In this paper, we apply two theoretical turbulence models, DIA and the recent GISS model, to study properties of a turbulent channel flow. Both models provide a turbulent kinetic energy spectral function E(k) as the solution of a nonlinear equation; the two models employ the same source function but different closures. The source function is characterized by a rate ns(k) which is derived from the complex eigenvalues of the Orr–Sommerfeld equation in which the basic flow is taken to be of a Poiseuille type. The Orr–Sommerfeld equation is solved for a variety of Reynolds numbers corresponding to available experimental data. A physical argument is presented whereby the central line velocity characterizing the basic flow, U0L, is not to be identified with the U0 appearing in the experimental Reynolds number. A renormalization is suggested which has the effect of yielding growth rates of magnitude comparable with those calculated by Orszag & Patera based on their study of a secondary instability. From the practical point of view, this renormalization frees us from having to solve the rather time-consuming equations describing the secondary instability. This point is discussed further in §13. In the present treatment, the shear plays only the role of a source of energy to feed the turbulence and not the possible additional role of an interaction between the shear of the mean flow and the eddy vorticity that would give rise to resonance effects when the shear is equal to or larger than the eddy vorticities. The inclusion of this possible resonance phenomenon, which is not expected to affect the large-eddy behaviour and thus the bulk properties, is left for a future study. The theoretical results are compared with two types of experimental data: (a) turbulence bulk properties, table 4, and (b) properties that depend strongly on the structure of the turbulence spectrum at low wavenumbers (i.e. large eddies), tables 5 and 6. The latter data are taken from recent experiments measuring the changes in the propagation of an electromagnetic wave through a turbulent channel flow. The fluctuations in the refractive index of the turbulent medium are thought to be due to pressure fluctuations whose spectral function Π(k) is contributed mostly by the interaction between the mean flow and the turbulent velocity. The spectrum Π(k) must be computed as a function of the wavenumber k, the position in the channel x2, and the width of the channel Δ. The only existing analytical expression for Π(k), due to Kraichnan, cannot be used in the present case because it applies to the case x2 = 0 and Δ = ∞, which corresponds to the case of a flat plate, not a finite channel. A general expression for Π(k, x2; Δ) is derived here for the first time and employed to calculate the fraction of incoherent radiation scattered out of a coherent beam. In §11, we treat anisotropy and show how to extend the previous results to include an arbitrary degree of anisotropy α in the sizes of the eddies. We show that the theoretical one-dimensional spectra yield a better fit to the data for a degree of anisotropy (α ≈ 4) that is within the range of experimental values. We also extend the expression for Π(k, x2; Δ) to Π(k, x2; Δ, α) and compute the pressure fluctuations for different values of α. Similarly, we evaluate the fraction of electromagnetic energy scattered by an anisotropic turbulent flow and find a good fit to the laboratory data for a value of α ≈ 4–6. Scaling formulae for the scattered fraction are presented in §12. These formulae reproduce the calculated results, both with and without the addition of anisotropy, to better than 5%.

Theoretical problems however remain which will require further study: among them, lack of backscatter (i.e. the transfer of energy from large to small wavenumbers) in the GISS model, possible resonance effects between the shear and eddy vorticity, behaviour of the one-dimensional spectral function at low wavenumbers, and the role of the secondary instability. These topics are now under investigation.

The starting flows past a two-dimensional oscillating and translating airfoil are investigated by visualization experiments and numerical calculations. The airfoil, elliptic in cross-section, is set in motion impulsively and subjected simultaneously to a steady translation and a harmonic oscillation in pitch. The incidence of the airfoil is variable between 0° and 45° and the Reynolds number based on the chord length is between 1500 and 10000. The main object of the present study is to reveal some marked characteristics of the unsteady vortices produced from the oscillating airfoil set at large incidences in excess of the static stall angle. Another purpose is to examine, in some detail, the respective and combined effects of the major experimental parameters on the vortex wake development. It is shown that, in general, the dominant parameter of the flow is the reduced frequency not only when the airfoil oscillates at incidences close to the static stall angle but also at larger incidences. It is also demonstrated that, as the pitching frequency is increased, the patterns of the vortex wake are dependent on the product of the reduced frequency and the amplitude rather than on the frequency itself. It is noted that the combined effect of a high reduced frequency and a large amplitude can give rise to cyclic superposition of leading-edge vortices from which a gradually expanding standing vortex is developed on the upper surface.

