statistics. Three different techniques are used to determine the phases of these coefficients: (i) a technique based on the bispectrum, (ii) a spatial compactness requirement, and (iii) a functional continuity argument. Results from these three techniques are found to be very similar. The implications of these techniques and the shot-noise expansion are discussed in the Appendix. The dominant eddy is found to contribute as much as 76% to the turbulent kinetic energy. In two and three dimensions, the characteristic eddies consist of an ejection region straddled by streamwise vortices which leave the wall in a very short streamwise distance of approximately 100 wall units.

Motion pictures taken at Duck Creek, a clear stream 6.5 m wide and 35 cm deep near Pinedale, Wyoming, provide detailed, quantitative information on both the modes of motion of individual bedload particles and the collective motions of large numbers of them. Bed shear stress was approximately 6 Pa (60 dynes cm−2), which was about twice the threshold for movement of the 4 mm median diameter fine gravel bed material; and transport was almost entirely as bedload. The displacements of individual particles occurred mainly by rolling of the majority of the particles and saltation of the smallest ones, and rarely by brief sliding of large, angular ones. Entrainment was principally by rollover of the larger particles and liftoff of the smaller ones, and infrequently by ejection caused by impacts, whereas distrainment was primarily by diminution of fluid forces in the case of rolling particles and by collisions with larger bed particles in the case of saltating ones. The displacement times averaged about 0.2−0.4 s and generally were much shorter than the intervening repose times. The collective motions of the particles were characterized by frequent, brief, localized, random sweep-transport events of very high rates of entrainment and transport, which in the aggregate transported approximately 70% of the total load moved. These events occurred 9% of the time at any particular point of the bed, lasted 1–2 s, affected areas typically 20–50 cm long by 10–20 cm wide, and involved bedload concentrations approximately 10 times greater than background. The distances travelled during displacements averaged about 15 times the particle diameter. Despite the differences in their dominant modes of movement, the 8–16 mm particles typically travelled only about 30% slower during displacement than the 2–4 mm ones, whose speeds averaged 21 cm s−1. Particles starting from the same point not only moved intermittently downstream but also dispersed both longitudinally and transversely, with diffusivities of 4.6 and 0.26 cm2 s−1, respectively. The bedload transport rates measured from the films were consistent with those determined conventionally with a bedload sampler. The 2–4 mm particles were entrained 6 times faster on finer areas of the bed, where 8–16 mm particles covered 6% of the surface area, than on coarser ones, where they covered 12%, even though 2–4 and 4–8 mm particles covered practically the same percentage areas in both cases. The 4–8 and 8–16 mm particles, in contrast, were entrained at the same rates in both cases. To within the statistical uncertainty, the rates of distrainment balanced the rates of entrainment for all three sizes, and were approximately proportional to the corresponding concentrations of bedload.

A general mathematical model which describes the motion of an interface between immiscible viscous fluids along a smooth homogeneous solid surface is examined in the case of small capillary and Reynolds numbers. The model stems from a conclusion that the Young equation, σ1 cos θ = σ2 − σ3, which expresses the balance of tangential projection of the forces acting on the three-phase contact line in terms of the surface tensions σi and the contact angle θ, together with the well-established experimental fact that the dynamic contact angle deviates from the static one, imply that the surface tensions of contacting interfaces in the immediate vicinity of the contact line deviate from their equilibrium values when the contact line is moving. The same conclusion also follows from the experimentally observed kinematics of the flow, which indicates that liquid particles belonging to interfaces traverse the three-phase interaction zone (i.e. the ‘contact line’) in a finite time and become elements of another interface – hence their surface properties have to relax to new equilibrium values giving rise to the surface tension gradients in the neighbourhood of the moving contact line. The kinematic picture of the flow also suggests that the contact-line motion is only a particular case of a more general phenomenon – the process of interface formation or disappearance – and the corresponding mathematical model should be derived from first principles for this general process and then applied to wetting as well as to other relevant flows. In the present paper, the simplest theory which uses this approach is formulated and applied to the moving contact-line problem. The model describes the true kinematics of the flow so that it allows for the ‘splitting’ of the free surface at the contact line, the appearance of the surface tension gradients near the contact line and their influence upon the contact angle and the flow field. An analytical expression for the dependence of the dynamic contact angle on the contact-line speed and parameters characterizing properties of contacting media is derived and examined. The role of a ‘thin’ microscopic residual film formed by adsorbed molecules of the receding fluid is considered. The flow field in the vicinity of the contact line is analysed. The results are compared with experimental data obtained for different fluid/liquid/solid systems.

Measurements are presented of the pressure fluctuations acting on a stationary squaresection cylinder, with the front face normal to the flow, and one forced to oscillate, transverse to a flow, at amplitudes up to 25% of the length of a side. The range of reduced velocities investigated, 4–13, includes the vortex lock-in regime. At lock-in the amplification of the coefficient of fluctuating lift is found to be much less than that found for a circular cylinder. The variation of the phase angle, between lift and displacement, is also different from that measured on a circular cylinder, and vortex-induced oscillations are possible only at the high-reduced-velocity end of the lock-in range. At reduced velocities sufficiently far below lock-in the natural vortex-shedding mode is suppressed and vortices are found to form over the side faces at the body frequency. Intermittent reattachment occurs over the side faces and, for an amplitude of oscillation equal to 10% of the length of a side face, the time-mean drag coefficient can be reduced to 60% of its fixed-cylinder value.

The organization of vortex clusters above the buffer layer of turbulent channels is analysed using direct numerical simulations at friction Reynolds numbers up to $\hbox{\it Re}_{\tau}\,{=}\,1900$. Especial attention is paid to a family of clusters that reach from the logarithmic layer to the near-wall region below $y^+\,{=}\,20$. These tall attached clusters are markers of structures of the turbulent fluctuating velocity that are more intense than their background. Their lengths and widths are proportional to their heights $\Delta_y$ and grow self-similarly with time after originating at different wall-normal positions in the logarithmic layer. Their influence on the outer region is measured by the variation of their volume density with $\Delta_y$. That influence depends on the vortex identification threshold, and becomes independent of the Reynolds number if the threshold is low enough. The clusters are parts of larger structures of the streamwise velocity fluctuations whose average geometry is consistent with a cone tangent to the wall along the streamwise axis. They form groups of a few members within each cone, with the larger individuals in front of the smaller ones. This behaviour is explained considering that the streamwise velocity cones are ‘wakes’ left behind by the clusters, while the clusters themselves are triggered by the wakes left by yet larger clusters in front of them. The whole process repeats self-similarly in a disorganized version of the vortex-streak regeneration cycle of the buffer layer, in which the clusters and the wakes spread linearly under the effect of the background turbulence. These results characterize for the first time the structural organization of the self-similar range of the turbulent logarithmic region.