Wentzel–Kramers–Brillouin theory was employed by Grimshaw (Geophys. Fluid Dyn., vol. 6, 1974, pp. 131–148) and Achatz et al. (J. Fluid Mech., vol. 210, 2010, pp. 120–147) to derive modulation equations for non-hydrostatic internal gravity wave packets in the atmosphere. This theory allows for wave packet envelopes with vertical extent comparable to the pressure scale height and for large wave amplitudes with wave-induced mean-flow speeds comparable to the local fluctuation velocities. Two classes of exact travelling wave solutions to these nonlinear modulation equations are derived here. The first class involves horizontally propagating wave packets superimposed over rather general background states. In a co-moving frame of reference, examples from this class have a structure akin to stationary mountain lee waves. Numerical simulations corroborate the existence of nearby travelling wave solutions under the pseudo-incompressible model and reveal better than expected convergence with respect to the asymptotic expansion parameter. Travelling wave solutions of the second class also feature a vertical component of their group velocity but exist under isothermal background stratification only. These waves include an interesting nonlinear wave–mean-flow interaction process: a horizontally periodic wave packet propagates vertically while draining energy from the mean wind aloft. In the process it decelerates the lower-level wind. It is shown that the modulation equations apply equally to hydrostatic waves in the limit of large horizontal wavelengths. Aside from these results of direct physical interest, the new nonlinear travelling wave solutions provide a firm basis for subsequent studies of nonlinear internal wave instability and for the design of subtle test cases for numerical flow solvers.

We consider the problem of acoustic radiation generated by a spatial instability wave on a weakly developing shear flow. Assuming a local WKBJ approximation for the instability wave near its maximum, we compute the acoustic pressure field by using a Fourier transform along the streamwise direction. When the instability wave is close to transonic near its maximum amplitude, approximations for this pressure field are obtained by a steepest descent method. A branch cut and several saddle points are shown possibly to contribute to the approximation. A detailed analysis of these contributions is provided. The modifications of the acoustic field when we pass from subsonic to supersonic are examined. In particular, the superdirective character of the acoustic field of subsonic instability waves and the directivity pattern of supersonic waves are shown to be both compatible with our mathematical description and associated with a single saddle-point contribution.

The acoustic near field is also shown to possess a caustic around which a specific approximation is derived. In a large region of the physical space, the near field is composed of two saddle-point contributions. Close to the shear flow, one of these contributions degenerates into a branch-point contribution which always becomes dominant over the instability wave downstream of a location that is computed. An interesting phenomenon is observed in certain regions downstream of the maximum: the transverse behaviour of the instability wave has to be exponentially growing far from the shear layer to match the acoustic field. We demonstrate that this phenomenon neither requires a branch-point contribution nor a supersonic instability wave.

A stratified body of fluid with compensating horizontal temperature and salinity gradients can undergo an interleaving instability which takes the form of almost horizontal intrusions. As the amplitude of these intrusions grows they can undergo secondary instabilities which eventually leads to the mixing of the fluid in the interior of the intrusions. A previous study of the secondary instabilities focused on two-dimensional disturbances. These corresponded to experimental observations of that time which all seemed to indicate that flows were indeed two-dimensional. Some more recent experiments have shown that the initial secondary instability can make the flow three-dimensional, with the secondary instabilities taking the form of rolls with their axes aligned with the direction of the flow in the intrusions. Here we present a three dimensional stability analysis of steady finite-amplitude intrusions and look at the circumstances which can lead to the three-dimensional instabilities being more likely to be observed.

