A fundamental relation is derived governing the Lagrangian kinematics of fluid particles on the surface of nonlinear ocean waves which may be known only stochastically. The horizontal trajectories of fluid particles on the free surface are shown to obey a pair of coupled nonlinear Ricatti-type ordinary differential equations driven by the temporal and spatial gradients of the free-surface elevation defined relative to an Eulerian frame. This equation is explicit in that it does not require the solution of a fully nonlinear potential flow free-surface problem and may be viewed as a deterministic or stochastic equation depending on the interpretation of the definition of the free-surface elevation. It is free of empirical corrections often used to estimate the particle kinematics above the calm water surface, is valid in potential flow and for waves of large steepness in two and three dimensions and in waters of all depths and may be used for the evaluation of the extreme unsteady loads exerted on surface piercing vertical circular cylinders by steep random waves.

A numerical experiment has been carried out to evaluate two of the methods available for finding the time-averaged mean velocity and the Reynolds stresses of a turbulent flow field using hot wires. The conventional method is based on the series expansion of the response equation, subsequent truncation of the series and time averaging. The improved method is based on squaring and time averaging without neglecting any terms. The method adopted to evaluate these two methods is based on the Monte Carlo simulation of a pseudo turbulent flow field using random-number generators and the corresponding hot-wire response, for a prescribed set of conditions, by assuming an appropriate model for the hot-wire response. The simulated hot-wire response and the calibration constants are then perturbed about their mean values to study the effects of errors in these quantities. The perturbed response is used to compute the time-averaged flow field by the two methods. The deviation of these values from the generated pseudo values, averaged over large number of trials, is used as the criterion to evaluate the methods. This procedure is also used to estimate the errors due to truncation in the conventional method, to study the effect of turbulence-intensity levels and to study the effects of measurement errors. The results indicate that the choice of the method for determining the time-averaged quantities should be based on the turbulence-intensity level and the measurement errors likely to be encountered. The conventional method yields reliable mean-velocity results for turbulence intensities as high as 50% with second-order turbulence correction. If measurement errors are within reasonable limits and the turbulence level is below 20%, the conventional method yields reliable results for Reynolds stresses. The improved method should be used to determine the time-averaged flow field for turbulence intensity above 40–50%. The error in the yaw sensitivity parameter k has an insignificant effect on the mean velocity and Reynolds stresses computed by both methods. By accurately determining the sensitivity s of the hot wire, the accuracy of the measured mean velocity and Reynolds stresses can be improved significantly. An improved method of carrying out the uncertainty analysis for measurements, based on the Monte Carlo technique, has also been outlined.

A second approximation is developed for the mass-transport velocity within the bottom boundary layer of cnoidal waves progressing over a smooth horizontal bed. Mass-transport profiles through the boundary layer are obtained by considering terms of up to third order in the perturbation parameter. A comparison with results based on a first approximation indicates that the effect of the third-order terms is to predict a smaller mass-transport velocity and that this difference is generally significant, particularly for waves extending to the intermediate depth range. The predicted correction to the first approximation is qualitatively supported by experimental evidence.

Exact coherent states in plane Couette flow are extended to an oblique parallelogram computational domain and the large-Reynolds-number asymptotic descriptions of the states are derived. When the tilt angle of the domain is increased, the states first obey vortex–wave interaction theory and then develop a new asymptotic structure. The latter asymptotic structure is characterized by a self-similar nonlinear interaction localized in the fluid layer. For the largest scale of the self-similar nonlinear structure, the theory predicts an inverse Reynolds number dependence for the tilt angle of the oblique pattern. That result is consistent with the numerical and experimental observations of the laminar–turbulent banded pattern in shear flows.

When a wave of permanent form is obliquely incident on an inclined plane, the wave pattern becomes stationary in a frame of reference which moves along the shore. This enables a simplified mathematical description of the problem which is used herein as a basis for efficient and accurate numerical simulations. First, a nonlinear and weakly dispersive set of Boussinesq equations for the downstream evolution of such stationary patterns is derived. In the hydrostatic approximation, streamline-based Lagrangian versions of the evolution equations are developed for automatic tracing of the shoreline. Both equation sets are, in their present form, developed for non-breaking waves only. Finite difference models for both equation sets are designed. These methods are then coupled dynamically to obtain a single nonlinear model with dispersive wave propagation in finite depth and an accurate runup representation. The models are tested by runup of waves at normal incidence and comparison with a more general model for the refraction of a solitary wave on a slope. Finally, a set of runup computations for oblique solitary waves is performed and compared with estimates of oblique runup heights obtained from a combination of an analytic solution for normal incidence and optics. We find that the runup heights decrease in proportion to the square of the angle of incidence for angles up to 45°, for which the height is reduced by around 12% relative to that of normal incidence. In Appendix A, the validity of the downstream formulation is discussed in the light of solitary wave optics and wave jumps.

