We derive effective equations for wave propagation in a bubbly liquid in a linearized low-frequency regime by a multiple-scale method. The effective equations are valid for finite volume fraction. For periodic bubble configurations, effective equations uniformly valid for small volume fraction are obtained. We compare the results to the ones obtained in a previous paper (Caflisch et al. 1985) for a nonlinear theory at small volume fraction.

A simple model is proposed for the two-dimensional injection of irrotational inviscid fluid from a slot into a free stream. In a certain range of values of the ratio of free-stream to injection total heads, the film thickness satisfies a nonlinear integral equation whose solution enables the mass flow in the film to be found. Some experiments are described which both agree with this theory when it is relevant and indicate its deficiencies at other values of the total head ratio.

This paper analyses effects of chemical reactions on reflected-shock flow fields in shock tubes. The method of linearized characteristics is applied to analyse gasdynamic disturbances due to chemical reactions. The analysis treats cases where combustible gas is highly diluted in inert gas, and assumes that flows are one-dimensional and that upstream flows in front of the reflected-shock waves are in the frozen state. The perturbed gasdynamic properties in the reflected-shock flow fields are shown to be expressible mainly in terms of a heat-release function for combustion process. In particular, simple relations are obtained between the heat-release function and the physical properties at the end wall of a shock tube. As numerical examples of the analysis, the present formulation is applied to calculate gasdynamic properties in the reflected-shock region in a H2–O2–Ar mixture. Procedures are demonstrated for calculation of the heat-release function by numerically integrating rate equations for chemical species. The analytical results are compared with rigorous solutions obtained numerically by use of a finite-difference method. It is shown that the formulation can afford exact solutions in cases where chemical behaviours are not essentially affected by gasdynamic behaviours. When the induction time of the combustion process is reduced to some extent owing to gasdynamic disturbances, some discrepancies appear between analytical results and rigorous solutions. An estimate is made of the induction-time reduction, and a condition is written down for applicability of the analysis.

The influence of weak periodic wall undulations on the structure of turbulent pipe flow has been studied in three ways: measurements in air flow using pressure probes and hot-wire techniques, visualizations in water flow and numerical predictions based on a turbulence (k-ε) model. The flows at Reynolds numbers of 30000 and 115000 have been particularly investigated. The flow characteristics proved to be very different from those observed in a straight pipe. Calculations and experiments agree well for the mean- and turbulent-energy fields; however the detailed behaviour of some local quantities such as anisotropy of the Reynolds stress is not well predicted particularly in the crest region. So the performances and the limitations of classical closure have been appraised. The existence of an unsteady reverse-flow region downstream of every crest suggested by measurements and calculations has been clearly confirmed by visualizations in water flow.

A rapidly scanning one-velocity-component directionally sensitive fringe-type laser-Doppler anemometer which scans the measurement volume perpendicular to the optical axis of the transmitting optics was used to investigate the flow structure of the steady freestream separated turbulent boundary layer of Simpson, Chew & Shivaprasad (1981a). Space–time correlations were obtained for the first time in a separated turbulent boundary layer and showed that the integral lengthscale Ly for the large eddies grows in size towards detachment, although the ratio of this lengthscale to the boundary-layer thickness remains constant. Results also indicate local dependence of the backflow on the middle and outer regions of the boundary layer at a given instant in time.

Early stages of unsteady viscous flows around a circular cylinder at Reynolds numbers of 3 × 103 and 9.5 × 103 are analysed numerically by direct integration of the Navier–Stokes equations – a fourth-order finite-difference scheme is used for the resolution of the stream-function equation and a second-order one for the vorticity-transport equation. Evolution with time of the flow structure is studied in detail. Some new phenomena are revealed and confirmed by experiments.

The influence of the grid systems and the downstream boundary conditions on the flow structure and the velocity profiles is reported. The computed results are compared qualitatively and quantitatively with experimental visualization and measurements. The comparison is found to be satisfactory.

