A perturbation expansion of the Boltzmann equation is used to derive constitutive relations for the granular flow of rough spheres in the limit where the energy dissipation in a collision is small compared to the energy of a particle. In the collision model, the post-collisional relative normal velocity at the point of contact is $-e_n$ times the pre-collisional normal velocity, and the post-collisional relative tangential velocity at the point of contact is $-e_t$ times the pre-collisional relative tangential velocity. A perturbation expansion is employed in the limit $(1 - e_n)\,{=}\,\varepsilon^2 \ll 1$, and $(1 - e_t^2) \propto \varepsilon^2 \ll 1$, so that $e_t$ is close to $\pm 1$. In the ‘rough’ particle model, the normal coefficient of restitution $e_n$ is close to 1, and the tangential coefficient of restitution $e_t$ is close to 1. In the ‘partially rough’ particle model, the normal coefficient of restitution $e_n$ is close to 1; and the tangential coefficient of restitution $e_t$ is close to $-1$ if the angle between the relative velocity vector and the line joining the centres of the particles is greater than the ‘roughness angle’ (chosen to be $(\upi/4)$ in the present calculation), and is close to 1 if the angle between the relative velocity vector and the line joining the centres is less than the roughness angle. The conserved variables in this case are mass and momentum; energy is not a conserved variable in the ‘adiabatic limit’ considered here, when the length scale is large compared to the ‘conduction length’. The results for the constitutive relations show that in the Navier–Stokes approximation, the form of the constitutive relation is identical to that for smooth particles, but the coefficient of shear viscosity for rough particles is 10%–50% higher than that for smooth particles. The coefficient of bulk viscosity, which is zero in the dilute limit for smooth particles, is found to be non-zero for rough and partially rough particles, owing to the transport of energy between the translational and rotational modes. In the Burnett approximation, there is an antisymmetric component in the stress tensor for rough and partially rough particles, which is not present for smooth particles.

The constitutive relations are used to analyse the ‘core region’ of a steady granular flow down an inclined plane, where there is a local balance between the production of energy due to the mean shear and the dissipation due to inelastic collisions. It is found that realistic results, such as the decrease in density upon increase in the angle of inclination near close packing, are obtained for the rough and partially rough particle models when the Burnett coefficients are included in the stress tensor, but realistic results are not obtained using the constitutive relations for smooth particles. This shows that the flow dynamics is sensitive to the numerical values of the viscometric coefficients, and provides an indication of the minimal model required to capture the flow dynamics.

The granular shear flow of rough inelastic particles driven by flat walls is considered in the high-Knudsen-number limit, where the frequency of particle collisions with the wall is large compared to the frequency of inter-particle collisions. An asymptotic analysis is used in the small parameter $\varepsilon = n d L$ in two dimensions and $\varepsilon = n d^2 L$ in three dimensions, where $n$ is the number of particles (per unit area in two dimensions and per unit volume in three dimensions), $d$ is the particle diameter and $L$ is the distance between the flat walls. In the collision model, the post-collisional velocity along the line joining the particle centres is $-e_n$ times the pre-collisional velocity, and the post-collisional velocity perpendicular to the line joining the particle centres is $-e_t$ times the pre-collisional value, where $e_n$ and $e_t$ are the normal and tangential coefficients of restitution. In the absence of binary collisions, a particle which has a random initial velocity tends to a final state where the translational velocities are zero, and the rotational velocity is equal to $(-2V_w/d)$, where $V_w$ is the wall velocity. When the effect of binary collisions is included, it is found that there are two possible final steady states, depending on the values of the tangential and normal coefficients of restitution. For certain parameter values, the final steady state is a stationary state, where the translational velocities of all the particles reduce to zero. For other parameter values, the final steady state is dynamic state where the translational velocity fluctuations are non-zero. In the dynamic state, the mean-square velocity has a power-law scaling with $\varepsilon$ in the limit $\varepsilon \rightarrow 0$. The exponents predicted by the theory are found to be in quantitative agreement with simulation results in two dimensions.

