The Tollmien-Schlichting waves appearing as a result of instability of laminar flows develop a three-dimensional configuration as the amplitude becomes large enough. A new explanation of this experimentally observed phenomenon is attempted on the basis of a resonance theory. It is shown that the existence of two-dimensional waves with finite amplitude can induce three-dimensional distortion with spanwise periodicity of the mean-flow field. Under a certain condition for resonance, the distortion grows, in proportion to the product of time and an exponential function of time, up to quite a large magnitude, and consequently interacts with the Tollmien-Schlichting waves to yield new three-dimensional travelling waves with the same streamwise wavenumber as the two-dimensional waves, and with the same spanwise wavenumber as the mean flow. The resulting flow field is of the peakvalley-splitting type, as observed often in experiments. The growth rate of the three-dimensional part in the mean flow depends significantly upon values of the spanwise wavenumber, suggesting that there is a preferred range of spanwise periodicity in the three-dimensional development of unstable laminar flows.

Two different numerical methods are used to demonstrate the existence of and calculate non-symmetric gravity waves on deep water. It is found that they appear via spontaneous symmetry-breaking bifurcations from symmetric waves. The structure of the bifurcation tree is the same as the one found by Zufiria (1987) for waves on water of finite depth using a weakly nonlinear Hamiltonian model. One of the methods is based on the quadratic relations between the Stokes coefficients discovered by Longuet-Higgins (1978a). The other method is a new one based on the Hamiltonian structure of the water-wave problem.

The interaction of a plane weak shock wave with a single discrete gaseous inhomogeneity is studied as a model of the mechanisms by which finite-amplitude waves in random media generate turbulence and intensify mixing. The experiments are treated as an example of the shock-induced Rayleigh-Taylor instability. or Richtmyer-Meshkov instability, with large initial distortions of the gas interfaces. The inhomogeneities are made by filling large soap bubbles and cylindrical refraction cells (5 cm diameter) whose walls are thin plastic membranes with gases both lighter and heavier than the ambient air in a square (8.9 cm side shock-tube text section. The wavefront geometry and the deformation of the gas volume are visualized by shadowgraph photography. Wave configurations predicted by geometrical acoustics, including the effects of refraction, reflection and diffraction, are compared to the observations. Departures from the predictions of acoustic theory are discussed in terms of gasdynamic nonlinearity. The pressure field on the axis of symmetry downstream of the inhomogeneity is measured by piezoelectric pressure transducers. In the case of a cylindrical or spherical volume filled with heavy low-sound-speed gas the wave which passes through the interior focuses just behind the cylinder. On the other hand, the wave which passes through the light high-sound-speed volume strongly diverges. Visualization of the wavefronts reflected from and diffracted around the inhomogeneities exhibit many features known in optical and acoustic scattering. Rayleigh-Taylor instability induced by shock acceleration deforms the initially circular cross-section of the volume. In the case of the high-sound-speed sphere, a strong vortex ring forms and separates from the main volume of gas. Measurements of the wave and gas-interface velocities are compared to values calculated for one-dimensional interactions and for a simple model of shock-induced Rayleigh-Taylor instability. The circulation and Reynolds number of the vortical structures are calculated from the measured velocities by modeling a piston vortex generator. The results of the flow visualization are also compared with contemporary numerical simulations.

We consider the flow of a suspension in a rotating, cylindrical container with inclined endwalls and a dividing harrier that hlocks any azimuthal motion around the axis. A boundary layer of clarified fluid appears when the influence of the Coriolis force is counteracted and although a bulk swirling motion is prevented by the meridional section, there is still an appreciable azimuthal flow in this thin purified-fluid layer. This flux produces an even more intense current on the leading side of the barrier (relative to the rotation direction) where the section meets the inclined wall.

The flow induced by a vortex ring approaching a plane wall on a trajectory normal to the wall is investigated for an incompressible fluid which is otherwise stagnant. The detailed characteristics of the interaction of the ring with the flow near the surface have been observed experimentally for a wide variety of laminar rings, using dye in water to visualize the flow in the ring as well as near the plane surface. Numerical solutions are obtained for the trajectory of the ring as well as for the unsteady boundary-layer flow that develops on the wall. The experimental and theoretical results show that an unsteady separation develops in the boundary-layer flow, in the form of a secondary ring attached to the wall. A period of explosive boundary-layer growth then ensues and a strong viscous-inviscid interaction occurs in the form of the ejection of the secondary vortex ring from the boundary layer. The primary ring then interacts with the secondary ring and in some cases was observed to induce the formation of a third, tertiary, ring near the wall. The details of this process are investigated over a wide Reynolds number range. The results clearly show how one vortex ring can produce another, through an unsteady interaction with a viscous flow near the wall.

