The dispersion relation and component phase speeds of surface gravity wavefields and modulated wavetrains are calculated. A parametric study is performed for a range of nonlinearity and spectral bandwidths. It is found that the amount of departure from linear theory increases with the ratio of nonlinearity to spectral bandwidth. The calculated results are compared quantitatively with laboratory and ocean measurements of wavetrains and wavefields with and without wind. The good agreement between theory and experiment suggests that the nonlinearity–dispersion balance is a likely candidate to account for the observed discrepancy between linear theory and data, as well as for the difference in behaviour between laboratory and oceanic wave measurements.

In this paper a model is presented that incorporates characteristic features of the turbulent structures as revealed by recent experimental observations. The principal characteristic features are the occurrence of periodicity and intermittency, not only at the edge of the boundary layer, but also close to the wall.

By an averaging procedure, equations are derived for the large-scale part of the turbulent motion. The unknown terms, representing the small-scale turbulent stress, are assumed to be zero except in the so-called burst regions which occupy only a small fraction of the total flow field and arise from local instability of the large-scale flow field. In the model distinction is made between a thin layer near the wall where viscous forces play an important role and the remaining part of the boundary layer where inviscid equations are valid. The momentum transport takes place in three different stages with different mechanisms. First of all the fluid in the wall region is retarded by viscous forces and collected in long narrow regions (streaks). After this a rapid exchange takes place in the burst regions where the low-momentum fluid is ejected into the outer region. Finally the large-scale structures in the outer region take over the transport.

It turns out that the transport properties of a turbulent boundary layer can be calculated reasonably well with this deterministic model. It can be concluded that the coherent part of the turbulent motion is very important in the transport process.

The results are reported of velocity measurements in oscillatory flow over rippled beds. Velocities were measured with a laser-doppler anemometer in both an oscillating tray rig and an oscillatory flow water channel. Both self-formed and artificial ripples were examined. In addition, some measurements were made with an apparently plane bed with intense sediment motion.

The experimental results were compared with the predictions of Kalkanis (1964), Kajiura (1968), Bakker (1973, 1974) and Sleath (1974). The closest agreement between theory and experiment was obtained with the method of Sleath.

Measurements of the Eulerian drift velocities showed drift towards the adjacent ripple crest in the immediate vicinity of the bed and away from the crest further out.

A lifting line theory of flapping wings in steady forward flight is presented in which the unsteady features of the flow are modelled. A detailed three-dimensional model of the vortex wake is used to evaluate the unsteadiness to first order. The method gives satisfactory agreement with well-known limiting cases. Relationships between the geometric and the kinematic parameters, and the forces and the power are predicted which are compatible with the limited experimental evidence. The theory is applied to the calculation of the power curve of specific birds. Important similarities and differences are observed between the present results and those of Pennycuick (1975) and Rayner (1979c).

An exact solution to the Navier–Stokes equations for the flow in a channel or tube with an accelerating surface velocity is presented. By means of a similarity transformation the equations of motion are reduced to a single ordinary differential equation for the similarity function which is solved numerically. For the two-dimensional flow in a channel, a single solution is found to exist when the Reynolds number R is less than 310. When R exceeds 310, two additional solutions appear and form a closed branch connecting two different asymptotic states at infinite R. The large R structure of the solutions consists of an inviscid fluid core plus an O(R−1) thin boundary layer adjacent to the moving wall. Matched-asymptotic-expansion techniques are used to construct asymptotic series that are consistent with each of the numerical solutions.

For the axisymmetric non-swirling flow in a tube, however, the situation is quite different. For R [Lt ] 10[sdot ]25, two solutions exist which form a closed branch. Beyond 10[sdot ]25, no similarity solutions exist within the range 10[sdot ]25 < R < 147. Once R exceeds 147, multiple solutions reappear, which form two closed branches that connect four different asymptotic states at infinite R. The possibility of an axisymmetric flow with swirl is considered, and two sets of swirling solutions are found to exist for all R > 0. These solutions, however, do not evolve from the R = 0 state nor do they bifurcate from the non-swirling solutions at any finite value of R.