Red blood cells or artificial vesicles may be conveniently represented by capsules, i.e. liquid droplets surrounded by deformable membranes. The aim of this paper is to assess the importance of viscoelastic properties of the membrane on the motion of a capsule freely suspended in a viscous liquid subjected to shear flow. A regular perturbation solution of the general problem is obtained when the particle is initially spherical and undergoing small deformations. With a purely viscous membrane (infinite relaxation time) the capsule deforms into an ellipsoid and has a continuous flipping motion. When the membrane relaxation time is of the same order as the shear time, the particle reaches a steady ellipsoidal shape which is oriented with respect to streamlines at an angle that varies between 45° and 0°, and decreases with increasing shear rates. Furthermore it is predicted that the deformation reaches a maximum value, which is consistent with experimental observations of red blood cells.

The convective behaviour of xenon gas in a vertical thermally conducting cylinder (height/radius = 6) heated from below was investigated. Convectively induced temperature fluctuations in the gas were analysed with digital signal-processing techniques over a range of Rayleigh number 0 ≤ Ra [lsim ] 2300. Quiescent, steady-state, periodic and weakly turbulent convective regimes were characterized. Bistability of steady states (mode switching) was observed in the range 400 [lsim ] Ra [lsim ] 700. At Ra = 1550 a strictly periodic flow developed. With increasing Ra two additional incommensurate frequencies appeared, leading to ‘turbulence’ at Ra ≈ 2000. This turbulence, characterized by a broadband power spectrum, intermittently showed periodic flow. A periodic window with a period-doubling sequence appeared between 2100 [lsim ] Ra [lsim ] 2200. The spectral features of this sequence can be followed into the broad band noise at higher Ra. Although these experiments were conducted quasistatically, a strong hysteresis was observed with decreasing Ra. Furthermore, it was demonstrated that the sequence of convective regimes can be fundamentally altered by minor perturbations (self-heating) from the flow sensors.

Measurements of the mean and fluctuating velocities have been obtained with pressure and hot-wire probes in the attached boundary layers and wakes of two airfoil models at a low Mach number. The first model is a conventional airfoil at zero incidence and the second an advanced supercritical airfoil at an angle of attack of 4°. The mean-flow and Reynolds-stress data and related quantities are presented with emphasis on the trailing-edge region. The results indicate that the flow around the conventional airfoil is a minor perturbation of a symmetric flat-plate flow with small wake curvature and weak viscous–inviscid interaction. The flow around the supercritical airfoil is in considerable contrast with strong streamwise pressure gradients, non-negligible normal pressure gradients, and large surface and streamline curvatures of the trailing-edge flow. The near wake is strongly curved and intense mixing occurs between the retarded upper-surface boundary layer and strongly accelerated lower-surface boundary layer.

Countercurrent flow may be induced by opposing buoyancy forces associated with compositional gradients and thermal gradients within a fluid. The occurrence and structure of such flows is investigated by solving the double-diffusive boundary-layer equations for steady laminar convection along a vertical wall of finite height. Non-similar solutions are derived using the method of matched asymptotic expansions, under the restriction that the Lewis and Prandtl numbers are both large. Two sets of asymptotic solutions are constructed, assuming dominance of one or the other of the buoyancy forces. The two sets overlap in the central region of the parameter space; each set matches up with neighbouring unidirectional similarity solutions at the respective borderlines of incipient counterflow.

Interaction between the buoyancy mechanisms is controlled by their relative strength R and their relative diffusivity Le. Flow in the outer thermal boundary layer deviates from single-diffusive thermal convection, depending upon the magnitude of the parameter RLe. Flow in the inner compositional boundary layer deviates from single-diffusive compositional convection, depending upon the magnitude of $RLe^{\frac{1}{3}}$.

The dynamics of a perfectly symmetric type-A multiple drop is studied. Up to first order in Reynolds number a force balance predicts the size ratios of the two constituents of such a drop to be unique for each system. Inertial effects are shown (a) to be destabilizing and (b) to exclude the possibility of obtaining perfectly concentric type-A droplets in a diffusion column. This latter conclusion is strengthened further by the sedimentation results.

A vortex-induced unsteady separation was investigated experimentally in the laminar boundary layer produced by an axisymmetric jet impinging normally onto a flat plate. By forcing the air jet, primary ring vortices were periodically generated in the jet shear layer. Phase-locked flow visualization showed that the wall-jet boundary layer separated periodically and evolved into a secondary vortex counter rotating with respect to the primary vortex. The unsteady separation is induced by the primary vortex and moves downstream in the radial mean-flow direction. Phase-averaged hot-wire measurements using a parallel-wire sensor in the vicinity of the unsteady separation provided data for locating the onset of separation in space and time. The data revealed that the unsteady separation originated from a local shear layer which was initiated by the unsteady adverse pressure gradient produced by the primary vortex.

