A temporal linear stability analysis has been carried out to predict the instability of a viscous liquid jet surrounded by a swirling air stream with three-dimensional disturbances. The effects of flow conditions and fluid properties on the instability of the liquid jet are investigated via a parametric study by varying axial Weber number axial velocity ratio of the gas to liquid phase, swirl Weber numbers, density ratio and the Ohnesorge number. It is observed that the relative axial velocity between the liquid and gas phases promotes the interfacial instability. As the axial Weber number increases, the growth rates of unstable waves, the most unstable wavenumber and the unstable range of wavenumbers increase. Meanwhile, the increasing importance of helical modes compared to the axisymmetric mode switches the breakup regime from the Rayleigh regime to the first wind-induced regime and on to the second wind-induced regime. The predicted range of wavenumbers in which the first helical mode has higher growth rates than the axisymmetric mode agrees very well with experimental data. Results show that the destabilizing effects of the density ratio and the axial Weber number are nearly the same. Liquid viscosity inhibits the disintegration process of the liquid jet by reducing the growth rate of disturbances and by shifting the most unstable wavenumber to a lower value. Moreover, it damps higher helical modes more significantly than the axisymmetric mode. Air swirl has a stabilizing effect on the liquid jet. As air swirl strength increases, the growth rates of helical modes are reduced more significantly than that of the axisymmetric mode. The air swirl profile is found to have a significant effect on the instability of the liquid jet. The global, as well as local, effects of the swirl profile are examined in detail.

This paper investigates finite Reynolds number effects in the problem of the propagation of an air finger into a liquid-filled flexible-walled two-dimensional channel. The study is motivated by the physiological problem of pulmonary airway reopening. A fully consistent model of the fluid–structure interaction is formulated and solved numerically using coupled finite element discretizations of the free-surface Navier–Stokes equations and the Lagrangian wall equations. It is shown that for parameter values which are representative of the conditions in the lung and in typical laboratory experiments, fluid inertia plays a surprisingly important role: even for relatively modest ratios of Reynolds and capillary numbers (Re/Ca ≈ 5–10), the pressure required to drive the air finger at a given speed increases significantly compared to the zero Reynolds number case. Fluid inertia leads to significant changes in the velocity and pressure fields near the bubble tip and is responsible for a noticeable change in the wall deformation pattern ahead of the bubble. For some parameter variations (such as variations in the wall tension), finite Reynolds number effects are shown to lead to qualitative changes in the system's behaviour. Finally, the implications of the result for pulmonary airway reopening are discussed.

The spatially developing region of a steady downslope current in transition from a hydraulically controlled, arrested wedge flow is examined through a set of laboratory experiments. The mixing and entrainment mechanisms at the shear interface are explored with the use of imaging techniques. The initial, unstable accelerating region, characterized by rapid development, low shear layer Richardson numbers and high entrainment rates, is limited by the effects of stratification which suppress the large-scale overturning. A subsequent high Richardson number region is characterized by weak entrainment and the collapse of turbulence from the initial region. The flow approaches a quasi-stable state as the mixed layer draws energy from the accelerating density current. Observed large-scale structure is attributed to an instability of the marginally unstable shear layer. Entrainment rates are calculated within each region for the various slopes considered and estimates for the extent of the transition regions are obtained.

This paper presents a model for quasi-two-dimensional MHD flows between two planes with small magnetic Reynolds number and constant transverse magnetic field orthogonal to the planes. A method is presented that allows three-dimensional effects to be taken into account in a two-dimensional equation of motion thanks to a model for the transverse velocity profile. This model is obtained by using a double perturbation asymptotic development both in the core flow and in the Hartmann layers arising along the planes. A new model is thus constructed that describes inertial effects in these two regions. Two separate classes of phenomena are found: one related to inertial effects in the Hartmann layer gives a model for recirculating flows and the other introduces the possibility of having a transverse dependence of the velocity profile in the core flow. The ‘recirculating’ velocity profile is then introduced in the transversally averaged equation of motion in order to provide an effective two-dimensional equation of motion. Analytical solutions of this model are obtained for two experimental configurations: isolated vortices generated by a point electrode and axisymmetric parallel layers occurring in the MATUR (MAgneticTURbulence) experiment. The theory is found to give a satisfactory agreement with the experiment so that it can be concluded that recirculating flows are actually responsible for both vortex core spreading and excessive dissipative behaviour of the axisymmetric sidewall layers.

