The linear stability of a two-dimensional buoyant plume is analysed by taking into account the transverse velocity component and the streamwise variations of the basic flow and of the disturbance waves. The solutions indicate the dependence of the spatial amplification rate and wavenumber on the disturbance flow quantity under consideration as well as on the streamwise and transverse coordinates. The use of the flow quantity relative to the basic flow leads to close agreement with the neutral curve according to quasi-parallel-stability theory, the usual method for treating nearly parallel flows. However, the amplification rate within an unstable region shows substantial deviation from that predicted by the quasi-parallel theory. The validity of this non-parallel theory is supported by the existing experimental data.

We examine the statistical mechanisms by which energy and scalar variance are cascaded to small scales for isotropic, three-dimensional turbulence. Two avenues are explored: (i) the traditional transfer function (defined by the nonlinear cascade that gives the time rate of change of the energy spectrum), and (ii) the bispectrum (the elementary triple-point correlation, averaged over directions perpendicular to three co-linear observation points). Our tools are direct numerical simulations (DNS), and the statistical theory of turbulence, here in the form of the test field model (TFM) (Kraichnan 1971). Comparison of the results indicates a fairly good quantitative agreement between DNS and the TFM at large Prandtl numbers (Pr ≥ 0.25), but substantial disagreement at lower Pr, where the transfer to small scales becomes too small. This disparity we trace to the Markovian aspect of the TFM; the more fundamental direct interaction approximation (DIA) (Kraichnan 1959) compares more favourably to DNS as Pr → 0. For Pr ∼ 1, we compare DNS and TFM bispectra for velocity and scalar fields in both Fourier and physical space. The physical space representation of bispectra serves as a useful means of discriminating between velocity and scalar transfer.

Effects of viscous damping on mass transport velocity in a two-layer fluid system are studied. A temporally decaying small-amplitude interfacial wave is assumed to propagate in the fluids. The establishment and the decay of mean motions are considered as an initial-boundary-value problem. This transient problem is solved by using a Laplace transform with a numerical inversion. It is found that thin ‘second boundary layers’ are formed adjacent to the interfacial Stokes boundary layers. The thickness of these second boundary layers is of O(ε1/2) in the non-dimensional form, where ε is the dimensionless Stokes boundary layer thickness defined as $\epsilon = \hat{k}\hat{\delta}=\hat{k}(2\hat{v}/\hat{\sigma})^{1/2}$ for an interfacial wave with wave amplitude â, wavenumber $\hat{k}$ and frequency $\hat{\sigma}$ in a fluid with viscosity $\hat{v}$. Inside the second boundary layers there exists a strong steady streaming of O(α2ε−1/2), where $\alpha = \hat{k}\hat{a}$ is the surface wave slope. The mass transport velocity near the interface is much larger than that in a single-layer system, which is O(α2) (e.g. Longuet-Higgins 1953; Craik 1982). In the core regions outside the thin second boundary layers, the mass transport velocity is enhanced by the diffusion of the mean interfacial velocity and vorticity. Because of vertical diffusion and viscous damping of the mean interfacial vorticity, the ‘interfacial second boundary layers’ diminish as time increases. The mean motions eventually die out owing to viscous attenuation. The mass transport velocity profiles are very different from those obtained by Dore (1970, 1973) which ignored viscous attenuation. When a temporally decaying small-amplitude surface progressive wave is propagating in the system, the mean motions are found to be much less significant, O(α2).

An analysis is constructed in order to estimate the dispersion relation for internal waves trapped in a layer and propagating linearly in a fluid of infinite depth with a rigid surface. The main interest is in predicting the structure of internal wave wakes, but the results are applicable to any internal waves. It is demonstrated that, in general 1/cp = 1/CpO + k/ωmax + ∈(k) where cp is the wave phase speed for a particular mode, CpO is the phase speed at k = 0, ωmax is the maximum possible wave angular frequency and ωmax ≤ Nmax where Nmax is the maximum buoyancy frequency. Also, ∈(0) = 0, ∈(k) = o(k) for k large, and is bounded for finite k. In particular, when ∈(k) can be neglected, the dispersion relation for a lowest mode wave is approximately 1/cp ≈ (∫∞0N2(y)ydy)-½ + k/ωmax. The eigenvalue problem is analysed for a class of buoyancy frequency squared functions N2(x) which is taken to be a class of realvalued functions of a real variable x where O ≤ x ∞ such that N2(x) = O(e-βx) as x → ∞ and 1/β is an arbitrary length scale. It is demonstrated that N2(x) can be represented by a power series in e-βx. The eigenfunction equation is constructed for such a function and it is shown that there are two cases of the equation which have solutions in terms of known functions (Bessel functions and confluent hypergeometric functions). For these two cases it is shown that ∈(k) can be neglected and that, in addition, ωmax = Nmax. More generally, it is demonstrated that when k → ∞ it is possible to approximate the equation uniformly in such a way that it can be compared with the confluent hypergeometric equation. The eigenvalues are then, approximately, zeros of the Whittaker functions. The main result which follows from this approach is that if N2(x) is O(e-βx) as x → ∞ and has a maximum value N2max then a sufficient condition for 1/cp ∼ k/Nmax to hold for large k for the lowest mode is that N2(t)/t is convex for O ≤ t ≤ 1 where t = e-βx.