The transition and the development of turbulence in the near wake of a circular cylinder are investigated using hot-wire anemometry and flow visualization. The formation zone of the large regular vortices is studied in the subcritical regime (2000 < U0D/v < 60000). with and without the introduction of a splitter plate. Two different regimes are identified in the interaction between the von Kármán vortices and those of the shear layer emerging from the separated boundary layer. Experimental evidence is given in support of the strong coupling at low Reynolds numbers characterized by phase modulations between the two types of structures. The interaction is weaker at high Reynolds numbers where the small-scale vortices are disconnected from the regular vortex shedding, giving rise to an intermittent pattern. Spectral properties are used to describe the different stages of the interaction between the shear-layer vortices and the alternating ones. Physical properties of the interaction are examined separately in a numerical simulation using a pressure-velocity formulation. Both unexcited and excited two-dimensional plane mixing layers are studied using streakline maps and time traces of the dynamical properties. The main features of the simulated vortex development are in agreement with the experimental results.

Separation from the leading edge of a delta wing with the subsequent roll-up into a vortex has been simulated in numerical solutions to the Euler equations. Such simulations raise a number of questions that are still outstanding, including the process of inviscid separation from a smooth edge, the role of artificial viscosity in the creation and capturing of vortex sheets, the roll-up mechanism and core features, losses in total pressure, and the stability of the vortical flow structures to three-dimensional disturbances. These matters are discussed in the context of two numerical experiments, both carried out in a sequence of three simulations that starts with a coarse-mesh discretization and ends with a fine mesh of over one million cells. The first experiment is for transonic flow, M∞ = 0.7, α = 10°, around a pure delta wing. The sequence converges to the expected classical steady vortex flow. In the second experiment, transonic flow, M∞ = 0.9, α = 8°, past a twisted cranked-and-cropped delta wing, the sequence does not converge. Instead the crank is observed in the fine-mesh solution to set off an instability in the vortex sheet that causes the vortex to burst into a thin chaotic vortical layer embedded in laminar flow. The mesh sequence suggests that it is the shortest waves resolved that are most unstable, but the energy contained in them comes from the large-scale motion and seems to be small.

Calculations of the growth and collapse of transient vapour cavities near a free surface when buoyancy forces may be important are made using the boundary-integral method described in Part 1. Bubble shapes, particle paths, pressure contours and centroid motion are used to illustrate the calculations. In the absence of buoyancy forces the bubble migrates away from the free surface during the collapse phase, yielding a liquid jet directed away from the free surface. When the bubble is sufficiently close to the free surface, the nonlinear response of the free surface produces a high-speed jet (‘spike’) that moves in the opposite direction to the liquid jet and, in so doing, produces a stagnation point in the fluid between the bubble and the free surface. For sufficiently large bubbles, buoyancy forces may be dominant, so that the bubble migrates towards the free surface with the resulting liquid jet in the same direction. The Kelvin impulse provides a reasonable estimate of the physical parameter space that determines the migratory behaviour of the collapsing bubbles.

Following ideas developed in the field of hydrodynamic stability of laminar flows (Stuart 1971) a predictive theory is proposed to determine the development of finite-amplitude alternate bars in straight channels with erodible bottoms. It is shown that an ‘equilibrium amplitude’ of bedforms is reached as t → ∞ within a wide range of values of the parameter (β − βc)/βc, where t is the time, β is the width ratio of the channel and βc is its ‘critical’ value below which bars would not form. The theory leads to relationships for the maximum height and the maximum scour of bars which compare satisfactorily with the experimental data of various authors. Moreover the experimentally detected tendency of the bed perturbation to form diagonal fronts is qualitatively reproduced.

This paper studies numerically the instabilities and quasi-steady states of thermal convection in a Boussinesq fluid within a unit square which is bounded rigidly on all sides, differentially heated on the bottom, insulated above, and rotating about a vertical axis. The flows for five ascending values of the thermal Rossby number β are studied for constant Prandtl number σ and infinitesimal Ekman number E. For small values of β, either transient or stationary spatial oscillations occur. The results agree with Daniels’ (1976) and Daniels & Stewartson's (1977, 1978) linear and weakly nonlinear theories in principle if not in detail. For large values of β. the flows are highly nonlinear. They undergo two distinct stages of instability and eventually settle down to steady states. It is shown that a viscous non-diffusive boundary layer can exist at steady state for large β. For the maximum value of β under investigation, only quasi-steady states have been reached in the interior. Inertial gravity waves are observed emanating from the unstable corner of the fluid. These numerical solutions establish that the initial instability is an exchange instability rather than a catastrophic one. The question of bifurcation has not been resolved.