Sound propagation in a dilute bubble–liquid mixture is studied by means of the Kramers–Kronig relationships, which relate the real and imaginary parts of the general susceptibility of a linear medium. These relationships are adopted for the case of acoustic waves, where they become coupled integral equations. A simple but approximate procedure is used to obtain from these equations the phase speed of sound waves for the case when the attenuation coefficient is independently known. The procedure can be used to obtain the speed of propagation of sound waves in acoustic media having internal dissipation, but is here applied only to fluids containing radially pulsating bubbles. Approximate results for the speed of propagation and for the attenuation per wavelength are obtained for this case on the basis of a first-order estimate for the attenuation coefficient. These results are the same as those derived previously on the basis of model equations for bubbly liquids. They therefore provide additional support for those equations, while indicating some of their limitations.

Experiment shows that the steady axisymmetric flow past a sphere becomes unstable in the range 120 < Re < 300. The resulting time-dependent non-axi-symmetric flow gives rise to non-axisymmetric vortex shedding at higher Reynolds numbers. The present work reports a computational investigation of the linear stability of the steady axisymmetric base flow. We use a spectral technique to represent the base flow. We then perform a linear stability analysis with respect to axisymmetric and non-axisymmetric disturbances. A spectral technique similar to that employed in the base-flow calculation is used to solve the linear-disturbance equations in stream-function form (for axisymmetric disturbances), and in a modified primitive variables form (for non-axisymmetric disturbances). The analysis shows that the axisymmetric base flow undergoes a Hopf bifurcation at Re = 175.1, with the critical disturbance having azimuthal wavenumber m = 1, and dimensionless frequency (non-dimensionalized as a Strouhal number, St) 0.0955. The critical Re, azimuthal mode number, and St are favourably compared to previous experimental work.

Mean flow and turbulence measurements have been made in three-dimensional turbulent boundary layers in curved ducts, simulating adverse pressure gradients on two ‘infinite’ swept curved wing surfaces with concave and convex curvature respectively. The ratio of the initial boundary-layer thickness to the surface radius of curvature in both cases is approximately 0.01, the value used in the earlier two-dimensional turbulent boundary-layer studies on the effects of concave and convex curvature by Hoffmann, Muck & Bradshaw (1985) and Muck, Hoffmann & Bradshaw (1985) respectively. The pressure-driven crossflow has nearly the same streamwise distribution as in the ‘infinite’ swept flat-surface experiment of Bradshaw & Pontikos (1985), which used a similar duct. The results of the present study show that the coupled effects of mean flow three-dimensionality and prolonged mild surface curvature of either sign have rather a weak influence on the turbulence structure, unlike the significant influence of the above extra strain rates when applied individually. In the concave case, the effect of the crossflow appears to oppose the destabilizing effect of curvature in addition to suppressing spanwise wavy inhomogeneities In contrast, the weak combined influence of convex curvature and crossflow, both of which, separately, tend to attenuate turbulence, implies that the interaction between the two effects is grossly nonlinear. Implications of the present results for turbulence modelling are briefly discussed.