We investigate the excitation and radiation of near-inertial internal gravity waves continuously excited by a latitudinally confined temporally fluctuating wind in a numerical model of a stratified ocean on a $\unicode[STIX]{x1D6FD}$-plane at mid-latitude. The surface wind forcing contains both high- and low-frequency components which excite propagating waves and a baroclinically unstable zonal jet respectively. Wentzel–Kramers–Brillouin (WKB) ray theory implies that near-inertial waves propagate strictly towards the equator. We seek to refine this view here by (i) adding the non-traditional Coriolis force (accounting for the horizontal component of the Earth’s rotation) into the equations of motion, in order to allow poleward sub-inertial propagation to occur, and (ii) relaxing the conceptual constraint of no zonal variability, to allow the zonal jet to undergo instability, to meander and to sustain an active field of mesoscale eddies, potentially impacting the excitation of near-inertial waves. The key results are that, while (i) permits weakly stratified waveguides with sub-inertial poleward wave propagation to develop in accord with theory, the sub-inertial energy flux observed is very small compared with the equatorward flux. Thus, in terms of energy radiated from the storm track, non-traditional effects are small for wind-driven near-inertial waves. The consequences of (ii) are much more pronounced. Refinement (ii) produces a radiating wave field that is bidirectional, i.e. with both poleward and equatorward components. We show that the presence of regions of significant background vorticity with horizontal scales significantly smaller than the width of the storm track provides the scale selection mechanism to excite waves with sufficiently super-inertial frequencies to propagate poleward distances of the order of 1000 km.

In the idealized problem of homogeneous isotropic stationary inertial-range turbulence the rate of relative dispersion of an ensemble of tracer pairs can be characterized by a constant C0. In order to compute this constant with random-flight equations, however, it is necessary first to know the values of two other constants, C1 and C2, that occur in the two-particle velocity-component relations of Lagrangian tracers (Faller 1992).

C1 and C2 are found by an elaborate trial and error procedure in a new two-tracer random-flight model of dispersion that matches input and output values of these two variates. The constant C0 is then computed using the Lagrangian relations and is found to be significantly smaller than when the Eulerian Kármán/Howarth correlations are used.

The probability density distribution of tracer separations has a kurtosis slightly larger than that of a comparable Gaussian distribution. At small spacings the frequency of tracer spacings is six to ten times larger than would be expected from a Gaussian distribution. The distribution function for the speed of separation of the Lagrangian tracers has a negative skewness similar to that found for two-point Eulerian velocities.

Turbulent buoyant plumes in cold fresh water are analysed, assuming a quadratic dependence of density on temperature. The model is based on the assumption that entrainment velocity is proportional to vertical velocity in the plume. Numerical and asymptotic solutions are obtained for both rising and descending plumes from virtual sources with all possible combinations of buoyancy, volume and momentum fluxes. Physical sources can be identified as points on trajectories of plumes from virtual sources.

The zero-buoyancy condition, at which the plume and the ambient have equal densities but their temperatures are on opposite sides of the temperature of maximum density, is of particular importance. If an upwardly buoyant plume rising through a body of water reaches the surface before passing through its zero-buoyancy level, it will form a surface gravity current; otherwise, the plume water will return to the source as a fountain. The height at which zero buoyancy is attained generally decreases as the source momentum flux increases: greater plume velocity produces greater entrainment and hence more rapid temperature change. Descending plumes, if ejected downwards against upward buoyancy, may be classified as strongly or weakly forced according to whether they reach the zero-buoyancy condition before being brought to rest. If they do, they continue to descend with favourable buoyancy; otherwise, they may form an inverted fountain. Once a descending plume has attained downward buoyancy, it can continue to descend indefinitely, ultimately behaving like a plume in a fluid with a linear equation of state. In contrast, a rising plume will eventually come to rest, however large its initial upward buoyancy and momentum fluxes are.

A simple idealized model is proposed to predict the appearance of alternate bottom forms in tidal channels. The model is based on the linear stability analysis of the flat seabed driven by tidal currents. The hydrodynamics is described by means of the full three-dimensional continuity and Reynolds equations. To quantify turbulent stresses, the Boussinesq assumption is introduced and an algebraic model is used for the eddy viscosity. The morphodynamics is described by the Exner equation and a simple sediment transport predictor. When applied to tidal channels, the model predicts the appearance of alternate bottom forms if the channel width is larger than a critical value. This finding agrees with previous analyses. However, the results obtained show the existence of two different modes. Close to the conditions of incipient sediment motion or when the suspended load is intense, the model suggests the appearance of an alternate sequence of shoals and pools (the first mode), characterized by a wavelength which might be comparable with the horizontal excursion of the tide. However, under such circumstances, to provide accurate quantitative results, the model should be extended to include the effects of the local acceleration and of the possible variations of the depth and width of the channel, which are neglected in the analysis. In all the other conditions, the model predicts the appearance of a second mode, presently termed tidal alternate bars. This mode is geometrically similar to the first mode, i.e. it is characterized by depositional and erosional areas which are found in an alternate arrangement, but it has significantly shorter wavelengths. In this case, the wavelength of the bedforms scales with the water depth. The physical mechanism generating tidal alternate bars appears to be the same as that generating tidal dunes, and it cannot be described by means of a depth-averaged approach.

