The phase distribution of a bimodal distribution of negatively buoyant particles in a low-Reynolds-number pressure-driven flow of a suspension in a horizontal pipe is measured using multi-frequency electrical impedance tomography (EIT). Suspensions of heavy silver-coated particles and slightly heavy PMMA particles exhibit different effective conductivities depending on the frequency of an applied electrical current. This difference allows the separate imaging of the phase distribution of each particle type and the composite suspension. At low flow rates the dense particles tend to distribute in the lower half of the pipe The particles are resuspended toward the centre as the flow rate is increased. The slightly heavy particles tend to accumulate closer to the centre of the pipe. The presence of the nearly neutrally buoyant particles enhances the resuspension of the heavy particles compared to that of a suspension of heavy particles alone at the same volume fraction. A suspension balance model is used to theoretically predict the distribution of particles in the flow assuming an ideal mixing rule for the particle partial pressures. The agreement between the predictions and the experimental observations is qualitatively correct and quantitatively fair.

The role of non-normality and nonlinearity in flame–acoustic interaction in a ducted diffusion flame is investigated in this paper. The infinite rate chemistry model is employed to study unsteady diffusion flames in a Burke–Schumann type geometry. It has been observed that even in this simplified case, the combustion response to perturbations of velocity is non-normal and nonlinear. This flame model is then coupled with a linear model of the duct acoustic field to study the temporal evolution of acoustic perturbations. The one-dimensional acoustic field is simulated in the time domain using the Galerkin technique, treating the fluctuating heat release from the combustion zone as a compact acoustic source. It is shown that the coupled combustion–acoustic system is non-normal and nonlinear. Further, calculations showed the occurrence of triggering; i.e. the thermoacoustic oscillations decay for some initial conditions whereas they grow for some other initial conditions. It is shown that triggering occurs because of the combined effect of non-normality and nonlinearity. For such a non-normal system, resonance or ‘pseudoresonance’ may occur at frequencies far from its natural frequencies. Non-normal systems can be studied using pseudospectra, as eigenvalues alone are not sufficient to predict the behaviour of the system. Further, both necessary and sufficient conditions for the stability of a thermoacoustic system are presented in this paper.

We demonstrate that data from direct numerical simulation of turbulent boundary layers at Mach 3 exhibit the same large-scale coherent structures that are found in supersonic and subsonic experiments, namely elongated, low-speed features in the logarithmic region and hairpin vortex packets. Contour plots of the streamwise mass flux show very long low-momentum structures in the logarithmic layer. These low-momentum features carry about one-third of the turbulent kinetic energy. Using Taylor's hypothesis, we find that these structures prevail and meander for very long streamwise distances. Structure lengths on the order of 100 boundary layer thicknesses are observed. Length scales obtained from correlations of the streamwise mass flux severely underpredict the extent of these structures, most likely because of their significant meandering in the spanwise direction. A hairpin-packet-finding algorithm is employed to determine the average packet properties, and we find that the Mach 3 packets are similar to those observed at subsonic conditions. A connection between the wall shear stress and hairpin packets is observed. Visualization of the instantaneous turbulence structure shows that groups of hairpin packets are frequently located above the long low-momentum structures. This finding is consistent with the very large-scale motion model of Kim & Adrian (1999).

Direct numerical simulation data of a Mach 2.9, 24○ compression ramp configuration are used to analyse the shock motion. The motion can be observed from the animated DNS data available with the online version of the paper and from wall-pressure and mass-flux signals measured in the free stream. The characteristic low frequency is in the range of (0.007–0.013) U∞/δ, as found previously. The shock motion also exhibits high-frequency, of O(U∞/δ), small-amplitude spanwise wrinkling, which is mainly caused by the spanwise non-uniformity of turbulent structures in the incoming boundary layer. In studying the low-frequency streamwise oscillation, conditional statistics show that there is no significant difference in the properties of the incoming boundary layer when the shock location is upstream or downstream. The spanwise-mean separation point also undergoes a low-frequency motion and is found to be highly correlated with the shock motion. A small correlation is found between the low-momentum structures in the incoming boundary layer and the separation point. Correlations among the spanwise-mean separation point, reattachment point and the shock location indicate that the low-frequency shock unsteadiness is influenced by the downstream flow. Movies are available with the online version of the paper.

