The fluctuating force model is developed and applied to the turbulent flow of a gas–particle suspension in a channel in the limit of high Stokes number, where the particle relaxation time is large compared to the fluid correlation time, and low particle Reynolds number where the Stokes drag law can be used to describe the interaction between the particles and fluid. In contrast to the Couette flow, the fluid velocity variances in the different directions in the channel are highly non-homogeneous, and they exhibit significant variation across the channel. First, we analyse the fluctuating particle velocity and acceleration distributions at different locations across the channel. The distributions are found to be non-Gaussian near the centre of the channel, and they exhibit significant skewness and flatness. However, acceleration distributions are closer to Gaussian at locations away from the channel centre, especially in regions where the variances of the fluid velocity fluctuations are at a maximum. The time correlations for the fluid velocity fluctuations and particle acceleration fluctuations are evaluated, and it is found that the time correlation of the particle acceleration fluctuations is close to the time correlations of the fluid velocity in a ‘moving Eulerian’ reference, moving with the mean fluid velocity. The variances of the fluctuating force distributions in the Langevin simulations are determined from the time correlations of the fluid velocity fluctuations and the results are compared with direct numerical simulations. Quantitative agreement between the two simulations are obtained provided the particle viscous relaxation time is at least five times larger than the fluid integral time.

The particle and fluid velocity fluctuations in a turbulent gas–particle suspension are studied experimentally using two-dimensional particle image velocimetry with the objective of comparing the experiments with the predictions of fluctuating force simulations. Since the fluctuating force simulations employ force distributions which do not incorporate the modification of fluid turbulence due to the particles, it is of importance to quantify the turbulence modification in the experiments. For experiments carried out at a low volume fraction of (mass loading is 0.19), where the viscous relaxation time is small compared with the time between collisions, it is found that the gas-phase turbulence is not significantly modified by the presence of particles. Owing to this, quantitative agreement is obtained between the results of experiments and fluctuating force simulations for the mean velocity and the root mean square of the fluctuating velocity, provided that the polydispersity in the particle size is incorporated in the simulations. This is because the polydispersity results in a variation in the terminal velocity of the particles which could induce collisions and generate fluctuations; this mechanism is absent if all of the particles are of equal size. It is found that there is some variation in the particle mean velocity very close to the wall depending on the wall-collision model used in the simulations, and agreement with experiments is obtained only when the tangential wall–particle coefficient of restitution is 0.7. The mean particle velocity is in quantitative agreement for locations more than 10 wall units from the wall of the channel. However, there are systematic differences between the simulations and theory for the particle concentrations, possibly due to inadequate control over the particle feeding at the entrance. The particle velocity distributions are compared both at the centre of the channel and near the wall, and the shape of the distribution function near the wall obtained in experiments is accurately predicted by the simulations. At the centre, there is some discrepancy between simulations and experiment for the distribution of the fluctuating velocity in the flow direction, where the simulations predict a bi-modal distribution whereas only a single maximum is observed in the experiments, although both distributions are skewed towards negative fluctuating velocities. At a much higher particle mass loading of 1.7, where the time between collisions is smaller than the viscous relaxation time, there is a significant increase in the turbulent velocity fluctuations by –2 orders of magnitude. Therefore, it becomes necessary to incorporate the modified fluid-phase intensity in the fluctuating force simulation; with this modification, the mean and mean-square fluctuating velocities are within 20–30 % of the experimental values.