An investigation is made into water-wave propagation through an array of vertical cylinders extending to infinity and periodic in both horizontal directions. Methods are presented for the calculation of the frequency ranges for which wave propagation without change of amplitude is possible (‘passing bands’), and for which propagation without change of amplitude is not possible (‘stopping bands’). Some of the techniques may be used to determine the change of wave amplitude for frequencies within the stopping bands. Approximate and numerical techniques are used to show how this infinite-array problem is related to trapped modes, Rayleigh–Bloch waves, and the problem of wave diffraction by a grating made up of a finite number of cylinder rows.

Simultaneous measurements of the velocity and scalar concentration fields have been made in the plume emitting from a two-dimensional line source at the wall. The source is one obstacle height, h, downstream of a two-dimensional square obstacle located on the wall of a turbulent boundary layer. These measurements were made in two fluid media: water and air. In both media particle image velocimetry (PIV) was used for the velocity field measurements. For the scalar concentration measurements laser-induced uorescence (LIF) was used for the water flow and Mie scattering diffusion (MSD) for the air flow. Profiles of the mean and root-mean-square streamwise and wall-normal velocity components, Reynolds shear stress and mean and root-mean-square scalar concentration were determined at x = 4h and 6h downstream of the obstacle in the recirculation region and above it in the mixing region. At these streamwise stations the scalar fluxes, uc and vc, were also determined from the simultaneous velocity and scalar concentration field data. Both of these fluxes change sign from negative to positive with increasing distance from the wall in the recirculating region at 4h.

A conditional analysis of the data was carried out by sorting them into the eight categories (octants) given by the sign combinations of the three variables: ±u, ±v and ±c. The octants with combinations of these three variables that correspond to types of scalar concentration flux motions that can be approximated by mean gradient scalar transport models are the octants that make the dominant contributions to uc and vc. However, in the recirculating zone, counter-gradient transport type motions also make significant contributions. Based on this conditional analysis, second-order mean gradient models of the scalar and the momentum uxes were constructed; they compare well to the measured values at 4h and 6h, particularly for the streamwise scalar flux, uc.

Additional measurements of the velocity and concentration fields were made further downstream of the reattachment location in the wake region of the air flow. The mean velocity deficit profile determined from these measurements at x = 20h compares quite well to a similarity solution profile obtained by Counihan, Hunt & Jackson (1974). Their analysis was extended in the present investigation to the concentration field. The similarity solution obtained for the mean concentration compares well to profiles measured at x = 12h, 15h, and 20h, up to about three obstacle heights above the wall.

The effect of a vertical gas flow on the dynamics of a coulombic granular material moving over a horizontal rigid porous surface has been studied experimentally and theoretically. The presence of a fluidizing gas significantly alters the granular flow dynamics. When the gas velocity, ug, is below the minimum fluidization velocity, umf, the effect of the gas is to reduce the angle of repose θ from the value measured in the absence of a gas flow. When material is poured from a point source onto a horizontal surface it forms a pile, which adjusts through episodic avalanching to a self-similar conical shape. Under these conditions, the development of the pile is determined by the local force balance on individual particles and its extent may be expressed in terms of the volume of particles added and the angle of repose. A volume of material is poured continuously from a point source onto a surface according to Qtα. Below the minimum uidization velocity, a quasi-static description gives the encroachment distance of the granular pile as l = (2Q/(2π/3)n−1 tan θ)1/n+1tα/n+1 where n = 1 for a planar release and n = 2 for an axisymmetric release.

