This article investigates the appearance of instabilities in two planar coflowing fluid sheets with different densities and viscosities via experiments, numerical simulation and linear stability analysis. At low dynamic pressure ratios a convective instability is shown to appear for which the frequency of the waves in the primary atomization region is influenced by both liquid and gas velocities. For large dynamic pressure ratios an asymptotic regime is obtained in which frequency is solely controlled by gas velocity and the instability becomes absolute. The transition from convective to absolute is shown to be influenced by the velocity defect induced by the presence of the separator plate. We show that in this regime the splitter plate thickness can also affect the nature of the instability if it is larger than the gas vorticity thickness. Computational and experimental results are in agreement with the predictions of a spatio-temporal stability analysis.

We used optical and X-ray imaging to observe the formation of jets from the impact of a single drop with a deep layer of the same liquid. For high Reynolds number there are two distinct jets: the thin, fast and early-emerging ejecta; and the slow, thick and late-emerging lamella. For low Reynolds number the two jets merge into a single continuous jet, the structure of which is determined by the distinct contributions of the lamella and the ejecta. We measured the emergence time, position and speed of the ejecta sheet, and find that these scale as power laws with the impact speed and the viscosity. We identified the origin of secondary droplets with the breakup of the lamella and the ejecta jets, and show that the size of the droplets is not a good indicator of their origin.

Experiments are described to study the motion and deformation of a synthetic, liquid-filled capsule that is freely suspended in hyperbolic extensional flow. The capsule is a composite particle consisting of a viscous liquid drop surrounded by a thin polymeric membrane. The method used to fabricate capsules suitable for macroscopic flow experiments is described. The deformation of the capsule is measured as a function of strain rate for an extensional flow generated in a four-roll mill. The data agree well with results from small-deformation theory developed by Barthes-Biesel and co-workers, provided the deformation of the capsule is not too large. Using the theory to correlate the experimental data produces an estimate for the elastic modulus of the membrane that agrees reasonably well with the elastic modulus obtained by an independent technique. However, for sufficiently large strain rates, the membrane exhibits strain hardening and a permanent change in its structure, both of which are reflected in the shape of the capsule.

The inertial migration of a small rigid spherical particle, suspended in a fluid flowing between two plane boundaries, is investigated theoretically to find the effect on the lateral motion. The channel Reynolds number is of order unity and thus both boundary-induced and Oseen-like inertial migration effects are important. The particle Reynolds number is small but non-zero, and singular perturbation techniques are used to calculate the component of the migration velocity which is directed perpendicular to the boundaries of the channel. The particle is non-neutrally buoyant and thus its buoyancy-induced motion may be either parallel or perpendicular to the channel boundaries, depending on the channel alignment. When the buoyancy results in motion perpendicular to the channel boundaries, the inertial migration is a first-order correction to the magnitude of this lateral motion, which significantly increases near to the boundaries. When the buoyancy produces motion parallel with the channel boundaries, the inertial migration gives the zeroth-order lateral motion either towards or away from the boundaries. It is found that those particles which have a velocity exceeding the undisturbed shear flow will migrate towards the boundaries, whereas those with velocities less than the undisturbed flow migrate towards the channel centreline. This calculation is of practical importance for various chemical engineering devices in which particles must be filtered or separated. It is useful to calculate the forces on a particle moving near to a boundary, through a shear flow. This study may also explain certain migration effects of bubbles and crystals suspended in molten rock flow flowing through volcanic conduits.

The paper presents a linear stability analysis of cohesionless granular materials during rapid shear flow. The analysis is based on the governing equations developed in the kinetic theory of Lun et al. (1984) for granular flows of smooth, nearly elastic, uniform spherical particles. The primary flow is taken to be a uniform, simple shear flow and the effects of small perturbations in velocity components, granular temperature and solids fraction are considered. The inelasticity of the particles is characterized by a constant coefficient of restitution which is assumed to be close to unity. Some permissible solutions are sinusoidal plane waves in which the wavenumber vector is continuously turned by the mean shear flow and its magnitude varied as time proceeds. The initial growth (or decay) rates for these perturbations are sought. The resulting linearized equations for the flow perturbations turn out to be exceedingly long and complex; they are determined by the use of computer algebra. It is found that, in general, long wavelengths are the most unstable and that short wavelengths are dampened by ‘viscous action’. ‘Instability’ increases with decreasing coefficient of restitution. Numerical results for initial growth rates were obtained for several values of mean solids fraction and particle coefficient of restitution. Flows tend to be more stable at both high and very low concentrations than at moderate concentrations. These results appear to be consistent in the main with recent computer simulations of granular flows of disk-like particles by Hopkins & Louge (1991).

