In the light of accumulated evidence in favour of Miles's mathematical theory of wave generation by wind, the author has thought it desirable to translate the theory into the form of a physical argument, which goes as follows. Travelling water waves in a wind produce, to a first approximation, airflow undulations with pressures least over crests and greatest over troughs. Hence, just above the ‘critical height’ where the airflow component in the direction of propagation equals the wave velocity, air after slowly overtaking a crest is turned back by the higher pressure near the trough, moves down to a lower level and back towards the crest. Similarly, behind crests, an upward movement at the critical height occurs. Quantitatively, the vertical velocity v is such that the ‘vortex force’ − ρων (where ρ is density and ω vorticity) balances the sinusoidal pressure gradient. Furthermore, since in turbulent boundary layers vorticity decreases with height, any downflow produces a local vorticity defect, and upflow a local vorticity excess, and hence the vortex force varies about a negative mean at the critical height (although at other levels, where air moves sinusoidally, with vertical displacement and velocity 90° out of phase, the mean vortex force is zero). The negative total mean force extracts wind energy, and transfers it to the wave, enabling it to grow exponentially. For pressure gradients adequate to initiate substantial energy transfer, the critical height must be fairly small compared with the wavelength, and waves can grow whenever their velocity and direction satisfies this condition, a conclusion supported by measurements (Longuet-Higgins 1962) of the directional spectrum of wind-generated waves.

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.

The motion of two cavitation bubbles near a rigid boundary is observed experimentally using a high-speed camera and compared against numerical solutions obtained using a boundary integral method. The comparisons are favourable with regard to both bubble shape history and centroid motion. The bubbles show a range of responses depending on the experimental configuration. Elongated bubbles, jets directed towards or away from the rigid boundary and bubble splitting phenomena are all observed and predicted for the given parameters. It is clear that nearby bubbles are equally as important as the presence of a rigid boundary in determining the behaviour of bubbles.

The flow between two finite rotating disks enclosed by a cylinder is investigated both numerically and experimentally. For this finite geometry the full stationary Navier–Stokes equations are solved numerically without similarity assumptions. Experimental results are obtained by means of stereophotography of small tracer particles. The results are in good agreement with the numerical solution. Owing to the presence of the cylinder sidewall, the solution is found to be unique for all values of the parameters considered. When the disks rotate in opposite senses with counter-rotation above 15%, a stagnation point appears at the slower-rotating disk. This stagnation point is associated with a two-cell structure in the meridional plane and is experimentally observed as a ring of particles at the slower-rotating disk. Near the axis of rotation the solution is found to satisfy similarity demands; for weak counter-rotation the solution is of Batchelor type near the axis of rotation, but for strong counter-rotation a Stewartson profile is found to be more adequate for the description of the tangential velocity near the axis.

Experiments are reported on the hydrodynamic performance of a flexible fin. The fin replicates some features of the pectoral fin of a batoid fish (such as a ray or skate) in that it is actuated in a travelling wave motion, with the amplitude of the motion increasing linearly along the span from root to tip. Thrust is found to increase with non-dimensional frequency, and an optimal oscillatory gait is identified. Power consumption measurements lead to the computation of propulsive efficiency, and an optimal efficiency condition is evaluated. Wake visualizations are presented, and a vortex model of the wake near zero net thrust is suggested. Strouhal number effects on the wake topology are also illustrated.

This note advocates a model of the steady flow about a bluff body at large Reynolds number which is different from the classical free-streamline model of Helmholtz and Kirchhoff. It is suggested that, although the free-streamline model may be a proper solution of the Navier-Stokes equation with μ = 0, it is unlikely to be the limit, as μ → 0, of the solution describing the steady flow due to the presence of a bluff body in an otherwise uniform stream. The limit solution proposed here is one which gives a closed wake.

A closed wake contains a standing eddy, or eddies, whose general features can be inferred from the results of an earlier investigation of steady flow in a closed region at large Reynolds number. In all cases, the drag (coefficient) on the body tends to zero as the Reynolds number tends to infinity. The proccedure for finding the details of the closed wake behind two-dimensional and axisymmetrical bodies is described, although no particular case has yet been worked out.