The three-dimensional steady laminar-boundary-layer equations have been cast in the appropriate form for semisimilar solutions, and it is shown that in this form they have the same structure as the semisimilar form of the two-dimensional unsteady laminar-boundary-layer equations. This similarity suggests that there may be a new type of singularity in solutions to the three-dimensional equations: a singularity that is the counterpart of the Stewartson singularity in certain solutions to the unsteady boundary-layer equations.

A family of simple three-dimensional laminar boundary-layer flows has been devised and numerical solutions for the development of these flows have been obtained in an effort to discover and investigate the new singularity. The numerical results do indeed indicate the existence of such a singularity. A study of the flow approaching the singularity indicates that the singularity is associated with the domain of influence of the flow for given initial (upstream) conditions as is prescribed by the Raetz influence principle.

Computations of two-dimensional solutions of the Navier–Stokes equations are carried out for finite-amplitude waves on steady unidirectional flow. Several cases are considered. The numerical method employs pseudospectral techniques in the streamwise direction and finite differences on a stretched grid in the transverse direction, with matching to asymptotic solutions when unbounded. Earlier results for Poiseuille flow in a channel are re-obtained, except that attention is drawn to the dependence of the minimum Reynolds number on the physical constraint of constant flux or constant pressure gradient. Attempts to calculate waves in Couette flow by continuation in the velocity of a channel wall fail. The asymptotic suction boundary layer is shown to possess finite-amplitude waves at Reynolds numbers orders of magnitude less than the critical Reynolds number for linear instability. Waves in the Blasius boundary layer and unsteady Rayleigh profile are calculated by employing the artifice of adding a body force to cancel the spatial or temporal growth. The results are verified by comparison with perturbation analysis in the vicinity of the linear-instability critical Reynolds numbers.

In two previous papers (Pullin & Grimshaw 1983a, b) we studied the wave profile and other properties of finite-amplitude interfacial progressive waves in a two-layer fluid. In this and the following paper (Pullin & Grimshaw 1985) we discuss the stability of these waves to small perturbations. In this paper we obtain anatytical results for the long-wavelength modulational instability of small-amplitude waves. Using a multiscale expansion, we obtain a nonlinear Schrödinger equation coupled to a wave-induced mean-flow equation to describe slowly modulated waves. From these coupled equations we determine the stability of a plane progressive wave. Our results are expressed by determining the instability bands in the (p, q)-plane, where (p, q) is the modulation wavenumber, and are obtained for a range of values of basic density ratio and undisturbed layer depths.

In the preceding paper (Grimshaw & Pullin 1985) we discussed the long-wavelength modulational instability of interfacial progressive waves in a two-layer fluid. In this paper we complement our analytical results by numerical results for the linearized stability of finite-amplitude waves. We restrict attention to the case when the lower layer is infinitely deep, and use the Boussinesq approximation. For this case the basic wave profile has been calculated by Pullin & Grimshaw (1983a, b). The linearized stability problem for perturbations to the basic wave is solved numerically by seeking solutions in the form of truncated Fourier series, and solving the resulting eigenvalue problem for the growth rate as a function of the perturbation wavenumber. For small or moderate basic wave amplitudes we show that the instabilities are determined by a set of low-order resonances. The lowest resonance, which contains the modulational instability, is found to be dominant for all cases considered. For higher wave amplitudes, the resonance instabilities are swamped by a local wave-induced Kelvin–Helmholtz instability.