Large-scale oceanic flows, ranging from large estuaries to basin-scale abyssal circulation, can be driven by imposed lateral density gradients and are significantly influenced by the Earth's rotation. Some essential features of such flows have been incorporated into a laboratory model by differentially heating and cooling the vertical endwalls of a shallow rectangular cavity, rotating about a vertical axis through its centre. In this paper, the flow observed after this system had been allowed to develop to a steady mean state is described. Measurements were made of the velocity and temperature fields, as well as the variability in temperature field associated with eddy motions. The temperature data were also used to calculate the potential vorticity distribution within the cavity. These results indicate that the heated and cooled end-walls acted not only as continuous sources of buoyancy, but also of anomalous potential vorticity. It is shown that the potential vorticity served as an indicator and tracer of the laboratory flow. The structure and location of boundary currents were largely determined by nonlinear potential vorticity dynamics which resulted in a mean circulation consisting of two basin-scale counter-rotating gyres. This flow differed markedly from the initial boundary currents set up by Kelvin waves.

The gravitationally driven collapse of a reservoir into an initially stationary layer of fluid, termed the tailwater, is studied using the nonlinear shallow water equations. The motion is tackled using the hodograph transformation of the governing equation which allows the solutions for velocity and depth of the shallow flowing layer to be constructed by analytical techniques. The front of the flow emerges as a bore across which the depth of the fluid jumps discontinuously to the tailwater depth. The speed of the front is initially constant, but progressively slows once the finite extent of the reservoir begins to influence the motion. There then emerges a variety of phenomena depending upon the depth of the tailwater relative to the initial depth of the reservoir. Provided that the tailwater is sufficiently deep, a region of quiescent fluid emerges adjacent to the rear wall of the reservoir, followed by a region within which the velocity is negative. Also it is shown that for non-vanishing tailwater depths, continuous solutions for the velocity and height of the flowing layer breakdown after a sufficient period and develop an interior bore, the location and time of inception of which are calculated directly from quasi-analytical solutions.

Resonance characteristics of tidal waves in a network are investigated with the linearized, one-dimensional shallow water equations. The network comprises a semi-enclosed main channel with an adjacent secondary channel at an arbitrary position. Water motion is forced by a prescribed incoming wave at the entrance of the main channel. The model is used to compute the ratio of sea surface height amplitude in the presence and absence of the secondary channel. Relevance lies in the possible construction of secondary channels to reduce tidal range in the main channel. When $\mu = 2\mathrm{\pi} { l}_{b}^{\ast } / { l}_{t}^{\ast } \gtrsim \mathrm{\pi} $ (${ l}_{b}^{\ast } $ being the length of the main channel, ${ l}_{t}^{\ast } $ the tidal wavelength) and friction is weak, it is found that reduction (amplification) of the tidal range occurs for secondary channels located less (more) than a quarter wavelength from the closed end of the main channel. Furthermore, a minimum is observed halfway between the closed end and the nodal point, and a maximum a quarter wavelength further seaward. With increasing friction and/or decreasing length of the main channel, amplitude ratios become less pronounced and depend weakly on the location of the secondary channel. The mechanism responsible for amplification or reduction of the sea surface height in the main channel is identified and explained in terms of the forced wave and waves radiating away from the secondary channel.