Turbulent boundary layer measurements above a smooth wall and sandpaper roughness are presented across a wide range of friction Reynolds numbers, ${\it\delta}_{99}^{+}$, and equivalent sand grain roughness Reynolds numbers, $k_{s}^{+}$ (smooth wall: $2020\leqslant {\it\delta}_{99}^{+}\leqslant 21\,430$, rough wall: $2890\leqslant {\it\delta}_{99}^{+}\leqslant 29\,900$; $22\leqslant k_{s}^{+}\leqslant 155$; and $28\leqslant {\it\delta}_{99}^{+}/k_{s}^{+}\leqslant 199$). For the rough-wall measurements, the mean wall shear stress is determined using a floating element drag balance. All smooth- and rough-wall data exhibit, over an inertial sublayer, regions of logarithmic dependence in the mean velocity and streamwise velocity variance. These logarithmic slopes are apparently the same between smooth and rough walls, indicating similar dynamics are present in this region. The streamwise mean velocity defect and skewness profiles each show convincing collapse in the outer region of the flow, suggesting that Townsend’s (The Structure of Turbulent Shear Flow, vol. 1, 1956, Cambridge University Press.) wall-similarity hypothesis is a good approximation for these statistics even at these finite friction Reynolds numbers. Outer-layer collapse is also observed in the rough-wall streamwise velocity variance, but only for flows with ${\it\delta}_{99}^{+}\gtrsim 14\,000$. At Reynolds numbers lower than this, profile invariance is only apparent when the flow is fully rough. In transitionally rough flows at low ${\it\delta}_{99}^{+}$, the outer region of the inner-normalised streamwise velocity variance indicates a dependence on $k_{s}^{+}$ for the present rough surface.

Rayleigh–Bénard convection in rotating spherical shells can be considered as a simplified analogue of many astrophysical and geophysical fluid flows. Here, we use three-dimensional direct numerical simulations to study this physical process. We construct a dataset of more than 200 numerical models that cover a broad parameter range with Ekman numbers spanning $3\times 10^{-7}\leqslant E\leqslant 10^{-1}$, Rayleigh numbers within the range $10^{3}<Ra<2\times 10^{10}$ and a Prandtl number of unity. The radius ratio $r_{i}/r_{o}$ is 0.6 in all cases and gravity is assumed to be proportional to $1/r^{2}$. We investigate the scaling behaviours of both local (length scales, boundary layers) and global (Nusselt and Reynolds numbers) properties across various physical regimes from onset of rotating convection to weakly rotating convection. Close to critical, the convective flow is dominated by a triple force balance between viscosity, Coriolis force and buoyancy. For larger supercriticalities, a small subset of our numerical data approach the asymptotic diffusivity-free scaling of rotating convection $Nu\sim Ra^{3/2}E^{2}$ in a narrow fraction of the parameter space delimited by $6\,Ra_{c}\leqslant Ra\leqslant 0.4\,E^{-8/5}$. Using a decomposition of the viscous dissipation rate into bulk and boundary layer contributions, we establish a theoretical scaling of the flow velocity that accurately describes the numerical data. In rapidly rotating turbulent convection, the fluid bulk is controlled by a triple force balance between Coriolis, inertia and buoyancy, while the remaining fraction of the dissipation can be attributed to the viscous friction in the Ekman layers. Beyond $Ra\simeq E^{-8/5}$, the rotational constraint on the convective flow is gradually lost and the flow properties continuously vary to match the regime changes between rotation-dominated and non-rotating convection. We show that the quantity $RaE^{12/7}$ provides an accurate transition parameter to separate rotating and non-rotating convection.

A theoretical model for the velocity field and the surface profile of bores and hydraulic jumps is developed. The turbulence is assumed to be concentrated in a wedge that originates at the toe of the front and spreads towards the bottom, and the turbulent closure used is a simplified k-ε model allowing for non-equilibrium in the turbulent kinetic energy. The flow equations are satisfied in depth-integrated form (method of weighted residuals), and measured deviations from static pressure are analysed and shown to have a negligible effect on the results. Comparison with measurements shows good agreement, but there is a clear need for further experimental results in the highly turbulent region near the free surface. Some basic mechanisms of the flow are discussed and explained from the theory.