We investigate the effect of boundary roughness on the dynamical properties of the flow in laminar Ekman boundary layers. The study considers wavy boundaries having both horizontal wavelength and vertical extent comparable in size with the boundary layer width. In the case of flat boundaries, Ekman layers are known to be active, i.e. to affect significantly the dynamics of the mainstream flow. We show how the layer modelling needs to be modified to account for such wavy boundaries. In particular, nonlinear terms enter the laminar description. This model can be linearized in the limit of small Reynolds numbers. The resulting equations are studied using both asymptotic expansions and full numerical simulations. We find that small-scale roughness significantly alters energy dissipation in the boundary layer. This can result in either a reduction or an increase of dissipation, depending on, in particular, the orientation of the mainstream flow with respect to boundary modulation. Agreement is obtained between theoretical and computational results.

A linearized analysis is made of the canonical problem of sound production during the convection of a line vortex of strength $\Gamma$ in steady flow of water over a ‘fence’ of height $h$ on a flat wall in the presence of a vacuous cavity in the wake of the fence. The cavity is assumed to extend sufficiently far downstream for sound waves to be regarded as launched above a non-compact, pressure-release surface. Additional vorticity is released from the tip of the fence in accordance with the Kutta condition, and is convected at the mean stream speed $U$ along the free streamline boundary of the cavity. Sound pressures of opposite phases are generated by the incident and the shed vorticity. The predicted radiation consists of a pressure pulse of amplitude proportional to $\rho_oU\Gamma/h$ ($\rho_o$ being the mean water density) and width approximately equal to fence height/mean flow Mach number, produced as the vortex passes the tip of the fence. The acoustic amplitude decreases rapidly at later times because of destructive interference between the sound generated by the impinging vortex and the shed vorticity.

A theory is described for propagation of vortical waves across alternate rigid and compliant panels. The structure in the fluid side at the junction of panels is a highly vortical narrow viscous structure which is idealized as a wave driver. The wave driver is modelled as a ‘half source cum half sink’. The incoming wave terminates into this structure and the outgoing wave emanates from it. The model is described by half Fourier–Laplace transforms respectively for the upstream and downstream sides of the junction. The cases below cutoff and above cutoff frequencies are studied. The theory completely reproduces the direct numerical simulation results of Davies & Carpenter (J. Fluid Mech., vol. 335, 1997, p. 361). Particularly, the jumps across the junction in the kinetic energy integral, the vorticity integral and other related quantities as obtained in the work of Davies & Carpenter are completely reproduced. Also, some important new concepts emerge, notable amongst which is the concept of the pseudo group velocity.

An analysis of the exact equations of the inviscid flow of a perfect gas over cusped concave bodies is described. The field is examined in the limit of infinite free-stream Mach number M∞. The slope of the shock wave in a small region adjacent to the leading edge is strongly dependent on M∞, while much further downstream the shock-wave slope is controlled primarily by the body slope. Consequently the region near the leading edge introduces into the field downstream a thin layer of gas, adjacent to the body, where the entropy is much lower than that of the gas above it. This layer is so dense that the gas velocity along it is not appreciably slowed by the pressure gradient along the body. However, it is so thin that there is little pressure change across it.

The well-known self-similar solutions to the hypersonic small-disturbance equations have previously only been used to study the flow on blunted slender convex surfaces. They are known to behave singularly at the body. It is shown that there is a region on concave power-law shapes where the self-similar solutions are the correct first approximation to the exact inviscid equations in the limit M∞ → ∞; and that, further, they predict the correct first-order surface pressure.

Numerical results for surface pressure from the similar solutions are presented, and comparisons are made with certain approximate theories available for more general shapes. Pressure measurements taken on a cubic surface in the Imperial College gun tunnel are presented and compared with the theoretical distributions.

