Field measurements of wave orbital velocities and pressure, collected in the lower part of the water column in 7 m depth with a three-component acoustic Doppler current meter and a co-located pressure transducer, are compared to the second-order theory for weakly nonlinear surface gravity waves in arbitrary water depth (Hasselmann 1962). Pressure and velocity spectra and cross-spectra are in excellent agreement with (linear) free wave transfer functions, even at (and higher than) twice the spectral peak frequency where nonlinearities (forced secondary waves) are expected to be important. Theoretical predictions show that although secondary waves sometimes contribute a significant fraction of the energy observed at double swell and sea frequencies, their effect on velocity-pressure transfer functions is small. However, forced waves are more apparent in deviations from Gaussian statistics. Good agreement between observed and predicted third-order statistics shows that Hasselmann's weakly nonlinear theory accurately describes the secondary pressure and orbital velocity (both horizontal and vertical components) field at double swell and sea frequencies, even for moderately large (0(0.1–0.2)) values of the nonlinear perturbation parameter. Only with near-breaking swell and relatively strong nonlinearities (perturbation parameter ≈ 0.22), do the observed third-order statistics diverge significantly from Hasselmann's theory.

This paper defines the linearized problem of free-surface flow past a shallow two-dimensional body in terms of complex variables and provides an asymptotic solution for the Froude number tending to zero. The simple case of flow past a rectangular box with a triangular extension in front is worked out as an example. The low-Froude-number solution for ‘double-body’ flow in a fully infinite fluid past the same box with its reflection in the upper half-plane is also calculated. The upstream free-surface height for the linear solution is compared to the equivalent free-surface height for the double-body flow. Experimental results for this body are also presented and compared.

Long-term variability in the ocean's thermohaline circulation has attracted considerable attention recently in the context of past and future climate change. Drastic circulation changes are documented in paleoceanographic data and have been simulated by general circulation models of the ocean. The mechanism of spontaneous, abrupt changes in thermohaline circulation is studied here in an idealized context, using a two-dimensional Boussinesq fluid in a rectangular container, over 5 decades of Rayleigh number.

When such a fluid is forced with a specified distribution of temperature and salinity at the surface — symmetric about a vertical axis - it attains a stable two-cell circulation, with the same symmetry. On the other hand, replacement of the specified salinity surface condition with an appropriate symmetric salt-flux condition leads to loss of stability of the symmetric circulation and gives rise to a new, asymmetric state. The extent of asymmetry depends on the magnitude of the thermal Rayleigh number, Ra, and on the strength of the salinity flux, γ. An approximate stability curve in the γ-Ra space, dividing the symmetric from the asymmetric states, is obtained numerically, and the entire range of asymmetric flows, from very slight dominance of one cell to its complete annihilation of the other cell, is explored. The physical mechanism of the pitchfork bifurcation from symmetric to asymmetric states is outlined. The effects of three other parameters of the problem are also discussed, along with implications of our results for glaciation cycles of the geological past and for interdecadal oscillations of the present ocean-atmosphere system.

The linear, inviscid reflection of a straight-crested surface wave from a vertical wall is determined on the hypothesis that the contact angle of the meniscus vanishes. The reflection coefficient is a function of the parameter γ ≡ k0l, where k0 is the wavenumber of the incident wave and l is the capillary length, and is approximated by R = exp (−4iγ2) for a gravity–capillary wave for which γ [Lt ] 1. The solution of this reflection problem is used to obtain matched-asymptotic approximations for standing waves in channels and circular cylinders. The meniscus-induced, fractional reduction of the frequency of the dominant mode in a deep circular cylinder is 0.77 γ2 (which exceeds the increase of ½γ2 associated with the capillary energy of the free surface). This decrement is within 2 mHz of the value inferred from the measurements of Cocciaro et al. (1991) after allowing for the reduction in frequency induced by the viscous boundary layers at the walls, but there are residual uncertainties (in this comparison) associated with the wetting process at the moving contact line and possible surface contamination.

