The leading-edge receptivity to acoustic waves of two-dimensional bodies is investigated using a spatial solution of the Navier–Stokes equations in vorticity/stream function form in general curvilinear coordinates. The free stream is composed of a uniform flow with a superposed periodic velocity fluctuation of small amplitude. The method follows that of Haddad & Corke (1998), in which the solution for the basic flow and the linearized perturbation flow are solved separately. The initial motivation for the work comes from past physical experiments for flat plates with elliptic leading edges, which indicated narrow frequency bands of higher neutral-curve Branch I receptivity. We investigate the same conditions in our simulations, as well as on a parabolic leading edge. The results document the importance of the leading edge, junction between the ellipse and flat plate, and pressure gradient to the receptivity coefficient at Branch I. Comparisons to the past experiments and other numerical simulations showed the influence of the elliptic leading-edge/flat-plate joint as an additional site of receptivity which, along with the leading edge, provides a wavelength selection mechanism which favours certain frequencies through linear superposition.

Direct numerical simulations (DNS) of a three-dimensional spatially-developing mixing layer (ML) laden with spherical gaseous bubbles are performed, with both one-way and two-way coupling between the two phases. Forcing is used to initialize the spanwise vortex roll-up and to create a pair of counter-rotating streamwise vortices, rendering the carrier flow three-dimensional. The characteristics of the resulting ML flow field are similar to those reported in numerous experimental and numerical studies. The volume fraction (or concentration) of the bubble phase is considered small enough to neglect bubble–bubble interactions. The no-slip fluid velocity condition is assumed at the bubble surface, and the bubble Reynolds number is less than 1 throughout the simulation time. The two-fluid formulation (TF) is used to compute the bubble-phase velocity and concentration and the two-way coupling source term in the fluid momentum equation. A Lagrangian–Eulerian mapping (LEM) solver is employed to solve the equations for the bubble velocity and concentration. LEM is capable of resolving the gradients of concentration created by the bubble preferential accumulation without numerical instabilities. Two different inflow profiles (Cref(z)) of bubble-phase concentration are considered: a uniform profile and a tanh-profile. In the latter case, the high-speed (upper) stream is devoid of bubbles, and the low-speed (lower) stream is uniformly laden with bubbles at the inflow plane.

The DNS results show that in addition to the well-known preferential accumulation of bubbles in the vortex centres, sheets of increased bubble concentration (C-sheets) develop in the rollers created by the vortex pairing in the ML core, with two local maxima of vorticity and an enhanced strain-rate field. The development of C-sheets is governed by the stretching and contraction along the principal axes of the local strain rate.

In the case of uniform Cref, the two-way coupling reduces the average ML vorticity thickness and the entrainment of the irrotational fluid into the ML core, as compared to the bubble-free case, upstream of the location of the first vortex pairing. However, both ML vorticity thickness is increased and entrainment is enhanced by the bubbles farther downstream, after the pairing. The fluid velocity fluctuations are reduced by the bubbles throughout the ML, as compared to the one-way coupling case.

In the case of the tanh-profile of Cref, the velocity fluctuations and the ML vorticity thickness are increased by the bubbles upstream the location of the first vortex pairing owing to the ‘unstable’ inflow bubble stratification (Druzhinin & Elghobashi 1998). On the other hand, the velocity fluctuations are reduced by the bubbles, and the ML vorticity thickness oscillates with the streamwise distance farther downstream.

This note addresses measurement of single bubble velocity in surfactant solutions and the physical model of the effect of surfactant concentration on the steady-state velocity. Continuous tracking of the bubble along the rise path is used as the measurement method. Using this technique, the steady-state velocity is found to be independent of surfactant concentration over a wide range, which contradicts the data frequently used to develop models.

A torsionally driven cavity, consisting of a fully enclosed cylinder with rotating bottom lid, is used to examine the confined swirling flow of low-viscosity Boger fluids for situations where inertia dominates the flow field. Flow visualization and the optical technique of particle image velocimetry (PIV) are used to examine the effect of small amounts of fluid elasticity on the phenomenon of vortex breakdown. Low-viscosity Boger fluids are used which consist of dilute concentrations of high molecular weight polyacrylamide or semi-dilute concentrations of xanthan gum in a Newtonian solvent. The introduction of elasticity results in a 20% and 40% increase in the minimum critical aspect ratio required for vortex breakdown to occur using polyacrylamide and xanthan gum, respectively, at concentrations of 45 p.p.m. When the concentrations of either polyacrylamide or xanthan gum are raised to 75 p.p.m., vortex breakdown is entirely suppressed for the cylinder aspect ratios examined. Radial and axial velocity measurements along the axial centreline show that the alteration in existence domain is linked to a decrease in the magnitude of the peak in axial velocity along the central axis. The minimum peak axial velocities along the central axis for the 75 p.p.m. polyacrylamide and 75 p.p.m. xanthan gum Boger fluids are 67% and 86% lower in magnitude, respectively, than for the Newtonian fluid at Reynolds number of Re ≈ 1500–1600. This decrease in axial velocity is associated with the interaction of elasticity in the governing boundary on the rotating base lid and/or the interaction of extensional viscosity in areas with high velocity gradients. The low-viscosity Boger fluids used in this study are rheologically characterized and the steady complex flow field has well-defined boundary conditions. Therefore, the results will allow validation of non-Newtonian constitutive models in a numerical model of a torsionally driven cavity flow.