The morphology and time-dependent integral properties of the multifluid compressible flow resulting from the shock–bubble interaction in a gas environment are investigated using a series of three-dimensional multifluid-Eulerian simulations. The bubble consists of a spherical gas volume of radius 2.54 cm (128 grid points), which is accelerated by a planar shock wave. Fourteen scenarios are considered: four gas pairings, including Atwood numbers −0.8 < A < 0.7, and shock strengths 1.1 < M ≤ 5.0. The data are queried at closely spaced time intervals to obtain the time-dependent volumetric compression, mean bubble fluid velocity, circulation and extent of mixing in the shocked-bubble flow. Scaling arguments based on various properties computed from one-dimensional gasdynamics are found to collapse the trends in these quantities successfully for fixed A. However, complex changes in the shock-wave refraction pattern introduce effects that do not scale across differing gas pairings, and for some scenarios with A > 0.2, three-dimensional (non-axisymmetric) effects become particularly significant in the total enstrophy at late times. A new model for the total velocity circulation is proposed, also based on properties derived from one-dimensional gasdynamics, which compares favourably with circulation data obtained from calculations, relative to existing models. The action of nonlinear-acoustic effects and primary and secondary vorticity production is depicted in sequenced visualizations of the density and vorticity fields, which indicate the significance of both secondary vorticity generation and turbulent effects, particularly for M > 2 and A > 0.2. Movies are available with the online version of the paper.

The nonlinear development of the interfacial-surfactant instability is studied for the semi-infinite plane Couette film flow. Disturbances whose spatial period is close to the marginal wavelength of the long-wave instability are considered first. Appropriate weakly nonlinear partial differential equations (PDEs) which couple the disturbances of the film thickness and the surfactant concentration are obtained from the strongly nonlinear lubrication-approximation PDEs. In a rescaled form each of the two systems of PDEs is controlled by a single parameter C, the ‘shear-Marangoni number’. From the weakly nonlinear PDEs, a single Stuart–Landau ordinary differential equation (ODE) for an amplitude describing the unstable fundamental mode is derived. By comparing the solutions of the Stuart–Landau equation with numerical simulations of the underlying weakly and strongly nonlinear PDEs, it is verified that the Stuart–Landau equation closely approximates the small-amplitude saturation to travelling waves, and that the error of the approximation converges to zero at the marginal stability curve. In contrast to all previous stability work on flows that combine interfacial shear and surfactant, some analytical nonlinear results are obtained. The Hopf bifurcation to travelling waves is supercritical for C < Cs and subcritical for C > Cs, where Cs is approximately 0.29. This is confirmed with a numerical continuation and bifurcation technique for ODEs. For the subcritical cases, there are two values of equilibrium amplitude for a range of C near Cs, but the travelling wave with the smaller amplitude is unstable as a periodic orbit of the associated dynamical system (whose independent variable is the spatial coordinate). By using the Bloch (‘Floquet’) disturbance modes in the linearized PDEs, it transpires that all the small-amplitude travelling-wave equilibria are unstable to sufficiently long-wave disturbances. This theoretical result is confirmed by numerical simulations which invariably show the large-amplitude saturation of the disturbances. In view of this secondary instability, the existence of small-amplitude periodic solutions (on the real line) bifurcating from the uniform flow at the marginal values of the shear-Marangoni number does not contradict the earlier conclusions that the interfacial-surfactant instability has a strongly nonlinear character, in the sense that there are no small-amplitude attractors such that the entire evolution towards them is captured by weakly-nonlinear equations. This suggests that, in general, for flowing-film instabilities that have zero wavenumber at criticality, the saturated disturbance amplitudes do not always have to decrease to zero as the control parameter approaches its value at criticality.

A three-layer intrusion flow is considered, in which all three layers are in motion, with different speeds, relative to the observer. Shear is present in the middle layer, and the lowest fluid may even move oppositely to the upper two (so giving an exchange flow). Two thin interfaces are present, above and below the moving middle layer. A linearized analysis is presented for small wave amplitudes. Nonlinear periodic solutions are then obtained using a Fourier technique, and reveal a range of nonlinear phenomena, including limiting waves, multiple solutions and resonances.