The relaxation of a dewetting contact line is investigated theoretically in the so-called ‘Landau–Levich’ geometry in which a vertical solid plate is withdrawn from a bath of partially wetting liquid. The study is performed in the framework of lubrication theory, in which the hydrodynamics is resolved at all length scales (from molecular to macroscopic). We investigate the bifurcation diagram for unperturbed contact lines, which turns out to be more complex than expected from simplified ‘quasi-static’ theories based upon an apparent contact angle. Linear stability analysis reveals that below the critical capillary number of entrainment, Cac, the contact line is linearly stable at all wavenumbers. Away from the critical point, the dispersion relation has an asymptotic behaviour σ∝|q| and compares well to a quasi-static approach. Approaching Cac, however, a different mechanism takes over and the dispersion evolves from ∼|q| to the more common ∼q2. These findings imply that contact lines cannot be described using a universal relation between speed and apparent contact angle, but viscous effects have to be treated explicitly.

The behaviour of non-wetting ganglia undergoing immiscible displacement in a porous medium is studied with the help of a theoretical simulator. The porous medium is represented by a network of randomly sized unit cells of the constricted-tube type. The fluid of a non-wetting ganglion is in contact with the wetting fluid at menisci which are assumed to be spherical cups. The flow in every constricted unit cell occupied by a single fluid is modelled as flow in a sinusoidal tube. The flow in every unit cell that contains a meniscus and portions of both fluids is treated with a combination of a Washburn-type analysis and a lubrication-theory approximation. The flow problem is thus reduced to a system of linear equations the solution of which gives the instantaneous pressures on the nodes, the flowrates through the unit cells, and the velocities of the menisci. The motion of a ganglion is determined by assuming quasi-static flow, taking a small time increment, updating the positions of the menisci, and iterating. The behaviour of solitary ganglia is studied under conditions of quasi-static displacement (Ca slightly larger than critical), as well as dynamic displacement (Ca substantially larger than critical). Shape evolution, rate of flow, mode of break-up, and stranding are examined. The stranding and break-up coefficients are determined as functions of the capillary number and the ganglion size for a 100 × 200 sandpack. The dependence of the average ganglion velocity on ganglion size, capillary number, viscosity ratio and dynamic contact angle is examined for the simple case of motion between straight rows of spheres. It is found, among other things, that when μo < μw the velocity of ganglia can be substantially larger than that of the displacing fluid.

A uniformly valid asymptotic solution for large Reynolds number is constructed for plane steady laminar flow of a liquid into the channel between two semi-infinite parallel plates. The entry condition is taken as either that for a cascade of plates in a uniform oncoming stream, or uniform flow directly at the inlet. A paradox in the standard solution of Schlichting—that near the inlet the flow due to displacement would not be the accelerated uniform core on which his expansion is based—is resolved by showing that his series for small as well as large distance actually applies only to conditions far downstream, and matches with another expansion valid near the inlet. Good agreement is found with three independent numerical solutions of the full Navier-Stokes equations, except for a discrepancy in one solution for uniform entry that is traced to erroneous neglect of inlet vorticity.

