Experiments on two-layer β-plane flows are described. The regime diagrams for both easterly and westerly forcing indicate complex scenarios by which baroclinically unstable flows can become chaotic as the forcing is increased. The transition sequence can involve as many as three different vacillation mechanisms, but also exhibits the periodic window phenomena prevalent in many model dynamical systems. The fractal dimension of the chaos at low rotational Froude number F is measurable and is somewhat less than 3. The dimension increases as F is raised. A six-wave low-order model, while successfully predicting some of the observed vacillations, gives a relatively poor description of the chaos.

In this work, we analyse the internal shear layer structures generated by the libration of an axisymmetric object in an unbounded fluid rotating at a rotation rate $\unicode[STIX]{x1D6FA}^{\ast }$ using direct numerical simulation and small Ekman number asymptotic analysis. We consider weak libration amplitude and libration frequency $\unicode[STIX]{x1D714}^{\ast }$ within the inertial wave interval $(0,2\unicode[STIX]{x1D6FA}^{\ast })$ such that the fluid dynamics is mainly described by a linear axisymmetric harmonic solution. The internal shear layer structures appear along the characteristic cones of angle $\unicode[STIX]{x1D703}_{c}=\text{acos}(\unicode[STIX]{x1D714}^{\ast }/(2\unicode[STIX]{x1D6FA}^{\ast }))$ which are tangent to the librating object at so-called critical latitudes. These layers correspond to thin viscous regions where the singularities of the inviscid solution are smoothed. We assume that the velocity field in these layers is described by the class of similarity solutions introduced by Moore & Saffman (Phil. Trans. R. Soc. Lond. A, vol. 264, 1969, pp. 597–634). These solutions are characterized by two parameters only: a real parameter $m$, which measures the strength of the underlying singularity, and a complex amplitude coefficient $C_{0}$. We first analyse the case of a disk for which a general asymptotic solution for small Ekman numbers is known when the disk is in a solid plane. We demonstrate that the numerical solutions obtained for a free disk and for a disk in a solid plane are both well described by the asymptotic solution and by its similarity form within the internal shear layers. For the disk, we obtain a parameter $m=1$ corresponding to a Dirac source at the edge of the disk and a coefficient $C_{0}\propto E^{1/6}$ where $E$ is the Ekman number. The case of a smoothed librating object such as a spheroid is found to be different. By asymptotically matching the boundary layer solution to similarity solutions close to a critical latitude on the surface, we show that the adequate parameter $m$ for the similarity solution is $m=5/4$, leading to a coefficient $C_{0}\propto E^{1/12}$, that is larger than for the case of a disk for small Ekman numbers. A simple general expression for $C_{0}$ valid for any axisymmetric object is obtained as a function of the local curvature radius at the critical latitude in agreement with this change of scaling. This result is tested and validated against direct numerical simulations.

Groups of short waves within a narrow frequency band are known to be accompanied by second-order long waves travelling at the group velocity of the predominant short waves. When the short waves are refracted by bottom topography, new long waves can be further radiated and propagated away from the topography at the shallow-water speed. Since over a long submarine ridge there can be trapped modes of long-period waves, incident groups of short waves can excite the trapped waves through a second-order mechanism. In this paper we study such excitations over a rectangular shelf which scatters the first-order short waves. By employing asymptotic methods we examine the transient excitation of the trapped long wave by both sinusoidal wave groups and wave packets. The effects of a small angular spread of the incident waves are also included.

The problem of gas motion in a tube closed at one end and driven at the other by an oscillating poston is studied theoretically. When the piston vibrates with a finite amplitude at the first acoustic resonance frequency, periodic shock waves are generated, travelling back and forth in the tube. A perturbation method, based on a small Mach number. M and a global mass conservation condition, is employed to formulate a solution of the problem in the form of two standing waves separated by a jump (shock front). By expanding the equations of motion in a series of a small parameter ε = M½, all hydrodynamic properties are predicted with an accuracy to second-order terms, i.e. to ε2. It is found that the first-order solution coincides with the previous theories of Betchov (1958) and Chester (1964), while additional terms predict a non-homogeneous time-averaged pressure along the tube. This prediction compares favourably with experimental results from the literature. The importance of the phenomenon is discussed in relation to different transport processes in resonance tubes.