While acoustic scatter from oceanic turbulence is sensitive to temperature–salinity covariations, there are unfortunately no published measurements of the turbulent temperature–salinity co-spectrum. Several models have been proposed for the form of the co-spectrum of two scalars in turbulence, but they all produce unsatisfactory results when applied to the turbulent scattering equations (either predicting negative scattering cross-sections in some regimes or predicting implausible levels of correlation between temperature and salinity at some scales). A new model is proposed and shown to give physically plausible scattering predictions in all density regimes. High-frequency acoustic data illustrate the importance of the co-spectrum for acoustic scattering, but were collected in a density regime where there is little difference between the co-spectrum models.

Generation and development of disturbances in a hypersonic viscous shock layer on a flat plate is studied both experimentally and numerically. The study is performed at the Mach number M∞ = 21 and the Reynolds number ReL = 1.44 × 105 and is aimed at elucidating the physical mechanisms that govern the receptivity and instability of the shock layer at extremely high hypersonic velocities. The experiments are conducted in a hypersonic nitrogen-driven wind tunnel. An electron-beam fluorescence technique, a Pitot probe and a piezoceramic transducer are used to measure the mean density and Mach number contours, as well as density and pressure fluctuations, their spectra and spatial distributions in the shock layer. Direct numerical simulations are performed by solving the Navier–Stokes equations with a high-order shock-capturing scheme in a computational domain including the leading and trailing edges of the plate, so that the bow shock wave and the wake behind the plate are also simulated. It is demonstrated that computational and experimental data characterizing the mean flow field, intensity of density fluctuations and their spatial distributions in the shock layer are in close agreement. It is found that excitation of the shock layer by external acoustic waves leads to generation of entropy–vortex disturbances with two maxima of density fluctuations: directly behind the shock wave and on the external edge of the boundary layer. At the same time, the pressure fluctuations decay inward into the shock layer, away from the shock, which agrees with the linear theory of interaction of shock waves with small perturbations. Thus, the entropy–vortex disturbances are shown to dominate in the hypersonic shock layer at very high Mach numbers, in contrast with the boundary layers at moderate hypersonic velocities where acoustic modes are most important. A parametric numerical study of wave processes in the shock layer induced by external acoustic waves is performed with variations of frequency, amplitude and angle of propagation of external disturbances. The amplitude of generated disturbances is observed to grow and decay periodically along the streamwise coordinate, and the characteristics of these variations depend on the frequency and direction of incident acoustic waves. The hypersonic shock layer excited by periodic blowing and suction near the leading edge is also investigated; in the experiments, this type of excitation is obtained by using an oblique-cut whistle. It is shown that blowing/suction generates disturbances resembling those generated by external acoustic waves, with similar spatial distributions and phase velocities. This result paves the way for active control of instability development in the shock layer by means of destructive interference of two types of disturbances. Numerical simulations are performed to show that instability waves can be significantly amplified or almost entirely suppressed, depending on the relative phase of blowing/suction and acoustic disturbances. Wind-tunnel experiments completely confirm this numerical prediction. Thus, the feasibility of delaying instability development in the hypersonic shock layer has been demonstrated for the first time.

Under the assumption that the local acceleration and inertial terms are to be neglected in the equation of motion, a comparison is made between the unsteadystate movement of a suspension of particles and the same problem in reversed (or negative) time. The formalized discussion confirms Bretherton's (1962) conclusion that, if in a steady unidirectional shear flow at small Reynolds number a rotationally symmetric particle twice assumes a position with its axis of rotation in the plane perpendicular to the flow, its orbit must be periodic. The reversibility phenomena observed by S. G. Mason (1963) in dilute and concentrated suspensions are explained as well.

Complex wave patterns caused by unsteady heat release due to cloud formation in confined compressible flows are discussed. Two detailed numerical studies of condensation-induced waves are carried out. First, the response of a flow of nitrogen in a slender Laval nozzle to a sudden addition of water vapour at the nozzle entrance is considered. Condensation occurs just downstream of the nozzle throat, which initially leads to upstream- and downstream-moving shocks and an expansion fan downstream of the condensation front. Then, the flow becomes oscillatory and the expansion fan disappears, while upstream and much weaker downstream shocks are repeatedly generated. For a lower initial humidity, only a downstream starting shock is formed and a steady flow is established. Second, homogeneous condensation in an unsteady expansion fan in humid nitrogen is considered. In the initial phase, two condensation-induced shocks are found, moving upstream and downstream. The upstream-moving shock changes the shape of the expansion fan and has a strong influence on the condensation process itself. It is even quenching the nucleation process locally, which leads to a renewed condensation process more downstream. This process is repeated with asymptotically decreasing strength. The repeated interaction of the condensation-induced shocks with the main expansion wave leads to a distortion of the expansion wave towards its shape that can be expected on the basis of phase equilibrium, i.e. a self-similar wave structure consisting of dry part, a plateau of constant state and a wet part. The strengths of the condensation-induced waves, as well for the Laval nozzle flow as for the expansion fan, appear to be in qualitative agreement with the results from the analytical Rayleigh–Bartlmä model.

