Transonic potential flow of dense gases of retrograde type around the leading edge of a thin airfoil with a parabolic nose is studied. The analysis follows the approach of Rusak (1993) for a perfect gas. Asymptotic expansions of the velocity potential function are constructed in terms of the airfoil thickness ratio in an outer region around the airfoil and in an inner region near the nose. The outer expansion consists of the transonic small-disturbance theory for dense gases, where a leading-edge singularity appears. Analytical expressions are given for this singularity by constructing similarity solutions of the governing nonlinear equation. The inner expansion accounts for the flow around the nose, where a stagnation point exists. A boundary value problem is formulated in the inner region for the solution of an oncoming uniform sonic flow with zero values of the fundamental derivative of gasdynamics (Γ=0) and the second nonlinearity parameter (Λ=0) around a parabola at zero angle of attack. The numerical solution of the inner problem results in a symmetric flow around the nose. The outer and inner expansions are matched asymptotically resulting in a uniformly valid solution on the entire airfoil surface. In the leading terms, the flow around the nose is symmetric and the stagnation point is located at the leading edge for every transonic Mach number, and small values of Γ and Λ of the oncoming flow and any shape and small angle of attack of the airfoil. Furthermore, analysis of the inner region in the immediate neighbourhood of the stagnation point reveals that the flow is purely subsonic, approaching critical conditions in the limit of large (scaled) distances, which excludes the formation of shock discontinuities in the nose region.

We examine the rotor–oscillator flow, a slow viscous flow between long parallel plates driven by the rotation of a slender cylinder (the rotor) and the longitudinal oscillation of one of the plates (the oscillator). For rotor locations of interest to us, this flow exhibits a hyperbolic mixing region, characterized by homoclinic tangling associated with a hyperbolic fixed point, and a degenerate mixing region, characterized by heteroclinic tangling associated with two degenerate fixed points on one of the boundary plates (normally the oscillator). These mixing regions are investigated both theoretically, by applying various dynamical tools to a mathematical model of the flow, and experimentally, by observing the advection of a passive tracer in a specially constructed apparatus. Although degenerate mixing regions have been largely ignored or undervalued in previous research on chaotic mixing, our results demonstrate that more mixing is associated with the degenerate mixing region than the hyperbolic one in many cases. We have also discovered a peculiar phenomenon, which we call Melnikov resonance, involving a rapid fluctuation in the size of the hyperbolic mixing region as the frequency of the oscillator is varied.

Vorticity filaments are characteristic structures of two-dimensional turbulence. The formation, persistence and effect of vorticity filaments are examined using a high-resolution direct numerical simulation (DNS) of the merging of two positive Gaussian vortices pushed together by a weaker negative vortex. Many intense spiral vorticity filaments are created during this interaction and it is shown using a wavelet packet decomposition that, as has been suggested, the coherent vortex stabilizes the filaments. This result is confirmed by a linear stability analysis at the edge of the vortex and by a calculation of the straining induced by the spiral structure of the filament in the vortex core. The time-averaged energy spectra for simulations using hyper-viscosity and Newtonian viscosity have slopes of −3 and −4 respectively. Apart from a much higher effective Reynolds number (which accounts for the difference in energy spectra), the hyper-viscous simulation has the same dynamics as the Newtonian viscosity simulation. A wavelet packet decomposition of the hyper-viscous simulation reveals that after the merger the energy spectra of the filamentary and coherent parts of the vorticity field have slopes of −2 and −6 respectively. An asymptotic analysis and DNS for weak external strain shows that a circular filament at a distance R from the vortex centre always reduces the deformation of a Lamb's (Gaussian) vortex in the region r[ges ]R. In the region r<R the deformation is also reduced provided the filament is intense and is in the vortex core, otherwise the filament may slightly increase the deformation. The results presented here should be useful for modelling the coherent and incoherent parts of two-dimensional turbulent flows.

Particle image velocitmetry (PIV) measurements and free-surface visualizations around a ship model focus on the flow within the attached liquid sheet, upstream of the point at which the bow wave separates from the model, the origin and structure of the bow wave and the flow downstream of the wave crest. The measurements are performed at Reynolds numbers ranging between 2.8×106 and 7.4×106 and Froude numbers between 0.17 and 0.45 (both are based on ship length L). Representative velocity and vorticity distributions at FrL=0.28 and FrL=0.45 demonstrate the characteristic structure of mild and steep waves, respectively. Very close to the bow the attached sheet is thin and quite unsteady. With increasing distance from the nose the sheet becomes thicker and its development involves considerable vorticity production. In the mild case this vorticity is originated at the free surface, whereas in the steep wave case, boundary layer separation occurs on the model, which also transports vorticity into the sheet. This vorticity and its associated induced lateral flow remain near the model downstream of the bow wave. By calculating the acceleration component tangent to the free surface of the sheet it is shown that the peaks in the near-surface vorticity appear in regions with high viscous flux of vorticity from the surface. Formation of a bow wave also involves considerable production of vorticity. Similar to two-dimensional breakers, the primary origin of this vorticity is at the toe of the breaker. However, unlike the two-dimensional cases, the region containing vorticity in the ship wave does not appear as an extended shear layer. Instead, this vorticity is convected out of the plane of the laser sheet in a series of distinct vortex filaments. The ship wave also has powerful counter-rotating vorticity concentrated near the wave crest that has been observed in two-dimensional waves, but not of the same strength. Breaking becomes weaker, i.e. there is less vorticity production, with increasing distance from the model, but it persists even at the ‘tail’ of the bow wave. The sites of vorticity entrainment of both signs are consistent with the computed near-surface acceleration. Estimates of the three-dimensional velocity distribution and head losses within the wave are also provided.