We investigate the flow field in an electrokinetic micromixer. The concept of the micromixer is based on the combination of an alternating electrical field applied to a pressure-driven base flow in a meander–channel geometry. The presence of the electrical field leads to an additional electro-osmotic velocity contribution, which results in a complex flow field within the meander bends. The velocity fields within the meander are measured by means of a microparticle-image velocimetry method. Furthermore, we introduce a mathematical model, describing the electrical and fluid-mechanical phenomena present within the device, and perform simulations comparable to the experiments. The comparison of simulations and experiments reveals good agreement, with minor discrepancies in flow topology, obviously caused by small but crucial differences between experimental and numerical geometries. In detail, simulations are performed for sharp corners of the bends, while in the experiments these corners are rounded due to the microfabrication process.

This paper describes an experimental study of the effects of surface roughness on the flow between two concentric cylinders, one of which rotates. The surface roughness has some effects on the coefficient of viscous frictional torque CM in the transient and in the fully developed turbulent region. In the fully developed rough turbulent flow, the value of CM depends on both the Reynolds number Rω and the relative roughness rm/k in the case where the outer cylinder rotates, but CM depends only on rm/k in the case where the inner cylinder rotates. The effect of the surface roughness of the inner cylinder is greater than that of the outer one for both cases.

A set of hydrodynamic equations has been applied to processes occurring in non-conductive fluids placed into magnetic fields. The equations are valid for equilibrium magnetization within the framework of a continuous medium. The ranges of physical parameters have been evaluated for which magnetization of a fluid should be taken into account in problems concerning the determination of equilibrium forms, and flows and their stability. The conclusion has been drawn that magnetization of natural fluids in these problems must be taken into consideration for fluids in high (exceeding 1 T) magnetic fields. As examples, the solutions of several typical problems concerning the equilibrium surface of capillary fluids and their stability in external magnetic fields have been considered. Hydrodynamic effects related to magnetization of natural capillary fluids in high magnetic fields are studied with due regard for the above solutions. Hydrodynamic effects in para- and diamagnetic fluids have been studied and the common and distinctive features of these effects discussed.

We report upon laboratory experiments and numerical simulations examining the evolution of an interfacial internal solitary wave incident upon a triangular ridge whose peak lies below the interface. If the ridge is moderately large, the wave is observed to shoal and break similar to solitary waves shoaling upon a constant slope, but interfacial waves are also observed to transmit over and reflect from the ridge. In laboratory experiments, by measuring the interface displacement as it evolves in time, we measure the relative transmission and reflection of available potential energy after the incident wave has interacted with the ridge. The numerical simulations of laboratory- and ocean-scale waves measure both the available potential and kinetic energy to determine the partition of incident energy into that which is transmitted and reflected. From shallow-water theory, we define a critical amplitude, $A_{c}$, above which interfacial waves are unstable. The transmission is found to decrease from one to zero as the ratio of the incident wave amplitude to $A_{c}$ increases from less than to greater than one. Empirical fits are made to analytic curves through measurements of the transmission and reflection coefficients.

Topographic Rossby waves in elongated basins on the f-plane are studied by transforming the linear boundary-value problem for the mass transport stream function into a class of two-point boundary-value problems of which the independent spatial variable is the (curved) basin axis. The procedure for deriving the substitute problems is the Method of Weighted Residuals. What emerges is a vector differential equation and associated boundary conditions, its dimension indicating the order of the approximate model. It is shown that each substitute problem in the class entails the qualitative features typical of topographic waves, and increasing the order of the model corresponds to higher-order approximations. Equations are explicitly presented for cross-sectional distributions of the lake topography which has a power-law representation and permits the analysis of weak and strong topographies.

Straight channels in which the depth profile does not change with position along the axis are studied in detail. The dispersion relation is discussed and dispersion curves are shown for the three lowest-order models. Convergence properties are thereby uncovered and phase speed and group velocity properties are found as they depend on wavenumber and topography. Further, for the lowest two modes, cross-channel stream-function distributions are presented. Apart from further convergence properties these distributions show that for U-shaped channels wave activity is nearer to the shore than for V-shaped channels, important information in the design of mooring systems.

An analysis of topographic Rossby wave reflection follows, which emphasizes the importance of the depth profile in the reflecting zone. Based on these results some lake solutions are presented.

