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Theory and experiments are used to investigate the water and sediment motion induced along a sea bed by travelling plane jets. Steadily moving jets are considered, and represent an idealization of the tools mounted on ships and remotely operated vehicles (ROVs) for injection dredging and trenching. The jet-induced turbulent currents simultaneously suspend sand from the bed and entrain water from the ambient. To describe these processes, a shallow-flow theory is proposed in which the turbulent current is assumed stratified into sediment-laden and sediment-free sublayers. The equations are written in curvilinear coordinates attached to the co-evolving bed profile. A sharp interface description is then adopted to account rigorously for mass and momentum exchanges between the bed, current and ambient, including their effects on the balance of mechanical energy. Travelling-wave solutions are obtained, in which the jet-induced current scours a trench of permanent form in a frame of reference moving with the jetting tool. Depending on the operating parameters, it is found that the sediment-laden current may remain supercritical throughout the trench, or be forced to undergo an internal hydraulic jump. These predictions are confirmed by laboratory experiments. For flows with or without jump in which the current remains attached to the bed, bottom profiles computed by the theory compare favourably with imaging measurements.
The stability of travelling wave Chapman–Jouguet and moderately overdriven detonations of Zeldovich–von Neumann–Döring type is formulated for a general system that incorporates the idealized gas and condensed-phase (liquid or solid) detonation models. The general model consists of a two-component mixture with a one-step irreversible reaction between reactant and product. The reaction rate has both temperature and pressure sensitivities and has a variable reaction order. The idealized condensed-phase model assumes a pressure-sensitive reaction rate, a constant-γ caloric equation of state for an ideal fluid, with the isentropic derivative γ=3, and invokes the strong shock limit. A linear stability analysis of the steady, planar, ZND detonation wave for the general model is conducted using a normal-mode approach. An asymptotic analysis of the eigenmode structure at the end of the reaction zone is conducted, and spatial boundedness (closure) conditions formally derived, whose precise form depends on the magnitude of the detonation overdrive and reaction order. A scaling analysis of the transonic flow region for Chapman–Jouguet detonations is also studied to illustrate the validity of the linearization for Chapman–Jouguet detonations. Neutral stability boundaries are calculated for the idealized condensed-phase model for one- and two-dimensional perturbations. Comparisons of the growth rates and frequencies predicted by the normal-mode analysis for an unstable detonation are made with a numerical solution of the reactive Euler equations. The numerical calculations are conducted using a new, high-order algorithm that employs a shock-fitting strategy, an approach that has significant advantages over standard shock-capturing methods for calculating unstable detonations. For the idealized condensed-phase model, nonlinear numerical solutions are also obtained to study the long-time behaviour of one- and two-dimensional unstable Chapman–Jouguet ZND waves.
We investigate the entrainment, deposition and motion of coarse spherical particles within a turbulent shallow water stream down a steep slope. This is an idealization of bed-load transport in mountain streams. Earlier investigations have described this kind of sediment transport using empirical correlations or concepts borrowed from continuum mechanics. The intermittent character of particle transport at low-water discharges led us to consider it as a random process. Sediment transport in this regime results from the imbalance between entrainment and deposition of particles rather than from momentum balance between water and particles. We develop a birth–death immigration–emigration Markov process to describe the particle exchanges between the bed and the water stream. A key feature of the model is its long autocorrelation times and wide, frequent fluctuations in the solid discharge, a phenomenon never previously explained despite its ubiquity in both nature and laboratory experiments. We present experimental data obtained using a nearly two-dimensional channel and glass beads as a substitute for sediment. Entrainment, trajectories, and deposition were monitored using a high-speed digital camera. The empirical probability distributions of the solid discharge and deposition frequency were properly described by the theoretical model. Experiments confirmed the existence of wide and frequent fluctuations of the solid discharge, and revealed the existence of long autocorrelation time, but theory overestimates the autocorrelation times by a factor of around three. Particle velocity was weakly dependent on the fluid velocity contrary to the predictions of the theoretical model, which performs well when a single particle is moving. For our experiments, the dependence of the solid discharge on the fluid velocity is entirely controlled by the number of moving particles rather than by their velocity. We also noted significant changes in the behaviour of particle transport when the bed slope or the water discharge was increased. The more vigorous the stream was, the more continuous the solid discharge became. Moreover, although 90% of the energy supplied by gravity to the stream is dissipated by turbulence for slopes lower than 10%, particles dissipate more and more energy when the bed slope is increased, but surprisingly, the dissipation rate is nearly independent of fluid velocity. A movie is available with the online version of the paper.