In this paper, we investigate the three-dimensional stability of the Moore–Saffman elliptical vortex in a rotating stratified fluid. By means of an asymptotic analysis for long vertical wavelength perturbations and small Froude number, we study the effects of Rossby number, external strain, and ellipticity of the vortex on the stability of azimuthal modes $m\,{=}\,1$ (corresponding to a bending instability) and $m\,{=}\,2$ (corresponding to a twisting instability).

In the case of a quasi-geostrophic fluid (small Rossby number), the asymptotic results are in striking agreement with previous numerical stability analyses even for vertical wavelengths of order one. For arbitrary Rossby number, the key finding is that the Rossby number has no effect on the domains of long-wavelength instability of these two modes: the two-dimensional or three-dimensional nature of the instabilities is controlled only by the background strain rate $\gamma$ and by the rotation rate $\Omega$ of the principal axes of the elliptical vortex relative to the rotating frame of reference.

For the $m=1$ mode, it is shown that when $\Omega < -\gamma$, the vortex is stable to any long-wavelength disturbances, when $-\gamma < \Omega \lesssim 0$, two-dimensional perturbations are most unstable, when $0 \lesssim \Omega < \gamma$, long-wavelength three-dimensional disturbances are the most unstable, and finally when $\gamma < \Omega$, short-wavelength three-dimensional perturbations are the most unstable. Similarly, the $m=2$ instability is two-dimensional or three-dimensional depending only on $\gamma$ and $\Omega$, independent of the Rossby number. This means that if a long-wavelength three-dimensional instability exists for a given elliptical vortex in a quasi-geostrophic fluid, a similar instability should be observed for any other Rossby number, in particular for infinite Rossby number (strongly stratified fluids). This implies that the planetary rotation plays a minor role in the nature of the instabilities observed in rotating strongly stratified fluids.

The present results for the azimuthal mode $m=1$ suggest that the vortex-bending instabilities observed previously in quasi-geostrophic fluids (tall-column instability) and in strongly stratified fluids (zigzag instability) are fundamentally related.

A self-consistent formalism to estimate baroclinic energy densities and fluxes resulting from the propagation of internal waves of arbitrary amplitude is derived using the concept of available potential energy. The method can be applied to numerical, laboratory or field data. The total energy flux is shown to be the sum of the linear energy flux $\int u'p'\,{\rm d}z$ (primes denote baroclinic quantities), plus contributions from the non-hydrostatic pressure anomaly and the self-advection of kinetic and available potential energy. Using highly resolved observations in Massachusetts Bay, it is shown that due to the presence of nonlinear internal waves periodically propagating in the area, $\int u'p'\,{\rm d}z$ accounts for only half of the total flux. The same data show that equipartition of available potential and kinetic energy can be violated, especially when the nonlinear waves begin to interact with the bottom.

The three-dimensional incompressible flow past a rectangular two-dimensional shallow cavity in a channel is investigated using large-eddy simulation (LES). The aspect ratio (length/depth) of the cavity is $L/D\,{=}\,2$ and the Reynolds number defined with the cavity depth and the mean velocity in the upstream channel is 3360. The sensitivity of the flow around the cavity to the characteristics of the upstream flow is studied by considering two extreme cases: a developing laminar boundary layer upstream of the cavity and when the upstream flow is fully turbulent. The two simulations are compared in terms of the mean statistics and temporal physics of the flow, including the dynamics of the coherent structures in the region surrounding the cavity. For the laminar inflow case it is found that the flow becomes unstable but remains laminar as it is convected over the cavity. Due to the three-dimensional flow instabilities and the interaction of the jet-like flow inside the recirculation region with the separated shear layer, the spanwise vortices that are shed regularly from the leading cavity edge are disturbed in the spanwise direction and, as they approach the trailing-edge corner, break into an array of hairpin-like vortices that is convected downstream the cavity close to the channel bottom. In the fully turbulent inflow case in which the momentum thickness of the incoming boundary layer is much larger compared to the laminar inflow case, the jittering of the shear layer on top of the cavity by the incoming near-wall coherent structures strongly influences the formation and convection of the eddies inside the separated shear layer. The mass exchange between the cavity and the main channel is investigated by considering the ejection of a passive scalar that is introduced instantaneously inside the cavity. As expected, it is found that the ejection is faster when the incoming flow is turbulent due to the interaction between the turbulent eddies convected from upstream of the cavity with the separated shear layer and also to the increased diffusion induced by the broader range of scales that populate the cavity. In the turbulent case it is shown that the eddies convected from upstream of the cavity can play an important role in accelerating the extraction of high-concentration fluid from inside the cavity. For both laminar and turbulent inflow cases it is shown that the scalar ejection can be described using simple dead-zone theory models in which a single-valued global mass exchange coefficient can be used to describe the scalar mass decay inside cavity over the whole ejection process.