The spin-up from rest of a homogeneous free-surface fluid contained in a rectangular tank with an inclined bottom has been studied in the laboratory. As in the case of a tank without bottom topography, it is found that in the spin-up process leading to the ultimate state of rigid-body rotation a number of stages can be distinguished, these being (i) the starting flow, characterized by zero absolute vorticity, (ii) the viscous generation of cyclonic vorticity at the lateral tank walls, leading to flow separation, and (iii) the formation of cyclonic and anticyclonic flow cells, which show a complicated interaction. When the topography steepness is small, these cells become organized in a regular array similar to what is observed in the non-sloping bottom case. For steeper topography, however, no organization into a regular cellular pattern is observed, and the relative fluid motion remains unsteady and irregular until eventually it has decayed owing to the spin-ip/spin-down mechanism provided by the Ekman layer at the tank bottom. During the first stage of the adjustment process the starting flow takes on the appearance of a large anticyclonic cell that fills the fluid domain entirely. Depending on the ratio of the horizontal and vertical lengthscales of the tank this cell is either symmetric or asymmetric, with a higher density of streamlines in the deeper part of the tank. The coupled vorticity equation, governing the depth-independent part of the starting flow, and the potential equation describing its depth-dependent part have been solved analytically, and the comparison between these results and observational data is generally good.

Direct numerical simulations are used to investigate potential enstrophy in stratified turbulence with small Froude numbers, large Reynolds numbers, and buoyancy Reynolds numbers ($R{e}_{b} $) both smaller and larger than unity. We investigate the conditions under which the potential enstrophy, which is a quartic quantity in the flow variables, can be approximated by its quadratic terms, as is often done in geophysical fluid dynamics. We show that at large scales, the quadratic fraction of the potential enstrophy is determined by $R{e}_{b} $. The quadratic part dominates for small $R{e}_{b} $, i.e. in the viscously coupled regime of stratified turbulence, but not when $R{e}_{b} \gtrsim 1$. The breakdown of the quadratic approximation is consistent with the development of Kelvin–Helmholtz instabilities, which are frequently observed to grow on the layerwise structure of stratified turbulence when $R{e}_{b} $ is not too small.

In many shear- and pressure-driven wall-bounded turbulent flows secondary motions spontaneously develop and their interaction with the main flow alters the overall large-scale features and transfer properties. Taylor–Couette flow, the fluid motion developing in the gap between two concentric cylinders rotating at different angular velocities, is not an exception, and toroidal Taylor rolls have been observed from the early development of the flow up to the fully turbulent regime. In this manuscript we show that under the generic name of ‘Taylor rolls’ there is a wide variety of structures that differ in the vorticity distribution within the cores, the way they are driven and their effects on the mean flow. We relate the rolls at high Reynolds numbers not to centrifugal instabilities, but to a combination of shear and anti-cyclonic rotation, showing that they are preserved in the limit of vanishing curvature and can be better understood as a pinned cycle which shows similar characteristics as the self-sustained process of shear flows. By analysing the effect of the computational domain size, we show that this pinning is not a product of numerics, and that the position of the rolls is governed by a random process with the space and time variations depending on domain size.

In Part 1 of this work, we derived general asymptotic results for the three-dimensional flow field and energy fluxes for flow within a tube whose walls perform prescribed small-amplitude periodic oscillations of high frequency and large axial wavelength. In the current paper, we illustrate how these results can be applied to the case of flow through a finite-length axially non-uniform tube of elliptical cross-section – a model of flow in a Starling resistor. The results of numerical simulations for three model problems (an axially uniform tube under pressure–flux and pressure–pressure boundary conditions and an axially non-uniform tube with prescribed flux) with prescribed wall motion are compared with the theoretical predictions made in Part 1, each showing excellent agreement. When upstream and downstream pressures are prescribed, we show how the mean flux adjusts slowly under the action of Reynolds stresses using a multiple-scale analysis. We test the asymptotic expressions obtained for the mean energy transfer E from the flow to the wall over a period of the motion. In particular, the critical point at which E = 0 is predicted accurately: this point corresponds to energetically neutral oscillations, the condition which is relevant to the onset of global instability in the Starling resistor.

A method is established for the calculation of the trajectories of shocks moving upward in the atmosphere, on the basis of the assumption that they are of the self-propagating type. The results of calculations for self-similar motions are given, and these are used to establish a propagation law based upon the concepts of the Chisnell, Chester and Whitham (CCW) approximation. This propagation law enters a characteristics law based upon that proposed by Whitham, but reformulated for the computation of axisymmetric shocks with varying density.

An asymptotic self-preserving shock shape is investigated, and is computed for the case γ = 1·4. A parabolic approximation scheme suggested by the self-preserving solution is developed, in which the solution near the axis is reduced to the solution of a system of ordinary differential equations. Finally, the governing equation for the general case without axial symmetry (but without winds) is presented.

