We study long-wave evolution and runup on piecewise linear one- and two-dimensional bathymetries analytically and experimentally with the objective of understanding certain coastal effects of tidal waves. We develop a general solution method for determining the amplification factor of different ocean topographies consisting of linearly varying and constant-depth segments to study how spectral distributions evolve over bathymetry, and apply our results to study the evolution of solitary waves. We find asymptotic results which suggest that solitary waves often interact with piecewise linear topographies in a counter-intuitive manner. We compare our analytical predictions with numerical results, with results from a new set of laboratory experiments from a physical model of Revere Beach, and also with the data on wave runup around an idealized conical island. We find good agreement between our theory and the laboratory results for the time histories of free-surface elevations and for the maximum runup heights. Our results suggest that, at least for simple piecewise linear topographies, analytical methods can be used to calculate effectively some important physical parameters in long-wave runup. Also, by underscoring the effects of the topographic slope at the shoreline, this analysis qualitatively suggests why sometimes predictions of field-applicable numerical models differ substantially from observations of tsunami runup.

All components of the fluctuating vorticity vector have been measured in decaying grid turbulence using a vorticity probe of relatively simple geometry (four X-probes, i.e. a total of eight hot wires). The data indicate that local isotropy is more closely satisfied than global isotropy, the r.m.s. vorticities being more nearly equal than the r.m.s. velocities. Two checks indicate that the performance of the probe is satisfactory. Firstly, the fully measured mean energy dissipation rate 〈ε〉 is in good agreement with the value inferred from the rate of decay of the mean turbulent energy 〈q2〉 in the quasi-homogeneous region; the isotropic mean energy dissipation rate 〈εiso〉 agrees closely with this value even though individual elements of 〈ε〉 indicate departures from isotropy. Secondly, the measured decay rate of the mean-square vorticity 〈ω2〉 is consistent with that of 〈q2〉 and in reasonable agreement with the isotropic form of the transport equation for 〈ω2〉. Although 〈ε〉≃〈εiso〉, there are discernible differences between the statistics of ε and εiso; in particular, εiso is poorly correlated with either ε or ω2. The behaviour of velocity increments has been examined over a narrow range of separations for which the third-order longitudinal velocity structure function is approximately linear. In this range, transverse velocity increments show larger departures than longitudinal increments from predictions of Kolmogorov (1941). The data indicate that this discrepancy is only partly associated with differences between statistics of locally averaged ε and ω2, the latter remaining more intermittent than the former across this range. It is more likely caused by a departure from isotropy due to the small value of Rλ, the Taylor microscale Reynolds number, in this experiment.

A new method for determining the slow and rapid pressure-strain rate terms directly from wind-tunnel experiments has been developed with the aid of a newly developed theoretical description of the kinematics of homogeneous axisymmetric turbulence. Both the straining and the return-to-isotropy process of homogeneous axisymmetric turbulence are studied with the aim of improving Reynolds stress closures. Direct experimental determination of the different terms in the transport equation for the Reynolds stress tensor plays a major role in the validation and development of turbulence models. For the first time it is shown that the pressure{strain correlation can be determined with good accuracy without balancing it out from the Reynolds stress transport equation (and without measuring the pressure). Instead it is determined through evaluation of integrals containing second- and third-order two-point velocity correlations. All the terms in the Reynolds stress equations are measured directly and balance is achieved.

Measurements obtained in boundary layers developing downstream of the highly turbulent, separated flow generated at the leading edge of a blunt flat plate are presented. Two cases are considered: first, when there is only very low (wind tunnel) turbulence present in the free-stream flow and, second, when roughly isotropic, homogeneous turbulence is introduced. With conditions adjusted to ensure that the separated region was of the same length in both cases, the flow around reattachment was significantly different and subsequent differences in the development rate of the two boundary layers are identified. The paper complements, but is much more extensive than, the earlier presentation of some of the basic data (Castro & Epik 1996), confirming not only that the development process is very slow, but also that it is non-monotonic. Turbulence stress levels fall below those typical of zero-pressure-gradient boundary layers and, in many ways, the boundary layer has features similar to those found in standard boundary layers perturbed by free-stream turbulence. It is argued that, at least as far as the turbulence structure is concerned, the inner layer region develops no more quickly than does the outer flow and it is the latter which essentially determines the overall rate of development of the whole flow. Some numerical computations are used to assess the extent to which current turbulence models are adequate for such flows.

We study the dynamics of internal gravity waves excited by parametric instability in a stably stratified medium, either at the interface between a water and a kerosene layer, or in brine with a uniform gradient of salinity. The tank has a rectangular section, and is narrow to favour standing waves with motion in the vertical plane. The fluid container undergoes vertical oscillations, and the resulting modulation of the apparent gravity excites the internal waves by parametric instability.