The gas transfer process across the air–water interface in a turbulent flow environment, with the turbulence generated in the water phase far away from the surface, was experimentally investigated for varying turbulent Reynolds numbers ReT ranging between 260 and 780. The experiments were performed in a grid-stirred tank using a combined particle image velocimetry – laser induced fluorescence (PIV-LIF) technique, which enables synoptic measurements of two-dimensional velocity and dissolved gas concentration fields. The visualization of the velocity and concentration fields provided direct insight into the gas transfer mechanisms. The high data resolution allowed detailed quantification of the gas concentration distribution (i.e. mean and turbulent fluctuation characteristics) within the thin aqueous boundary layer as well as revealing the near-surface hydrodynamics. The normalized concentration profiles show that as ReT increases, the rate of concentration decay into the bulk becomes slower. Independent benchmark data for the transfer velocity KL were obtained and their normalized values (KLSc0.5/uHT) depend on ReT with exponent −0.25. The spectra of the covariance term c′ w′ indicate that the contribution of c′ w′ is larger in the lower-frequency region for cases with small ReT, whereas for the other cases with higher ReT, the contribution of c′ w′ appears to be larger in the higher-frequency region (small eddies). These interrelated facts suggest that the gas transfer process is controlled by a spectrum of different eddy sizes and the gas transfer at different turbulence levels can be associated with certain dominant eddy sizes. The normalized mean turbulent flux profiles increase from around 0 at the interface to about 1 within a depth of approximately 2δe, where δe is the thickness of the gas boundary layer. The measured turbulent flux (c′ w′) is of the same order as the total flux (j), which shows that the contribution of c′ w′ to the total flux is significant.

The influence of wind on extreme wave events in deep water is investigated experimentally and numerically. A series of experiments conducted in the Large Air–Sea Interactions Facility (LASIF-Marseille, France) shows that wind blowing over a short wave group due to the dispersive focusing of a longer frequency-modulated wavetrain (chirped wave packet) may increase the time duration of the extreme wave event by delaying the defocusing stage. A detailed analysis of the experimental results suggests that extreme wave events may be sustained longer by the air flow separation occurring on the leeward side of the steep crests. Furthermore it is found that the frequency downshifting observed during the formation of the extreme wave event is more important when the wind velocity is larger. These experiments have pointed out that the transfer of momentum and energy is strongly increased during extreme wave events.

Two series of numerical simulations have been performed using a pressure distribution over the steep crests given by the Jeffreys sheltering theory. The first series corresponding to the dispersive focusing confirms the experimental results. The second series which corresponds to extreme wave events due to modulational instability, shows that wind sustains steep waves which then evolve into breaking waves. Furthermore, it was shown numerically that during extreme wave events the wind-driven current could play a significant role in their persistence.

The linear instability of a barotropic flow with uniform horizontal shear in a stratified rotating fluid is investigated with respect to perturbations invariant in the alongflow direction. The flow can be inertially unstable if there is sufficiently strong anticyclonic shear, but only for sufficiently high Reynolds numbers Re. We determine the critical Reynolds numbers required for amplification of the instability as a function of Prandtl number, strength of the stratification and magnitude of the shear. The vertical scales at the onset of the instability are calculated. For Prandtl number P < 1.44 instability always sets in through stationary overturning motions, for P > 1.44 it may also commence through overstable (oscillatory) motions. For Re exceeding the critical value, we determine the vertical scale of the most rapidly amplifying modes and the corresponding growth rates and how they vary with Re, P, the shear and the strength of stratification.

Reliable information on forces on a finite-sized particle in a turbulent boundary layer is lacking, so workers continue to use standard drag and lift correlations developed for a laminar flow to predict drag and lift forces. Here we consider direct numerical simulations of a turbulent channel flow over an isolated particle of finite size. The size of the particle and its location within the turbulent channel are systematically varied. All relevant length and time scales of turbulence, attached boundary layers on the particle, and particle wake are faithfully resolved, and thus we consider fully resolved direct numerical simulations. The results from the direct numerical simulation are compared with corresponding predictions based on the standard drag relation with and without the inclusion of added-mass and shear-induced lift forces. The influence of turbulent structures, such as streaks, quasi-streamwise vortices and hairpin packets, on particle force is explored. The effect of vortex shedding is also observed to be important for larger particles, whose Re exceeds a threshold.

The influence of surfactant on the breakup of a prestretched bubble in a quiescent viscous surrounding is studied by a combination of direct numerical simulation and the solution of a long-wave asymptotic model. The direct numerical simulations describe the evolution toward breakup of an inviscid bubble, while the effects of small but non-zero interior viscosity are readily included in the long-wave model for a fluid thread in the Stokes flow limit.