Stability of the steady motion of a fluid confined between two differentially heated rigid vertical plates is considered. When a stable, constant vertical salinity gradient is also present, the steady mean velocity in the vertical direction and the mean lateral salinity gradient are characterized by the solute Rayleigh number, Rs. Experimental investigations (Elder 1965; Hart 1970) show that when Rs = 0 the instability is induced by shear and occurs in the form of two-dimensional convection cells. However, at moderate values of Rs, these shear instabilities are replaced by double-diffusive cellular convection (Thorpe, Hutt & Soulsby 1969; Paliwal & Chen 1980a). It is generally believed that the instability is stationary and cellular for all values of Rs (Hart 1971; Paliwal & Chen 1980b). We have solved the general eigenvalue problem, and our results indicate that, during transition from the stationary shear-induced instability to stationary double-diffusive cellular convection, overstable motion occurs. Furthermore, in this transition region, over a range of moderately small values of Rs, there is no preferred wavelength at the onset of instability.

The theory of rotating open-channel flow between two basins is presented and experimentally verified for three cases. The first case considered is the submerged weir-flow, i.e. the height of the channel floor is below the free-surface height of the fluid in the lower basin. Flow for this case is found to depend on the fluid level in the downstream basin when the fluid level there is high and is independent of the fluid level downstream as soon as the level falls below a critical height; at that point the flow in the channel becomes critical (controlled). The dependence of the flow on upstream potential vorticity and on rotation rate is also examined. It is shown that transport of the flow decreases whenever rotation rate or upstream potential vorticity is increased. The second case studied is controlled flow through a channel of irregular cross section (the truncated channel). Transport of this flow is shown to increase at some moderate rotations if the channel floor has a cross-channel slope similar to that of the surface of the current. Otherwise, the transport always decreases with rotation rate. The final case is concerned with supercritical separation of the current downstream of the control section, i.e. the current separates from the left wall of the channel (looking downstream). Visual evidence and measurements of formation of such a boundary current are obtained. Comparisons between the theory and the experiment for the three cases are generally reasonable except when flow separates in the control section; the theory is found inapplicable to such separation.

The response of an electromagnetic flowmeter to the flow rate through it should ideally be independent of the velocity distribution of the liquid within it. In general such flowmeters do not exhibit this ideal behaviour and the variation in response with different velocity distributions, for a given flow rate, can be very large. The sensitivity of a flowmeter to velocity distribution is here described in terms of a ‘worst flow’, which is a particular velocity distribution peculiar to a given flowmeter. The ‘worst flow’ is described mathematically in terms of functions which depend upon the design parameters of the flowmeter. The flowmeter response to its ‘worst flow’, for a given kinetic energy of motion, is formulated. This enables a criterion (ε) to be defined which describes the sensitivity of a given flowmeter to velocity profile effects and which permits different flowmeters to be compared in this respect. Methods of evaluating ε are discussed and its value is calculated for a conventional flowmeter, which employs small electrodes and an approximately uniform magnetic field. It is shown that it is possible actually to generate this worst flow in a given flowmeter, by operating the flowmeter in the role of an electromagnetic pump. This could lead to the direct measurement of ε The possibility is discussed of other definitions of ε based on different boundary conditions and constraints and which might lead to a less severe criterion.

Based on theoretical analysis and laboratory data, we proposed a unified two-parameter wave spectral model as $\phi(n) = \frac{\beta g^2}{n^m n_0^{5-m}} {\rm exp} \left\{-\frac{m}{4}\left(\frac{n_0}{n}\right)^4\right\}$ with β and m as functions of the internal parameter, the significant slope η of the wave field which is defined as $\sect = \frac{(\overline{\zeta^2})^{\frac{}1{2}}}{\lambda_0},$ where $\overline{\zeta^2}$ is the mean squared surface elevation, and λ0, n0 are the wavelength and frequency of the waves at the spectral peak. This spectral model is independent of local wind. Because the spectral model depends only on internal parameters, it contains information about fluid-dynamical processes. For example, it maintains a variable bandwidth as a function of the significant slope which measures the nonlinearity of the wave field. And it also contains the exact total energy of the true spectrum. Comparisons of this spectral model with the JONSWAP model and field data show excellent agreements. Thus we established an alternative approach for spectral models. Future research efforts should concentrate on relating the internal parameters to the external environmental variables.