In a companion paper (Part 1, Jameson et al. J. Fluid Mech. vol. 649, 2010, 19–43), the discovery of a new type of water bell was reported. When a vertical liquid jet impacts on the underside of a large horizontal plate, the resulting thin film spreads radially along the plate to an unspecified abrupt departure point, from whence it falls away from the plate of its own accord. The departure radius of the fluid from the plate is seen to depend strongly on the volumetric flow rate. The falling liquid may then coalesce to form a water bell. Here we present a theoretical analysis and explanation of this phenomenon. A force balance determining the maximum radial extension of the thin film flow along the plate is considered as a mechanism for fluid departure from the plate, for which an analytical model is developed. This model gives good predictions of the measured radius of departure. When a water bell has been formed, and the flow rate is altered, many interesting shapes are produced that depend on the shapes at previous flow rates. We discuss the origin of this hysteresis, and also present a leading order theory for the bell shape under a regime of changing flow rate. The models are compared with experimental results spanning two orders of magnitude in viscosity.

This paper presents the force coefficients (added mass and damping) for a circular cylinder oscillating horizontally in a uniformly stratified fluid of limited depth and in a continuously stratified fluid with a smooth pycnocline. The frequency-dependent added mass and damping are evaluated from Fourier transforms of the experimental records of impulse response functions. The stratification is shown to have a strong effect on the fluid–body interaction. It is found that, when the characteristic vertical extent of stratification (depth of uniformly stratified fluid or pycnocline thickness) decreases, the power radiated with internal waves is reduced and the maximum of the frequency spectrum of wave power shifts toward lower frequency. The results of experiments are compared with available theoretical predictions.

A fluctuating-force model is developed for representing the effect of the turbulent fluid velocity fluctuations on the particle phase in a turbulent gas–solid suspension in the limit of high Stokes number, where the particle relaxation time is large compared with the correlation time for the fluid velocity fluctuations. In the model, a fluctuating force is incorporated in the equation of motion for the particles, and the force distribution is assumed to be an anisotropic Gaussian white noise. It is shown that this is equivalent to incorporating a diffusion term in the Boltzmann equation for the particle velocity distribution functions. The variance of the force distribution, or equivalently the diffusion coefficient in the Boltzmann equation, is related to the time correlation functions for the fluid velocity fluctuations. The fluctuating-force model is applied to the specific case of a Couette flow of a turbulent particle–gas suspension, for which both the fluid and particle velocity distributions were evaluated using direct numerical simulations by Goswami & Kumaran (2010). It is found that the fluctuating-force simulation is able to quantitatively predict the concentration, mean velocity profiles and the mean square velocities, both at relatively low volume fractions, where the viscous relaxation time is small compared with the time between collisions, and at higher volume fractions, where the time between collisions is small compared with the viscous relaxation time. The simulations are also able to predict the velocity distributions in the centre of the Couette, even in cases in which the velocity distribution is very different from a Gaussian distribution.

The paper considers the refraction of a plane shock wave at an interface between two streams of different Mach number. Particular attention is paid to the irregular wave systems. It is found that when the interface is slow-fast, that is when the speed of sound a0 in the first or incident wave medium is less than the speed of sound aB in the second or transmitted wave medium, then there are two irregular systems, one being a double Mach reflexion type and the other being a four-wave confluence type. There are also two irregular systems when the refraction is fast-slow; these are a single Mach reflexion type and an expansion wave type. This last system has a central expansion wave when the flow is steady and a continuous band expansion wave when the flow is self-similar. Only two of the irregular wave systems have been observed experimentally in the fully developed state. Possible degeneracies are discussed.

The large eddy simulation (LES) technique will soon be 50 years old. Since Deardorff’s first papers in 1970 introducing this approach, major advances in the theory of LES and its computational implementation have been made and widely adopted. However, in terms of validation, LES studies continue to largely focus on the first- and second-order statistics, which in fact are the same tests that Deardorff conducted 45 years ago. Further advances in LES and wider adoption for new flows require advanced and more challenging tests to be developed and documented to serve as benchmarks. The paper by Stevens, Wilczek & Meneveau (J. Fluid Mech., 2014, vol. 757, pp. 888–907) does precisely that. The authors demonstrate the ability of LES to capture the recently established log-law of streamwise velocity variance and the related log-laws for even-order statistics up to order 10, as well as the departure of these statistics from a Gaussian distribution. The paper also provides key insights into the role of grid resolution on the computed turbulence field.