We revisit and supplement the description of gravity waves based on perturbation expansions in Lagrangian coordinates. A general analytical framework is developed to derive a second-order Lagrangian solution to the motion of arbitrary surface gravity wave fields in a compact and vectorial form. The result is shown to be consistent with the classical second-order Eulerian expansion by Longuet-Higgins (J. Fluid Mech., vol. 17, 1963, pp. 459–480) and is used to improve the original derivation by Pierson (1961 Models of random seas based on the Lagrangian equations of motion. Tech. Rep. New York University) for long-crested waves. As demonstrated, the Lagrangian perturbation expansion captures nonlinearities to a higher degree than does the corresponding Eulerian expansion of the same order. At the second order, it can account for complex nonlinear phenomena such as wave-front deformation that we can relate to the initial stage of horseshoe-pattern formation and the Benjamin–Feir modulational instability to shed new light on the origins of these mechanisms.

It is well-known that fluidized beds are usually unstable to small perturbations and that this leads to the primary bifurcation of vertically travelling plane wavetrains. These one-dimensional periodic waves have been shown recently to be unstable to two-dimensional perturbations of large transverse wavelength in gas-fluidized beds. Here, this result is generalized to include liquid-fluidized beds and to compare typical beds fluidized with either air or water. It is shown that the instability mechanism remains the same but there are big differences in the ratio of the primary and secondary growth rates in the two cases. The tendency is that the secondary growth rates, scaled with the amplitude of a fully developed plane wave, are of similar magnitude for both gas- and liquid-fluidized beds, while the primary growth rate is much larger in the gas-fluidized bed. This means that the secondary instability is accordingly stronger than the primary instability in the liquid-fluidized bed, and consequently sets in at a much smaller amplitude of the primary wave. However, since the waves in the liquid-fluidized bed develop on a larger time and length scale, the primary perturbations need longer time and thereby travel farther until they reach the critical amplitude. Which patterns are more amenable to being visually recognized depends on the magnitude of the initially imposed disturbance and the dimensions of the apparatus. This difference in scale plays a key role in bringing about the differences between gas- and liquid-fluidized beds; it is produced mainly by the different values of the Froude number.

The supersonic flow field past a circular cone of semi-apex angle 7·5° at incidences up to 26° has been investigated experimentally at a free-stream Mach number of 2·94. The experiments were made using a five-hole conical-head directional probe. Since the flow past the cone was found to be conical, the flow phenomena may be described in a plane perpendicular to the cone axis. This paper gives the direction of the conical streamlines, the conical sonic line and the conical Mach number in such a plane, all of which may be deduced from the measurements. At incidences exceeding twice the cone semi-apex angle a separated flow regime was observed which consisted of two main vortices on either side of the leeward plane of symmetry, and probably also two secondary vortices close to the cone surface. From incidences of 17° onwards, an embedded region of conical supersonic flow terminated by a shock wave was revealed. At 22°, approximately, a second embedded shock wave was measured close to the cone surface and extended slightly across the leeward plane of symmetry. This shock wave may have been generated by the vortex system, which induced supersonic cross-flow components towards the cone.

The stability of fluid flow past a membrane of infinitesimal thickness is analysed in the limit of zero Reynolds number using linear and weakly nonlinear analyses. The system consists of two Newtonian fluids of thickness R* and HR*, separated by an infinitesimally thick membrane, which is flat in the unperturbed state. The dynamics of the membrane is described by its normal displacement from the flat state, as well as a surface displacement field which provides the displacement of material points from their steady-state positions due to the tangential stress exerted by the fluid flow. The surface stress in the membrane (force per unit length) contains an elastic component proportional to the strain along the surface of the membrane, and a viscous component proportional to the strain rate. The linear analysis reveals that the fluctuations become unstable in the long-wave (α→0) limit when the non-dimensional strain rate in the fluid exceeds a critical value Λt, and this critical value increases proportional to α2 in this limit. Here, α is the dimensionless wavenumber of the perturbations scaled by the inverse of the fluid thickness R*−1, and the dimensionless strain rate is given by Λt = (γ˙*R*η*/Γ*), where η* is the fluid viscosity, Γ* is the tension of the membrane and γ˙* is the strain rate in the fluid. The weakly nonlinear stability analysis shows that perturbations are supercritically stable in the α→0 limit.