The behaviour of concentric double emulsion droplets in linear flows is examined analytically, for the case when both fluid–fluid interfaces remain nearly spherical, and numerically, for the effect of finite interface deformation. The theoretical analysis is used to calculate the velocity fields interior and exterior to the particle, the first effects of flow-induced deformation, and the effective viscosity of a dilute emulsion of compound droplets. The numerical simulations allow for a complete investigation of the finite deformation of both the outer drop and the encapsulated particle. For concentric multiphase particles, there appear to be two distinct mechanisms of globule breakup: (i) continuous extension of the globule corresponding to non-existence of a steady particle shape or (ii) contact of the two interfaces at the globule centre, owing to incompatibility of the steady inner and outer interface shapes, even though the globule is only modestly deformed. Finally, the effect of different flow-types, i.e. uniaxial or biaxial extensional flows, is shown, in one example, to suggest breakup of the inner droplet even though the outer droplet maintains a steady shape.

The structures of the flow induced by a circular cylinder performing sinusoidal oscillations in a fluid at rest are investigated by means of flow visualization. The experiments are carried out at Keulegan–Carpenter numbers between 1.6 and 15 and at Stokes numbers between 5 and 160. Above a certain value of Keulegan–Carpenter number, depending on the Stokes number, some asymmetry appears in the flow separation and the associated vortex development behind the cylinder. The two vortices that are developed in a half cycle differ in strength and may be convected in different directions. This results in a fascinating set of flow patterns. Eight different regimes of flow can be identified within the ranges of Keulegan–Carpenter number and Stokes number studied. Furthermore, most of the resulting flows show a three-dimensional instability along the axis of the cylinder. Measurements of the wavelength of these disturbances are presented.

Oscillating flows over periodic ripples are of practical as well as scientific interest because of their relevance to beach processes. When either the ripples are sufficiently steep or the amplitude of ambient oscillations large, streamlines of a viscous flow are no longer parallel to the ripple surface. Circulation cells are formed which can help redistribute suspended sediments. Here we study theoretically these cells for a low-viscosity fluid such as pure water over rigid ripples. In particular we have calculated cells whose dimensions are as large as the ripple wavelength and therefore represent viscous effects far above the usual Stokes boundary layer. An idea of Stuart which was originated for stationary mean circulations around a cylinder is extended here. For large ambient amplitude, large oscillating vortices drifting with the ambient flow are found by seeking the stationary cells in a moving coordinate system.

The interaction of a Kármán vortex street with an elliptical edge is investigated experimentally. Basic types of interaction, as a function of scale and transverse displacement of the incident vortex street, are revealed using flow visualization. Unsteady pressure fields induced by these interactions are measured by a phase-averaging technique and correlated with the visualized flow patterns for basic classes of interactions.

For a generic vortex–edge interaction, measurements of the phase-averaged velocity field allow construction of streamlines and vorticity contours showing the details of the interaction, including distortion of the vortical structures near the edge. The pressure field is calculated from the measured velocity field and interpreted in relation to the vortical structures.

Simulation of flow visualization using the measured velocity field demonstrates possible misinterpretations related to the underlying vorticity field.

A turbulent vortex ring having a relatively thin core is formed in water by a momentary jet discharge from an orifice in a submerged plate. The necessary impulse is provided by a pressurized reservoir and is controlled by a fast programmable solenoid valve. The main aim of the research is to verify the similarity properties of the mean flow, as defined by ensemble averaging, and to find the distribution of mean vorticity, turbulent energy, and other quantities in the appropriate non-steady similarity coordinates. The velocity field of the vortex is measured for numerous realizations with the aid of a two-channel tracking laser-Doppler velocimeter. The problem of dispersion in the trajectories of the individual rings is overcome by development of a signature-recognition technique in two variables. It is found that the turbulence intensity is largest near the vortex core and that at least the radial component is not negligible in the near wake. The slow growth of the ring structure is controlled by a slight excess of entrainment over de-entrainment. An important inference is that the growth process and the process of turbulence production probably involve secondary vortices wrapped around the core in azimuthal planes.