A torsionally driven cavity has been used to examine the influence of elasticity on the swirling flow of constant-viscosity elastic liquids (Boger fluids). A wealth of phenomena is observed as the degree of inertia, elasticity and viscous forces are varied by using a range of low- to high-viscosity flexible polyacrylamide Boger fluids and a semi-rigid xanthan gum Boger fluid. As the inertia is decreased and elasticity increased by using polyacrylamide Boger fluids, the circulation rates for a ‘Newtonian-like’ secondary flow decreases until flow reversal occurs owing to the increasing magnitude of the primary normal stress difference. For each polyacrylamide fluid, the flow becomes highly unstable at a critical combination of Reynolds number and Weissenberg number resulting in a new time-dependent elastic instability. Each fluid is characterized by a dimensionless elasticity number and a correlation with Reynolds number is found for the occurrence of the instability. In the elasticity dominated flow of the polyacrylamide Boger fluids, the instability disrupts the flow dramatically and causes an increase in the peak axial velocity along the central axis by as much as 400%. In this case, the core vortex spirals with the primary motion of fluid and is observed in some cases at Reynolds numbers much less than unity. Elastic ‘reverse’ flow is observed for the xanthan gum Boger fluid at high Weissenberg number. As the Weissenberg number decreases, and Reynolds number increases, counter-rotating vortices flowing in the inertial direction form on the rotating lid. The peak axial velocity decreases for the xanthan gum Boger fluid with decreasing Weissenberg number. In addition, several constitutive models are used to describe accurately the rheological properties of the fluids used in this work in shear and extensional flow. This experimental investigation of a complex three-dimensional flow using well-characterized fluids provides the information necessary for the validation of non-Newtonian constitutive models through numerical analysis of the torsionally driven cavity flow.

The origin of shock unsteadiness in a Mach 2.9 turbulent reattaching shear layer was investigated experimentally using temporally resolved flow visualization and measurements of wall pressure fluctuations. In this flow, the separation point of a turbulent boundary layer is essentially fixed at a backward-facing step, and the reattachment point is free to move along a ramp. In order to examine the influence of disturbances originating in the incoming shear layer, artificial disturbances were introduced into the flow through steady air injection in the vicinity of separation. The effect on the reattachment shock system was dramatic: the intensity of the pressure fluctuations and the amplitude of the shock motion increased substantially, and power spectra of the pressure fluctuations showed a distinct shift to lower frequency. The spectra collapsed onto a common curve in non-dimensional coordinates based on a length scale derived from two-point cross-correlations of the flow visualization data and a convection velocity derived from cross-correlations of the pressure measurements. The data were compared to a theory developed by Plotkin (1975), which is based on perturbation of a shock by random fluctuations in the incoming turbulent flow. Plotkin's model mimics the manner in which relatively broad-band perturbations in the incoming turbulent flow lead to relatively low-frequency motion of the separation bubble and its associated shock system. It is an excellent fit to separation shock motion, such as that generated in a blunt fin flow (briefly illustrated here). In the present shear layer flow, this low-frequency motion was detectable in the spectra near reattachment, but contained considerably less energy relative to the shock motions caused by direct perturbations by the incoming turbulent structures. These results indicate that the shock motion in the reattaching shear layer is primarily caused by organized structures in the incoming turbulent flow.

The present paper addresses the direct numerical simulation of turbulent zero-pressure-gradient boundary layers on a flat plate at Mach numbers 3, 4.5 and 6 with momentum-thickness Reynolds numbers of about 3000. Simulations are performed with an extended temporal direct numerical simulation (ETDNS) method. Assuming that the slow streamwise variation of the mean boundary layer is governed by parabolized Navier–Stokes equations, the equations solved locally in time with a temporal DNS are modified by a distributed forcing term so that the parabolized Navier–Stokes equations are recovered for the spatial average. The correct mean flow is obtained without a priori knowledge, the streamwise mean-flow evolution being approximated from its upstream history. ETDNS reduces the computational effort by up to two orders of magnitude compared to a fully spatial simulation.