Three-dimensional highly resolved simulations are presented for cylindrical density currents using the Boussinesq approximation for small density difference. Three Reynolds numbers (Re) are investigated (895, 3450 and 8950, which correspond to values of the Grashof number of 105, 1.5 × 106 and 107, respectively) in order to identify differences in the flow structure and dynamics. The simulations are performed using a fully de-aliased pseudospectral code that captures the complete range of time and length scales of the flow. The simulated flows present the main features observed in experiments at large Re. As the current develops, it transitions through different phases of spreading, namely acceleration, slumping, inertial and viscous Soon after release the interface between light and heavy fluids rolls up forming Kelvin–Helmholtz vortices. The formation of the first vortex sets the transition between acceleration and slumping phases. Vortex formation continues only during the slumping phase and the formation of the last Kelvin–Helmholtz vortex signals the departure from the slumping phase. The coherent Kelvin–Helmholtz vortices undergo azimuthal instabilities and eventually break up into small-scale turbulence. In the case of planar currents this turbulent region extends over the entire body of the current, while in the cylindrical case it only extends to the regions of Kelvin–Helmholtz vortex breakup. The flow develops three-dimensionality right from the beginning with incipient lobes and clefts forming at the lower frontal region. These instabilities grow in size and extend to the upper part of the front. Lobes and clefts continuously merge and split and result in a complex pattern that evolves very dynamically. The wavelength of the lobes grows as the flow spreads, while the local Re of the flow decreases. However, the number of lobes is maintained over time. Owing to the high resolution of the simulations, we have been able to link the lobe and cleft structure to local flow patterns and vortical structures. In the near-front region and body of the current several hairpin vortices populate the flow. Laboratory experiments have been performed at the higher Re and compared to the simulation results showing good agreement. Movies are available with the online version of the paper.

The wake transition of the flow around two circular cylinders placed in staggered arrangements with fixed streamwise separation of 5D and cross-stream separation varying from 0 to 3D has been studied. The wake transition is compared to that of a single isolated cylinder. Linear stability analysis utilizing Floquet theory and direct numerical simulations using a spectral/hp element spatial discretization were carried out. The unstable modes that first appear in the wake transition of the flow around a single cylinder, which are the long-spanwise-wavelength mode A and the short-spanwise-wavelength mode B, are also found in the flow around the staggered arrangements. However, a third mode, referred to as mode C, is also present in the wake transition of the flow around staggered arrangements, depending on the relative positioning of the cylinders. This mode has an intermediate spanwise wavelength and period-doubling character. The structure and onset characteristics of mode C are analysed and the nonlinear character of the bifurcation for this mode is investigated.

The starting flows past a two-dimensional NACA 0012 airfoil translating and oscillating at large incidences are investigated by visualization experiments and numerical calculations. The airfoil model is set in motion impulsively and subjected simultaneously to a constant translation and harmonic oscillation in pitch. The evolution of the vortex wake is followed in a sequence of streamline visualizations and the wake pattern generated is analysed. The parameters varied in the visualization experiment are the Reynolds number ranging from 1500 to 10000, the reduced frequency from 0.1 to 1.0, the mean incidence 30° or 15° and the angular amplitude 15° or 7°. There are also two additional parameters of special interest: the airfoil cross-section and the pitching axis. The effects of these parameters are discussed in relation to the resultant wake patterns. Some comparison is made with the results of earlier experiments.

The scaling of the scour depth of equilibrium at the base of a solid cylinder immersed within an erodible granular bed and impinged by a turbulent shear flow is investigated here, for the first time, by means of the phenomenological theory of turbulence. The proposed theory allows the derivation of a predictive formula that (i) includes all the relevant non-dimensional parameters controlling the process, and (ii) contrary to commonly employed empirical formulae, is free from scale issues. Theoretical predictions agree very well with experimental data, shed light on unresolved issues on the physics of the problem, and clarify the effects of various dimensionless parameters controlling the scouring process.

The late stages of transition, from the Λ-vortex stage up to turbulence, are investigated by postprocessing data from a direct numerical simulation of the complete K-type transition process in plane channel flow at a Reynolds number of 5000 (based on channel half-width and laminar centreline velocity). The deterministic roll-up of the high-shear layer that forms around the Λ-vortices is examined in detail. The new vortices arising from this process are visualized by plotting three-dimensional surfaces of constant pressure. Five vortices are observed, with one of these developing into a strong hairpin-shaped vortex. Interactions between the different vortices, and between the two channel halves, are found to be important. In the very last stage of transition second-generation shear layers are observed to form and roll up into new vortices. It is postulated that at this stage a sustainable mechanism of wall-bounded turbulence exists in an elementary form. The features which are locally present include high wall shear, sublayer streaks, ejections and sweeps. Large-scale energetic vortices are found to be an important part of the mechanism by which the turbulence spreads to other spanwise positions. The generality of the findings are discussed with reference to data from simulations of H-type and mixed-type transition.