In the present work, we study the transverse vortex-induced vibrations of an elastically mounted rigid cylinder in a fluid flow. We employ a technique to accurately control the structural damping, enabling the system to take on both negative and positive damping. This permits a systematic study of the effects of system mass and damping on the peak vibration response. Previous experiments over the last 30 years indicate a large scatter in peak-amplitude data ($A^*$) versus the product of mass–damping ($\alpha$), in the so-called ‘Griffin plot’.

A principal result in the present work is the discovery that the data collapse very well if one takes into account the effect of Reynolds number ($\mbox{\textit{Re}}$), as an extra parameter in a modified Griffin plot. Peak amplitudes corresponding to zero damping ($A^*_{{\alpha}{=}0}$), for a compilation of experiments over a wide range of $\mbox{\textit{Re}}\,{=}\,500-33000$, are very well represented by the functional form $A^*_{\alpha{=}0} \,{=}\, f(\mbox{\textit{Re}}) \,{=}\, \log(0.41\,\mbox{\textit{Re}}^{0.36}$). For a given $\mbox{\textit{Re}}$, the amplitude $A^*$ appears to be proportional to a function of mass–damping, $A^*\propto g(\alpha)$, which is a similar function over all $\mbox{\textit{Re}}$. A good best-fit for a wide range of mass–damping and Reynolds number is thus given by the following simple expression, where $A^*\,{=}\, g(\alpha)\,f(\mbox{\textit{Re}})$: \[ A^* \,{=}\,(1 - 1.12\,\alpha + 0.30\,\alpha^2)\,\log (0.41\,\mbox{\textit{Re}}^{0.36}). \] In essence, by using a renormalized parameter, which we define as the ‘modified amplitude’, $A^*_M\,{=}\,A^*/A^*_{\alpha{=}0}$, the previously scattered data collapse very well onto a single curve, $g(\alpha)$, on what we refer to as the ‘modified Griffin plot’. There has also been much debate over the last three decades concerning the validity of using the product of mass and damping (such as $\alpha$) in these problems. Our results indicate that the combined mass–damping parameter ($\alpha$) does indeed collapse peak-amplitude data well, at a given $\mbox{\textit{Re}}$, independent of the precise mass and damping values, for mass ratios down to $m^*\,{=}\,1$.

We present the results of direct numerical simulations (DNS) of the evolution of nonlinear random water wave fields. The aim of the work is to validate the hypotheses underlying the statistical theory of nonlinear dispersive waves and to clarify the role of exactly resonant, nearly resonant and non-resonant wave interactions. These basic questions are addressed by examining relatively simple wave systems consisting of a finite number of wave packets localized in Fourier space. For simulation of the long-term evolution of random water wave fields we employ an efficient DNS approach based on the integrodifferential Zakharov equation. The non-resonant cubic terms in the Hamiltonian are excluded by the canonical transformation. The proposed approach does not use a regular grid of harmonics in Fourier space. Instead, wave packets are represented by clusters of discrete Fourier harmonics.

The simulations demonstrate the key importance of near-resonant interactions for the nonlinear evolution of statistical characteristics of wave fields, and show that simulations taking account of only exactly resonant interactions lead to physically meaningless results. Moreover, exact resonances can be excluded without a noticeable effect on the field evolution, provided that near-resonant interactions are retained. The field evolution is shown to be robust with respect to the details of the account taken of near-resonant interactions. For a wave system initially far from equilibrium, or driven out of equilibrium by an abrupt change of external forcing, the evolution occurs on the ‘dynamical’ time scale, that is with quadratic dependence on nonlinearity $\varepsilon$, not on the $O(\varepsilon^{-4})$ time scale predicted by the standard statistical theory. However, if a wave system is initially close to equilibrium and evolves slowly in the presence of an appropriate forcing, this evolution is in quantitative accordance with the predictions of the kinetic equation. We suggest a modified version of the kinetic equation able to describe all stages of evolution.