Incompressible flow in toroidal pipes of circular cross-section was investigated by three-dimensional, time-dependent numerical simulations using a finite volume method. The computational domain included a whole torus and was discretized by up to nodes. Two curvatures (radius of the cross-section/radius of the torus), namely 0.3 and 0.1, were examined; a streamwise forcing term was imposed, and its magnitude was made to vary so that the bulk Reynolds number ranged between and . The results were processed by different techniques in order to confirm the spatio-temporal structure of the flow. Consecutive transitions between different flow regimes were found, from stationary to periodic, quasi-periodic and chaotic flow. At low Reynolds number, stationary flow was predicted, exhibiting a symmetric couple of Dean vortex rings and a strong shift of the streamwise velocity maximum towards the outer wall. For , between and a first transition occurred from stationary to periodic flow, associated with a supercritical Hopf bifurcation and giving rise to a travelling wave which took the form of a varicose streamwise modulation of the Dean vortex ring intensity. A further transition, associated with a secondary Hopf bifurcation, occurred between and and led to a quasi-periodic flow characterized by two independent fundamental frequencies associated with distinct travelling waves, the first affecting mainly the Dean vortex rings and similar to that observed in purely periodic flow, the second localized mainly in the secondary flow boundary layers and manifesting itself as an array of oblique vortices produced at the edge of the Dean vortex regions. Both the periodic and the quasi-periodic regimes were characterized by an instantaneous anti-symmetry of the oscillatory components with respect to the equatorial midplane of the torus. For , between and a direct (‘hard’) transition from steady to quasi-periodic flow occurred. Hysteresis was also observed: starting from a quasi-periodic solution and letting the Reynolds number decrease, both quasi-periodic and periodic stable solutions were obtained at Reynolds numbers below the critical value. A further decrease in led to steady-state solutions. This behaviour suggests the existence of a subcritical Hopf bifurcation followed by a secondary Hopf bifurcation. The resulting periodic and quasi-periodic flows were similar to those observed for the higher curvature, but the travelling modes were now instantaneously symmetric with respect to the equatorial midplane of the torus. Also, the further transition from quasi-periodic to chaotic flow occurred with different modalities for the two curvatures. For , a centrifugal instability of the main flow in the outer region occurred abruptly between and , while a further increase of up to 13 180 did not cause any relevant change in the distribution and intensity of the fluctuations. For the transition to chaotic flow was gradual in the range to 8160 and affected mainly the inner region; only a further increase of to 14 700 caused fluctuations to appear also in the outer region.

We consider laminar compositional convection of buoyant melt released by ablation of a vertical solid surface into a two-component fluid. Asymptotic solutions are used to describe separate cases: the ablation rate is either controlled by thermal transport, corresponding to melting, or by solutal transport, corresponding to dissolution. Melting is faster and generates a stronger flow than dissolving. We determine the temperature and solute concentration conditions leading to either melting or dissolving and find that these conditions do not vary with the strength of the buoyancy that drives convective flow.

This paper presents an experimental study devoted to investigating the effects of permeability on wall turbulence. Velocity measurements were performed by means of laser Doppler anemometry in open channel flows over walls characterized by a wide range of permeability. Previous studies proposed that the von Kármán coefficient associated with mean velocity profiles over permeable walls is significantly lower than the standard values reported for flows over smooth and rough walls. Furthermore, it was observed that turbulent flows over permeable walls do not fully respect the widely accepted paradigm of outer-layer similarity. Our data suggest that both anomalies can be explained as an effect of poor inner–outer scale separation if the depth of shear penetration within the permeable wall is considered as the representative length scale of the inner layer. We observed that with increasing permeability, the near-wall structure progressively evolves towards a more organized state until it reaches the condition of a perturbed mixing layer where the shear instability of the inflectional mean velocity profile dictates the scale of the dominant eddies. In our experiments such shear instability eddies were detected only over the wall with the highest permeability. In contrast attached eddies were present over all the other wall conditions. On the basis of these findings, we argue that the near-wall structure of turbulent flows over permeable walls is regulated by a competing mechanism between attached and shear instability eddies. We also argue that the ratio between the shear penetration depth and the boundary layer thickness quantifies the ratio between such eddy scales and, therefore, can be used as a diagnostic parameter to assess which eddy structure dominates the near-wall region for different wall permeability and flow conditions.

A librating cylinder consists of a rotating cylinder whose rate of rotation is modulated. When the mean rotation rate is large compared with the viscous damping rate, the flow may support inertial waves, depending on the frequency of the modulation. The modulation also produces time-dependent boundary layers on the cylinder endwalls and sidewall, and the sidewall boundary layer flow in particular is susceptible to instabilities which can introduce additional forcing on the interior flow with time scales different from the modulation period. These instabilities may also drive and/or modify the inertial waves. In this paper, we explore such flows numerically using a spectral-collocation code solving the Navier–Stokes equations in order to capture the dynamics involved in the interactions between the inertial waves and the viscous boundary layer flows.

The interaction of a dipolar vortex with topography is examined using a combination of analytical solutions and idealized numerical models. It is shown that an anticyclonic vortex may generate along-topography flow with sufficient speeds to excite hydraulic control with respect to local Kelvin waves. A critical condition for Kelvin wave hydraulic control is found for the simplest case of a 1.5-layer shallow water model. It is proposed that in the continuously stratified case this mechanism may allow an interaction between low mode vortices and higher mode Kelvin waves, thereby generating rapidly converging isopycnals and hydraulic jumps. Thus, Kelvin wave hydraulic control may contribute to the flux of energy from mesoscale to smaller, unbalanced, scales of motion in the ocean.