This investigation describes the effects of an adverse pressure gradient on a flat plate supersonic turbulent boundary layer (Mf ≈ 2.9, βx ≈ 5.8, Reθ, ref ≈ 75600). Single normal hot wires and crossed wires were used to study the Reynolds stress behaviour, and the features of the large-scale structures in the boundary layer were investigated by measuring space–time correlations in the normal and spanwise directions. Both the mean flow and the turbulence were strongly affected by the pressure gradient. However, the turbulent stress ratios showed much less variation than the stresses, and the essential nature of the large-scale structures was unaffected by the pressure gradient. The wall pressure distribution in the current experiment was designed to match the pressure distribution on a previously studied curved-wall model where streamline curvature acted in combination with bulk compression. The addition of streamline curvature affects the turbulence strongly, although its influence on the mean velocity field is less pronounced and the modifications to the skin-friction distribution seem to follow the empirical correlations developed by Bradshaw (1974) reasonably well.

Velocity and passive-scalar spectra for turbulent fields generated by a forced three-dimensional simulation with 1283 grid points and Taylor-microscale Reynolds numbers up to 83 are shown to have convective and diffusive spectral regimes. One-and three-dimensional spectra are compared with experiment and theory. If normalized by the Kolmogorov dissipation scales and scalar dissipation, velocity spectra and scalar spectra for given Prandtl numbers collapse to single curves in the dissipation regime with exponential tails. If multiplied by k⅗ the velocity spectra show an anomalously high Kolmogorov constant that is consistent with low Reynolds number experiments. When normalized by the Batchelor scales, the scalar spectra show a universal dissipation regime that is independent of Prandtl numbers from 0.1 to 1.0. The time development of velocity spectra is illustrated by energy-transfer spectra in which distinct pulses propagate to high wavenumbers.

The mathematical formulation of the linearized potential theory for a slowly translating body undergoing oscillations in infinitely deep water is derived based on a perturbation series in terms of forward speed. It is shown that the hydrodynamic force on the oscillating body can be obtained from the solution of the velocity potential without forward speed, if second-order terms in forward speed are neglected. An example of a submerged circular cylinder is discussed. The numerical results are compared with the general solution of the linearized potential theory by a coupled finite-element method (Wu & Eatock Taylor 1987) which is not restricted to low forward speeds. Very good agreement is found. The nonlinear effect of the steady potential on the hydrodynamic forces is also discussed and is illustrated for a floating semicircular cylinder.

Henry & Roux have recently conducted extensive numerical studies on the interaction of Soret separation with convection in cylindrical geometry. Many of their solutions exhibit parallel flow away from endwalls. We show that their parallel flow results can be matched by closed-form solutions. We find that solutions are non-unique in some parameter regions. Disappearance of one branch of solutions correlates with a sudden transition of Henry & Roux's results from a separated to a well-mixed flow.

Experiments by Huppert (1982) have demonstrated that a finite volume of fluid placed on an inclined plane will elongate into a thin sheet of fluid as it slides down the plane. If the fluid is initially placed uniformly across the plane, the sheet retains its two-dimensionality for some time, but when it has become sufficiently long and thin, the leading edge develops a spanwise instability. A similarity solution for this motion was derived by Huppert, without taking account of the edge regions where surface tension is important. When these regions are examined, it is found that the conditions at the edges can be satisfied, but only when the singularity associated with the moving contact line is removed. When the sheet is sufficiently elongated, the profile of the free surface shows an upward bulge near the leading edge. It is suggested that the observed instability of the shape of the leading edge is a result of the dynamics of the fluid in this bulge. The related problem of a ridge of fluid sliding down the plane is examined and found to be linearly unstable. The spanwise lengthscale of this instability is, however, dependent on the width of the channel occupied by the fluid, which is at variance with the observed nature of the instability. Preliminary numerical solutions for the nonlinear development of a small disturbance to the position of a straight leading edge show the incipient development of a finger of fluid with a width that does not depend on the channel size, in accordance with the observations.

A new Lyapunov stability condition is formulated for the shallow-water equations, using a gauge-variable formalism. This sufficient condition is derived for the class of perturbations that conserve the total mass. It is weaker than existing stability criteria, i.e. it applies to a wider class of flows. Formal stability to infinitesimally small perturbations of arbitrary shape is obtained for two classes of large-scale geophysical flows: pseudo-eastward flow with constant shear, and localized coherent structures of modon type.