Each internal wave mode is amplified for an excitation frequency close to twice its natural frequency, when the excitation amplitude is sufficient to overcome viscous damping (these conditions define an ‘instability tongue’ in the parameter space frequency-amplitude). In the interfacial case, each mode is well separated from the others in frequency, and behaves like a simple pendulum. The case of a continuous stratification is more complex as different modes have overlapping instability tongues. In both cases, the growth rates and saturation amplitudes behave as predicted by the theory of parametric instability for an oscillator. However, complex friction effects are observed, probably owing to the development of boundary-layer instabilities.

In the uniformly stratified case, the excited standing wave is unstable via a secondary parametric instability: a wave packet with small wavelength and half the primary wave frequency develops in the vertical plane. This energy transfer toward a smaller scale increases the maximum slope of the iso-density surfaces, leading to local turning and rapid growth of three-dimensional instabilities and wave breaking. These results illustrate earlier stability analyses and numerical studies. The combined effect of the primary excitation mechanism and wave breaking leads to a remarkable intermittent behaviour, with successive phases of growth and decay for the primary wave over long timescales.

The convective flows which arise in shallow cavities filled with low-Prandtl-number fluids when subjected to a horizontal temperature gradient are studied numerically with a finite element method. Attention is focused on a rigid cavity with dimensions 4×2×1, for which experimental data are available. The three-dimensional results indicate that, after a relative concentration of the initial Hadley circulation, a transition to time-dependent flows occurs in the form of a roll oscillation with a purely dynamical origin. This transition corresponds to a Hopf bifurcation with a breaking of symmetry that gives some specific properties to the time evolution of the flow: these properties are shown to be the result of the general behaviour of the dynamical systems. Calculations performed in the case of mercury compare well with the experiments with similar power spectra of the temperature, and this validates the analysis of the nature of the global flow performed in the limiting case Pr=0. All these results are discussed with respect to the linear and nonlinear analyses and to other computational experiments. Numerical results obtained in the corresponding two-dimensional situation give a different transition to the time-dependent flow: it is shown that in the three-dimensional cavity this type of two-dimensional transition is less probable than the observed transition with breaking of symmetry.

Viscous flow about a circular cylinder that is submerged beneath free-surface travelling waves is considered. The wave amplitude is assumed small and results are presented for a wide range of Reynolds number. Particular attention is focused on the second-order time-averaged flow that manifests itself as a circulatory motion about the cylinder. The paper complements earlier work on this problem by Yan & Riley (1996) in the large Reynolds number, boundary-layer, regime and Riley & Yan (1996) in the inviscid flow limit, and makes a comparison with experimental work by Chaplin (1984) possible.

We consider the transport of a tracer substance in Poiseuille flow through a pipe lined with a thin, fixed wall layer in which the tracer is soluble. A formal solution is given for the variation of concentration with time at a fixed downstream position following an initial release of tracer. Asymptotic approximations are derived assuming that: (i) the Péclet number is large; (ii) the time scale for diffusion across the wall layer is much larger than that for diffusion across the fluid phase and (iii) the dimensionless distance downstream of the point of release, z, is large. This means that the transverse concentration variation is small within the fluid phase, so that transport is dominated by the exchange of tracer between the phases and radial diffusion within the wall layer. The character of the concentration transient is found to be determined by two dimensionless numbers, an absorption parameter κ and an effective wall layer thickness ν (both rescaled to take account of the ratio of diffusivities in the two phases); by assumption (ii), ν is large. Several different regimes are possible, according to the values of κ, ν and z. At sufficiently large distances, a Gaussian approximation, analogous to Taylor's solution, is applicable. At intermediate distances, provided κ is not too large, a highly skewed transient is predicted. If κ is small, there exists another region further upstream where the effect of the wall is negligible, and Taylor's Gaussian approximation applies. More complicated behaviour occurs in the zones of transition between these three regions. The behaviour described is expected to be typical of a range of similar systems. In particular, it may be shown that the basic form of the skewed approximation is insensitive to the geometry of the system, and also applies when the Péclet number is of order unity.

