This experimental and theoretical study considers the dynamics and the instability of a Lamb–Oseen vortex in a stably stratified fluid. In a companion paper, it was shown that tilting the vortex axis with respect to the direction of stratification induces the formation of a rim of strong axial flow near a critical radius when the Froude number of the vortex is larger than one.

Here, we demonstrate that this tilt-induced flow is responsible for a three-dimensional instability. We show that the instability results from a shear instability of the basic axial flow in the critical-layer region. The theoretical predictions for the wavelength and the growth rate obtained by a local stability analysis of the theoretical critical-layer profile are compared to experimental measurements and a good agreement is observed. The late stages of the instability are also analysed experimentally. In particular, we show that the tilt-induced instability does not lead to the destruction of the vortex, but to a sudden decrease of its Froude number, through the turbulent diffusion of its core size, when the initial Froude number is close to 1. A movie is available with the online version of the paper.

Using molecular dynamic simulations, we investigate the characteristics of dense flows of model cohesive grains. We describe their rheological behaviour and its origin at the scale of the grains and of their organization. Homogeneous plane shear flows give access to the constitutive law of cohesive grains which can be expressed by a simple friction law similar to the case of cohesionless grains, but intergranular cohesive forces strongly enhance the resistance to the shear. Then we show the consequence on flows down a slope: a plugged region develops at the free surface where the cohesion intensity is the strongest. Moreover, we measure various indicators of the microstructure within flows which evidence the aggregation of grains owing to cohesion and we analyse the properties of the contact network (force distributions and anisotropy). This provides new insights into the interplay between the local contact law, the microstructure and the macroscopic behavior of cohesive grains. Movies are available with the online version of the paper.

The flow over cylinders of varying fineness ratio (length to diameter) aligned with the free stream was examined using a magnetic suspension and balance system in order to avoid model support interference. The drag coefficient variation of a right circular cylinder was obtained for a wide range of fineness ratios. Particle image velocimetry (PIV) was used to examine the flow field, particularly the behaviour of the leading-edge separation shear layer and its effect on the wake. Reynolds numbers based on the cylinder diameter ranged from 5×104 to 1.1×105, while the major portion of the experiment was conducted at ReD=1.0×105. For moderately large fineness ratio, the shear layer reattaches with subsequent growth of the boundary layer, whereas over shorter cylinders, the shear layer remains detached. Differences in the wake recirculation region and the immediate wake patterns are clarified in terms of both the mean velocity and turbulent flow fields, including longitudinal vortical structures in the cross-flow plane of the wake. The minimum drag corresponded to the fineness ratio for which the separated shear layer reattached at the trailing edge of the cylinder. The base pressure was obtained with a telemetry technique. Pressure fields and aerodynamic force fluctuations are also discussed.

Instability and transition to turbulence in a magnetohydrodynamic channel flow are studied numerically for the case of a uniform magnetic field imposed along the spanwise direction. Optimal perturbations and their maximum amplifications over finite time intervals are computed in the framework of the linear problem using an iterative scheme based on direct and adjoint governing equations. It is shown that, at sufficiently strong magnetic field, the maximum amplification is no longer provided by classical streamwise rolls, but rather by rolls oriented at an oblique angle to the basic flow direction. The angle grows with the Hartmann number Ha and reaches the limit corresponding to purely spanwise rolls at Ha between 50 and 100 depending on the Reynolds number. Direct numerical simulations are applied to investigate the transition to turbulence at a single subcritical Reynolds number Re = 5000 and various Hartmann numbers. The transition is caused by the transient growth and subsequent breakdown of optimal perturbations, which take the form of one or two symmetric optimal modes (streamwise, oblique or spanwise modes depending on Ha) with low-amplitude three-dimensional noise added at the moment of strongest energy amplification. A sufficiently strong magnetic field (Ha larger than approximately 30) is found to completely suppress the instability. At smaller Hartmann numbers, the transition is observed but it is modified in comparison with the pure hydrodynamic case.