Experimental and theoretical studies of sloshing waves in a rectangular channel in the vicinity of the second cutoff frequency are presented. The experiments were performed in a wave tank which is 1.2 m wide, 18 m long and 0.9 m deep. Sloshing waves were generated by a computer-controlled segmented wavemaker consisting of four independent modules. A sharp transition between two wave patterns, which exhibited hysteresis-type behaviour, was observed. At lower forcing frequencies a steady wave regime was obtained, while at higher frequencies modulation on a long timescale appeared. At stronger forcing, solitons were generated periodically at the wavemaker and then propagated away with a seemingly constant velocity. Experimental results are compared with numerical solutions of the appropriate nonlinear Schrödinger equation, a derivation of which is also presented. The importance of dissipation on the physical processes of wave evolution is discussed, and a simple dissipative model is suggested and incorporated in the governing equations.

The problem of wave generation by sources near the ocean surface is examined. The model is used to deduce that in the ocean, at low frequencies, both sound and surface gravity waves are possibly induced by the same sources. Sound radiation from particular sources is significantly influenced, at low frequencies, by surface waves which themselves are almost unmodified by fluid compressibility. The wave fields and energy levels are calculated analytically in a simple model of point-source excitation, which reveals considerable character. The theory is used to interpret measurements from the real ocean in the case where the source is a finite distribution of surface pressure fluctuations caused by winds. Good consistency is found in both structure and order of magnitude. Our main conclusions are that surface waves and wave-related sound are not necessarily cause and effect; a considerable proportion of the wave-associated underwater sound in the ocean is very likely generated by the same sources that produced the surface waves themselves.

The theory of sound generation by weakly nonlinearly interacting ocean surface waves is examined. The main conclusion is that this mechanism may not be a strong generator of ocean sound. It is shown that at low frequency and with small wind the sound generated by this mechanism is weak in comparison with that directly radiated by the turbulent airflow, the flow which is also the cause of surface waves. With increasing frequency and/or wind speed, the sound from surface-wave interactions becomes appreciable, but it is found that the condition for this sound overwhelming the aerial turbulence radiation implies the precise condition at which fully nonlinear surface motions occur. In that case processes such as splashing of water sprays by breaking waves become the main cause of ocean noise. In fact it seems that the weakly nonlinear mechanism proposed by Brekhovskikh is never an important source of sound in the real ocean.

Various flow processes resulting from the breaking of ocean surface waves are examined in order to determine their relative efficiencies as sources of sound. Momentum fluctuation arising from splashing water sprays is identified as the major contributor to the underwater sound. It is shown that the splashing is more efficient in radiating sound than other processes, such as unsteady foaming that entrains air bubbles into water, and turbulent motions in the surface layer associated with compressibility of the entrained bubbles. A model is presented to estimate the sound power radiated in terms of parameters of the wind and surface wave field. Comparison of theory with measurements is made and good agreement is discovered.

The general dynamical problem for bubbles moving in an infinite expanse of perfect liquid is discussed from the standpoint of Hamiltonian theory, which is appreciated as a basis for linking symmetries with conservation laws and for identifying variational principles that describe steady motions. Allowance is made for surface tension and for an arbitrary gas law relating the pressure and volume of the bubble contents, but particular attention is paid to models where the volume is constant.

In §, the most detailed part of the paper, a comprehensive theory is developed which represents the free surface parametrically and so applies globally in time. Conservation laws for energy and for linear and angular components of impulse are shown to follow simply from respective symmetries; consequences of Galilean in variance and of a scaling symmetry are also explored. Finally in §, variational characterizations of steady translational, spinning and spiralling motions are explained. In §3 a formally simpler Hamiltonian theory is shown to derive from the mildly restrictive assumption that the free surface can be represented in an orthogonal coordinate system; and some special details attending the use of cylindrical coordinates are noted. For bubbles steadily translating along an axis of symmetry, approximate calculations supported by Rayleigh's principle are presented in §4.1. Steadily spiralling motions are treated in §5; estimates based on spheroidal approximations to shape are presented in §5.1; and some speculations about stability are discussed in §5.2. A brief account of generalizations dealing with multiply connected bubbles is given in §6.

The potential flow of an inviscid ferromagnetic fluid is formulated as a Hamiltonian system: the conjugate coordinates are the normal displacement of the free surface and the velocity potential on the free surface. If we take the equation of the magnetic field as a constraint, we can see that the flow of a magnetic liquid is a control-theory problem; the potential of the magnetic self-field is the control variable.