A continuum description of fluidized granular flow has been developed by vertically averaging the mass and momentum conservation equations and including the momentum exchange between the gas and granular flow. The bulk movement is driven along the ground by horizontal gradients of particle pressure and is resisted by a viscous drag force due to the particles moving horizontally through a vertical gas flow. Above the minimum fluidization velocity the character of the granular flow is significantly altered and takes on fluid-like properties. The model predicts the shape of the fluidized granular pile and that the encroachment distance grows as l = λnα (Q(ug + umf) / ε)1/n+2tα+1/n+2, where ε is the void fraction in the bed and λnα is a constant. Below the conditions for fluidization (ug < umf), the pile of granular material grows quasi-statically when t > t∗, where t∗ ∼ (εn+1Qug + umf) / μ2+n (umf − ug)2+n) 1/1+n−α corresponds to the critical time when frictional forces are comparable to gradients of particle pressure and the drag force. Numerical solutions describing the granular flow are presented.

Experimental observations of the shape and extent of planar and axisymmetric granular flows when α = 1 compare well with theoretical predictions for various values of particle volume flux Q, time t, and gas flow rate ug. The mathematical description of fluidized granular flows along rigid surfaces indicates a strong analogy with buoyancy-driven flows in a porous medium. This analogy permits extension of our description to include flows down slopes and the effect of internal stratification.

A theoretical model of axisymmetric turbulent fountains in both homogeneous and stratified fluids is developed. The model quantifies the entrainment of ambient fluid into the initial fountain upflow, and the entrainment of fluid from both the upflow and environment into the subsequently formed downflow. Four different variations of the model are considered, comprising the two most reasonable formulations of the body forces acting on the ‘double’ structure and two formulations of the rate of entrainment between the flows. The four model variations are tested by comparing the predictions from each of them with experimental measurements of fountains in homogeneous and stratified fluids.

The turbulent flow in a cylindrical pipe oscillating around its longitudinal axis is studied via direct numerical solution of the Navier–Stokes equations, and compared to the reference turbulent flow in a fixed pipe and in a pipe with steady rotation. The maximum amount of drag reduction achievable with appropriate oscillations of the pipe wall is found to be of the order of 40%, hence comparable to that of similar flows in planar geometry. The transverse shear layer due to the oscillations induces substantial modifications to the turbulence statistics in the near-wall region, indicating a strong effect on the vortical structures. These modifications are illustrated, together with the implications for the drag-reducing mechanism. A conceptual model of the interaction between the moving wall and a streamwise vortex is discussed.

In a plane-channel displacement flow of two visco-plastic fluids, it is possible for there to be a static residual layer of the displaced fluid left stuck to the walls of the channel. This phenomenon provides an idealized model for the formation of a wet micro-annulus, due to poor mud removal, during the primary cementing of an oil well. Using a lubrication approximation, it is shown that sufficient conditions for the non-existence of a static wall layer can be computed simply in terms of two dimensionless parameters: the Bingham number for the displacing fluid (B1) and the ratio of the yield stresses of the two fluids (ϕY). When these conditions are not met, it is possible to compute the maximum possible static wall layer thickness hmax, which depends on B1, ϕY and on a third dimensionless parameter ϕB, a buoyancy to yield stress ratio.

On computing displacements using the lubrication approximation, the interface is observed to asymptotically approach the maximum static layer thickness as t → ∞. Results from fully two-dimensional displacement computations are also presented. These indicate that the displacement front propagates at a steady speed along the channel, leaving behind a static layer which is significantly thinner than hmax. Surprisingly, the computed static layer thickness is observed to decrease with a parametric increase in the dimensionless yield stress of the displaced fluid. To explain these results we analyse the streamline configuration close to a steadily advancing displacement front. We demonstrate heuristically that the local visco-plastic dissipation functional will be approximately minimized by a critical layer thickness at which the displaced fluid begins to recirculate ahead of the displacement front. Comparison of the critical recirculation limit with the static layer thickness computed from the fully transient model gives a very close agreement, suggesting that a form of energy minimization is responsible in this case for selecting the static layer thickness.