The results of an experimental investigation dealing with finite-amplitude internal solitary waves in a two-fluid system are presented. Particular attention is paid to characterizing solitons in terms of their shape and amplitude–wavelength scale relationship. Two cases are considered, viz., a shallow- and a deep-water configuration, in order to study the depth effect upon the propagational characteristics of the waves. Comparisons are made between the experimental results and existing internal-wave theories. In addition, discussion is presented describing how these existing theories may be extended to include higher-order nonlinear and viscous effects.

A linear mechanism for growth of three-dimensional perturbations on plane Poiseuille flow is investigated. The mechanism, resonant forcing of vertical vorticity waves by Tollmien–Schlichting waves, leads to an algebraic growth for small times. Eventually, viscous damping becomes dominant and the disturbance decays. The resonance occurs only at discrete points in the wave-number space. Nine resonances have been investigated. For these, the phase velocities range from 0[sdot ]67 to 0[sdot ]81 of the centre-line velocity. The lowest Reynolds number for which the resonance can occur is 25. The strongest resonance appears only above a Reynolds number of 341. Also, two cases of degeneracy in the Orr–Sommerfeld dispersion relationship have been found.

An approximate solution is developed for two-dimensional, steady, inviscid supersonic flow over an airfoil. This approximation produces accurate results for a wide range of Mach numbers and airfoil thicknesses. It is used as the starting point for a rapidly convergent iterative numerical solution of the exact equations. A co-ordinate system consisting of the principal characteristics and streamlines is employed. Examples computed for a symmetric airfoil reveal several interesting features in the tail shock and the flow behind the airfoil.

The development of a laminar boundary layer on a twisted helical blade is described. An appropriate co-ordinate system is developed in which the boundary-layer equations have a relatively simple form. The choice of blade geometry and the free-stream conditions result in a constant-pressure flow. This permits the flow to be considered the analogue, in a rotating frame, of the zero-pressure-gradient flat-plate boundary layer in a stationary frame. The boundary-layer equations are solved using a double series expansion in powers of distance from the leading edge and the cosine of the blade twist angle. Chordwise and spanwise velocity profiles are calculated. The variation in the skin friction coefficients is calculated as a function of position on the blade.

At low Reynolds numbers, three-dimensional features are frequently observed in the vortices shed behind a basically two-dimensional circular cylinder. This paper deals with the dependence of the configuration of the vortices on various end constructions. The cylinder is towed at a uniform speed in a water tank and simple flow visualization is used. It is found that the three-dimensional structure of the wake depends strongly on the flow configuration at each end of the cylinder. The boundary condition imposed on the nascent vortex lines determines the subsequent behaviour of the shed vortices. Consequently, the vortex street can be rendered more nearly two-dimensional by allowing the vortices to link outside the boundary as they approach that boundary normally. This is the case for the water–air interface when the water surface is clean. In the case of a contaminated water surface or of a solid surface acting as a boundary to the vortex street, the vortices link between themselves underneath the water surface and a strong interaction takes place behind the end of the cylinder. The subsequent effect is a bowing of the vortices towards the end of the cylinder. The free-end effect at the bottom end of the cylinder induces a strong bowing of the vortices towards that end and causes the wake to contract. It follows from the effect of surface contamination that the study of vortex wakes by the spreading of some surface contaminants might not necessarily show the true behaviour of the wake below the surface. It is postulated that slantwise shedding arises from a difference in the two end effects.

When viscous fluid is contained in the corner between two planes intersecting at an angle α, a flow may be ‘forced’ either by relative motion of the two planes keeping α constant (the ‘paint-scraper’ problem – Taylor 1960) or by relative rotation of the planes about their line of intersection (the hinged-plate problem – Moffatt 1964). In either case, a similarity solution is available describing the flow sufficiently near the corner, where inertia forces are negligible. In this paper, we investigate the effects of inertia forces, by constructing regular perturbation series for the stream function, of which the leading term is the known similarity solution. The first-order inertial effect is obtained analytically, and, for the Taylor problem with α = ½π, 25 terms of the perturbation series for the wall stress are generated numerically. Analysis of the coefficients suggests that the radius of convergence of the series is given by [rscr ] where r is distance from the corner, U is the relative speed of the planes, and v is the kinematic viscosity of the fluid. For the hinged-plate problem, discussed in § 5, the unsteadiness of the flow contributes to an inertial effect which is explicitly incorporated in the analysis. For both problems, streamline plots are presented which indicate the first influence of inertia forces at distances from the corner at which these become significant.