Spatial length scales of the rate of dissipation, χ, of fluctuations of a conserved scalar, Z, are inferred numerically using a DNS database of a turbulent planar jet flame. The Taylor-scale Reynolds numbers lie in the range of 38 to 58 along the centreline of the simulated jet flame. Three different methods are used to study the spatial length scales associated with χ. First, analysis of the one-dimensional dissipation spectra shows an expected Reδ−3/4 (Kolmogorov) scaling with the outer-scale Reynolds number, Reδ. Secondly, thin sheet-like three-dimensional scalar dissipation structures have been investigated directly. Such structures were identified within the computational domain using level-sets of the χ-field, and their thicknesses were subsequently computed. The study shows, in accordance with experimental studies, that the captured dissipation-layer thickness also shows a Kolmogorov scaling with Reδ. Finally, spatial filters of varying widths were applied to the instantaneous Z field in order to model the averaging effect that takes place with some experimental measurement techniques. The filtered scalar dissipation rate was then calculated from the filtered scalar field. The peaks in the instantaneous filtered χ-profiles are observed to decrease exponentially with increasing filter width, yielding estimates of the true value of χ. Unlike the dissipation length scales obtained from the spectral analysis and the level-set method, the length-scale estimates from the spatial-filtering method are found to be proportional to Reδ−1. This is consistent with the small-scale intermittency of χ which cannot be captured by techniques that just resolve the conventional Batchelor/Obukhov–Corrsin scale. These results have implications when considering resolution requirements for measuring scalar dissipation length scales in experimental flows.

The transition of the flow in a duct of square cross-section is studied. Like in the similar case of the pipe flow, the motion is linearly stable for all Reynolds numbers; this flow is thus a good candidate to investigate the ‘bypass’ path to turbulence. Initially the so-called ‘linear optimal perturbation problem’ is formulated and solved, yielding optimal disturbances in the form of longitudinal vortices. Such optimals, however, fail to elicit a significant response from the system in the nonlinear regime. Thus, streamwise-inhomogeneous sub-optimal disturbances are focused upon; nonlinear quadratic interactions are immediately caused by such initial perturbations and an unstable streamwise-homogeneous large-amplitude mode rapidly emerges. The subsequent evolution of the flow, at a value of the Reynolds number at the boundary between fully developed turbulence and relaminarization, shows the alternance of patterns with two pairs of large-scale vortices near opposing parallel walls. Such edge states bear a resemblance to optimal disturbances.

We consider the problem of weakly nonlinear buoyant convection in horizontal mushy layers with permeable mush–liquid interface during the solidification of binary alloys. We analyse the effects of several parameters of the problem on the stationary modes of convection in the form of either a hexagonal pattern or a non-hexagonal pattern such as rolls, rectangles and squares. No assumption is made on the thickness of the mushy layer, and a number of simplifying assumptions made in previous theoretical investigations of the problem are relaxed here in order to study the problem based on a more realistic model. Using both analytical and numerical methods, we determine the steady solutions for the nonlinear problem in a range of the Rayleigh number R near its critical value. Both the nonlinear basic state and variable permeability of the present problem favour hexagon-pattern convection. The results of the analyses and computations indicate that depending on the range of values of the parameters, bifurcation to hexagonal or non-hexagonal convection can be either supercritical or subcritical. However, among all the computed solutions in the particular range of values of the parameters that are most relevant to those of the experiments, only convection in the form of down-hexagons with downflow at the cell centres and upflow at the cell boundaries, was found to be realizable, in the sense that its amplitude increases with R.

We consider the linear evolution of a localized vortex with Gaussian potential vorticity that is superposed on a horizontal Couette flow in a rapidly rotating strongly stratified fluid. The Rossby number, defined as the ratio of the shear of the Couette flow to the Coriolis frequency, is assumed small. Our focus is on the inertia–gravity waves that are generated spontaneously during the evolution of the vortex. These are exponentially small in the Rossby number and hence are neglected in balanced models such as the quasi-geostrophic model and its higher-order generalizations. We develop an exponential-asymptotic approach, based on an expansion in sheared modes, to give an analytic description of the three-dimensional structure of the inertia–gravity waves emitted by the vortex. This provides an explicit example of the spontaneous radiation of inertia–gravity waves by localized balanced motion in the small-Rossby-number regime.

The inertia–gravity waves are emitted as a burst of four wavepackets propagating downstream of the vortex. The approach employed reduces the computation of inertia–gravity-wave fields to a single quadrature, carried out numerically, for each spatial location and each time. This makes it possible to unambiguously define an initial state that is entirely free of inertia–gravity waves, and circumvents the difficulties generally associated with the separation between balanced motion and inertia–gravity waves.

When the acoustic force inside a cavity balances the gravitational force on a particle the result is known as acoustic levitation. Using the lattice Boltzmann equation method we find the acoustic force acting on a rounded particle for two different single-axis acoustic levitators in two dimensions, the first with plane waves, the second with a rounded reflector that enhances the acoustic force. With no gravitational force, a particle oscillates around a pressure node; in the presence of gravity the oscillation is shifted a small vertical distance below the pressure node. This distance increases linearly as the density ratio between the solid particle and fluid grows. For both cavities, the particle oscillates with the frequency of the sound source and its harmonics and in some cases there is a much smaller second dominant frequency. When the momentum of the acoustic source changes, the oscillation around the average vertical position can have both frequencies mentioned above. However, if this quantity is large enough, the oscillations of the particle are aperiodic in the cavity with a rounded reflector.