Although the dynamic time scale of quintet interactions $\varepsilon^{-3}$ is smaller than the kinetic time scale $\varepsilon^{-4}$, they are not included in the existing statistical theory, and their effect on the evolution of wave spectra is unknown. We show that these interactions can significantly affect the spectrum evolution, although on a time scale much larger than $O(\varepsilon^{-4})$. However, for waves of high but still realistic steepness $\varepsilon\,{\sim}\,0.25$, the scales of evolution are no longer separated. By tracing the evolution of high statistical moments of the wave field, we directly verify one of the main assumptions used in the derivation of the kinetic equation: the quasi-Gaussianity of the wave holds throughout the evolution, both with and without accounting for quintet interactions.

The conclusions are not confined to water waves and are applicable to a generic weakly nonlinear dispersive wave field with prohibited triad interactions.

Steady-state distributions of water potential and salt concentration in coastal aquifers are typically modelled by the Henry problem, which consists of a fully coupled system of flow and transport equations. Coupling arises from the dependence of water density on salt concentration. The physical behaviour of the system is fully described by two dimensionless groups: (i) the coupling parameter $\alpha$, which encapsulates the relative importance of buoyancy and viscous forces, and (ii) the Péclet number $\mbox{\textit{Pe}}$, which quantifies the relative importance of purely convective and dispersive transport mechanisms. We provide a systematic analytical analysis of the Henry problem for a full range of the Péclet number. For moderate $\mbox{\textit{Pe}}$, analytical solutions are obtained through perturbation expansions in $\alpha$. This allows us to elucidate the onset of density-driven vertical flux components and the dependence of the local hydraulic head gradients on the coupling parameter. The perturbation solution identifies the regions where salt concentration is most pronounced and relates their spatial extent to the development of a convection cell. Next, we compare our solution to a solution of the pseudo-coupled model, wherein flow and transport are coupled only via the boundary conditions. This enables us to isolate the effects caused by density-dependent processes from those induced by external forcings (boundary conditions). For small $\mbox{\textit{Pe}}$, we develop a perturbation expansion around the exact solution corresponding to $\mbox{\textit{Pe}}\,{=}\,0$, which sheds new light on the interpretation of processes observed in diffusion experiments with variable-density flows in porous media. The limiting case of infinite Péclet numbers is solved exactly for the pseudo-coupled model and compared to numerical simulations of the fully coupled problem for large $\mbox{\textit{Pe}}$. The proposed perturbation approach is applicable to a wide range of variable-density flows in porous media, including seawater intrusion into coastal aquifers and temperature or pressure-driven density flows in deep aquifers.

A straight front separating two semi-infinite regions of uniform potential vorticity (PV) in a rotating shallow-water fluid gives rise to a localized fluid jet and a geostrophically balanced shelf in the free surface. The linear stability of this configuration, consisting of the simplest non-trivial PV distribution, has been studied previously, with ambiguous results. We revisit the problem and show that the flow is weakly unstable when the maximum Rossby number ${\textsfi R}\,{>}\,1$. The instability is surprisingly weak, indeed exponentially so, scaling like $\exp[-4.3/({\textsfi R} - 1)]$ as ${\textsfi R}\,{\to}\,1$. Even when ${\textsfi R}\,{=}\,\sqrt{2}$ (when the maximum Froude number ${\textsfi F}\,{=}\,1$), the maximum growth rate is only $7.76\,{\times}\,10^{-6}$ times the Coriolis frequency. Its existence nonetheless sheds light on the concept of ‘balance’ in geophysical flows, i.e. the degree to which the PV controls the dynamical evolution of these flows.