The operation of an expansion tube is investigated with particular attention given to the test flow disturbances which have limited their utility in the past. Theoretical bounds for the duration of uniform test flow are first explored using one-dimensional ideal-gas relations, together with shock-tube boundary-layer entrainment effects. It is seen that test flow duration is limited either by the arrival of the downstream edge of the test-gas unsteady expansion or by the arrival of the upstream edge of this expansion after it has been reflected from the driver–test gas interface. These bounds are seen to be in good agreement with measurements made with large driver-gas expansion ratios. For small expansion ratios additional disturbances are observed in the test gas. Similar disturbances are also observed in the driver gas. It is postulated that these disturbances first appear in the driver gas and are transmitted into the test gas before the test gas is expanded. These disturbances remain with the test gas as it is expanded and subsequently produce unsteady conditions at the test section. Theoretical calculations for the range of frequencies which occur in the test gas before the expansion are obtained by modelling the disturbances as acoustic waves. It is shown that only the high-frequency components of the disturbances in the driver gas can penetrate the driver–test gas interface and this provides a mechanism for suppressing disturbances in the test gas. Additional analytical calculations for the shift in frequency produced as an acoustic wave traverses an unsteady expansion are also presented and it is shown that all frequencies of a given acoustic wave mode converge to one frequency. This focusing of frequencies is seen to occur in three different facilities.

The nonlinear resonant-triad interaction, proposed by Raetz (1959). Craik (1971), and others for a Blasius boundary layer, is analysed here for an adverse-pressure-gradient boundary layer. We assume that the adverse pressure gradient is in some sense weak and, therefore, that the instability growth rate is small. This ensures that there is a well-defined critical layer located somewhere within the flow and that the nonlinear interaction is effectively confined to that layer. The initial interaction is of the parametric resonance type, even when the modal amplitudes are all of the same order. This means that the oblique instability waves exhibit faster than exponential growth and that the growth rate of the two-dimensional mode remains linear. However, the interaction and the resulting growth rates become fully coupled, once oblique-mode amplitudes become sufficiently large, but the coupling terms are now quartic, rather than quadratic as in the Craik (1971) analysis. More importantly, however, new nonlinear interactions, which were not present in the Craik-type analyses, now come into play. These interactions eventually have a dominant effect on the instability wave development.

The nonlinear evolution of high-frequency disturbances in high-Reynolds-number Stokes layers is studied. The disturbances are composed of a two-dimensional wave (2α, 0) of magnitude δ, and a pair of oblique waves (α, ± β) of magnitude ε, where α, β are the streamwise and spanwise wavenumbers respectively. We assume that β = √3α so that the waves form a resonant triad when they are nearly neutral. It is shown that the growth rate of the disturbance is controlled by nonlinear interactions inside ‘critical layers’. In order for there to be a nonlinear feedback mechanism between the two-dimensional and the three-dimensional waves, the former is required to have a smaller magnitude than the latter, namely $\delta \sim O(\epsilon^{\frac{4}{3}})$. The timescale of the nonlinear evolution is $O(\epsilon^{-\frac{1}{3}})$.

As in Goldstein & Lee (1992), the amplitude equations turn out to be significantly different from those of Raetz (1959), Craik (1971) and Smith & Stewart (1987) in two respects. Firstly, they are integro-differential equations, i.e. the local growth rate depends on the whole history of the evolution. Secondly the back reaction of the oblique waves on the two-dimensional wave is represented by two cubic terms and one quartic term, rather than by one quadratic term. Our numerical investigations show that the amplitudes of the two- and three-dimensional waves can develop a finite-time singularity, a result of some importance. The structure of the finite-time singularity is identified, and it is found that the two-dimensional wave has a ‘more singular’ structure than the three-dimensional waves. The finite-time singularity implies that explosive growth is induced by nonlinear effects. We suggest that this nonlinear blow-up of high-frequency disturbances is related to the bursting phenomena observed in oscillatory Stokes layers and can lead to transition to turbulence.