In this paper we consider a suspension of elastic solid particles in a viscous liquid. The particles are assumed to be neo-Hookean and can undergo finite elastic deformations. A polarization technique, originally developed for analogous problems in linear elasticity, is used to establish a theory for describing the finite-strain, time-dependent response of an ellipsoidal elastic particle in a viscous fluid flow under Stokes flow conditions. A set of coupled, nonlinear, first-order ODEs is obtained for the evolution of the uniform stress fields in the particle, as well as for the shape and orientation of the particle, which can in turn be used to characterize the rheology of a dilute suspension of elastic particles in a shear flow. When applied to a suspension of cylindrical particles with initially circular cross-section, the theory confirms the existence of steady-state solutions, which can be given simple analytical expressions. The two-dimensional, steady-state solutions for the particle shape and orientation, as well as for the effective viscosity and normal stress differences in the suspension, are in excellent agreement with direct numerical simulations of multiple-particle dispersions in a shear flow obtained by using an arbitrary Lagrangian–Eulerian (ALE) finite element method (FEM) solver. The corresponding solutions for the evolution of the microstructure and the rheological properties of suspensions of initially spherical (three-dimensional) particles in a simple shear flow are also obtained, and compared with the results of Roscoe (J. Fluid Mech., vol. 28, 1967, pp. 273–293) in the steady-state regime. Interestingly, the results show that sufficiently soft elastic particles can be used to reduce the effective viscosity of the suspension (relative to that of the pure fluid).

When a laminar diffusion flame is established over a spinning, thermoplastic, polymer fuel disc in a quiescent, oxidizing environment, the polymer melts and flows radially outwards, causing some fuel to be lost and not transported to the diffusion flame. The viscosity of the liquid in the melt layer retards the radial flow, thereby determining the amount of fuel lost. The melt layer is analysed here for two limiting cases, namely one in which the liquid viscosity depends strongly on temperature, leading to an asymptotic analysis involving two zones in the liquid, and one in which the liquid viscosity is constant, independent of temperature, so that there is only one zone in the liquid. The utility of these two limits is assessed by comparing their predictions with those of full numerical integrations for poly(methyl methacrylate) (PMMA) discs burning in air at atmospheric pressure.

A method is proposed for computing the low-Reynolds-number hydrodynamic forces on particles comprising a suspension confined by two parallel, no-slip walls. This is constructed via the two-dimensional analogue of Hasimoto’s solution (J. Fluid Mech., vol. 5, 1959, pp. 317–328) for a periodic array of point forces in a viscous, incompressible fluid, and, like Hasimoto, the summation of interactions is accelerated by substitution and superposition of ‘Ewald-like’ forcing. This method is akin to the accelerated Stokesian dynamics technique (J. Fluid Mech., vol. 448, 2001, pp. 115–146) and models the suspension dynamics with log–linear computational scaling. The effectiveness of this approach is demonstrated with a calculation of the high-frequency dynamic viscosity of a colloidal dispersion as function of volume fraction and channel width. Similarly, the short-time self-diffusivity for and the sedimentation rate of spherical particles in a confined suspension are determined. The results demonstrate the influence of confining geometry on the transport of small particles, which is becoming increasingly important for micro- and biofluidics.

A new experimental investigation of decaying turbulence generated by a low-blockage space-filling fractal square grid is presented. We find agreement with previous works by Seoud & Vassilicos (Phys. Fluids, vol. 19, 2007, 105108) and Mazellier & Vassilicos (Phys. Fluids, vol. 22, 2010, 075101) but also extend the length of the assessed decay region and consolidate the results by repeating the experiments with different probes of increased spatial resolution. It is confirmed that this moderately high Reynolds number turbulence (up to here) does not follow the classical high Reynolds number scaling of the dissipation rate and does not obey the equivalent proportionality between the Taylor-based Reynolds number and the ratio of integral scale to the Taylor microscale . Instead we observe an approximate proportionality between and during decay. This non-classical behaviour is investigated by studying how the energy spectra evolve during decay and examining how well they can be described by self-preserving single-length-scale forms. A detailed study of homogeneity and isotropy is also presented which reveals the presence of transverse energy transport and pressure transport in the part of the turbulence decay region where we take data (even though previous studies found mean flow and turbulence intensity profiles to be approximately homogeneous in much of the decay region). The exceptionally fast turbulence decay observed in the part of the decay region where we take data is consistent with the non-classical behaviour of the dissipation rate. Measurements with a regular square mesh grid as well as comparisons with active-grid experiments by Mydlarski & Warhaft (J. Fluid Mech., vol. 320, 1996, pp. 331–368) and Kang, Chester & Meveneau (J. Fluid Mech., vol. 480, 2003, pp. 129–160) are also presented to highlight the similarities and differences between these turbulent flows and the turbulence generated by our fractal square grid.