A generalized similarity formulation extending the work of Terrill (1967) for Couette–Poiseuille flow in the annulus between concentric cylinders of infinite extent is given. Boundary conditions compatible with the formulation allow a study of the effects of inner and outer cylinder transpiration, rotation, translation, stretching and twisting, in addition to that of an externally imposed constant axial pressure gradient. The problem is governed by η, the ratio of inner to outer radii, a Poiseuille number, and nine Reynolds numbers. Single-cylinder and planar problems can be recovered in the limits η→0 and η→1, respectively. Two coupled primary nonlinear equations govern the meridional motion generated by uniform mass flux through the porous walls and the azimuthal motion generated by torsional movement of the cylinders; subsidiary equations linearly slaved to the primary flow govern the effects of cylinder translation, cylinder rotation, and an external pressure gradient. Steady solutions of the primary equations for uniform source/sink flow of strength F through the inner cylinder are reported for 0[les ]η[les ]1. Asymptotic results corroborating the numerical solutions are found in different limiting cases. For F<0 fluid emitted through the inner cylinder fills the gap and flows uniaxially down the annulus; an asymptotic analysis leads to a scaling that removes the effect of η in the pressure parameter β, namely β=π2R*2, where R*=F(1−η)/(1+η). The case of sink flow for F>0 is more complex in that unique solutions are found at low Reynolds numbers, a region of triple solutions exists at moderate Reynolds numbers, and a two-cell solution prevails at large Reynolds numbers. The subsidiary linear equations are solved at η=0.5 to exhibit the effects of cylinder translation, rotation, and an axial pressure gradient on the source/sink flows.

This paper deals with the stability and transition to turbulence of wall-bounded unsteady velocity profiles with reverse flow. Such flows occur, for example, during unsteady boundary layer separation and in oscillating pipe flow. The main focus is on results from experiments in time-developing flow in a long pipe, which is decelerated rapidly. The flow is generated by the controlled motion of a piston. We obtain analytical solutions for laminar flow in the pipe and in a two-dimensional channel for arbitrary piston motions. By changing the piston speed and the length of piston travel we cover a range of values of Reynolds number and boundary layer thickness. The velocity profiles during the decay of the flow are unsteady with reverse flow near the wall, and are highly unstable due to their inflectional nature. In the pipe, we observe from flow visualization that the flow becomes unstable with the formation of what appears to be a helical vortex. The wavelength of the instability ≃3δ where δ is the average boundary layer thickness, the average being taken over the time the flow is unstable. The time of formation of the vortices scales with the average convective time scale and is ≃39/(Δū/δ), where Δu=(umax−umin) and umax, umin and δ are the maximum velocity, minimum velocity and boundary layer thickness respectively at each instant of time. The time to transition to turbulence is ≃33/(Δū/δ). Quasi-steady linear stability analysis of the velocity profiles brings out two important results. First that the stability characteristics of velocity profiles with reverse flow near the wall collapse when scaled with the above variables. Second that the wavenumber corresponding to maximum growth does not change much during the instability even though the velocity profile does change substantially. Using the results from the experiments and the stability analysis, we are able to explain many aspects of transition in oscillating pipe flow. We postulate that unsteady boundary layer separation at high Reynolds numbers is probably related to instability of the reverse flow region.

We consider steady compressible Euler flow corresponding to the compressible analogue of the well-known incompressible Hill's spherical vortex (HSV). We first derive appropriate compressible Euler equations for steady homentropic flow and show how these may be used to define a continuation of the HSV to finite Mach number M∞=U∞/C∞, where U∞, C∞ are the fluid velocity and speed of sound at infinity respectively. This is referred to as the compressible Hill's spherical vortex (CHSV). It corresponds to axisymmetric compressible Euler flow in which, within a vortical bubble, the azimuthal vorticity divided by the product of the density and the distance to the axis remains constant along streamlines, with irrotational flow outside the bubble. The equations are first solved numerically using a fourth-order finite-difference method, and then using a Rayleigh–Janzen expansion in powers of M2∞ to order M4∞. When M∞>0, the vortical bubble is no longer spherical and its detailed shape must be determined by matching conditions consisting of continuity of the fluid velocity at the bubble boundary. For subsonic compressible flow the bubble boundary takes an approximately prolate spheroidal shape with major axis aligned along the flow direction. There is good agreement between the perturbation solution and Richardson extrapolation of the finite difference solutions for the bubble boundary shape up to M∞ equal to 0.5. The numerical solutions indicate that the flow first becomes locally sonic near or at the bubble centre when M∞≈0.598 and a singularity appears to form at the sonic point. We were unable to find shock-free steady CHSVs containing regions of locally supersonic flow and their existence for the present continuation of the HSV remains an open question.