This work focuses on the mass transfer around unconfined bubbles in cylindrical microchannels when they are arranged in a train. We characterise how the mass transfer, quantified by the Sherwood number, $Sh$, is affected by the channel and bubble sizes, distance between bubbles, diffusivity, mean flow velocity, deformation of the bubble, the presence of surfactants in the limit of rigid interface and off-centred positions of the bubbles. We analyse the influence of the dimensionless numbers and especially the distance between bubbles and the Péclet number, $Pe$, which we vary over eight decades, identifying five different mass transfer regimes. We show different concentration patterns and the dependence of the Sherwood numbers. These regimes can be classified by either the importance of the diffusion along the streamlines or the interaction between bubbles. For small $Pe$ the diffusion along the streamlines is not negligible as compared to convection, whereas for large $Pe$ convection dominates in the streamlines direction and, thus, crosswind diffusion becomes crucial in governing the mass transfer through boundary layers or the remaining wake behind the bubbles. Interaction of bubbles occurs for very small $Pe$ where the mass transfer is purely diffusive, or for very large $Pe$ where long wakes eventually reach the following bubble. We also observe that the bubble deformability mainly affects the $Sh$ in the regime for very large $Pe$ in which bubbles interaction matters, and also that the rigid interface affects the boundary layer and the remaining wake. The effect of off-centred position of the bubble, determined by the transverse force balance, is also limited to large $Pe$. The boundary layers on rigid bubble surfaces are thicker than those on stress-free bubble surfaces, and thus the mass transfer is weaker. For centred bubbles, the influence of inertia on the mass transfer is negligible. Finally, we discuss the implication of our results on the dissolution of bubbles.

When a gas bubble rises in an impure liquid, its surface often has an upper spherical cap with negligible shear stress, a lower spherical cap with negligible tangential velocity, and a very small transition region between the two caps.

This paper gives the diffusion boundary-layer theory for the distribution of surfactant around a stagnant-cap bubble, allowing for slowness of both adsorption and diffusion. The resulting singular Volterra integro-differential equations are solved numerically for creeping flow (small Reynolds number). The main result is the relation between the surface pressure of surfactant in the bulk solution, the cap angle and Péclet number of the bubble, and the adsorption depth and adsorption speed of the surfactant. The values of the latter two parameters affect the validity of the approximations much more than the numerical results.

Large-amplitude atmospheric flows past mountain ridges are investigated. The flows are assumed to be steady and two-dimensional. Diffusive and viscous effects are neglected but static compressibility is taken into account.

The larger part of the investigation is devoted to the study of waves in the lee of mountain ridges. The major contribution consists in the treatment of the large-amplitude motion. The flows are governed by an equation which turns out to be linear for certain upstream conditions. These conditions impose some restrictions on the wind profile and stratification of the entropy and specific energy far upstream. However, flow patterns representing realistic upstream conditions have been obtained.

A comparison between a compressible flow and an incompressible flow with equivalent upstream conditions is included.

When a fluid of low conductivity flows parallel to a plane wall from behind which projects a magnetic field, an electric potential field is established throughout the fluid. In this paper the potential field is obtained explicitly in terms of the velocity field when the latter is unidirectional and depends only on the coordinate normal to the wall. Experiments with a variety of velocity profiles are described, and the agreement with the theory is satisfactory. The effect of slow variation of the profiles in the direction of the flow is considered, and is shown to be unimportant under the conditions of the experiments.