In the dense-inertial regime of granular flow, the stresses scale inertially, but the flow is dominated by clusters of particles. This paper describes observations of cluster development in this regime. Clusters were seen to form for both elastic and inelastic reasons: elastic when the shear rate pushes the particles together faster than the contacts can elastically disperse them, and inelastic as large energy dissipation leads to cluster formation. Furthermore, large particle surface friction leads to cluster formation both for structural reasons, because it generates stronger clusters, and for energetic reasons, as friction dissipates energy. However, the most intriguing result of this work is that clusters appear to have little effect on the rheology of the dense inertial regime, which suggests that one can model the dense inertial regime with entirely collisional hard sphere models, and not have to worry about the complexities of modelling clusters. But at the same time it presents a physical puzzle, as one would normally expect the rheology to be strongly dependent on microstructural features such as clusters, particularly as they present an elastic pathway for internal momentum transport. There is no completely satisfying explanation for why the clusters can be ignored, but two possibilities suggest themselves. Because the clusters are short-lived, it is possible that they do not survive long enough to make a significant contribution to the momentum transport. And it is also possible for the granular temperature that governs transport between clusters to act as a rate-limiting bottleneck that is in overall control of the momentum transport rate.

In this paper, on the basis of the classical Levi-Civita formulae for hydrodynamic forces exerted on any profile in an infinite cavity flow, we deduce new representations for the lift and drag. In these representations the forces are expressed only in terms of the velocity distribution along the profile surface. So, the representations are analogous to the well-known Kutta–Joukowskii theorem. By means of the new representations we find optimum velocity distributions which provide the maximum lift or maximum drag of cavitating profiles and determine corresponding optimum shapes.

The flow in an axisymmetric contraction fitted to a fully developed pipe flow is experimentally and numerically studied. The reduction in turbulence intensity in the core region of the flow is discussed on the basis of the budgets for the various turbulent stresses as they develop downstream. The contraction generates a corresponding increase in energy in the near-wall region, where the sources for energy production are quite different and of opposite sign compared to the core region, where these effects are caused primarily by vortex stretching. The vortices in the pipe become aligned with the flow as the stretching develops through the contraction. Vortices which originally have a spanwise component in the pipe are stretched into pairs of counter-rotating vortices which become disconnected and aligned with the mean flow. The structures originating in the pipe which are inclined at an angle with respect to the wall are rotated towards the local mean streamlines. In the very near-wall region and the central part of the contraction the flow tends towards two-component turbulence, but these structures are different. The streamwise and azimuthal stresses are dominant in the near-wall region, while the lateral components dominate in the central part of the flow. The two regions are separated by a rather thin region where the flow is almost isotropic.

We explore the relevance of the idealized Jeffery–Hamel similarity solution to the practical problem of flow in a diverging channel of finite (but large) streamwise extent. Numerical results are presented for the two-dimensional flow in a wedge of separation angle , bounded by circular arcs at the inlet/outlet and for a net radial outflow of fluid. In particular, we show that in a finite domain there is a sequence of nested neutral curves in the plane, each corresponding to a midplane symmetry-breaking (pitchfork) bifurcation, where is a Reynolds number based on the radial mass flux. For small wedge angles we demonstrate that the first pitchfork bifurcation in the finite domain occurs at a critical Reynolds number that is in agreement with the only pitchfork bifurcation in the infinite-domain similarity solution, but that the criticality of the bifurcation differs (in general). We explain this apparent contradiction by demonstrating that, for , superposition of two (infinite-domain) eigenmodes can be used to construct a leading-order finite-domain eigenmode. These constructed modes accurately predict the multiple symmetry-breaking bifurcations of the finite-domain flow without recourse to computation of the full field equations. Our computational results also indicate that temporally stable, isolated, steady solutions may exist. These states are finite-domain analogues of the steady waves recently presented by Kerswell, Tutty, & Drazin (J. Fluid Mech., vol. 501, 2004, pp. 231–250) for an infinite domain. Moreover, we demonstrate that there is non-uniqueness of stable solutions in certain parameter regimes. Our numerical results tie together, in a consistent framework, the disparate results in the existing literature.