The direct numerical simulations use a specific but realizable and representative initial bubble shape to compare the evolution toward breakup of a clean or surfactant-free bubble and a bubble that is coated with insoluble surfactant. A distinguishing feature of the evolution in the presence of surfactant is the interruption of bubble breakup by formation of a slender quasi-steady thread of the interior fluid. This forms because the decrease in surface area causes a decrease in the surface tension and capillary pressure, until at a small but non-zero radius, equilibrium occurs between the capillary pressure and interior fluid pressure.

The long-wave asymptotic model, for a thread with periodic boundary conditions, explains the principal mechanism of the slender thread's formation and confirms, for example, the relatively minor role played by the Marangoni stress. The large-time evolution of the slender thread and the precise location of its breakup are, however, influenced by effects such as the Marangoni stress and surface diffusion of surfactant.

The influence of different wing kinematic models on the aerodynamic performance of a hovering insect is investigated by means of two-dimensional time-dependent Navier–Stokes simulations. For this, simplified models are compared with averaged representations of the hovering fruit fly wing kinematics. With increasing complexity, a harmonic model, a Robofly model and two more-realistic fruit fly models are considered, all dynamically scaled at Re = 110. To facilitate the comparison, the parameters of the models were selected such that their mean quasi-steady lift coefficients were matched. Details of the vortex dynamics, as well as the resulting lift and drag forces, were studied.

The simulation results reveal that the fruit fly wing kinematics result in forces that differ significantly from those resulting from the simplified wing kinematic models. In addition, light is shed on the effect of different characteristic features of the insect wing motion. The angle of attack variation used by fruit flies increases aerodynamic performance, whereas the deviation is probably used for levelling the forces over the cycle.

This paper examines intrusive Boussinesq gravity currents, propagating into a continuously stratified fluid. We develop a model, based on energy arguments, to predict the front speed of such an intrusive gravity current from a lock release. We find that the depth at which the intrusion occurs, which corresponds to the level of neutral buoyancy (i.e. the depth where the intrusion density equals the stratified fluid density), affects the front speed. The maximum speeds occur when the intrusion travels along the top and bottom boundaries and the minimum speed occurs at mid-depth. Experiments and numerical simulations were conducted to compare to the theoretically predicted values, and good agreement was found.

We examine the effect of local thermal non-equilibrium on the infiltration of a hot fluid into a cold porous medium. The temperature fields of the solid porous matrix and the saturating fluid are governed by separate, but coupled, parabolic equations, forming a system governed by three dimensionless parameters. A scale analysis and numerical simulations are performed to determine the different manners in which the temperature fields evolve in time. These are supplemented by a large-time analysis showing that local thermal equilibrium between the phases is eventually attained. It is found that the thickness of the advancing thermal front is a function of the governing parameters rather than being independent of them. This has the implication that local thermal equilibrium is not equivalent to a single equation formulation of the energy equation as might have been expected. When the velocity of the infiltrating fluid is sufficiently large, the equations reduce to a hyperbolic system and a thermal shock wave is formed within the fluid phase. The strength of the shock decays exponentially with time, but the approach to local thermal equilibrium is slower and is achieved algebraically in time.

Three different approaches were used in the present study to predict the influence of roughness on laminar flow in microchannels. Experimental investigations were conducted with rough microchannels 100 to 300μm in height (H). The pressure drop was measured in test-sections prepared with well-controlled wall roughness (periodically distributed blocks, relative roughness k* =k/0.5H≈0.15) and in test-sections with randomly distributed particles anchored on the channel walls (k* ≈0.04–0.13). Three-dimensional numerical simulations were conducted with the same geometry as in the test-section with periodical roughness (wavelength L). A one-dimensional model (RLM model) was also developed on the basis of a discrete-element approach and the volume-averaging technique. The numerical simulations, the rough layer model and the experiments agree to show that the Poiseuille number Po increases with the relative roughness and is independent of Re in the laminar regime (Re<2000). The increase in Po observed during the experiments is predicted well both by the three-dimensional simulations and the rough layer model. The RLM model shows that the roughness effect may be interpreted by using an effective roughness height keff. keff/k depends on two dimensionless local parameters: the porosity at the bottom wall; and the roughness height normalized with the distance between the rough elements. The RLM model shows that keff/k is independent of the relative roughness k* at given k/L and may be simply approximated by the law: keff/k = 1 − (c(ϵ)/2π)(L/k) for keff/k>0.2, where c decreases with the porosity ϵ.