The spreading and sediment deposit of a two-dimensional, unsteady, laminar mud flow from a constant-volume source on a relatively steep slope is studied theoretically and experimentally. The mud under consideration has the rheological properties of a Herschel–Bulkley fluid. The flow is of low-Reynolds-number type and has a well-formed wave front moving a substantial distance downslope. Due to the nonlinear rheological characteristics, a set of nonlinear partial differential equations is needed for this transient problem. Depth-integrated continuity and momentum equations are derived by applying von Kármán's momentum integral method. A matched-asymptotic perturbation method is implemented analytically to get asymptotic solutions for both the outer region away from, and the inner region near, the wave front. The outer solution gives accurate results for spreading characteristics, while the inner solution, which is shown to agree well with experimental results of Liu & Mei (1989) for a Bingham fluid, predicts fairly well the free-surface profile near the wave front. A composite solution uniformly valid over the whole spreading length is then achieved through a matching of the inner and outer solutions in an overlapping region. The range of accuracy of the solution and the size of the inner and overlapping regions are quantified by physical scaling analyses. Rheological and dynamic measurements are obtained through laboratory experiments. Theoretical predictions are compared with experimental results, showing reasonable agreement. The impact of shear thinning on the runout characteristics, free-surface profiles and final deposit of the mud flow is examined. A mud flow with shear thinning spreads beyond the runout distance estimated by a Bingham model, and has a long and thin deposit.

Following the finite-time collapse of an unsteady interacting boundary layer (step 1), shortened length and time scales are examined here in the near-wall dynamics of transitional-turbulent boundary layers or during dynamic stall. The next two steps are described, in which (step 2) normal pressure gradients come into operation along with a continuing nonlinear critical-layer jump and then (step 3) vortex formation is induced typically. Normal pressure gradients enter in at least two ways, depending on the internal or external flow configuration. This yields for certain internal flows an extended KdV equation with an extra nonlinear integral contribution multiplied by a coefficient which is proportional to the normal rate of change of curvature of the velocity profile locally and whose sign turns out to be crucial. Positive values of the coefficient lead to a further finite-time singularity, while negative values produce a rapid secondary instability phenomenon. Zero values in contrast allow an interplay between solitary waves and wave packets to emerge at large scaled times, this interplay eventually returning the flow to its original, longer, interactive, boundary-layer scales but now coupled with multiple shorter-scale Euler regions. In external or quasi-external flows more generally an extended Benjamin–Ono equation holds instead, leading to a reversal in the roles of positive and negative values of the coefficient. The next step, 3, typically involves the strong wind-up of a local vortex, leading on to explosion or implosion of the vortex. Further discussion is also presented, including the three-dimensional setting, the computational implications, and experimental links.

A separated turbulent boundary layer over a flat plate was investigated by direct numerical simulation of the incompressible Navier–Stokes equations. A suction-blowing velocity distribution was prescribed along the upper boundary of the computational domain to create an adverse-to-favourable pressure gradient that produces a closed separation bubble. The Reynolds number based on inlet free-stream velocity and momentum thickness is 300. Neither instantaneous detachment nor reattachment points are fixed in space but fluctuate significantly. The mean detachment and reattachment locations determined by three different definitions, i.e. (i) location of 50% forward flow fraction, (ii) mean dividing streamline (ψ=0), (iii) location of zero wall-shear stress (τw=0), are in good agreement. Instantaneous vorticity contours show that the turbulent structures emanating upstream of separation move upwards into the shear layer in the detachment region and then turn around the bubble. The locations of the maximum turbulence intensities as well as Reynolds shear stress occur in the middle of the shear layer. In the detached flow region, Reynolds shear stresses and their gradients are large away from the wall and thus the largest pressure fluctuations are in the middle of the shear layer. Iso-surfaces of negative pressure fluctuations which correspond to the core region of the vortices show that large-scale structures grow in the shear layer and agglomerate. They then impinge on the wall and subsequently convect downstream. The characteristic Strouhal number St=fδ*in/U0 associated with this motion ranges from 0.0025 to 0.01. The kinetic energy budget in the detachment region is very similar to that of a plane mixing layer.

Experiments have been undertaken to investigate dam-break flows where a thin plate separating water at different levels is withdrawn impulsively in a vertically upwards direction. Depth ratios of 0, 0.1 and 0.45 were investigated for two larger depth values of 10 cm and 36 cm. The resulting sequence of surface profiles is shown to satisfy approximately Froude scaling. For the dry-bed case a horizontal jet forms at small times and for the other cases a vertical, mushroom-like jet occurs, none of which have been observed previously. We analyse the initial-release problem in which the plate is instantaneously removed or dissolved. Although this shows singular behaviour, jet-like formations are predicted. Artificially smoothing out the singularity enables a fully nonlinear, potential-flow computation to follow the jet formation for small times. There is qualitative agreement between theory and experiment.

In the experiments, after a bore has developed downstream as a result of highly complex flow interactions, the surface profiles agree remarkably well with exact solutions of the shallow-water equations which assume hydrostatic pressure and uniform velocity over depth.