We present results for a constant wall temperature Tw chosen to be equal to its laminar adiabatic value, which is about 2.5 T∞, 4.4 T∞ and 7 T∞, respectively, where T∞ is the free-stream temperature for the three Mach numbers considered. The simulations are initialized with transition-simulation data or with re-scaled turbulent data at different parameters. We find that the ETDNS results closely match experimental mean-flow data. The van Driest transformed velocity profiles follow the incompressible law of the wall with small logarithmic regions.

Of particular interest is the significance of compressibility effects in a Mach number range around the limit of M∞ ≃ 5, up to which Morkovin's hypothesis is believed to be valid. The results show that pressure dilatation and dilatational dissipation correlations are small throughout the considered Mach number range. On the other hand, correlations derived from Morkovin's hypothesis are not necessarily valid, as is shown for the strong Reynolds analogy.

An experimental investigation of oscillatory shallow-water flow around islands has been undertaken to determine the dependence of wake formation on Keulegan–Carpenter number, KC = UoT/D, and stability parameter, S = cfD/h, where Uo is amplitude of velocity oscillation, T is oscillation period, D is a representative island diameter, cf is friction coefficient and h is water depth. Two geometries are investigated: a vertical cylinder and a conical island with a small side slope of 8°. Existing experimental results for current flow around the same geometries have shown the influence of the stability parameter. Predominantly laminar flows are investigated and the flows are subcritical.

Four modes of wake formation have been identified for both geometries: one with symmetric attached counter-rotating vortices only forming in each half-cycle, one with vortex pairs forming symmetrically in addition in each half-cycle, one with vortex pairs forming with some asymmetry and one with complex vortex shedding. The last results from one of the attached vortices crossing to the opposite side of the body during flow reversal; in the other cases the attached vortices are convected back on the same sides. For convenience these formations are called: symmetric without pairing, symmetric with pairing, sinuous with pairing and vortex shedding. They are shown on KC/S planes for both geometries. Numerical modelling of the flows for the conical island, based on the three-dimensional shallow-water equations with the hydrostatic pressure assumption, is undertaken in Part 2 (Stansby & Lloyd 2001).

A three-dimensional numerical model for shallow-water flow with the assumption of hydrostatic pressure has been applied to oscillatory laminar flow around a conical island of small side slope (8°). The model is (mainly) second-order accurate in space and time with advection in conservation form. This form of model is effectively a boundary-layer solution with imposed depth-integrated continuity. To reproduce experimental conditions particular care was required for boundary conditions to account for seiching in the flume. The model predicted the wide range of wake structures observed experimentally and described in Part 1 (Lloyd, Stansby & Chen 2001), from symmetric with weak vortex pairing to complex vortex shedding.

The interaction of weak shock waves with small heterogeneities in gaseous media is studied. It is first shown that various linear theories proposed for this problem lead to pathological breakdowns or singularities in the solution near the wavefront and necessarily fail to describe this interaction. Then, a nonlinear small-disturbance model is developed. The nonlinear theory is uniformly valid and accounts for the competition between the near-sonic speed of the wavefront and the small variations of vorticity and sound speed in the heterogeneous media. This model is an extension of the transonic small-disturbance problem, with additional terms accounting for slight variations in the media. The model is used to analyse the propagation of the sonic-boom shock wave through the turbulent atmospheric boundary layer. It is found that, in this instance, the nonlinear model accounts for the observed behaviour. Various deterministic examples of interaction phenomena demonstrate good agreement with available experimental data and explain the main observed phenomena in Crow (1969).

We examine small deviations from axial symmetry in a solid-propellant rocket motor, and describe a ‘bath-tub-vortex’ effect, in which substantial axial vorticity is generated in a neighbourhood of the chamber centreline. The unperturbed flow field is essentially inviscid at modest Reynolds numbers, even at the chamber walls, as has long been known, but the inviscid perturbed flow is singular at the centreline, and viscous terms are required to regularize it. We examine perturbations sufficiently small that a linear analysis is valid everywhere (εRe small, where ε is a measure of the perturbation amplitude and Re is a Reynolds number), and larger perturbations in which a nonlinear patch is created near the centreline of radius O(√ε). Our results provide an explanation of swirl experimentally observed by others, and a cautionary note for those concerned with numerical simulations of these flows, whether laminar or turbulent.