It is now well established that two distinct types of motion occur in geophysical turbulence: slow motions associated with potential vorticity advection and fast oscillations due to inertia–gravity waves (or acoustic waves). Many studies have theorized the existence of a flow for which the entire motion is controlled by the potential vorticity (or one ‘master variable’) – this is known as balance. In real geophysical flows, deviations from balance in the form of inertia–gravity waves or ‘imbalance’ have often been found to be small. Here we examine the extent to which balance holds in rotating stratified turbulence which is nearly balanced initially.

Using the non-hydrostatic fluid dynamical equations under the Boussinesq approximation, we analyse properties of rotating stratified turbulence spanning a range of Rossby numbers (Ro≡|ζ|max/f) and the frequency ratios (c≡N/f) where ζ is the relative vertical vorticity, f is the Coriolis frequency and N is the buoyancy frequency. Using a recently introduced diagnostic procedure, called ‘optimal potential vorticity balance’, we extract the balanced part of the flow in the simulations and assess how the degree of imbalance varies with the above parameters.

We also introduce a new and more efficient procedure, building upon a quasi-geostrophic scaling analysis of the complete non-hydrostatic equations. This ‘nonlinear quasi-geostrophic balance’ procedure expands the equations of motion to second order in Rossby number but retains the exact (unexpanded) definition of potential vorticity. This proves crucial for obtaining an accurate estimate of balanced motions. In the analysis of rotating stratified turbulence at Ro≲1 and N/f≫1, this procedure captures a significantly greater fraction of the underlying balance than standard (linear) quasi-geostrophic balance (which is based on the linearized equations about a state of rest). Nonlinear quasi-geostrophic balance also compares well with optimal potential vorticity balance, which captures the greatest fraction of the underlying balance overall.

More fundamentally, the results of these analyses indicate that balance dominates in carefully initialized simulations of freely decaying rotating stratified turbulence up to O(1) Rossby numbers when N/f≫1. The fluid motion exhibits important quasi-geostrophic features with, in particular, typical height-to-width scale ratios remaining comparable to f/N.

In order to gain a better insight into flow, vortical and turbulence structure and their correlation with the local heat transfer in impinging flows, we performed large-eddy simulations (LES) of a round normally impinging jet issuing from a long pipe at Reynolds number Re = 20000 at the orifice-to-plate distance H = 2D, where D is the jet-nozzle diameter. This configuration was chosen to match previous experiments in which several phenomena have been detected, but the underlying physics remained obscure because of limitations in the measuring techniques applied. The instantaneous velocity and temperature fields, generated by the LES, revealed interesting time and spatial dynamics of the vorticity and eddy structures and their imprints on the target wall, characterized by tilting and breaking of the edge ring vortices before impingement, flapping, precessing, splitting and pairing of the stagnation point/line, local unsteady separation and flow reversal at the onset of radial jet spreading, streaks pairing and branching in the near-wall region of the radial jets, and others. The LES data provided also a basis for plausible explanations of some of the experimentally detected statistically-averaged flow features such as double peaks in the Nusselt number and the negative production of turbulence energy in the stagnation region. The simulations, performed with an in-house unstructured finite-volume code T-FlowS, using second-order-accuracy discretization schemes for space and time and the dynamic subgrid-scale stress/flux model for unresolved motion, showed large sensitivity of the results to the grid resolution especially in the wall vicinity, suggesting care must be taken in interpreting LES results in impinging flows.

Particle-size segregation is a common feature of dense gravity-driven granular free-surface flows, where sliding and frictional grain–grain interactions dominate. Provided that the diameter ratio of the particles is not too large, the grains segregate by a process called kinetic sieving, which, on average, causes the large particles to rise to the surface and the small grains to sink to the base of the avalanche. When the flowing layer is brought to rest this stratification is often preserved in the deposit and is known by geologists as inverse grading. Idealized experiments with bi-disperse mixtures of differently sized grains have shown that inverse grading can be extremely sharp on rough beds at low inclination angles, and may be modelled as a concentration jump or shock. Several authors have developed hyperbolic conservation laws for segregation that naturally lead to a perfectly inversely graded state, with a pure phase of coarse particles separated from a pure phase of fines below, by a sharp concentration jump. A generic feature of these models is that monotonically decreasing sections of this concentration shock steepen and eventually break when the layer is sheared. In this paper, we investigate the structure of the subsequent breaking, which is important for large-particle recirculation at the bouldery margins of debris flows and for fingering instabilities of dry granular flows. We develop an exact quasi-steady travelling wave solution for the structure of the breaking/recirculation zone, which consists of two shocks and two expansion fans that are arranged in a ‘lens’-like structure. A high-resolution shock-capturing numerical scheme is used to investigate the temporal evolution of a linearly decreasing shock towards a steady-state lens, as well as the interaction of two recirculation zones that travel at different speeds and eventually coalesce to form a single zone. Movies are available with the online version of the paper.