The local entrainment rate in the initial region of axisymmetric turbulent air jets has been measured by a novel method, which is an adaptation of the ‘porous-wall’ technique used by Ricou & Spalding (1961). The local entrainment rate, which is independent of the nozzle Reynolds number for values greater than 6 × 104, is strongly dependent upon the axial distance. At an axial distance of one nozzle diameter the local entrainment rate is only about one-third of that in the fully developed jet; the entrainment rate increases with increasing axial distance to reach the fully developed value at an axial distance of about thirteen nozzle diameters.

We discuss the procedure for the exact solution of the Riemann problem in special relativistic magnetohydrodynamics (MHD). We consider both initial states leading to a set of only three waves analogous to the ones in relativistic hydrodynamics, as well as generic initial states leading to the full set of seven MHD waves. Because of its generality, the solution presented here could serve as an important test for those numerical codes solving the MHD equations in relativistic regimes.

A model for predicting the behaviour of a compressible flow laden with shocks interacting with granular material has been developed and tested. The model consists of two sets of coupled Euler equations, one for the gas phase and the other for the granular phase. Drag, convective, heat transfer and non-conservative terms couple the two sets of governing equations. Intergranular stress acting on the grains is modelled using granular kinetic theory in dilute regimes where particle collisions are dominant and frictional–collisional pressure in dense regions where layers of granular material slide over one another. The two-phase granular–gaseous model, as a result, is valid from dilute to densely packed granular regimes. The solution of these nonlinearly coupled Euler equations is challenging due to the presence of the non-conservative nozzling and work terms. A numerical technique, based on Godunov’s method, was designed for solving these equations. This method takes advantage of particle incompressibility to simplify the nozzling terms. It also uses the observation that a Riemann problem is valid in the region where gas can flow between particles and can be used to provide a physically accurate approximation of the non-conservative terms. The model and solution method are verified by comparisons to test problems involving granular shocks and two-phase shock-tube problems, and they are validated against experimental measurements of shock and dense particle-curtain interactions and transmitted oblique granular shocks.

It is demonstrated experimentally and theoretically that through the use of an active (feedback) controller one can dramatically modify the nature of the flow in a toroidal thermal convection loop heated from below and cooled from above. In particular, we show how a simple control strategy can be used to suppress (laminarize) the naturally occurring chaotic motion or induce chaos in otherwise time-independent flow. The control strategy consists of sensing the deviation of fluid temperatures from desired values at a number of locations inside the loop and then altering the wall heating to either counteract or enhance such deviations.

Spanwise variations of surface shear stress, amounting to 10% or more, may be produced in nominally two-dimensional boundary layers by a spatial instability of the flow through the wind-tunnel damping screens: boundary layers are shown theoretically to be very sensitive to variations of free-stream direction. Screens with open-area ratios more than about 0·57 do not produce appreciable spanwise variations in the boundary layer, and it is recommended that tunnels for boundary-layer measurements should be fitted with such screens. The critical open-area ratio does not seem to depend noticeably on Reynolds number in the usual range encountered in wind tunnels, but the figure of 0·57 should not be taken as general until measurements have been made in representative selection of wind tunnels in other laboratories.

In certain parameter regimes, it is possible to derive third-order sets of ordinary differential equations that are asymptotically exact descriptions of weakly nonlinear double convection and that exhibit chaotic behaviour. This paper presents a unified approach to deriving such models for two-dimensional convection in a horizontal layer of Boussinesq fluid with lateral constraints. Four situations are considered: thermosolutal convection, convection in an imposed vertical or horizontal magnetic field, and convection in a fluid layer rotating uniformly about a vertical axis. Thermosolutal convection and convection in an imposed horizontal magnetic field are shown here to be governed by the same sets of model equations, which exhibit the period-doubling cascades and chaotic solutions that are associated with the Shil'nikov bifurcation (Proctor & Weiss 1990). This establishes, for the first time, the existence of chaotic solutions of the equations governing two-dimensional magneto-convection. Moreover, in the limit of tall thin rolls, convection in an imposed vertical magnetic field and convection in a rotating fluid layer are both modelled by a new third-order set of ordinary differential equations, which is shown here to have chaotic solutions that are created in a homoclinic explosion, in the same manner as the chaotic solutions of the Lorenz equations. Unlike the Lorenz equations, however, this model provides an accurate description of convection in the parameter regime where the chaotic solutions appear.