We present an experimental study of an axisymmetric turbulent fountain in a two-layer stratified environment. Interacting with the interface, the fountain is observed to exhibit three regimes of flow. It may penetrate the interface, but nonetheless return to the source where it spreads as a radially propagating gravity current; the return flow may be trapped at the interface where it spreads as a radially propagating intrusion or it may do both. These regimes have been classified using empirically determined regime parameters which govern the relative initial momentum of the fountain and the relative density difference of the fountain and the ambient fluid. The maximum vertical distance travelled by the fountain in a two-layer fluid has been theoretically determined by extending the theory developed for fountains in a homogeneous environment. The theory compares favourably with experimental measurements. We have also developed a theory to analyse the initial speeds of the resulting radial currents. The spreading currents exhibited two different flow regimes: a constant-velocity regime and an inertia-buoyancy regime in which the front position, R, scales with time, t, as R ∼ t3/4. These regimes were classified using a critical Froude number which characterized the competing effects of momentum and buoyancy in the currents.
We describe the results of a numerical investigation of the dynamics of breakup of streams of immiscible fluids in the confined geometry of a microfluidic T-junction. We identify three distinct regimes of formation of droplets: squeezing, dripping and jetting, providing a unifying picture of emulsification processes typical for microfluidic systems. The squeezing mechanism of breakup is particular to microfluidic systems, since the physical confinement of the fluids has pronounced effects on the interfacial dynamics. In this regime, the breakup process is driven chiefly by the buildup of pressure upstream of an emerging droplet and both the dynamics of breakup and the scaling of the sizes of droplets are influenced only very weakly by the value of the capillary number. The dripping regime, while apparently homologous to the unbounded case, is also significantly influenced by the constrained geometry; these effects modify the scaling law for the size of the droplets derived from the balance of interfacial and viscous stresses. Finally, the jetting regime sets in only at very high flow rates, or with low interfacial tension, i.e. higher values of the capillary number, similar to the unbounded case.
A perfectly conducting spherical particle is suspended within an electrolyte solution and is exposed to a uniformly applied electric field. Using a weak-field approximation, the electro-kinetic flow is analysed for arbitrary Debye-layer thickness, the commonly employed thin-layer model emerging as a special case. We identify a scalar property which quantifies the global strength of the quadrupolar flow structure.
The theory of turbulent resistivity in ‘wavy’ magnetohydrodynamic turbulence in two dimensions is presented. The goal is to explore the theory of quenching of turbulent resistivity in a regime for which the mean field theory can be rigorously constructed at large magnetic Reynolds number Rm. This is achieved by extending the simple two-dimensional problem to include body forces, such as buoyancy or the Coriolis force, which convert large-scale eddies into weakly interacting dispersive waves. The turbulence-driven spatial flux of magnetic potential is calculated to fourth order in wave slope – the same order to which one usually works in wave kinetics. However, spatial transport, rather than spectral transfer, is the object here. Remarkably, adding an additional restoring force to the already tightly constrained system of high Rm magnetohydrodynamic turbulence in two dimensions can actually increase the turbulent resistivity, by admitting a spatial flux of magnetic potential which is not quenched at large Rm, although it is restricted by the conditions of applicability of weak turbulence theory. The absence of Rm-dependent quenching in this wave-interaction-driven flux is a consequence of the presence of irreversibility due to resonant nonlinear three-wave interactions, which are independent of collisional resistivity. The broader implications of this result for the theory of mean field electrodynamics are discussed.
We numerically investigate turbulent thermal convection driven by a horizontal surface of constant heat flux and compare the results with those of constant temperature. Below Ra ≈ 109, where Ra is the Rayleigh number, when the flow is smooth and regular, the heat transport in the two cases is essentially the same. For Ra > 109 the heat transport for imposed heat flux is smaller than that for constant temperature, and is close to experimental data. We provide a simple dimensional argument to indicate that the unsteady emission of thermal plumes renders typical experimental conditions closer to the constant heat flux case.