The simultaneous effect of small deformation and short-range van der Waals attraction on the coalescence efficiency of two different-sized slowly sedimenting drops is considered. For spherical drops, it has been shown previously that the tangential mobility of drop surfaces makes collision possible even without van der Waals attraction; on the other hand, even a small amount of deformation precludes drops from coming into contact unless van der Waals attraction is accounted for. In the present work, the conditions are delineated when these two small-scale factors, acting in opposite directions, have a considerable combined effect on the coalescence efficiency. The problem is solved by matched asymptotic expansions valid for small capillary numbers (Ca). The outer solution, for two spherical drops moving in apparent contact without van der Waals attraction, determines the contact force as a function of time. This force is used as the driving force for the inner solution of the relevant integro-differential thin-film equations (coupling the flow in the small-gap region to that inside the drops) to determine whether coalescence occurs during the apparent contact motion. The initial gap profile for the inner solution is provided by matching with the outer trajectory for spherical drops approaching contact.

The analysis shows that, for Ca[Lt ]1, the near-contact deformation is mainly axisymmetric, greatly simplifying the inner solution; nevertheless, determination of the critical horizontal offsets leading to coalescence and the parametric analysis are computationally very intensive. To facilitate these tasks, a substantially new, highly efficient, and absolutely stable numerical method for solving stiff thin-film equations is developed. Unlike for spherical drops, when the upstream intersection area is a circle, the existence of a second coalescence zone for deformable drops is found over much of the parameter space. Results are mapped out for a range of four dimensionless parameters (capillary number, size and drop-to-medium viscosity ratios, dimensionless Hamaker parameter). As a physical application, predicted coalescence efficiencies are shown for a system of ethyl salicylate drops in diethylene glycol.

The present solution extends the range of drop sizes where the coalescence efficiencies are known theoretically and can be used in drop population dynamics. Comparison with full three-dimensional boundary-integral calculations for deformable drops without van der Waals attraction is also made to demonstrate that, when the drop-to-medium viscosity ratio is of the order of unity, the present asymptotic approach is valid in a wide range of small and moderately small capillary numbers.

The trajectory of a non-isolated monopole on the beta-plane is calculated as an asymptotic expansion in the ratio of the strength of the vortex to the beta-effect. The method of matched asymptotic expansions is used to solve the equations of motion in two regions of the flow: a near field where the beta-effect enters as a first-order forcing in relative vorticity, and a wave field in which the dominant balance is a linear one between the beta-effect and the rate of change of relative vorticity. The resulting trajectory is computed for Gaussian and Rankine vortices.

The primary goal of the paper is to demonstrate that a circular jet of inviscid liquid, which issues from a nozzle at an angle to the gravitational field, can sustain a self-excited linear global mode. The mode is found for the flow that is locally convectively unstable at every point, i.e. has no region of local absolute instability. A simple experiment shows that the mode can evolve into a wave of finite amplitude, which is localized in space near the nozzle.

Section 2 of the paper summarizes some recent developments in the linear theory of stability of weakly inhomogeneous flows.

The ensemble-averaged characteristics of the turbulent near-wake flow around two side-by-side identical square cylinders at a Reynolds number ≈23100 have been studied using a two-component laser-Doppler velocimeter system. The work focuses on a single case with a gap/diameter ratio of 2, for which the resulting individual vortex streets are coupled so as to yield a flow predominantly symmetric about the line midway between the two cylinders. Data sorting or conditioning according to phase was performed with the aid of pressure signals taken from taps on a sidewall of each cylinder. The two-cylinder results are compared in detail to results from a previous study of the one-cylinder case. Vortex structures shed on the side towards the flow centreline, termed inner structures, are distinguished from those shed on the free-stream side, termed outer structures, and the differences between the features associated with the two different structures are examined. The circulation associated with outer structures evolves downstream in a manner similar to that observed in the one-cylinder case, but the circulation of the inner structures is found to decrease dramatically downstream. This not only gives support to previous theoretical predictions but also reconciles these with previously apparently conflicting experimental observations. Information regarding vortex structure motion and the relevant length and time scales is obtained. Differences between momentum and vorticity transport, particularly across the flow centreline are pointed out, and effective turbulent vorticity fluxes are defined. Similarities in local flow topologies in one- and two-cylinder cases are discussed, and the role of local velocity-gradient invariants and their relationship to critical points and turbulence statistics are examined.