Simultaneous measurements of time-resolved velocity and temperature have been obtained by laser-Doppler anemometry and numerically compensated fine-wire thermocouples in the near wake of a premixed flame stabilized on a disk baffle located on the axis, and at the exit, of a confining pipe. The diameter of the disk was 0.056 m, the diameter of the pipe was 0.080 m, the volumetric equivalence ratio with natural gas as the fuel was 0.79 and the Reynolds number, based on pipe diameter and upstream pipe bulk velocity of 9 m/s, was 46 800. The purpose of the measurements is to quantify the relative magnitudes of terms involving the mean pressure gradient and Reynolds stresses in the balance of turbulent kinetic energy and heat flux in a strongly sheared, high-Reynolds-number, reacting flow. The latter term has been associated with non-gradient diffusion in other flows. Source terms involving the mean pressure gradient are large in the conservation of turbulent heat flux but not in the conservation of Reynolds stress. The ‘thin-flame’ model of burning suggests that the sign and magnitude of the heat flux is closely related to the conditioned mean velocities. The mean axial velocity of the reactants is larger (by up to 0.27 of the reference velocity) than that of the products on the low-velocity side of the shear layer that surrounds the recirculation bubble but the reverse is true on the high-velocity side. These observations are related to the sign of the axial pressure gradient, which is associated with the streamline curvature, and the consequent preferential acceleration of the low-density products. Generally, the Reynolds stresses of the products are higher than those of the reactants and, in contrast to previously reported measurements, the contribution to the unconditioned stresses by the difference in the mean velocity between products and reactants, the so-called ‘intermittent’ contribution, is small. This is a consequence of the high Reynolds number of our flow.

In the course of viscometric measurements of concentrated suspensions of spheres in Newtonian fluids using a Couette device, Gadala-Maria & Acrivos (1980) observed a decrease in the suspension viscosity after long periods of shearing even though the viscosity of the pure suspending fluid remained constant under identical conditions. In the present work we demonstrate that this phenomenon is due to the shear-induced migration of particles out of the sheared Couette gap and into the fluid reservoir, which reduces the particle concentration in the gap and thereby the observed viscosity. We show further that this rate of viscosity decrease is consistent with a gap-limited shear-induced diffusion process normal to the plane of shear, with the relevant diffusion coefficient being proportional to $a^2\dot{\gamma}$, where a is the particle radius and $\dot{\gamma}$ is the applied shear rate.

Additional experiments also uncovered a new phenomenon - a short-term increase in the viscosity upon initial shearing of a suspension in a Couette device - which was attributed to the diffusive migration of particles across the width of the Couette gap and thus was used to infer values of the corresponding diffusion coefficient within the plane of shear parallel to gradients in fluid velocity.

In the theoretical part we demonstrate that the particle migrations that led to these observed phenomena may be explained in terms of the irreversible interparticle interactions that occur in these suspensions. From simple arguments, these interactions are shown to lead to effective diffusivities both normal to the plane of shear and normal to the direction of fluid motion within the plane of shear whose estimated magnitudes are comparable with those that were inferred from the experimental measurements. Furthermore, these interactions should induce, within a shear flow, particle drifts from regions of high to low shear stress, which are estimated to be of sufficient intensity to account for the observed initial viscosity increase mentioned above.

Compressible flows with r.m.s. velocities of the order of the speed of sound are studied with direct numerical simulations using a pseudospectral method. We concentrate on turbulent homogeneous flows in the two-dimensional case. The fluid obeys the Navier-Stokes equations for a perfect gas, and viscous terms are included explicitly. No modelling of small scales is used. We show that the behaviour of the flow differs sharply at low compared with high r.m.s. Mach number Ma, with a transition at Ma = 0.3. In the large scales, temporal exchanges between longitudinal and solenoidal modes of energy retain an acoustical character; they lead to a slowing down of the decrease of the Mach number with time, which occurs with interspersed plateaux corresponding to quiescent periods. When the flow is initially supersonic, the small scales are dominated by shocks behind which vortices form. This vortex production is particularly prominent when two strong shocks collide, with the onset of shear turbulence in the region downstream of the collision. However, at the resolutions reached by our code on a 256 × 256 uniform grid, this mechanism proves insufficient to bring vortices into equipartition with shocks in the small-scale tail of the energy spectrum.

The analytical solution of a pseudo-steady Mach reflection was considered. It was found that the solution of the well-known perfect-gas conservation equations of a pseudo-steady Mach reflection - the three-shock theory - failed to accurately predict the angles between the incident, reflected and Mach stem shock waves. The disagreement between theory and experiments was not settled even when real-gas effects were accounted for. However, the inclusion of real-gas effects did improve the analytical predictions. In order to improve the analytical model, the boundary layers developing on both sides of the slipstream were integrated into the analysis. Using these boundary layers, the displacement thickness as a function of distance along the slipstream from the triple point was calculated. The displacement thickness was then related to the angular displacement of the slipstream, as a function of that distance. Finally it was shown that the displacement, taken at a distance equivalent to the incident-shock-wave thickness, could be used to obtain computed results which agree with experimentally measured data.