We consider the steady supercritical flow of a fluid layer. The layer is bounded above by a free surface and below by a rigid no-slip base. The base is in two parts: the downstream part of the base is stationary, while the upstream part translates in the streamwise direction with a uniform speed; there is an abrupt transition. At high Reynolds number, a boundary layer forms in the fluid above the base downstream of the transition point. The displacement due to this boundary layer creates a perturbation to the outer flow and therefore to the free surface. We show that the Blasius boundary layer solution, which applies in an infinitely deep fluid, also applies at high Froude numbers. The Blasius solution no longer applies for flows that are just supercritical, as the outer flow is strongly affected by the presence of the boundary layer. We outline possible applications of this work to depth-averaged models of gravity currents.

The transition from steady to oscillatory three-dimensional convection in a cube of saturated porous material is calculated to occur at Rayleigh number R = 584 due to seven pairs of thermal blobs which circulate around the cube. This travelling wave instability is shown to be closely related, first as regards structural characteristics and then as regards mechanism of instability, to an analogous instability in two dimensions. The correspondence with the two-dimensional flow is established via a correspondence with a nonlinear base flow in a box of square planform of a different aspect ratio (l/√2) and ultimately derives from the symmetries of the base flow in the cube.

A class of constants of motion is obtained for the time-dependent form of the Rayleigh stability equation, which provides stability criteria for velocity profiles with multiple inflexion points.

The possibility of acoustic control of the two-dimensional instabilities of a lossless plane shear layer of vanishing thickness is studied. The shear layer is formed from a body of incompressible fluid sliding over another fluid at rest. It is unstable through the generation of Kelvin–Helmholtz waves. We consider the possibility of adding to this linearly unstable flow a simple source, driven in such a way that its field interferes destructively with the instability to render the flow stable. The required strength of the unsteady control source is determined in terms of the fluctuating velocity at some fixed position in the moving fluid. We show that no unstable Kelvin–Helmholtz wave could survive the action of such a source. Next, we examine the scope for constructing the control signal from a measurement of the flow velocity at some fixed position. The source is a linear functional of the monitored velocity and we give the transfer function that would be required for the instabilities to be controlled. We prove that such control action would completely stabilize the otherwise unstable vortex sheet, and that other alternative sensor/actuator arrangements could also be effective. We go on to show that our particular very simple arrangement could not in fact be realized because, if required to work at all frequencies, it would not be causal. If we insisted on causality the vortex sheet would then only be stabilized over most frequencies. That would of course make the controlled flow completely different from the vortex sheet whose instabilities are so well known – and troublesome. We conjecture that there will exist some variations of the basic control arrangement described here that are both physically realizable and effective over the required frequency range. From our study of the initial value problem we have concluded that short perturbations would be attenuated very rapidly.

A numerical solution is presented for predicting the flow and heat transfer due to free convection in the entry region of a cooled vertical pipe, open at both ends. Following Lighthill (1953) the Kármán-Pohlhausen method is used in this analysis. Velocity and temperature profiles are assumed satisfying the physical boundary conditions, and the integrated forms of the equations of motion and the equations themselves at the axis and the walls of the pipe are used to calculate the various parameters involved in the profiles assumed. Two cases of constant wall temperature and linearly decreasing temperature with height were considered. Cooling decelerates the fluid in the boundary layer that forms at the wall and the fluid in the core is accelerated by virtue of continuity. As cooling progresses, at Rayleigh number O(103) the fluid in the boundary layer starts reversing at first and then assumes progressively increasing velocities. Graphs are presented for the development of the predicted velocity and temperature profiles and the other parameters involved, for a situation when the boundary layer fills the whole pipe.