Recently a mathematical model was proposed (Sutera, Maeder & Kestin 1963) to demonstrate that vorticity amplification by stretching was an important mechanism underlying the sensitivity of stagnation-point heat transfer on cylinders to free-stream turbulence. According to the model, vorticity of a scale larger than a certain neutral scale and appropriately oriented can undergo amplification as it is convected towards the boundary layer. Such vorticity, present in the oncoming flow with small intensity, can reach the boundary layer with a greatly magnified intensity and induce substantial three-dimensional effects therein. The mean temperature profile was shown to be much more responsive to these effects than the mean velocity profile and very large increases in the wall-heat-transfer rate were calculated for Prandtl numbers 0·74 and 7·0.

In this work a second, more general, case is treated in which the approaching flow carries vorticity of scale 1·5 times the neutral. By means of iterative procedures applied on an electronic analogue computer, an approximate solution to the full Navier–Stokes equation is generated. The heat-transfer problem is solved simultaneously for Pr = 0·70, 7·0 and 100. It is found that a vorticity input which increases the wall-shear rate by less than 3% is capable of increasing the wall-heat-transfer rate by as much as 40%. The sensitivity of the thermal boundary layer depends on Prandtl number. In the three cases investigated it is greatest for Pr = 7·0 and least for Pr = 100.

A jet of dyed water is directed into a large tank of water. As the flow rate is increased several successive modes of breakdown of the dyed column are observed:

1. tiny puffs of dye near the nozzle;

2. axisymmetric condensations well away from the nozzle;

3. sinuous undulations of long wave length far from the nozzle;

4. formation of foot-shaped pockets of dye;

5. confused breakup near the nozzle with rapid spread of dye.

Superhydrophobic surfaces can entrap gas pockets within their grooves when submerged in water. Such a mixed-phase boundary is shown to result in an effective slip velocity on the surface, and has promising potential for drag reduction and energy-saving in hydrodynamic applications. The target flow regime, in most such applications, is a turbulent flow. Previous analyses of this problem involved direct numerical simulations of turbulence with the superhydrophobic surface modelled as a flat boundary, but with a heterogeneous mix of slip and no-slip boundary conditions corresponding to the surface texture. Analysis of the kinematic data from these simulations has helped to establish the magnitude of drag reduction for various texture topologies. The present work is the first investigation that, alongside a kinematic investigation, addresses the robustness of superhydrophobic surfaces by studying the load fields obtain from data from direct numerical simulations (DNS). The key questions at the focus of this work are: does a superhydrophobic surface induce a different pressure field compared to a flat surface? If so, how does this difference scale with system parameters, and when does it become significant that it can deform the air–water interface and potentially rapture the entrapped gas pockets? To this end, we have performed DNS of turbulent channel flows subject to superhydrophobic surfaces over a wide range of texture sizes spanning values from $L^{+}=6$ to $L^{+}=155$ when expressed in terms of viscous units. The pressure statistics at the wall are decomposed into two contributions: one coherent, caused by the stagnation of slipping flow hitting solid posts, and one time-dependent, caused by overlying turbulence. The results show that the larger texture size intensifies the contribution of stagnation pressure, while the contribution from turbulence is essentially insensitive to $L^{+}$. The two-dimensional stagnation pressure distribution at the wall and the pressure statistics in the wall-normal direction are found to be self-similar for different $L^{+}$. The scaling of the induced pressure and the consequent deformations of the air–water interface are analysed. Based on our results, an upper bound on the texture wavelength is quantified that limits the range of robust operation of superhydrophobic surfaces when exposed to high-speed flows. Our results indicate that when the system parameters are expressed in terms of viscous units, the main parameters controlling the problem are $L^{+}$ and a Weber number based on inner dimensions; We obtain good collapse when all our results are expressed in wall units, independently of the Reynolds number.