A temporal linear instability analysis of a charged coflowing jet with two immiscible viscous liquids in a radial electric field is carried out for axisymmetric disturbances. According to the magnitude of the liquid viscosity relative to the ambient air viscosity, two generic cases are considered. The analytical dimensionless dispersion relations are derived and solved numerically. Two unstable modes, namely the para-sinuous mode and the para-varicose mode, are identified in the Rayleigh regime. The para-sinuous mode is found to always be dominant in the jet instability. Liquid viscosity clearly stabilizes the growth rates of the unstable modes, but its effect on the cut-off wavenumber is negligible. The radial electric field has a dual effect on the modes, stabilizing them when the electrical Euler number is smaller than a critical value and destabilizing them when it exceeds that value. Moreover, the electrical Euler number and Weber number increase the dominant and cut-off wavenumbers significantly. Based on the Taylor–Melcher leaky dielectric theory, two limit cases, i.e. the small electrical relaxation time limit (SERT) and the large electrical relaxation time limit (LERT), are discussed. For coflowing jets having a highly conducting outer liquid, SERT may serve as a good approximation. In addition, the dispersion relations under the thin layer approximation are derived, and it is concluded that the accuracy of the thin layer approximation is closely related to the values of the dimensionless parameters.

A theoretical study is presented of convection in a horizontal fluid layer heated from below or above which is periodically accelerated in its plane. The analysis is based on Galerkin methods as well as on direct numerical simulations of the underlying Boussinesq equations.

Shaking in a fixed direction breaks the original isotropy of the layer. At onset of convection and at small acceleration, we find longitudinal rolls, where the roll axis aligns parallel to the acceleration direction. With increasing acceleration amplitude, a shear instability takes over and transverse rolls with the axis perpendicular to the shaking direction nucleate at onset. In the nonlinear regime, the longitudinal rolls become unstable against transverse modulations very close to onset which leads to a kind of domain chaos between patches of symmetry degenerated oblique rolls.

In the case of circular shaking, the system is isotropic in the time average sense, however, with a broken chiral symmetry. The onset of convection corresponds to the transverse roll case studied before with the roll axis selected spontaneously. With increasing Rayleigh number, a heteroclinic cycle is observed with the roll changing its orientation periodically in time. At even higher Rayleigh number, this heteroclinic cycle becomes chaotic similarly to the case of the Küppers–Lortz instability.

We study the dynamics of a mushy layer in directional solidification for the case of a thin near-eutectic mush with a deformable and permeable mush–liquid interface. We examine the onset of convection using linear stability analysis, and the weakly nonlinear growth of liquid inclusions that signal the onset of chimneys. This analysis is compared to past analyses in which the mush–liquid interface is replaced by a rigid impermeable lid. We find qualitative agreement between the two models, but the rigid-lid approximation gives substantially different quantitative behaviour.

In linear theory, the rigid-lid approximation leads to an over-estimate of the critical Rayleigh number and wavenumber of the instability. The condition for the onset of oscillatory instability is also changed by a factor of about 5 in composition number C. In the weakly nonlinear theory, the location of the onset of liquid inclusions is near the undisturbed front for the free-boundary analysis, whereas it lies at the centre of the mushy layer when the rigid-lid approximation is used. For hexagonal patterns, the boundary between regions of parameter space in which up and down hexagons are stable, shifts as a result of coupling between the liquid and mush regions.