A numerical investigation of the onset of three-dimensional states in Taylor–Couette flow for aspect ratio one is presented. Two main branches exist, one preserving and the other breaking the reflection symmetry about the mid-plane. Both branches become three-dimensional via Hopf bifurcations to rotating waves with different azimuthal wavenumbers. Moreover, the symmetric branch exhibits secondary Hopf bifurcations and transitions to complex spatio-temporal dynamics at Reynolds numbers $\hbox{\it Re}\,{\sim}\,1000$. The analysis of the three-dimensional solutions shows that the dynamics is driven by the jet of angular momentum erupting from the inner cylinder boundary layer and its interactions with the sidewall and endwall layers. The various solutions are organized by a lattice of spatial and spatio-temporal symmetry subgroups which provides a framework for the relationships between the solution types and for the symmetry-breaking bifurcations. The results obtained agree with previous experimental results and help clarify many aspects of the mode competition at the higher $\hbox{\it Re}$ values.

Direct numerical simulation results are presented for turbulent channel flows with two-dimensional roughness elements of different shapes. The focus is mainly on a geometry where the separation between consecutive roughness elements is small and for which the rate of change of the roughness function with respect to the separation between consecutive elements is large. Roughness elements are placed either along the flow direction or orthogonally to it. In the latter case, the drag is increased. For the former case, the possibility of drag reduction reflects the different relative contributions from viscous and Reynolds shear stresses. The Reynolds shear stress depends on the shape of the surface more than the viscous stress and is closely related to the near-wall structures. For orthogonal elements, there is no satisfactory correlation between the roughness function and parameters describing the roughness geometry. On the other hand, a satisfactory collapse of the data is achieved when the roughness function is plotted against the root mean square wall-normal velocity averaged over the plane of the roughness crests. Relative to a smooth wall surface, the Reynolds stress tensor near the wall tends to become more isotropic when the elements are orthogonal to the flow and less isotropic when the elements are aligned with the flow. The interdependencies between the departure from isotropy in the wall region, the organization of the wall structures, and the magnitude of the drag are assessed by examining the rotational component of the turbulent kinetic energy production and the probability density function of the helicity density.

Fluid injection into the deep subsurface, such as injection of carbon dioxide (CO$_2$) into deep saline aquifers, often involves two-fluid flow in confined geological formations. Similarity solutions may be derived for these problems by assuming that a sharp interface separates the two fluids, by imposing a suitable no-flow condition along both the top and bottom boundaries, and by including an explicit solution for the pressure distribution in both fluids. When the injected fluid is less dense and less viscous than the resident fluid, as is the case for CO$_{2}$ injection into a resident brine, gravity override produces a fluid flow system that is captured well by the similarity solutions. The similarity solutions may be extended to include slight miscibility between the two fluids, as well as compressibility in both of the fluid phases. The solutions provide the location of the interface between the two fluids, as well as drying fronts that develop within the injected fluid. Applications to cases of supercritical CO$_{2}$ injection into deep saline aquifers demonstrate the utility of the solutions, and comparisons to solutions from full numerical simulations show the ability to predict the system behaviour.

The organization of vortex clusters above the buffer layer of turbulent channels is analysed using direct numerical simulations at friction Reynolds numbers up to $\hbox{\it Re}_{\tau}\,{=}\,1900$. Especial attention is paid to a family of clusters that reach from the logarithmic layer to the near-wall region below $y^+\,{=}\,20$. These tall attached clusters are markers of structures of the turbulent fluctuating velocity that are more intense than their background. Their lengths and widths are proportional to their heights $\Delta_y$ and grow self-similarly with time after originating at different wall-normal positions in the logarithmic layer. Their influence on the outer region is measured by the variation of their volume density with $\Delta_y$. That influence depends on the vortex identification threshold, and becomes independent of the Reynolds number if the threshold is low enough. The clusters are parts of larger structures of the streamwise velocity fluctuations whose average geometry is consistent with a cone tangent to the wall along the streamwise axis. They form groups of a few members within each cone, with the larger individuals in front of the smaller ones. This behaviour is explained considering that the streamwise velocity cones are ‘wakes’ left behind by the clusters, while the clusters themselves are triggered by the wakes left by yet larger clusters in front of them. The whole process repeats self-similarly in a disorganized version of the vortex-streak regeneration cycle of the buffer layer, in which the clusters and the wakes spread linearly under the effect of the background turbulence. These results characterize for the first time the structural organization of the self-similar range of the turbulent logarithmic region.