The stability of a liquid film flowing down an inclined plane is considered when the film is contaminated by an insoluble surfactant and subjected to a uniform normal electric field. The liquid is treated as a perfect conductor and the air above the film is treated as a perfect dielectric. Previous studies have shown that, when acting in isolation, surfactant has a stabilizing influence on the flow while an electric field has a destabilizing influence. The competition between these two effects is the focus of the present study. The linear stability problem is formulated and solved at arbitrary parameter values. An extended form of Squire's theorem is presented to argue that attention may be confined to two-dimensional disturbances. The stability characteristics for Stokes flow are described exactly; the growth rates of the normal modes at finite Reynolds number are computed numerically. We plot the neutral curves dividing regions of stability and instability, and trace how the topology of the curves changes as the intensity of the electric field varies both for a clean and for a contaminated film. With a sufficiently strong electric field, the neutral curve for a clean film consists of a lower branch trapping an area of stable modes around the origin, and an upper branch above which the flow is stable. With surfactant present, a similar situation obtains, but with an additional island of stable modes disjoint from the upper and lower branches.
We propose a method for the dynamic simulation of a collection of self-propelled particles in a viscous Newtonian fluid. We restrict attention to particles whose size and velocity are small enough that the fluid motion is in the creeping flow regime. We propose a simple model for a self-propelled particle, and extended the Stokesian Dynamics method to conduct dynamic simulations of a collection of such particles. In our description, each particle is treated as a sphere with an orientation vector p, whose locomotion is driven by the action of a force dipole Sp of constant magnitude S0 at a point slightly displaced from its centre. To simplify the calculation, we place the dipole at the centre of the particle, and introduce a virtual propulsion force Fp to effect propulsion. The magnitude F0 of this force is proportional to S0. The directions of Sp and Fp are determined by p. In isolation, a self-propelled particle moves at a constant velocity u0 p, with the speed u0 determined by S0. When it coexists with many such particles, its hydrodynamic interaction with the other particles alters its velocity and, more importantly, its orientation. As a result, the motion of the particle is chaotic. Our simulations are not restricted to low particle concentration, as we implement the full hydrodynamic interactions between the particles, but we restrict the motion of particles to two dimensions to reduce computation. We have studied the statistical properties of a suspension of self-propelled particles for a range of the particle concentration, quantified by the area fraction φa. We find several interesting features in the microstructure and statistics. We find that particles tend to swim in clusters wherein they are in close proximity. Consequently, incorporating the finite size of the particles and the near-field hydrodynamic interactions is of the essence. There is a continuous process of breakage and formation of the clusters. We find that the distributions of particle velocity at low and high φa are qualitatively different; it is close to the normal distribution at high φa, in agreement with experimental measurements. The motion of the particles is diffusive at long time, and the self-diffusivity decreases with increasing φa. The pair correlation function shows a large anisotropic build-up near contact, which decays rapidly with separation. There is also an anisotropic orientation correlation near contact, which decays more slowly with separation. Movies are available with the online version of the paper.
An analysis is presented of parametric instability in a finite-length rotating cylinder subject to periodic axial compression by small sinusoidal oscillations of one of its ends (the ‘piston’). The instability is due to resonant interactions between inertial-wave (Kelvin) modes of the cylinder and the oscillatory compression and is resisted by viscosity, acting both through thin boundary layers and throughout the volume, the two mechanisms proving crucial for a satisfactory description. Instability is found to take the form of either a single axisymmetric mode with frequency near to half that of compression, or a pair of non-axisymmetric modes of the same azimuthal and axial orders and oppositely signed frequencies, whose difference is close to the compression frequency. Thus, in the axisymmetric case, instability leads to spontaneous growth of a system of one or more oscillating toroidal vortices encircling the cylinder axis, while the differing frequencies of the two modes of non-axisymmetric instability implies an oscillatory aperiodic flow. The neutral curves giving the threshold for instability are determined for all modes/mode pairs. For a given mode or mode pair, the neutral curve shows a critical piston amplitude dependent on rotational Reynolds number and cylinder aspect ratio, below which instability does not occur, and above which there is instability provided the compression frequency is chosen to lie in a band centred on the exact resonance condition.