The dynamics of inviscid-limit, incompressible and axisymmetric swirling flows in finite-length, diverging or contracting, long circular pipes is studied through global analysis techniques and numerical simulations. The inlet flow is described by the profiles of the circumferential and axial velocity together with a fixed azimuthal vorticity while the outlet flow is characterized by a state with zero radial velocity. A mathematical model that is based on the Squire–Long equation (SLE) is formulated to identify steady-state solutions of the problem with special conditions to describe states with separation zones. The problem is then reduced to the columnar (axially-independent) SLE, with centreline and wall conditions for the solution of the outlet flow streamfunction. The solution of the columnar SLE problem gives rise to the existence of four types of solutions. The SLE problem is then solved numerically using a special procedure to capture states with vortex-breakdown or wall-separation zones. Numerical simulations based on the unsteady vorticity circulation equations are also conducted and show correlation between time-asymptotic states and steady states according to the SLE and the columnar SLE problems. The simulations also shed light on the stability of the various steady states. The uniqueness of steady-state solutions in a certain range of swirl is proven analytically and demonstrated numerically. The computed results provide the bifurcation diagrams of steady states in terms of the incoming swirl ratio and size of pipe divergence or contraction. Critical swirls for the first appearance of the various types of states are identified. The results show that pipe divergence promotes the appearance of vortex-breakdown states at lower levels of the incoming swirl while pipe contraction delays the appearance of vortex breakdown to higher levels of swirl and promotes the formation of wall-separation states.

A problem with possible physiological applications concerns the escape of a viscous fluid through a small hole in a wall. The solution presented here is for a line source of sinusoidally pulsating strength located at the origin x = y = 0, where the plane y = 0 is a rigid wall and the fluid is at rest at y = + ∞. The linearized Navier—Stokes equations are solved, and results in the form of streamline plots are shown and discussed.

We consider a spectral problem that describes the time-harmonic small-amplitude motion of the mechanical system that consists of a three-dimensional water layer of constant depth and a body (either surface-piercing or totally submerged), freely floating in it. This coupled boundary-value problem contains a spectral parameter – the frequency of oscillations – in the boundary conditions as well as in the equations governing the body motion. It is proved that the total energy of the water motion is finite and the equipartition of energy of the whole system is established. Under certain restrictions on body's geometry the problem is proved to have only a trivial solution for sufficiently large values of the frequency. The uniqueness frequencies are estimated from below.

Shock waves have been observed travelling in a closed gas-filled tube when the gas is excited by a piston operating at half the fundamental frequency of the tube. Linear theory predicts a continuous periodic solution, while its first correction in a regular expansion is unbounded at such a quadratic resonant frequency. To take account of the intrinsic nonlinearity of travelling waves, a finite-rate theory of resonance is necessary. The periodic motion is then calculated from discontinuous solutions of a functional equation. Two of the three weak-shock conditions and the entropy condition are inherent in the functional equation, and hence the addition of the equal-area rule to fit shocks ensures uniqueness of the solutions.

The paper is concerned with the problem of the linear stability of an arbitrary inviscid zonal flow on a β-plane. Based on the analysis of integral relations following from the linear boundary-value problem, new evaluations, considerably more exact than the previously known ones, of the parameter region of unstable disturbances are derived. Some new relations among these bounds are established.

The propagation of linearized waves on a non-uniform slowly varying potential current is studied by converting the equations of flow into a Schrödinger ordinary differential equation in the complex plane. This equation, which is solved by the WKB method, indicates the existence of current barriers which allow partial energy transmission while reflecting the complementary part. The classical result of total transmission (Longuet-Higgins & Stewart 1961) as well as that of complete reflexion (Peregrine 1976) are recovered as limiting cases by the present, more general approach.

The scattering of plane surface water waves by both surface-piercing and submerged axisymmetric bed formations has been investigated by a number of authors. We extend previous contributions in two ways: we employ the recently derived modified mild-slope equation to approximate the fluid motion and we consider arbitrary axisymmetric topography, subject only to the restriction imposed by the mild-slope approximation. We thereby derive a robust method for calculating the scattered wave field for the small seabed slopes typical of most real situations and make a contribution to the phenomena of near-resonance and near-trapping over submerged axisymmetric shoals, which have previously been detected for only one idealized bedform.

Nutrient uptake is one of the most important factors in cell growth. Despite the biological importance, little is known about the effect of cell–cell hydrodynamic interactions on nutrient uptake in a suspension of swimming micro-organisms. In this study, we numerically investigate the nutrient uptake in an infinite suspension of squirmers. In the dilute limit, our results are in good agreement with a previous study by Magar et al. (Q. J. Mech. Appl. Maths, vol. 56, 2003, pp. 65–91). When we increased the volume fraction of squirmers, the nutrient uptake of individual cells was enhanced by the hydrodynamic interactions. The average nutrient concentration in the suspension decayed exponentially as a function of time, and the relaxation time could be scaled using the Sherwood number, the Péclet number and the volume fraction of cells. We propose a fitting function for the Sherwood number, which is useful in predicting nutrient uptake in the non-dilute regime. Furthermore, we analyse the swimming energy consumed by individual cells. The results indicate that both the energetic cost and the nutrient uptake increased as the volume fraction of cells was increased, and that the uptake per unit energy was not significantly affected by the volume fraction. These findings are important in understanding the mass transport and metabolism of swimming micro-organisms in nature and for industrial applications.