We report an experimental investigation of the wake of free falling super-hydrophobic spheres. The mutual interaction between the air layer (plastron) encapsulating the super-hydrophobic spheres and the flow is emphasised by studying the hydrodynamic performance. It is found that the air plastron adapts its shape to the flow-induced stresses which compete with the surface tension. This competition is characterised by introducing the Weber number ${\mathcal{W}}e$, whilst the plastron deformation is estimated via the aspect ratio $\unicode[STIX]{x1D712}$. While noticeable distortions are locally observed, the plastron becomes more and more spherical on average (i.e. $\unicode[STIX]{x1D712}\rightarrow 1$) as far as ${\mathcal{W}}e$ increases. The study of the falling motion reveals that the plastron compliance has a sizeable influence on the wake development. Investigating the lift force experienced by the super-hydrophobic spheres, the onset of wake instabilities is found to be triggered earlier than for smooth spheres used as reference. Surprisingly, it is also observed that the early promotion of the wake instabilities is even more pronounced beyond a critical Weber number, ${\mathcal{W}}e_{c}$, which corresponds to a critical aspect ratio $\unicode[STIX]{x1D712}_{c}$. Furthermore, the magnitude of the hydrodynamic loads is found to be dependent on the average deformation of the gas/liquid interface. Indeed, in comparison to the reference spheres, the high deformation achieved for $\unicode[STIX]{x1D712}>\unicode[STIX]{x1D712}_{c}$ (oblate shape) leads to lift and drag increase, whereas the low deformation obtained for $\unicode[STIX]{x1D712}<\unicode[STIX]{x1D712}_{c}$ (spherical shape) yields lift and drag mitigation. Accordingly, taking into account the plastron deformation provides an attractive way to explain the somehow discordant results reported in other studies at comparable Reynolds numbers. These results suggest that the amount of vorticity produced at the body surface and then released in the wake is strongly impacted by the plastron compliance. If confirmed by additional studies and extrapolated to other flow configurations, our findings would imply that plastron compliance and its feedback on the flow, which are currently neglected in most theoretical works and numerical simulations, must be accounted for to design super-hydrophobic surfaces and/or predict their performance.

The steady-state solution for the stratified flow over a bounded obstacle in a channel of finite height is obtained through the use of the inviscid unsteady Oseen equations; the body streamline encloses and is produced by a planar momentum sink. The calculations are compared with the experimental observations of Wei, Kao & Pao (1975) and Baines (1977). In a case where theoretical and experimental values match, good agreement was found. This suggests that the Oseen model with its columnar, wavelike and decaying solutions may provide a reasonable alternative to Long's model for subcritical flows.

Two-dimensional steady laminar flows past multiple thin blades positioned in near or exact sequence are examined for large Reynolds numbers. Symmetric configurations require solution of the boundary-layer equations alone, in parabolic fashion, over the successive blades. Non-symmetric configurations in contrast yield a new global inner–outer interaction in which the boundary layers, the wakes and the potential flow outside have to be determined together, to satisfy pressure-continuity conditions along each successive gap or wake. A robust computational scheme is used to obtain numerical solutions in direct or design mode, followed by analysis. Among other extremes, many-blade analysis shows a double viscous structure downstream with two streamwise length scales operating there. Lift and drag are also considered. Another new global interaction is found further downstream. All the interactions involved seem peculiar to multi-blade flows.

This paper considers hydraulic control and upstream influence in systems where the only wave propagation mechanism arises from the variation of vorticity or potential vorticity. These systems include two-dimensional shear flows as well as many simple paradigms for large-scale geophysical flows. The simplest is a flow in which the vorticity or potential vorticity is piecewise constant. We consider such a flow confined to a rotating channel and disturbed by a topographic perturbation. We analyse the behaviour of the system using steady nonlinear long-wave theory and demonstrate that it exhibits behaviour analogous to open-channel hydraulics, with the possibility of different upstream and downstream states. The manner by which the system achieves such states is examined using time-dependent long-wave theory via integration along characteristics and using full numerical solution via the contour-dynamics technique.