A weakly nonlinear analysis is presented of parametric instability in a rotating cylinder subject to periodic axial compression by small sinusoidal oscillations of one of its ends (‘the piston’). Amplitude equations are derived for the pair of parametrically resonant (primary) inertial modes which were found to arise from linear instability in Part 1. These equations introduce an infinity of geostrophic mode amplitudes, representing a nonlinear modification of the mean flow, for which evolution equations are also derived. Consequences of the total system of equations are investigated for axisymmetric modes. Different possible outcomes are found at large times: (a) a fixed point, representing a saturated state in which the oscillatory toroidal vortices of the primary mode are phase-locked to the piston motion with half its frequency; (b) a limit cycle or chaotic attractor, corresponding to slow-time oscillations of the primary mode; or (c) exponential divergence of the amplitudes to infinity. The latter outcome, a necessary condition for which is derived in the form of a threshold piston amplitude for divergence, invalidates the theory, inducing a gross change in the character of the flow and providing a route out of the weakly nonlinear regime. Non-zero fixed-point branches arise via bifurcations from both sides of the linear neutral curve, where the basic flow changes local stability. The lower-amplitude branch is shown to be unstable, while the upper one may lose local stability, resulting in a Hopf bifurcation to a limit cycle, which can subsequently become aperiodic via a series of further bifurcations. Typically, during the resulting oscillations, whether periodic or not, the perturbation first grows from small amplitude owing to basic-flow instability, then nonlinear detuning of the parametric resonance causes decay back to small amplitude in the second half of the cycle, which then restarts.
Mean flow measurements are obtained in a commercial steel pipe with krms/D = 1/26 000, where krms is the roughness height and D the pipe diameter, covering the smooth, transitionally rough, and fully rough regimes. The results indicate a transition from smooth to rough flow that is much more abrupt than the Colebrook transitional roughness function suggests. The equivalent sandgrain roughness was found to be 1.6 times the r.m.s. roughness height, in sharp contrast to the value of 3.0 to 5.0 that is commonly used. The difference amounts to a reduction in pressure drop for a given flow rate of at least 13% in the fully rough regime. The mean velocity profiles support Townsend's similarity hypothesis for flow over rough surfaces.
The stretched spiral vortex is identified using direct numerical simulation (DNS) data for homogeneous isotropic turbulence and its properties are studied. Its genesis, growth and annihilation are elucidated, and its role in the generation of turbulence is shown. Aside from the two symmetric modes of configurations with regard to the vorticity alignment along two spiral sheets and the vortex tube in the core region studied in previous works, a third asymmetric mode is found. One of the two symmetric modes and the asymmetric mode are created not by a conventional rolling-up of a single vortex sheet but through the interaction among several sheets. The stagnation flow caused by the two sheets converges to form recirculating flow through its interaction with the vortex along the third sheet. This recirculating flow strains and stretches the sheets. The vortex tube is formed by axial straining, lowering of pressure and the intensification of the swirling motion in the recirculating region. As a result of the differential rotation induced by the tube and that self-induced by the sheet, the vortex sheets are entrained by the tube and form spiral turns. The transition between the three modes is examined. The initial configuration is in one of two symmetric modes, but it is transformed into another set of two modes due to the occurrence of reorientation in the vorticity direction along the stretched sheets. The symmetric mode tends to be more persistent than the asymmetric mode, among the two transformed modes. The tightening of the spiral turns of the spiral sheets produces a cascade of velocity fluctuations to smaller scales and generates a strongly intermittent dissipation field. To precisely capture the spiral turns, a grid resolution with at least (kmax is the largest wavenumber, is the averaged Kolmogorov scale) is required. At a higher Reynolds number, self-similar spiral vortices are successively produced by the instability cascade along the stretched vortex sheets. A cluster consisting of spiral vortices with an extensive range of length scales is formed and this cluster induces an energy cascade.
The aim of this work is to investigate the fully nonlinear dynamics of mixed convection in porous media heated non-uniformly from below and through which an axial flow is maintained. Depending on the choice of the imposed inhomogeneous temperature profile, two cases prove to be of interest: the base flow displays an absolute instability region either detached from the inlet or attached to it. Results from a combined direct numerical simulations and linear stability approach have revealed that in the first case, the nonlinear solution is a steep nonlinear global mode, with a sharp stationary front located at a marginally absolutely unstable station. In the second configuration, the scaling laws for the establishment of a nonlinear global mode quenched by the inlet are found to agree perfectly with the theory. It is also found that in both configurations, the global frequency of synchronized oscillations corresponds to the local absolute frequency determined by linear criterion, even far from the threshold of global instability.