The linear stability of an infinite fluid layer with a deformable free surface covered by an insoluble surfactant and bounded below by a horizontal rigid plate oscillating in its own plane is studied based on the Floquet theory. The differential system governing the stability problem for perturbations of arbitrary wavenumbers is solved numerically by a Chebyshev collocation method. Stability boundaries are obtained in a wide range of amplitude and frequency of the modulation as well as surfactant elasticity. Results show that the presence of the surfactant may significantly stabilize (destabilize) the flow by raising (lowering) the critical Reynolds number associated with the onset of instability. The effect of the surfactant plays a stabilizing role for small surfactant elasticity and a destabilizing one for relatively large surfactant elasticity. The destabilizing effect of the surfactant on the stability of flows with a zero-shear surface is found for the first time. The disturbance modes in the form of travelling waves may be induced by the surfactant and dominate the instability of the flow.

Some simple but exact general expressions are derived for the viscous stresses required at the surface of irrotational capillary–gravity waves of periodic or solitary type on deep water in order to maintain them in steady motion. These expressions are applied to nonlinear capillary waves, and to capillary–gravity waves of solitary type on deep water. In the case of pure capillary waves some algebraic expressions are found for the work done by the surface stresses, from which it is possible to infer the viscous rate of decay of free, nonlinear capillary waves.

Similar calculations are carried out for capillary–gravity waves of solitary type on deep water. It is shown that the limiting rate of decay of a solitary wave at low amplitudes is just twice that for linear, periodic waves. This is due to the spreading out of the wave envelope at low wave steepnesses. At large wave steepnesses the dissipation increases by an order of magnitude, owing to the sharply increased curvature in the wave troughs. The calculated rates of decay are in agreement with recent observations.

A vertical cylindrical tube is partially immersed in a water-filled container and pressurized to lower the fluid level inside the tube. A sudden release of the pressure in the tube creates a singularity on top of the rising free surface. At the very beginning of the process a jet emerges at the centre of the surface, the strength of which strongly depends on the initial shape of the meniscus. Here, the time-evolution of the complex shape of the free surface and the flow around the cylindrical tube are analysed using high-speed imaging, particle image velocimetry, and numerical simulations. The tubular jet is found to be created by the following series of events, which eventually lead to the flow focusing at the tube's centre. A circular surface wave, produced by the funnelling of flow into the tube, is pushed inwards by the radial flow directly underneath the surface. As the wave moves inward and eventually collapses at the centre of the tube, a bump of fluid grows in the centre due to the converging flow in the bulk. This converging flow continues to feed the jet after the circular wave has collapsed. The singularity of the wave collapse is manifested in the initial sharp tip of the jet. All of the above events are traced back to a single origin: the convergence of the flow as it enters the tube. Movies are available with the online version of the paper.

The standard resonance conditions for Bragg scattering as well as weakly nonlinear wave triads have been traditionally derived in the absence of any background velocity. In this paper, we have studied how these resonance conditions get modified when uniform, as well as various piecewise linear velocity profiles, are considered for two-layered shear flows. Background velocity can influence the resonance conditions in two ways: (i) by causing Doppler shifts, and (ii) by changing the intrinsic frequencies of the waves. For Bragg resonance, even a uniform velocity field changes the resonance condition. Velocity shear strongly influences the resonance conditions since, in addition to changing the intrinsic frequencies, it can cause unequal Doppler shifts between the surface, pycnocline and the bottom. Using multiple scale analysis and Fredholm alternative, we analytically obtain the equations governing both the Bragg resonance and the wave triads. We have also extended the higher-order spectral method, a highly efficient computational tool usually used to study triad and Bragg resonance problems, to incorporate the effect of piecewise linear velocity profile. A significant aspect, both on the theoretical and numerical fronts, has been extending the potential flow approximation, which is the basis of the study of these kinds of problems, to incorporate piecewise constant background shear.

The uniqueness condition that was utilized by the author (Concus 1962) is considered. The condition, which excludes certain fluid depths, is shown to be physically unacceptable because it is essentially impossible to satisfy in practice. The resulting in validation of the perturbation method is discussed, and a revision is presented, which invokes the presence of viscosity and allows retention of the previously obtained solutions. The revision may also be applied to the work of other authors who utilized the same method to solve other standing-wave problems.

The dynamic equation for the second-order moment of a passive scalar increment is investigated in the context of DNS data for decaying isotropic turbulence at several values of the Schmidt number Sc, between 0.07 and 7. When the terms of the equation are normalized using Kolmogorov and Batchelor scales, approximate independence from Sc is achieved at sufficiently small r/ηB (r is the separation across which the increment is estimated and ηB is the Batchelor length scale). The results imply approximate independence of the mixed velocity-scalar derivative skewness from Sc and underline the importance of the non-stationarity. At small r/ηB, the contribution from the non-stationarity increases as Sc increases.