We consider the finite-amplitude instability of incompressible spherical Couette flow between two concentric spheres of radii $R_1$ and $R_2$$({>}R_1)$ in the narrow-gap limit, $\varepsilon\,{\equiv}\,(R_2-R_1)/R_1\,{\ll}\,1$, caused by rotating them both about a common axis with distinct angular velocities $\Omega_1$ and $\Omega_2$ respectively. In this limit it is well-known that the onset of (global) linear instability is manifested by Taylor vortices of roughly square cross-section close to the equator. According to linear theory this occurs at a critical Taylor number $T_{\rm crit}$ which, remarkably, exceeds the local value $T_c$ obtained by approximating the spheres as cylinders in the vicinity of the equator even as $\varepsilon\,{\downarrow}\,0$. Previous theoretical work on this problem has concentrated on the case of almost co-rotation with $\delta\,{\approx}\,(\Omega_1\,{-}\,\Omega_2)/\Omega_1\,{=}\,\OR(\varepsilon^{1/2})$ for which $T_{\rm crit}\,{=}\,T_c\,{+}\,\OR(\delta^2)\,{+}\,\OR(\varepsilon)$. In this limit the amplitude equation that governs the spatio-temporal modulation of the vortices on the latitudinal extent $\OR(\varepsilon^{1/2}R_1)$ gives rise to an interesting bifurcation sequence. In particular, the appearance of global bifurcations heralds the onset of complicated subcritical time-dependent finite-amplitude solutions.

Here we switch attention to the case when $\varepsilon^{1/2}\,{\ll}\,\delta\,{\le}\, 1$. We show that for Taylor numbers $T\,{=}\,T_c+\OR((\delta\varepsilon)^{2/3})$ there exists a locally unstable region of width $\OR((\delta\varepsilon)^{1/3}R_1)$ within which the amplitude equation admits solutions in the form of pulse-trains. Each pulse oscillates at a frequency proportional to its distance from the equatorial plane and consists of a wave propagating towards the equator under an envelope. The pulse drifts at a slow speed (relative to the wave velocity) proportional to its distance (and away) from the equator. Both the wavelength and the envelope width possess the same relatively short length scale $\OR((\varepsilon^{2}/\delta)^{1/3}R_1)$. The appropriate theory of spatially periodic pulse-trains is developed and numerical solutions found. Significantly, these solutions are strongly subcritical and have the property that $T\to T_c$ as $\varepsilon\,{\downarrow}\,0$.

Two particular limits of our theory are examined. In the first, $\varepsilon^{1/2}\,{\ll}\,\delta\,{\ll}\,1$, the spheres almost co-rotate and the pulse drift velocity is negligible. A comparison is made of the pulse-train predictions with previously obtained numerical results pertaining to large (but finite) values of $\delta/\varepsilon^{1/2}$. The agreement is excellent, despite the complicated long-time behaviour caused by inhomogeneity across the relatively wide unstable region.

Our second special case $\delta\,{=}\,1$ relates to the situation when the outer sphere is at rest. Now the poleward drift of the pulses leads to a slow but exponential increase of their separation with time. This systematic pulse movement, over and above the spatial inhomogeneity just mentioned, necessarily leads to complicated and presumably chaotic spatio-temporal behaviour across the wide unstable region of width $\OR(\varepsilon^{1/3}R_1)$ on its associated time scale, which is $\OR(\varepsilon^{-1/3})$ longer than the wave period. In view of the several length and time scales involved only qualitative comparison with experimental results is feasible. Nevertheless, the pulse-train structure is robust and likely to provide the building block of the ensuing complex dynamics.

Stratified, inviscid channel flow over a thin barrier or into an abrupt contraction is considered on the hypotheses that the upstream dynamic pressure and density gradient are constant (Long's model) for those parametric régimes in which the hypotheses are tenable for finite-amplitude disturbances, namely k < 2 for the barrier and k < 1 for the contraction, where k = NH/πU is an inverse Froude number based on the Vaisälä frequency N, the channel height H, and the upstream velocity U. Reverse flow in the neighbourhood of the forward stagnation point, which implies the formation of an upstream separation bubble, is found for certain critical ranges of k. The maximum barrier height for which the dominant lee-wave mode can exist without reversed flow either upstream or downstream of the barrier is 0·34H. The limiting case of a half space is considered briefly, and forward separation is found for κ = Nh/U > κs, where κs = 2·05 for a thin barrier and 1·8 for a semi-circular barrier. The corresponding values for reverse flow in the lee-wave field are κc = 1·73 and 1·3, respectively.