Experiments were performed in a vertical channel to study the behaviour of a monodisperse bubble suspension for which the dual limit of large Reynolds number and small Weber number was satisfied. Measurements of the liquid-phase velocity fluctuations were obtained with a hot-wire anemometer. The gas volume fraction, bubble velocity, bubble velocity fluctuations and bubble collision rate were measured using a dual impedance probe. Digital image analysis was performed to quantify the small polydispersity of the bubbles as well as the bubble shape.

A rapid decrease in bubble velocity with bubble concentration in very dilute suspensions is attributed to the effects of bubble–wall collisions. The more gradual subsequent hindering of bubble motion is in qualitative agreement with the predictions of Spelt & Sangani (1998) for the effects of potential-flow bubble–bubble interactions on the mean velocity. The ratio of the bubble velocity variance to the square of the mean is O(0.1). For these conditions Spelt & Sangani predict that the homogeneous suspension will be unstable and clustering into horizontal rafts will take place. Evidence for bubble clustering is obtained by analysis of video images. The fluid velocity variance is larger than would be expected for a homogeneous suspension and the fluid velocity frequency spectrum indicates the presence of velocity fluctuations that are slow compared with the time for the passage of an individual bubble. These observations provide further evidence for bubble clustering.

The evolution and stability of two-dimensional, large-amplitude, non-hydrostatic internal wavepackets are examined analytically and by numerical simulations. The weakly nonlinear dispersion relation for horizontally periodic, vertically compact internal waves is derived and the results are applied to assess the stability of weakly nonlinear wavepackets to vertical modulations. In terms of Θ, the angle that lines of constant phase make with the vertical, the wavepackets are predicted to be unstable if [mid ]Θ[mid ] < Θc, where Θc = cos−1 (2/3)1/2 ≃ 35.3° is the angle corresponding to internal waves with the fastest vertical group velocity. Fully nonlinear numerical simulations of finite-amplitude wavepackets confirm this prediction: the amplitude of wavepackets with [mid ]Θ[mid ] > Θc decreases over time; the amplitude of wavepackets with [mid ]Θ[mid ] < Θc increases initially, but then decreases as the wavepacket subdivides into a wave train, following the well-known Fermi–Pasta–Ulam recurrence phenomenon.

If the initial wavepacket is of sufficiently large amplitude, it becomes unstable in the sense that eventually it convectively overturns. Two new analytic conditions for the stability of quasi-plane large-amplitude internal waves are proposed. These are qualitatively and quantitatively different from the parametric instability of plane periodic internal waves. The ‘breaking condition’ requires not only that the wave is statically unstable but that the convective instability growth rate is greater than the frequency of the waves. The critical amplitude for breaking to occur is found to be ACV = cot Θ (1 + cos2 Θ)/2π, where ACV is the ratio of the maximum vertical displacement of the wave to its horizontal wavelength. A second instability condition proposes that a statically stable wavepacket may evolve so that it becomes convectively unstable due to resonant interactions between the waves and the wave-induced mean flow. This hypothesis is based on the assumption that the resonant long wave–short wave interaction, which Grimshaw (1977) has shown amplifies the waves linearly in time, continues to amplify the waves in the fully nonlinear regime. Using linear theory estimates, the critical amplitude for instability is ASA = sin 2Θ/(8π2)1/2. The results of numerical simulations of horizontally periodic, vertically compact wavepackets show excellent agreement with this latter stability condition. However, for wavepackets with horizontal extent comparable with the horizontal wavelength, the wavepacket is found to be stable at larger amplitudes than predicted if Θ [lsim ] 45°. It is proposed that these results may explain why internal waves generated by turbulence in laboratory experiments are often observed to be excited within a narrow frequency band corresponding to Θ less than approximately 45°.

The dynamics of a thick layer of viscous liquid flowing down a thin vertical fibre is investigated. Three qualitatively different regimes of the interfacial patterns in the form of beads were observed experimentally. Two typical regimes at relatively small flow rate are described reasonably well by the creeping-flow model equation proposed here.

A laboratory study of the exchange of two fluids of different density through a constant-width channel with an underwater sill has enabled us to study Holmboe's instability in greater detail than has been possible in mixing-layer experiments. The internal hydraulics of the exchange flow are such that we have been able to observe the initiation of instability, the development and behaviour of both symmetric and asymmetric Holmboe instabilities, and the suppression of the instability at bulk Richardson numbers above about 0.7. A number of stability criteria resulting from previous numerical investigations have been verified experimentally. Our laboratory measurements are consistent with theoretical predictions of wave speed and wavenumber.