Lateral variations in the thermal boundary conditions at the horizontal boundary of an otherwise stably stratified fluid layer drive circulatory motion. For a rapidly rotating electrically conducting fluid, the introduction of a background vertical magnetic field reduces the dimensionless strength of the thermal wind from $O(1)$ to $O(E^{1/4})$ for prescribed temperature variation or to $O(E^{1/2})$ for prescribed heat-flux variation, where $E$ is the Ekman number. This is a significant effect when considering the spatially variable cooling of the Earth's core by the mantle. A general discussion of linear hydromagnetic flows identifies a large number of lengthscales inherent in the differential system as functions of the Ekman, Elsasser and stratification numbers, and shows that other scalings arise from the boundary conditions.

Relative equilibria of two and three satellite vortices in a rotating shallow water layer have been recorded via particle image velocimetry (PIV) and their autorotation speed was estimated. This study shows that these equilibria retain the fundamental characteristics of Kelvin’s equilibria, and could be adequately described by the classical idealized point vortex theory. The same conclusion can also be inferred using the experimental dataset of Bergmann et al. (J. Fluid Mech., vol. 679, 2011, pp. 415–431; J. Fluid Mech., vol. 691, 2012, pp. 605–606) if the assigned field’s contribution to pattern rotation is included.

The deformation of the Mach stem in pseudo-steady shock wave reflections is investigated numerically and theoretically. The numerical simulation provides the typical flow patterns of Mach stem deformation and reveals the differences caused by high-temperature gas effects. The results also show that the wall jet, which causes Mach stem deformation, can be regarded as a branch of the mainstream from the first reflected shock. A new theoretical model for predicting the Mach stem deformation is developed by considering volume conservation. The theoretical predictions agree well with the numerical results in a wide range of test conditions. With this model, the wall-jet velocity and the inflow velocity from the Mach stem are identified as the two dominating factors that convey the influence of high-temperature thermodynamics. The mechanism of high-temperature gas effects on the Mach stem deformation phenomenon are then discussed.

In this paper, we have found two new nonlinear travelling wave solutions in pipe flows. We investigate possible asymptotic structures at large Reynolds number $R$ when wavenumber is independent of $R$ and identify numerically calculated solutions as finite $R$ realizations of a nonlinear viscous core (NVC) state that collapses towards the pipe centre with increasing $R$ at a rate $R^{-1/4}$. We also identify previous numerically calculated states as finite $R$ realizations of a vortex wave interacting (VWI) state with an asymptotic structure similar to the ones in channel flows studied earlier by Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). In addition, asymptotics suggests the possibility of a VWI state that collapses towards the pipe centre like $R^{-1/6}$, though this remains to be confirmed numerically.

From shaping cosmic structures in space to producing intense sounds in aircraft engines, shock waves in fluids ineluctably convert entropy fluctuations into swirling motions and sound waves. Studies of the corresponding conversion from internal energy to kinetic energy have so far been restricted to ideal (or idealised) fluids. Yet, many substances do not obey the ideal-gas law (including those in the above two examples). The present work demonstrates that non-ideal thermodynamic properties provide a remarkable degree of control over the conversion to solenoidal and dilatational kinetic energies. Of particular interest is the ability to suppress much of the emitted acoustic field whilst promoting mixing downstream of the shock. This is made possible by exploiting the convexity (or lack thereof) of the shock adiabats. Whilst illustrated here using dense vapours near the thermodynamic critical point, this ability to design and control specific shock-induced energy transfers extends beyond near-critical-point phenomena; e.g. shocked mixtures (high-speed dusty flows on Mars, nanoparticle formation in supersonic expanders for drug manufacturing), reacting fronts (supersonic combustion, rocket propulsion), ionising shocks (reentry systems, inertial confinement fusion) or fronts in active fluids (bacterial and crowd flows). This theoretical work, which demonstrates the predictive capabilities of linear theory, lays the foundation for future experimental investigations ultimately aimed at delivering novel shock-based flow-control strategies exploiting the thermodynamic properties of the fluid.