The stack-driven flow between two interconnected rooms produced by a single heat source is studied. In particular, the features of the transient flow for different positions and areas of two openings in the shared vertical wall are analysed. An analytical model provides the time evolution of the stratified flows in rooms of any size. The concept of an equivalent layer representing a non-uniform density profile, which is useful in other contexts, is included in the theoretical approach and provides physical insight and aids the mathematical solution of the problem. New salt-bath experiments are performed to simulate the thermal forcing between the rooms, to validate the model and to analyse the mixing generated and the effects of a source of volume in the configuration studied.

The results of numerical simulations of convection in a rotating layer are used to compute the α-effect of mean-field electrodynamics. The computations are carried out for different system sizes. It is found that the outcomes can depend critically on the system size, and that physically meaningful results can only be obtained if the system size is large compared with the typical eddy size.

Results are presented for a numerical simulation of vortex-induced vibrations of a circular cylinder of low non-dimensional mass (m* = 10) in the laminar flow regime (60 < Re < 200). The natural structural frequency of the oscillator, fN, matches the vortex shedding frequency for a stationary cylinder at Re = 100. This corresponds to fND2/ν = 16.6, where D is the diameter of the cylinder and ν the coefficient of viscosity of the fluid. A stabilized space–time finite element formulation is utilized to solve the incompressible flow equations in primitive variables form in two dimensions. Unlike at high Re, where the cylinder response is known to be associated with three branches, at low Re only two branches are identified: ‘initial’ and ‘lower’. For a blockage of 2.5% and less the onset of synchronization, in the lower Re range, is accompanied by an intermittent switching between two modes with vortex shedding occurring at different frequencies. With higher blockage the jump from the initial to lower branch is hysteretic. Results from free vibrations are compared to the data from experiments for forced vibrations reported earlier. Excellent agreement is observed for the critical amplitude required for the onset of synchronization. The comparison brings out the possibility of hysteresis in forced vibrations. The phase difference between the lift force and transverse displacement shows a jump of almost 180° at, approximately, the middle of the synchronization region. This jump is not hysteretic and it is not associated with any radical change in the vortex shedding pattern. Instead, it is caused by changes in the location and value of the maximum suction on the lower and upper surface of the cylinder. This is observed clearly by comparing the time-averaged flow for a vibrating cylinder for different Re. While the mean flow for Re beyond the phase jump is similar to that for a stationary cylinder, it is associated with a pair of counter-rotating vortices in the near wake for Re prior to the phase jump. The phase jump appears to be one of the mechanisms of the oscillator to self-limit its vibration amplitude.

Wave-activity conservation laws are key to understanding wave propagation in inhomogeneous environments. Their most general formulation follows from the Hamiltonian structure of geophysical fluid dynamics. For large-scale atmospheric dynamics, the Eliassen–Palm wave activity is a well-known example and is central to theoretical analysis. On the mesoscale, while such conservation laws have been worked out in two dimensions, their application to a horizontally homogeneous background flow in three dimensions fails because of a degeneracy created by the absence of a background potential vorticity gradient. Earlier three-dimensional results based on linear WKB theory considered only Doppler-shifted gravity waves, not waves in a stratified shear flow. Consideration of a background flow depending only on altitude is motivated by the parameterization of subgrid-scales in climate models where there is an imposed separation of horizontal length and time scales, but vertical coupling within each column. Here we show how this degeneracy can be overcome and wave-activity conservation laws derived for three-dimensional disturbances to a horizontally homogeneous background flow. Explicit expressions for pseudoenergy and pseudomomentum in the anelastic and Boussinesq models are derived, and it is shown how the previously derived relations for the two-dimensional problem can be treated as a limiting case of the three-dimensional problem. The results also generalize earlier three-dimensional results in that there is no slowly varying WKB-type requirement on the background flow, and the results are extendable to finite amplitude. The relationship between pseudoenergy and pseudomomentum , where c is the horizontal phase speed in the direction of symmetry associated with , has important applications to gravity-wave parameterization and provides a generalized statement of the first Eliassen–Palm theorem.