Kinetic energy transfer in compressible isotropic turbulence is studied using numerical simulations with solenoidal forcing at turbulent Mach numbers ranging from 0.4 to 1.0 and at a Taylor Reynolds number of approximately 250. The pressure dilatation plays an important role in the local conversion between kinetic energy and internal energy, but its net contribution to the average kinetic energy transfer is negligibly small, due to the cancellation between compression and expansion work. The right tail of probability density function (PDF) of the subgrid-scale (SGS) flux of kinetic energy is found to be longer at higher turbulent Mach numbers. With an increase of the turbulent Mach number, compression motions enhance the positive SGS flux, and expansion motions enhance the negative SGS flux. Average of SGS flux conditioned on the filtered velocity divergence is studied by numerical analysis and a heuristic model. The conditional average of SGS flux is shown to be proportional to the square of filtered velocity divergence in strong compression regions for turbulent Mach numbers from 0.6 to 1.0. Moreover, the antiparallel alignment between the large-scale strain and the SGS stress is observed in strong compression regions. The inter-scale transfer of solenoidal and compressible components of kinetic energy is investigated by Helmholtz decomposition. The SGS flux of solenoidal kinetic energy is insensitive to the change of turbulent Mach number, while the SGS flux of compressible kinetic energy increases drastically as the turbulent Mach number becomes larger. The compressible mode persistently absorbs energy from the solenoidal mode through nonlinear advection. The kinetic energy of the compressible mode is transferred from large scales to small scales through the compressible SGS flux, and is dissipated by viscosity at small scales.

We develop a theory of nonlinear geostrophic adjustment of arbitrary localized (i.e. finite-energy) disturbances in the framework of the non-dissipative rotating shallow-water dynamics. The only assumptions made are the well-defined scale of disturbance and the smallness of the Rossby number Ro. By systematically using the multi-time-scale perturbation expansions in Rossby number it is shown that the resulting field is split in a unique way into slow and fast components evolving with characteristic time scales f−10 and (f0Ro)−1 respectively, where f0 is the Coriolis parameter. The slow component is not influenced by the fast one and remains close to the geostrophic balance. The algorithm of its initialization readily follows by construction.

The scenario of adjustment depends on the characteristic scale and/or initial relative elevation of the free surface ΔH/H0, where ΔH and H0 are typical values of the initial elevation and the mean depth, respectively. For small relative elevations (ΔH/H0 = O(Ro)) the evolution of the slow motion is governed by the well-known quasi-geostrophic potential vorticity equation for times t [les ] (f0Ro)−1. We find modifications to this equation for longer times t [les ] (f0Ro2)−1. The fast component consists mainly of linear inertia–gravity waves rapidly propagating outward from the initial disturbance.

For large relative elevations (ΔH/H0 [Gt ] Ro) the slow field is governed by the frontal geostrophic dynamics equation. The fast component in this case is a spatially localized packet of inertial oscillations coupled to the slow component of the flow. Its envelope experiences slow modulation and obeys a Schrödinger-type modulation equation describing advection and dispersion of the packet. A case of intermediate elevation is also considered.

As an acid flows through porous rock, it etches the rock and so increases the permeability. This propagating reaction front suffers an instability, rather like the viscous fingering instability, in which the acid prefers to follow high-permeability channels which it has already etched. We have examined the linear stability, obtaining analytic results for small and large wavenumbers and for small variations of the permeability, and obtaining numerical results in other cases.

Using small-amplitude expansions, we discuss nonlinear effects in the reflection from a sloping wall of a time-harmonic (frequency $\omega$) plane-wave beam of finite cross-section in a uniformly stratified Boussinesq fluid with constant buoyancy frequency $N_{0}$. The linear solution features the incident and a reflected beam, also of frequency $\omega$, that is found on the same (opposite) side to the vertical as the incident beam if the angle of incidence relative to the horizontal is less (greater) than the wall inclination. As each of these beams is an exact nonlinear solution, nonlinear interactions are confined solely in the vicinity of the wall where the two beams meet. At higher orders, this interaction region acts as a source of a mean and higher-harmonic disturbances with frequencies $n\omega$ ($n\,{=}\,2,3,\ldots$); for $n\omega\,{<}\,N_{0}$ the latter radiate in the far field, forming additional reflected beams along $\sin^{-1}(n\omega/N_{0})$ to the horizontal. Depending on the flow geometry, higher-harmonic beams can be found on the opposite side of the vertical from the primary reflected beam. Using the same approach, we also discuss collisions of two beams propagating in different directions. Nonlinear interactions in the vicinity of the collision region induce secondary beams with frequencies equal to the sum and difference of those of the colliding beams. The predictions of the steady-state theory are illustrated by specific examples and compared against unsteady numerical simulations.