We describe flow-visualization experiments and theory on the two-dimensional unsteady flow of an incompressible fluid in a channel with a time-dependent indentation in one wall. There is steady Poiseuille flow far upstream, and the indentation moves in and out sinusoidally, its retracted position being flush with the wall. The governing parameters are Reynolds number Re, Strouhal number (frequency parameter) St and amplitude parameter ε (the maximum fraction of the channel width occupied by the indentation); most of the experiments were performed with ε ≈ 0.4. For St ≤ 0.005 the flow is quasi-steady throughout the observed range of Re (360 < Re < 1260). For St > 0.005 a propagating train of waves appears, during every cycle, in the core flow downstream of the indentation, and closed eddies form in the separated flow regions on the walls beneath their crests and above their troughs. Later in the cycle, a second, corotating eddy develops upstream of the first in the same separated-flow region (‘eddy doubling’), and, later still, three-dimensional disturbances appear, before being swept away downstream to leave undisturbed parallel flow at the end of the cycle. The longitudinal positions of the wave crests and troughs and of the vortex cores are measured as functions of time for many values of the parameters; they vary with St but not with Re. Our inviscid, long-wavelength, small-amplitude theory predicts the formation of a wavetrain during each cycle, in which the displacement of a core-flow streamline satisfies the linearized Kortewegde Vries equation downstream of the indentation. The waves owe their existence to the non-zero vorticity gradient in the oncoming flow. Eddy formation and doubling are not described by the theory. The predicted positions of the wave crests and troughs agree well with experiment for the larger values of St used (up to 0.077), but less well for small values. Analysis of the viscous boundary layers indicates that the inviscid theory is self-consistent for sufficiently small time, the time of validity increasing as St increases (for fixed ε).

The Navier-Stokes equations are solved numerically for the steady separated flow past a normal flat plate for Reynolds numbers in the range 0.1 ≤ R ≤ 20. Eddy dimensions together with the position of the vortex centre are presented and compared with the few other estimates and predictions available. Streamlines and equivorticity lines are also given. The main result of interest is the extremely good comparison with experimental results over this range of Reynolds numbers. The method of solution is based on an artificial time-dependent procedure using a distorted time. Results are given only for the steady-state flow.

The spatial interactions between a fundamental instability wave and its subharmonics in a turbulent round jet are studied for ‘natural’ or forced exit conditions. Time-averaging and conditional-averaging techniques are used to split each flow component into a mean one, a random turbulence one and several wave-like coherent-structure components at fundamental and subharmonic frequencies. The energy equations for the flow components are derived and integrated across the jet. Shape assumptions regarding the radial distributions of each flow component are used to obtain a set of nonlinear ordinary differential equations representing the energy interactions between the coherent components, while interacting with the mean flow and with the background turbulence. Vortex pairing is viewed here as occurring when the subharmonic absorbs energy from the fundamental and from the mean flow and exceeds the fundamental's level to become the dominant instability component. At the proper initial phase difference between the subharmonic and fundamental only the first subharmonic was found to amplify if the fundamental Strouhal number based on diameter is in the range of 0.6–1.0. For higher Strouhal numbers, several subharmonics can amplify. The pairing location moves closer to the nozzle exit with increasing excitation Strouhal number. The time-averaged coherent Reynolds stresses exhibit regions of sign change, indicating a reversal in the direction of energy transfer between the mean flow and the coherent components.

Synthetic turbulent boundary layers were constructed on a flat plate by generating systematic moving patterns of turbulent spots in a laminar flow. The experiments were carried out in a wind tunnel at a Reynolds number based on plate length of 1.7 × 106. Spots were generated periodically in space and time near the leading edge to form a regular hexagonal pattern. The disturbance mechanism was a camshaft that displaced small pins momentarily into the laminar flow at frequencies up to 80 Hz. The main instrumentation was a rake of 24 single hot wires placed across the flow in a line parallel to the surface. The main measured variable was local intermittency; i.e. the probability of observing turbulent flow at a particular point in space and time. The results are reported in numerous (x, z, t)-diagrams showing the evolution of various synthetic flows along the plate. The dimensionless celerity or phase velocity of the large eddies was found to be very nearly 0.88, independent of eddy scale. All patterns with sufficiently small scales eventually showed loss of coherence as they moved downstream. A novel phenomenon called eddy transposition was observed in several flows that contained appreciable laminar regions. The original large eddies were replaced by new eddies at new positions, intermediate to the original ones, while preserving the hexagonal pattern. The present results, together with some empirical properties of a turbulent spot, were used to estimate the best choice of scales for constructing a synthetic boundary layer suitable for detailed study as a model for a natural flow. The values recommended are: spanwise period/thickness ≈ 2.5, streamwise period/thickness ≈ 8.0.