Hydromagnetic Taylor-vortex flows of liquid metals, subjected to an axially applied magnetic field, are considered. The supercritical state of flow is treated by means of perturbation method with multiple scale expansions in time and space. Results show that the supercritical state of flow is significantly affected by the imposed magnetic field. Calculations on finite equilibrium amplitude of Taylor vortices are compared with experiment. In the case of Taylor–Couette flows between insulating cylinders, good agreement is found between theory and experiment, while, in the case of electro-magnetically driven flows between steady conducting cylinders, disagreement occurs once magnetic field effects become significant. These results are discussed.

The circular hydraulic jump commonly forms on a horizontal plate struck by a vertical jet of liquid. New observations of this phenomenon are described.

A previously unreported instability of the jump is examined. This is shown to arise when the local Reynolds number Rj just ahead of the jump exceeds a critical value of 147. Prior to this instability, the flow behind the jump contains a closed eddy, the length of which decreases to zero as Rj increases towards its critical value. Physical explanations for this flow structure and instability are proposed.

Accurate measurements of liquid depths were made using a light-absorption technique, in which a laser was shone through water containing a strong dye. Liquid depths ahead of and behind the jump were so determined and depth profiles of the jump in the stable regime were obtained.

As the outer depth was increased, the jump closed in on the jet and eventually disappeared: this extinction of the jump is also investigated.

Instability of the alternate-bar type in straight channels has long been identified as the cause of fluvial meandering. The condition of inerodible sidewalls, however, does not allow a meandering channel to develop. Herein a stability analysis of a sinuous channel with erodible banks allows for delineation of a ‘bend’ instability that does not occur in straight channels, and differs from the alternate-bar instability.

In the case of alluvial meanders, the two mechanisms are shown to operate at similar characteristic wavelengths. This provides a rationale for the continuous evolution of alternate bars into true bends such that each bend contains one alternate bar.

The same bend instability applies to incised meanders. A mechanism for incised alternate bars which differs from that for the alluvial case appears to operate at different characteristic wavelengths than that of bend instability. Analysis of data suggests that meandering in supraglacial meltwater streams is primarily due to the alternate bar mechanism, whereas the meandering of rills incised in cohesive material and of caves is likely due to the bend mechanism.

The meander wavelength of incised reaches of meandering streams is often longer than that of adjacent alluvial reaches. An explanation is offered in terms of bend instability.

The applicability of the Taylor hypothesis to large-scale coherent structures in turbulent shear flows has been evaluated by comparing the actual spatial distributions of the structure properties with those deduced through the use of the hypothesis. This study has been carried out in the near field of a 7[sdot ]62 cm circular air jet at a jet Reynolds number of 3[sdot ]2 x 104, where the coherent structures and their interactions have been organized through controlled excitation. Actual distributions of the structure properties have been obtained through phase-average hot-wire data, the measurements having been repeated at different spatial points over the extents of the structure crosssections at a fixed phase. The corresponding ‘spatial’ distributions of these properties obtained (by using the Taylor hypothesis) from the temporal data at appropriate phases and locations, show that the hypothesis works quite well for an isolated coherent structure if a constant convection velocity, equal to the structure centre velocity, is used in the hypothesis everywhere across the shear flow. The popular use of the local time-average or even the instantaneous streamwise velocity produces unacceptably large distortions. When structure interactions like pairing are involved, no convection velocity can be found with which the hypothesis works. Distributions of the terms in the Navier–Stokes equation contributing to the phase-average vorticity, but neglected by the hypothesis, have been quantitatively determined. These show that the terms associated with the background turbulence field, but not those associated with the coherent motion field, can be neglected. In particular, the pressure term due to the coherent motion field is large and cannot be neglected.

The motion of a very thin vortex filament is investigated using the localized induction equation. A family of vortex filaments which move without change of form are obtained. They are expressed in terms of elliptic integrals of the first, second and third kinds. In general they do not close and have infinite lengths. In some particular cases they take the form of closed coils which wind a doughnut. There exist a family of closed vortex filaments which do not travel in space but only rotate around a fixed axis. Our solutions include various well-known shapes such as the circular vortex ring, the helicoidal filament, the plane sinusoidal filament, Euler's elastica and the solitary-wave-type filament. It is shown that they correspond to the travelling wave solution of a nonlinear Schrödinger equation.