Three-dimensional Navier–Stokes simulations of viscously unstable, miscible Hele-Shaw displacements are discussed. Quasisteady fingers are observed whose tip velocity increases with the Péclet number and the unfavourable viscosity ratio. These fingers are widest near the tip, and become progressively narrower towards the root. The film of resident fluid left behind on the wall decreases in thickness towards the finger tip. The simulations reveal the detailed mechanism by which the initial spanwise vorticity of the base flow, when perturbed, gives rise to the cross-gap vorticity that drives the fingering instability in the classical Darcy sense. Cross-sections at constant streamwise locations reveal the existence of a streamwise vorticity quadrupole along the length of the finger. This streamwise vorticity convects resident fluid from the wall towards the centre of the gap in the cross-gap symmetry plane of the finger, while it transports injected fluid laterally away from the finger centre within the mid-gap plane. In this way, it results in the emergence of a longitudinal, inner splitting phenomenon some distance behind the tip that has not been reported previously. This inner splitting mechanism, which leaves the tip largely intact, is fundamentally different from the familiar tip-splitting mechanism. Since the inner splitting owes its existence to the presence of streamwise vorticity and cross-gap velocity, it cannot be captured by gap-averaged equations. It is furthermore observed that the role of the Péclet number in miscible displacements differs in some ways from that of the capillary number in immiscible flows. Specifically, larger Péclet numbers result in wider fingers, while immiscible flows display narrower fingers for larger capillary numbers. Furthermore, while higher capillary numbers are known to promote tip-splitting, inner splitting is delayed for larger Péclet numbers.

We study the effects of emergent coastal forests on the propagation of long surface waves of small amplitude. The forest is idealized by an array of vertical cylinders. Simple models are employed to represent bed friction and to simulate turbulence generated by flow through the tree trunks. A multi-scale (homogenization) analysis similar to that for seepage flows is carried out to deduce the effective equations on the macro-scale. The effective coefficients are calculated by numerically solving the micro-scale problem in a unit cell surrounding one or several cylinders. Analytical and numerical solutions for wave attenuation on the macro-scale for different bathymetries and coastal forest configurations are presented. For a transient incident wave, analytical results are discussed for the damping of a leading tsunami. For comparison series of laboratory data for periodic and transient incident waves are also presented. Good agreement is found even though some of the measured waves are short or nonlinear.

A possible route to finite-time singularity in the quasigeostrophic system, via a cascade of filament instabilities of geometrically decreasing spatial and temporal scales, is investigated numerically using a high-resolution hybrid contour dynamical algorithm. A number of initial temperature distributions are considered, of varying degrees of continuity. In all cases, primary, secondary, and tertiary instabilities are apparent before the algorithm loses accuracy due to limitations of finite resolution. Filament instability is also shown to be potentially important in the closing saddle scenario investigated in many previous studies. The results do not provide a rigorous demonstration of finite-time singularity, but suggest avenues for further investigation.

Non-modal analysis determines the potential for energy amplification in stable flows. The latter is quantified in the frequency domain by the singular values of the resolvent operator. The present work extends previous analysis on the effect of base-flow modifications on flow stability by considering the sensitivity of the flow non-modal behaviour. Using a variational technique, we derive an analytical expression for the gradient of a singular value with respect to base-flow modifications and show how it depends on the singular vectors of the resolvent operator, also denoted the optimal forcing and optimal response of the flow. As an application, we examine zero-pressure-gradient boundary layers where the different instability mechanisms of wall-bounded shear flows are all at work. The effect of the component-type non-normality of the linearized Navier–Stokes operator, which concentrates the optimal forcing and response on different components, is first studied in the case of a parallel boundary layer. The effect of the convective-type non-normality of the linearized Navier–Stokes operator, which separates the spatial support of the structures of the optimal forcing and response, is studied in the case of a spatially evolving boundary layer. The results clearly indicate that base-flow modifications have a strong impact on the Tollmien–Schlichting (TS) instability mechanism whereas the amplification of streamwise streaks is a very robust process. This is explained by simply examining the expression for the gradient of the resolvent norm. It is shown that the sensitive region of the lift-up (LU) instability spreads out all over the flat plate and even upstream of it, whereas it is reduced to the region between branch I and branch II for the TS waves.

Double-diffusive density stratified systems are well studied and have been shown to display a rich variety of instability behaviour. However double-diffusive systems where the inhomogeneities in solute concentration are manifested in terms of stratified viscosity rather than density have been studied far less and, to the best of the authors’ knowledge, not in high-Reynolds-number shear flows. In a simple geometry, namely the two-fluid channel flow of such a system, we find a new double-diffusive mode of instability. The instability becomes stronger as the ratio of diffusivities of the two scalars increases, even in a situation where the net Schmidt number decreases. The double-diffusive mode is destabilized when the layer of viscosity stratification overlaps with the critical layer of the perturbation.