In this paper, we consider a viscous instability of a stratified boundary layer that is a form of the familiar Tollmien–Schlichting (T-S) waves modified by a stable density stratification. As with the usual T-S waves, the triple-deck formalism was employed to provide a self-consistent description of linear and nonlinear instability properties at asymptotically large Reynolds numbers. The effect of stratification on the temporal and spatial linear growth rates is studied. It is found that stratification reduces the maximum spatial growth rate, but enhances the maximum temporal growth rate. This viscous instability may offer a possible alternative explanation for the origin of certain long atmospheric waves, whose characteristics are not well predicted by inviscid instabilities. In the high-frequency limit, the nonlinear evolution of the disturbances is shown to be governed by a nonlinear amplitude equation, which is an extension of the well-known Benjamin–Davis–Ono equation. Numerical solutions indicate that as a spatially isolated disturbance evolves, it radiates a beam of long gravity waves, and meanwhile small-scale ripples develop on its front to form a well-defined wavepacket. It is also shown that for jet-like velocity profiles, the standard triple-deck theory must be adjusted to account for both the displacement and transverse pressure variation induced by the inviscid flow in the main layer. The nonlinear evolution of high-frequency disturbances is governed by a mixed KdV–Benjamin–Davis–Ono equation.
The aim of this paper is to show that the viscous shear instability identified in Part 1 is intrinsically coupled with internal gravity waves when a localized surface topography is present within a boundary layer. The coupling involves two aspects: receptivity and radiation. The former refers to excitation of shear instability modes by gravity waves, and the latter to emission of gravity waves by instability modes. Both physical processes are studied using triple-deck theory. In particular, the radiated gravity waves are found to produce a leading-order back action on the source, and this feedback effect, completely ignored in the acoustic analogy type of approach, is naturally taken into account by the triple-deck formalism. A by-product is that for certain incident angles, gravity waves are over-reflected by the boundary layer, i.e. the reflected waves are stronger than the incident waves.
The phenomenon of backflow in the capillary wave region of laminar falling liquid films is studied in detail. For the first time, the mechanism leading to the origination of the phenomenon is identified and explained. It is shown that backflow forms as the result of a separation eddy developing at the bounding wall similar to the case of classical flow separation. Results show that the adverse pressure distribution causing the separation of the flow in the capillary wave region is induced by the strong third-order deformation (i.e. change in curvature) of the liquid–gas free surface there. This deformation acts on the interfacial pressure jump, and thereby the wall pressure distribution, as a result of surface tension forces. It is shown that only the capillary waves, owing to their short wavelength and large curvature, impose a pressure distribution satisfying the conditions for flow separation. The effect of this capillary separation eddy on momentum and heat transfer is investigated from the perspective of modelling approaches for falling liquid films. The study is centred on a single case of inclined liquid film flow in the visco-capillary regime with surface waves externally excited at a single forcing frequency. Investigations are based on temporally and spatially highly resolved numerical data obtained by solving the Navier–Stokes equations for both phases. Computation of phase distribution is performed with the volume of fluid method and the effect of surface tension is modelled using the continuum surface force approach. Numerical data are compared with experimental data measured in the developed region of the flow. Laser-Doppler velocimetry is used to measure the temporal distribution of the local streamwise velocity component, and confocal chromatic imaging is employed to measure the temporal distribution of film thickness. Excellent agreement is obtained with respect to film thickness and reasonable agreement with respect to velocity.
The linear stability of an infinite fluid layer with a deformable free surface covered by an insoluble surfactant and bounded below by a horizontal rigid plate oscillating in its own plane is studied based on the Floquet theory. The differential system governing the stability problem for perturbations of arbitrary wavenumbers is solved numerically by a Chebyshev collocation method. Stability boundaries are obtained in a wide range of amplitude and frequency of the modulation as well as surfactant elasticity. Results show that the presence of the surfactant may significantly stabilize (destabilize) the flow by raising (lowering) the critical Reynolds number associated with the onset of instability. The effect of the surfactant plays a stabilizing role for small surfactant elasticity and a destabilizing one for relatively large surfactant elasticity. The destabilizing effect of the surfactant on the stability of flows with a zero-shear surface is found for the first time. The disturbance modes in the form of travelling waves may be induced by the surfactant and dominate the instability of the flow.
The stability and transition of flow past a pair of circular cylinders in a side-by-side arrangement are investigated by numerical simulations and linear stability analyses. Various flow patterns around the cylinders have been reported to appear due to an instability of the steady symmetric flow that is realized at small Reynolds numbers, among which deflected oscillatory flow is particularly noticeable. The physical origin of the flow is explored by bifurcation analyses of the numerical data. We found that the deflected oscillatory flow arises from the steady symmetric flow through sequential instabilities due to stationary and oscillatory unstable modes. Steady asymmetric flow with respect to the streamwise centreline between the two cylinders was also found to be induced by the instability due to a stationary mode in a very narrow range of the gap width between the two cylinders. We classify the instability modes of the steady symmetric flow into four groups in the parameter space of the gap width, and evaluate the critical Reynolds number for each mode of instability.