A quasi-one-dimensional model is used to examine the steady flow of granular materials through a wedge-shaped hopper with smooth, steep walls. Hybrid frictional–kinetic equations are used in an attempt to overcome some of the dificulties faced by earlier works, which were based on frictional equations. Owing to computational difficulties, two different solution procedures are used: (i) in the upper region, where frictional effects dominate, and (ii) the lower region which includes the exit slot and a part of the particle jet below the hopper, where kinetic and frictional effects are expected to be comparable. The equations are integrated numerically in (i). In (ii), they are linearized, and a semi-analytical solution is constructed. In contrast to the works of Kaza & Jackson (1982a) and Prakash & Rao (1991), the density varies smoothly across the exit slot. The density profile is qualitatively similar to the data of Fickie, Mehrabi & Jackson (1989). However, the range of density variation is much smaller than that observed. Owing to the approximations used, and perhaps also to the form of the kinetic constitutive equations, kinetic effects are dominated by frictional effects, except close to the downstream boundary.

Self-excited oscillations of a confined flame, burning in the wake of a bluff-body flame-holder, are considered. These oscillations occur due to interaction between unsteady combustion and acoustic waves. According to linear theory, flow disturbances grow exponentially with time. A theory for nonlinear oscillations is developed, exploiting the fact that the main nonlinearity is in the heat release rate, which essentially ‘saturates’. The amplitudes of the pressure fluctuations are sufficiently small that the acoustic waves remain linear. The time evolution of the oscillations is determined by numerical integration and inclusion of nonlinear effects is found to lead to limit cycles of finite amplitude. The predicted limit cycles are compared with results from experiments and from linear theory. The amplitudes and spectra of the limit-cycle oscillations are in reasonable agreement with experiment. Linear theory is found to predict the frequency and mode shape of the nonlinear oscillations remarkably well. Moreover, we find that, for this type of nonlinearity, describing function analysis enables a good estimate of the limit-cycle amplitude to be obtained from linear theory.

Active control has been successfully applied to eliminate these oscillations. We demonstrate the same effect by adding a feedback control system to our nonlinear model. This theory is used to explain why any linear controller capable of stabilizing the linear flow disturbances is also able to stabilize finite-amplitude oscillations in the nonlinear limit cycles.

Coalescence and fragmentation of equal and unequal liquid-drop pairs are studied using a new experimental technique in which mercury drops collide while sliding on a horizontal glass surface. The limits for coalescence measured as a function of the incident relative velocity and impact parameter are found to be similar to what has been reported for free-moving drops of other liquids, while new correlations are found to occur among the number, size, speed and angular distribution of fragmentation residues. The predictions of various models, including a dynamic theory originally developed for nuclear reactions, and specifically modified by us for macroscopic applications, are compared with the observations.

In this paper the solution to the three-dimensional and unsteady interacting boundary-layer equations for a vortex approaching a cylinder is calculated. The flow is three-dimensional and unsteady. The purpose of this paper is to enhance the understanding of the structure in three-dimensional unsteady boundary-layer separation commonly observed in a high-Reynolds-number flow. The short length scales associated with the boundary-layer eruption process are resolved through an efficient and effective moving adaptive grid procedure. The results of this work suggest that like its two-dimensional counterpart, the three-dimensional unsteady interacting boundary layer also terminates in a singularity at a finite time. Furthermore, the numerical calculations confirm the theoretical analysis of the singular structure in two dimensions for the interacting boundary layer due to Smith (1988).

The phenomenon of shelf generation by long nonlinear internal waves in stratified flows is investigated. The problem of primary interest is the case of a uniformly stratified Boussinesq fluid of finite depth. In analysing the transient evolution of a finite-amplitude long-wave disturbance, the expansion procedure of Grimshaw & Yi (1991) breaks down far downstream, and it proves expedient to follow a matched-asymptotics procedure: the main disturbance is governed by the nonlinear theory of Grimshaw & Yi (1991) in the ‘inner’ region, while the ‘outer’ region comprises multiple small-amplitude fronts, or shelves, that propagate downstream and carry O(1) mass. This picture is consistent with numerical simulations of uniformly stratified flow past an obstacle (Lamb 1994). The case of weakly nonlinear long waves in a fluid layer with general stratification is also examined, where it is found that shelves of fourth order in wave amplitude are generated. Moreover, these shelves may extend both upstream and downstream in general, and could thus lead to an upstream influence of a type that has not been previously considered. In all cases, transience of the main nonlinear wave disturbance is a necessary condition for the formation of shelves.

The streamwise velocity profiles of low-velocity isothermal axisymmetric jets from nozzles of different diameters were measured and compared with previous experimental data. The objective of the measurements was to examine the dependence of the diffusion of the jet on the outlet conditions. As the outlet velocity was decreased, the centreline velocity decay coefficient began to decrease at an outlet velocity of about 6 m s−1.