Direct numerical simulations (DNS) are conducted of turbulent flow passing an infinitely thin trailing edge. The objective is to investigate the turbulent flow field in the vicinity of the trailing edge and the associated broadband noise generation. To generate a turbulent boundary layer a short distance from the inflow boundary, high-amplitude lifted streaks and disturbances that can be associated with coherent outer-layer vortices are introduced at the inflow boundary. A rapid increase in skin friction and a decrease in boundary layer thickness and pressure fluctuations is observed at the trailing edge. It is demonstrated that the behaviour of the hydrodynamic field in the vicinity of the trailing edge can be predicted with reasonable accuracy using triple-deck theory if the eddy viscosity is accounted for. Point spectra of surface pressure difference are shown to vary considerably towards the trailing edge, with a significant reduction of amplitude occurring in the low-frequency range. The acoustic pressure obtained from the DNS is compared with predictions from two- and three-dimensional acoustic analogies and the classical trailing-edge theory of Amiet. For low frequencies, two-dimensional theory succeeds in predicting the acoustic pressure in the far field with reasonable accuracy due to a significant spanwise coherence of the surface pressure difference and predominantly two-dimensional sound radiation. For higher frequencies, however, the full three-dimensional theory is required for an accurate prediction of the acoustic far field. DNS data are used to test some of the key assumptions invoked by Amiet for the derivation of the classical trailing-edge theory. Even though most of the approximations are shown to be reasonable, they collectively lead to a deviation from the DNS results, in particular for higher frequencies. Moreover, because the three-dimensional acoustic analogy does not provide significantly improved results, it is suggested that some of the discrepancies can be attributed to the approach of evaluating the far-field sound using a Kirchhoff-type integration of the surface pressure difference.

In this paper we consider the propagation of acoustic waves on top of an inviscid steady flow along a curved hollow or annular duct with hard or lined walls. The curvature of the duct centreline (which is not restricted to being small) and the wall radii vary slowly along the duct, allowing application of an asymptotic multiple-scales analysis. The modal wavenumbers and mode shapes are determined locally as modes of a torus with the same local curvature, while the amplitude of the modes evolves as the mode propagates along the duct. The duct modes are found explicitly at each axial location using a pseudospectral numerical method.

Unlike the case of a straight duct carrying uniform flow, there is a fundamental asymmetry between upstream and downstream propagating modes, with some mode shapes tending to be concentrated on either the inside or outside of the bend depending on the direction of propagation, curvature and steady-flow Mach number. The interaction between the presence of wall lining and curvature is also significant; for instance, in a representative case it is found that the curvature causes the first few acoustic modes to be more heavily damped by the duct boundary than would be expected for a straight duct.

Using ray theory, we suggest explanations of these features. For the lowest azimuthal-order modes, three distinct regimes are found in which the modes are localized in different parts of the duct cross-section. This phenomenon is explained by a balance between whispering-gallery effects along the duct and refraction by the steady flow. At the opposite extreme, strongly spinning modes are investigated, and are seen to be due to a different whispering-gallery effect across the duct cross-section.

The pair trajectories of neutrally buoyant rigid spheres immersed in finite-inertia simple-shear flow are described. The trajectories are obtained using the lattice-Boltzmann method to solve the fluid motion, with Newtonian dynamics describing the sphere motions. The inertia is characterized by the shear-flow Reynolds number , where μ and ρ are the viscosity and density of the fluid respectively, is the shear rate and a is the radius of the larger of the pair of spheres in the case of unequal sizes; the majority of results presented are for pairs of equal radii. Reynolds numbers of 0 ≤ Re ≤ 1 are considered with a focus on inertia at Re = O(0.1). At finite inertia, the topology of the pair trajectories is altered from that predicted at Re = 0, as closed trajectories found in Stokes flow vanish and two new forms of trajectories are observed. These include spiralling and reversing trajectories in addition to largely undisturbed open trajectories. For Re = O(0.1), the limits of the various regions in pair space yielding open, reversing and spiralling trajectories are roughly defined.

The dynamics of particle–particle collisions and the bouncing motion of a particle colliding with a wall in a viscous fluid is numerically investigated. The dependence of the effective coefficient of restitution on the Stokes number and surface roughness is analysed. A distributed Lagrange multiplier-based computational method in a solid–fluid system is developed and an efficient method for predicting the collision between particles is presented. A comparison between this method and previous collision strategies shows that the present approach has some significant advantages over them. Comparison of the present methodology with experimental studies for the bouncing motion of a spherical particle onto a wall shows very good agreement and validates the collision model. Finally, the effect of the coefficient of restitution for a dry collision on the vortex dynamics associated with this problem is discussed.

The velocity fluctuations of particles in a low-Reynolds-number fluidized bed have important similarities and differences with the velocity fluctuations in a low-Reynolds-number sedimenting suspension. We show that, like sedimentation, the velocity fluctuations in a fluidized bed are described well by the balance between density fluctuations due to Poisson statistics and Stokes drag. However, unlike sedimentation, the correlation length of the fluctuations in a fluidized bed increases with volume fraction. We argue that this difference arises because the relaxation time of density fluctuations is completely different in the two systems.