Three-dimensional direct numerical simulations (DNS) of the nonlinear dynamics and a route to chaos in a rotating fluid subjected to lateral heating are presented here and discussed in the context of laboratory experiments in the baroclinic annulus. Following two previous preliminary studies, the fluid used is air rather than a liquid as used in all other previous work. This study investigates a bifurcation sequence from the axisymmetric flow to a number of complex flows.

The transition sequence, on increase of the rotation rate, from the axisymmetric solution via a steady fully developed baroclinic wave to chaotic flow, followed a variant of the classical quasi-periodic bifurcation route, starting with a subcritical Hopf and associated saddle-node bifurcation. This was followed by a sequence of two supercritical Hopf-type bifurcations, first to an amplitude vacillation, then to a three-frequency quasi-periodic modulated amplitude vacillation (MAV), and finally to a chaotic (MAV). In the context of the baroclinic annulus this sequence is unusual as the vacillation is usually found on decrease of the rotation rate from the steady wave flow.

Further transitions of a steady wave with a higher wavenumber pointed to the possibility that a barotropic instability of the sidewall boundary layers and the subsequent breakdown of these barotropic vortices may play a role in the transition to structural vacillation and, ultimately, geostrophic turbulence.

It is known that perfectly trapped surface waves exist at certain eigenfrequencies near a vertical cylinder in a long channel, or an infinite and periodic array of vertical cylinders, and excitation by incident waves of the same frequency is not possible according to the linear theory. We present a nonlinear theory whereby a trapped wave near a cylinder in a channel is excited subharmonically by an incident wave of twice the eigenfrequency. The effects of geometrical parameters on the initial growth of resonance and the final amplification are studied in detail.

A linear stability analysis of finite-amplitude periodic progressive gravity waves on water of finite depth has extended existing results to steeper waves and shallower water. Some new types of instability are found for shallow water. When the water depth decreases, higher-order resonances lead to the dominant instabilities. In contrast with the deep water case, we have found that in shallow water the dominant instabilities are usually associated with resonant interactions between five, six, seven and eight waves. For small steepness, dominant instabilities are quasi two-dimensional. For moderate and large steepness, the dominant instabilities are three-dimensional and phased-locked with the unperturbed nonlinear wave. At the margin of instability diagrams, these results suggest the existence of new bifurcated three-dimensional steady waves.

Numerical simulations are presented of shear flow past two-dimensional droplets adhering to a wall, at moderate Reynolds numbers. The results were obtained using a level-set method to track the interface, with measures to eliminate any errors in the conservation of mass of droplets. First, the case of droplets whose contact lines are pinned is considered. Data are presented for the critical value of the dimensionless shear rate (Weber number, $\hbox{\it We}$), beyond which no steady state is found, as a function of Reynolds number, $\hbox{\it Re}$. $\hbox{\it We}$ and $\hbox{\it Re}$ are based on the initial height of the droplet and shear rate; the range of Reynolds numbers simulated is $\hbox{\it Re} \leq 25$. It is shown that, as $\hbox{\it Re}$ is increased, the critical value $\hbox{\it We}_c$ changes from $\hbox{\it We}_c\propto \hbox{\it Re}$ to $\hbox{\it We}_c\approx$ const., and that the deformation of droplets at $\hbox{\it We}$ just above $\hbox{\it We}_c$ changes fundamentally from a gradual to a sudden dislodgement. In the second part of the paper, drops are considered whose contact lines are allowed to move. The contact-line singularity is removed by using a Navier-slip boundary condition. It is shown that macroscale contact angles can be defined that are primarily functions of the capillary number based on the contact-line speed, instead of the value of $\hbox{\it We}$ of the shear flow. It is shown that a Cox–Voinov-type expression can be used to describe the motion of the downstream contact line. A qualitatively different relation is tested for the motion of the upstream contact line. In a third part of this paper, results are presented for droplets moving on a wall with position-dependent sliplength or contact-angle hysteresis window, in an effort to stabilize or destabilize the drop.

When a laser beam of high intensity interacts with a dense material, an ablation front appears in the high-temperature plasma resulting from the interaction. Such a front can be used to accelerate and compress the dense material. The dynamics of the ablation front is strongly coupled to that of the absorption front where the laser energy is absorbed. The present paper determines analytical solutions of the front internal structure in the fully compressible case.