The mechanisms by which turbulent shear flow causes waves on a gas–liquid interface are studied analytically, with a critical assessment of the possible role played by wave-induced Reynolds stresses (WIRSs). First, turbulent flow past a corrugated surface of a small slope is analysed; the surface can either be stationary or support a travelling wave. This problem serves as a useful model because direct numerical simulation (DNS) and experimental data are available to test the analysis, and because this picture is itself a model for the fully coupled two-layer problem. It is demonstrated that the WIRSs play no significant role in shear-driven turbulent flow past a moving wavy wall, and that they alter the structure of the flow only in a quantitative fashion in the pressure-driven case. In the shear-driven case in particular, excellent agreement is obtained with previously reported DNS results. Two closure assumptions are made in our model: the first concerns the wave-induced dissipation of turbulent kinetic energy; the second concerns the importance of rapid distortion. The results of our calculations are sensitive to the assumptions used to close the wave-induced dissipation but are insensitive to the details of the rapid-distortion modelling. Finally, the fully coupled two-layer problem is addressed in the setting of waves of small amplitude, where it is demonstrated that the WIRSs do not play a significant role in the growth of interfacial waves, even at relatively high Reynolds numbers. Again, good agreement is obtained between data from experiments and DNS.

Generalised Navier–Stokes (GNS) equations describing three-dimensional active fluids with flow-dependent narrow spectral forcing have been shown to possess numerical solutions that can sustain significant energy transfer to larger scales by realising chiral Beltrami-type chaotic flows. To rationalise these findings, we study here the triad truncations of polynomial and Gaussian GNS models focusing on modes lying in the energy injection range. Identifying a previously unknown cubic invariant for the triads, we show that their asymptotic dynamics reduces to that of a forced rigid body coupled to a particle moving in a magnetic field. This analogy allows us to classify triadic interactions by their asymptotic stability: unstable triads correspond to rigid-body forcing along the largest and smallest principal axes, whereas stable triads arise from forcing along the middle axis. Analysis of the polynomial GNS model reveals that unstable triads induce exponential growth of energy and helicity, whereas stable triads develop a limit cycle of bounded energy and helicity. This suggests that the unstable triads dominate the initial relaxation stage of the full hydrodynamic equations, whereas the stable triads determine the statistically stationary state. To test whether this hypothesis extends beyond polynomial dispersion relations, we introduce and investigate an alternative Gaussian active turbulence model. Similar to the polynomial case, the steady-state chaotic flows in the Gaussian model spontaneously accumulate non-zero mean helicity while exhibiting Beltrami statistics and upward energy transport. Our results suggest that self-sustained Beltrami-type flows and an inverse energy cascade may arise generically in the presence of flow-dependent narrow spectral forcing.

Rayleigh's (1926, Appendix A) method for the approximate calculation of potential flow from the open end of a semi-infinite flanged cylinder is applied to obtain analytical representations of Green's function describing the generation of sound waves within a flanged cylinder by sources located in the neighbourhood of the open end. Detailed results are given for the circular cylinder considered by Rayleigh and extension made to a flanged cylinder of rectangular cross-section. The validity of various approximations is assessed by comparison with the exact solution available (by the method of conformal transformation) for potential flow from a two-dimensional, flanged duct. The results are used to compare the profiles and the pressure gradients of compression waves generated when a high-speed train enters tunnels of circular and rectangular cross-sections.

It is well known that in any conservative system that admits resonant triad interactions, a uniform (test) wavetrain that participates in a single triad is unstable if it has the highest frequency in the triad, and neutrally stable otherwise. We show that this result changes significantly in the presence of coupled triads: with coupling, the test wave can be unstable to a high-frequency perturbation. The coupling sends energy from the (weak) high-frequency source into particular low-frequency waves that grow even though they had zero amplitudes initially. This mechanism thereby selects these low-frequency waves from the spectrum of low-frequency waves available for triad interactions. Moreover, the instability persists in the presence of weak damping, provided the wave amplitudes exceed two thresholds. First, the initial amplitude of the test wavetrain must be large enough for the instability to dominate the damping. Secondly, the (small) initial amplitudes of the high-frequency perturbations must exceed a threshold in order for the low-frequency waves to grow to a prescribed amplitude.