The 245th Euromech Colloquium, on the effect of background rotation on fluid motions, was held in Cambridge from 10–13 April 1989 with the two authors acting as chairmen. There were sixty-five participants with widely different backgrounds. The striking feature of the Colloquium was the recent advances on nonlinear processes. A wide range of nonlinear phenomena was presented and particular emphasis was on the formation and dynamics of coherent vortices. The similarities between the processes in rotating, curved and stratified flows, which lead to anisotropic motions and long-lived coherent structures alongside linear wave motions was a feature of many of the presentations. Fifty-one papers were presented, covering engineering, geophysical and astrophysical applications of rotating fluids. These papers are summarized in this report with the purpose of giving an up-to-date view of current research in rotating fluids.

Landau–Ginzburg equations are derived and used to study the three-dimensional stability of convection in a layered porous medium of infinite horizontal extent. Criteria for the stability of convection with banded or square planform are determined and results are presented for two-layer and symmetric three-layer systems. In general the neutral curve is uni-modal and parameter space is divided into regions where either rolls or square cells are stable. For certain ranges of parameters, however, the neutral curve is bimodal and there exists a locus of parameters where two modes with different wavenumbers have simultaneous onset.

In reviewing the current status of our understanding of the mechanisms underlying wind-wave generation, it is apparent that existing theories and models are not applicable to situations where the sea surface is disturbed by breaking waves, and that the available experimental data on this question are sparse. In this context, this paper presents the results of a detailed study of the effects of wave breaking on the aerodynamic surface pressure distribution and consequent wave-coherent momentum flux, as well as its influence on the total wind stress.

Two complementary experimental configurations were used to focus on the details and consequences of the pressure distribution over breaking waves under wind forcing. The first utilized a stationary breaking wave configuration and confirmed the presence of significant phase shifting, due to air flow separation effects, between the surface pressure and surface elevation (and slope) distributions over a range of wind speeds. The second configuration examined the pressure distribution, recorded at a fixed height above the mean water surface just above the crest level, over short mechanically triggered waves which were induced to break almost continuously under wind forcing. This allowed a very detailed comparison of the form drag for actively breaking waves and for waves of comparable steepness just prior to breaking (‘incipiently’ breaking waves). For these propagating steep-wave experiments, the pressure phase shifts and distributions closely paralleled the stationary configuration findings. Moreover, a large increase (typically 100%) in the total windstress was observed for the breaking waves, with the increase corresponding closely to the comparably enhanced form drag associated with the actively breaking waves.

In addition to further elucidating some fundamental features of wind-wave interactions for very steep wind waves, this paper provides a useful data set for future model calculations of wind flow over breaking waves. The results also provide the basis for a parameterization of the wind input source function applicable for a wave field undergoing active breaking, an important result for numerical modelling of short wind waves.

A generalized lifting-line theory is developed in inviscid, incompressible, steady flow for curved, swept wings of large aspect ratio. It is shown in this paper that by using the integral formulation of the problem instead of the partial differential equation formulation, it is possible to circumvent the algebraic complications encountered by the previous approaches using the method of the matched asymptotic expansions. At each approximation order the problem is reduced to inverting a classical Carleman type integral equation. The asymptotic solution in terms of circulation is found up to A−1 and A−1 In (A−1). It is very convenient for illustrating the major three-dimensional effects induced on the flow by curvature and yaw angle. The concept of the finite part integrals, introduced by Hadamard (1932), is shown to be very useful for handling elegantly singularities like 1/x|x| or 1/|x| which occur in the course of our developments. Comparisons of the new, simple approach with lifting-surface theories reveal excellent agreements in terms of circulation. Furthermore, a consistent calculation of the three components of the total force acting on the wing is done in the lifting-line context without re-introducing the inner geometry of the wing.