The full integrations agree well with the hydraulic interpretation of the steady-state theory. One aspect of the behaviour of the system that is not seen in open-channel hydraulics is that for strong subcritical flows there is a critical topographic amplitude beyond which information from the control cannot propagate far upstream. Instead flow upstream of the topographic perturbation adjusts until the long-wave speed is zero, the control moves to the leading edge of the obstacle and flow downstream of the control is supercritical, with a transition from one supercritical branch to another on the downstream slope of the obstacle.

This paper investigates numerically and through an asymptotic approach the three-dimensional stability of steady vertical vortex arrays in a stratified and rotating fluid. Three classical vortex arrays are studied: the Kármán vortex street, the symmetric double row and the single row of co-rotating vortices. The asymptotic analysis assumes well-separated vortices and long-wavelength bending perturbations following Billant (J. Fluid Mech., vol. 660, 2010, p. 354) and Robinson & Saffman (J. Fluid Mech., vol. 125, 1982, p. 411). Very good agreement with the numerical stability analysis is found even for finite wavelength and relatively close vortices. For a horizontal Froude number Fh ≤ 1 and for a non-rotating fluid, it is found that the Kármán vortex street for a street spacing ratio (the distance h between the rows divided by the distance b between vortices in the same row) κ ≤ 0.41 and the symmetric double row for any spacing ratio are most unstable to a three-dimensional instability of zigzag type that vertically bends the vortices. The most amplified vertical wavenumber scales like 1/(bFh) and the growth rate scales with the strain Γ/(2πb2), where Γ is the vortex circulation. For the Kármán vortex street, the zigzag instability is symmetric with respect to the midplane between the two rows while it is antisymmetric for the symmetric double row. For the Kármán vortex street with well-separated vortex rows κ > 0.41 and the single row, the dominant instability is two-dimensional and corresponds to a pairing of adjacent vortices of the same row. The main differences between stratified and homogeneous fluids are the opposite symmetry of the dominant three-dimensional instabilities and the scaling of their most amplified wavenumber. When Fh > 1, three-dimensional instabilities are damped by a viscous critical layer. In the presence of background rotation in addition to the stratification, symmetric and antisymmetric modes no longer decouple and cyclonic vortices are less bent than anticyclonic vortices. However, the dominant instability remains qualitatively the same for the three vortex arrays, i.e. quasi-symmetric or quasi-antisymmetric and three-dimensional or two-dimensional. The growth rate continues to scale with the strain but the most unstable wavenumber of three-dimensional instabilities decreases with rotation and scales like Ro/(bFh) for small Rossby number Ro, in agreement with quasi-geostrophic scaling laws.

Sonic-boom noise penetrating under a deep ocean is affected by its time-dependent interaction with the surface waves, which can significantly influence the perceived sound pressure level and tonal content of the disturbances at depth far greater than expected from the flat-ocean (Sawyers) model. The present theory assumes a small surface slope and a high water-to-air density ratio; the ocean surface in the analysis is modelled by a sinusoidal surface-wave train. The analysis shows that a distinct acoustic wave mode in the form of a packet of wavelets emerges in the sound field far below the surface and attenuates with increasing distance in a manner similar to the cylindrical spreading of monochromatic waves. The latter feature renders the surface waviness influence an effect of first-order importance, overwhelming the primary noise field at large depth. Detailed properties of the deep-water wave fields are examined and illustrated for the case of an incident N-wave, for which an explicit, analytic solution is obtained. The result reveals a similarity structure of the wave field with two distinct time scales and the invariance characteristics of the cylindrically spreading waves, in accord with the group-velocity concept of dispersive waves. An example is given of the interaction, illustrating the underwater waveform, sound-pressure and frequency levels.

An axisymmetric flow due to a submerged sink in water of infinite depth is considered, with a stagnation point on the free surface above the sink. Forbes & Hocking (1990) calculated numerically a flow for each value of the Froude number F smaller than a critical value Fc. For F close to Fc there is a ring-shaped bump on the free surface. At F=Fc, the crest of the bump becomes a ring of stagnation points. We use the numerical procedure of Hocking & Forbes to show that the bump is the first crest of a train of axisymmetric waves. The wave amplitude decreases with increasing distance from the source. Then we give a local analysis of axisymmetric free-surface flows with a circular ring of stagnation points. We find flows in which the surface has a discontinuity in slope with an enclosed angle of 120° all along the ring. This behaviour is consistent with the numerical solution for F=Fc near the crest of the bump.