Measurements in a zero-pressure-gradient turbulent boundary layer over a mesh-screen rough wall indicate several differences, in both inner and outer regions, in comparison to a smooth-wall boundary layer. The mean velocity distribution indicates that, apart from the expected k-type roughness function shift in the inner region, the strength of the rough-wall outer region ‘wake’ is larger than on a smooth wall. Normalizing on the wall shear stress, there is a significant increase in the normal turbulence intensity and a moderate increase in the Reynolds shear stress over the rough wall. The longitudinal turbulence intensity distribution is essentially the same for both surfaces. Normalized contributions to the Reynolds shear stress from the second (Q2) and fourth (Q4) quadrants are greater over the rough wall. The data indicate that not only are Q2 and Q4 events stronger on the rough wall but their frequency of occurrence is nearly twice as large for the rough wall as for the smooth wall. Comparison between smooth- and rough-wall spectra of the normal velocity fluctuation suggests that the strength of the active motion may depend on the nature of the surface.

New power spectra computed from LDA measurements of the fluctuating u- and v-velocity signals in a turbulent channel flow with and without drag-reducing polymer (polyethylene oxide) injection are presented. LDA data rates were sufficiently high to reconstruct the simultaneous time-dependent u- and v-velocity signals along with the time-dependent Reynolds stress signal. Time-averaged statistics of the turbulent flow are presented in conjunction with the power spectral measurements which show a dramatic reduction in both the v-velocity fluctuations and Reynolds stress fluctuations throughout the channel over all frequencies. There is also a redistribution of energy in the u-velocity fluctuations from high frequencies to low frequencies throughout the channel. Different injection conditions were examined: different polymer concentrations were injected at different flow rates such that the total amount of polymer in the channel remained constant. For certain polymer concentrations, ‘large’ negative Reynolds stress, -〈uv〉/uτ2 ≈ − 0.2, was measured in the near-wall region. In addition, there is a marked difference in the u-velocity spectra and the Reynolds stress spectra close to the wall for the different injection conditions.

The role of streamwise vortex structures in the near-field (x/d < 10) evolution of a round jet is examined. In free shear layers the streamwise vorticity develops into Bernal-Roshko structures which are streamwise vortex pairs. Similar structures are shown to exist in round jets. These structures, which evolve and amplify in the braid region between primary vortical structures, are shown to drastically alter the entrainment process in the near field and to increase the rate at which fluid is entrained into the jet. As the flow evolves downstream, the efficiency of the streamwise vorticity in entraining fluid increases relative to that of the azimuthal vorticity. Beyond the end of the potential core regime, the entrainment process is mainly controlled by streamwise vorticity. These processes are identified via flow visualization and confirmed by detailed global entrainment measurements.

We investigate the swirling flow of liquid metal in an axisymmetric cavity of arbitrary profile, generated by a rotating magnetic field. In addition to the primary swirling motion, a recirculation is generated by the Bödewadt-like boundary layers on the inclined sides of the cavity. As in the classic problem of ‘spin-up’ in a cylinder, this secondary flow has a dominating effect over the distribution of angular momentum. It is shown that, in the inviscid core, the angular momentum is independent of z, the axial coordinate, and that the applied body force is balanced by the Coriolis force. The bulk of the streamlines pass through both the core and the boundary layer, picking up energy in one region and losing it in the other. By matching the angular momentum and recirculating mass flux in the core to that in the boundary layer, a single governing equation is established for the swirl distribution. This second-order ordinary differential equation is valid for any axisymmetric shape, but is solved here for two cases; those of flow in a truncated cylinder and in a hemisphere. The former of these is compared with previously published experimental data, and with a full numerical simulation. Finally, we extend some of these ideas to buoyancy-driven flow. Here we take advantage of the analogy between centrifugal and thermally stratified flows to model natural convection of liquid metal in a cavity.

The evolution of a linear vortex filament close to a rigid sphere is investigated at high Reynolds number. The limiting evolution in an ideal flow, is analysed using a cutoff method and the results are compared with those of a singular vortex approach able to account for a viscous effect on the vortex structure evolution. The computed results show the creation of a closed vortex structure in ideal flow and also, at low Reynolds number, an unrealistic reattachment of the vortex to the surface of the body. The nature of the boundary-layer development, when the no-slip condition is satisfied, is calculated near the symmetry plane. The solutions show the development of an unsteady, vortex-driven, separating boundary layer with the three-dimensional separations dependent on the initial distance of the filament from the wall. All the solutions ultimately show a rapid growth of the secondary vorticity field near the surface and suggest an ejection from the boundary layer, followed by a strong viscous-inviscid interaction.