An extended Korteweg–de Vries (KdV) equation is derived that describes the evolution and propagation of long interfacial gravity waves in the presence of a strong, space–time varying background. Provision is made in the derivation for a spatially varying lower depth so that some topographic effects can also be included. The extended KdV model is applied to some simple scenarios in basins of constant and varying depths, using approximate expressions for the variable coefficients derived for the case when the background field is composed of a moderate-amplitude ultra-long wave. The model shows that energy can be transferred either to or from the evolving wave packet depending on the relative phases of the evolving waves and the background variation. Comparison of the model with laboratory experiments confirms its applicability and usefulness in examining the evolution of weakly nonlinear waves in natural systems where the background state is rarely uniform or steady.

The propagation of sound across a shear layer of finite thickness is studied using exact solutions of the acoustic wave equation for a shear flow with hyperbolic-tangent velocity profile. The wave equation has up to four regular singularities: two corresponding to the upper and lower free streams; one corresponding to a critical layer, where the Doppler-shifted frequency vanishes if the free streams are supersonic; and a fourth singularity which is always outside the physical region of interest. In the absence of a critical layer the matching of the two solutions, around the upper and lower free streams, specifies exactly the acoustic field across the shear layer. For example, for a sound wave incident from below (i.e. upward propagation in the lower free stream), the reflected wave (i.e. downward propagating in the lower free stream) and the transmitted wave (i.e. upward propagating in upper free stream) are specified by the continuity of acoustic pressure and vertical displacement. Thus the reflection and transmission coefficients, which are generally complex, i.e. involve amplitude and phase changes, are plotted versus angle of incidence for several values of free stream Mach number, and ratio of thickness of the shear layer to the wavelength; the vortex sheet is the particular case when the latter parameter is zero. The modulus and phase of the total acoustic field are also plotted versus the coordinate transverse to the shear flow, for several values of angle of incidence, Mach number and shear layer thickness. The analysis and plots in the present paper demonstrate significant differences between sound scattering by a shear layer of finite thickness, and the limiting case of the vortex sheet.

Sound pressure fields generated by coaxial collisions of two vortex rings with equal/unequal strengths are simulated numerically. The axisymmetric, unsteady, compressible Navier–Stokes equations are solved by a finite difference method, not only for a near field but also for a far field. The sixth-order-accurate compact Padé scheme is used for spatial derivatives, together with the fourth-order-accurate Runge–Kutta scheme for time integration. The results show that the generation of sound is closely related to the change of direction of the vortex ring motion induced by the mutual interaction of the two vortex rings. For the case of equal strength (head-on collision), the change of direction is associated with stretching of the vortex rings. Generated sound waves consist of compression parts and rarefaction parts, and have a quadrupolar nature. For the case of unequal strengths, the two vortex rings pass through each other; the weaker vortex ring moves outside the stronger vortex ring which shows a loop motion. The number of generated waves depends on the relative strength of the two vortex rings. The sound pressure includes dipolar and octupolar components, in addition to monopolar and quadrupolar components which are observed for the case of a head-on collision.

The estimation of the growth of propagating instability waves in laminar boundary layers is considered when the Reynolds number is sufficiently large for the mean flow to deviate only slightly from a truly parallel flow. An approximate solution for the linear perturbation is sought in the form of a scaled solution of the related locally parallel flow problem. The amplitude scaling is chosen so as to satisfy the full linearized perturbation equations as closely as possible by making the mean-square deviation of the remainder a minimum. By re-arranging the terms in the equations so that some of the small correction terms arising from the non-parallel mean flow are contained in the ordinary differential equation (ODE) defining the quasi-parallel flow solution, a useful simplification is obtained for the scaling function. Then a modified Orr–Sommerfeld equation defines the base solution and the differential expression for the scaling that can be integrated forms a simple conservation relation.