Two problems involving the unsteady motion of two-dimensional vortex sheets are considered. The first is the roll-up of an initially plane semi-infinite vortex sheet while the second is the power-law starting flow past an infinite wedge with separation at the wedge apex modelled by a growing vortex sheet. In both cases well-known similarity solutions are used to transform the time-dependent problem for the sheet motion into an integro-differential equation. Finite-difference numerical solutions to these equations are obtained which give details of the large-scale structure of the rolled-up portion of the sheet. For the semi-infinite sheet good agreement with Kaden's asymptotic spiral solution is obtained. However, for the starting-flow problem distortions in the sheet shape and strength not predicted by the leading-order asymptotic solutions were found to be significant.

Experimental results are reported on the structure of gravity-driven film flow along an inclined periodic wall with rectangular corrugations. A fluorescence imaging method is used to capture the evolution of film height in space and time with accuracy of a few microns. The steady flow is found to exhibit a statically deformed free surface, as predicted by previous asymptotic and numerical studies. Though usually unstable, its characteristics determine much of the subsequent non-stationary dynamics. Travelling disturbances are observed to evolve into solitary multi-peaked humps, and pronounced differences from the respective phenomena along a flat wall are noted. Finally, a remarkable stabilization of the flow at high Reynolds numbers is documented, which proceeds through the development of a three-dimensional flow structure and leads to a temporary decrease in film thickness and recession of solitary waves.

We compute the optimal response of the turbulent Couette mean flow to initial conditions, harmonic and stochastic forcing at Re = 750. The equations for the coherent perturbations are linearized near the turbulent mean flow and include the associated eddy viscosity. The mean flow is found to be linearly stable but it has the potential to amplify steamwise streaks from streamwise vortices. The most amplified structures are streamwise uniform and the largest amplifications of the energy of initial conditions and of the variance of stochastic forcing are realized by large-scale streaks having spanwise wavelengths of 4.4h and 5.2h respectively. These spanwise scales compare well with the ones of the coherent large-scale streaks observed in experimental realizations and direct numerical simulations of the turbulent Couette flow. The optimal response to the harmonic forcing, related to the sensitivity to boundary conditions and artificial forcing, can be very large and is obtained with steady forcing of structures with larger spanwise wavelength (7.7h). The optimal large-scale streaks are furthermore found proportional to the mean turbulent profile in the viscous sublayer and up to the buffer layer.

Gravity waves occur on the surface of a liquid such as water, and the manner in which they propagate depends upon its depth. Although this dependence is described in principle by the equations of the ‘exact linear theory’ of surface waves, these equations have not been solved except in some special cases. Therefore, oceanographers have been unable to use the theory to describe surface wave propagation in water whose depth varies in a general way. Instead they have employed a simplified geometrical optics theory for this purpose (see, for example, Sverdrup & Munk (1944)). It has been used very successfully, and consequently various attempts, only partially successful, have been made to deduce it from the exact linear theory. It is the purpose of this article to present a derivation which appears to be satisfactory and which also yields corrections to the geometrical optics theory.

The dynamics and control of two-dimensional disturbances in the spatially evolving boundary layer on a flat plate are investigated from an input–output viewpoint. A set-up of spatially localized inputs (external disturbances and actuators) and outputs (objective functions and sensors) is introduced for the control design of convectively unstable flow configurations. From the linearized Navier–Stokes equations with the inputs and outputs, controllable, observable and balanced modes are extracted using the snapshot method. A balanced reduced-order model (ROM) is constructed and shown to capture the input–output behaviour of the linearized Navier–Stokes equations. This model is finally used to design a 2-feedback controller to suppress the growth of two-dimensional perturbations inside the boundary layer.