Stokesian Dynamics has been used to investigate the origins of shear thickening in concentrated colloidal suspensions. For this study, we considered a monolayer suspension composed of charge-stabilized non-Brownian monosized rigid spheres dispersed at an areal fraction of ϕa=0.74 in a Newtonian liquid. The suspension was subjected to a linear shear field. In agreement with established experimental data, our results indicate that shear thickening in this system is associated with an order–disorder transition of the suspension microstructure. Below the critical shear rate at which this transition occurs, the suspension microstructure consists of two-dimensional analogues of experimentally observed sliding layer configurations. Above this critical shear rate, suspensions are disordered, contain particle clusters, and exhibit viscosities and microstructures characteristic of suspensions of non-Brownian hard spheres. In addition, suspensions possessing the sliding layer microstructure at the beginning of supercritical shearing tend to retain this microstructure for a period of time before disordering. The onset of this disorder is due to the formation of particle doublets within the suspension. Once formed, these doublets rotate, due to the bulk motion, and disrupt the long-range order of the suspension. The cross-stream component of the centre-to-centre separation vector associated with the two particles forming a doublet, which is zero when the doublet is perfectly aligned with the bulk velocity vector, grows exponentially with time. This strongly suggests that the evolution of these doublets is due to a change in the stability of the sliding layer configurations, with this type of ordered microstructure being linearly unstable above a critical shear rate. This contention is supported by results of a stability analysis. The analysis shows that a single string of particles is subject to a linear instability leading to the formation of particle doublets. Simulations were repeated with different numbers of particles in the computational domain, with the results found to be qualitatively independent of system size.

Measurements are reported of the pressure differences ΔP existing at large distances above and below a ball settling along the axis of a circular cylinder filled with an otherwise quiescent viscous Newtonian liquid in which identical particles, comparable in size to the settling ball, are suspended. The suspensions ranged in solids volume fraction ϕ from 0.30 to 0.50 and consisted of 0.635 cm diameter spheres density-matched to the suspending oil. The settling balls varied in diameter from 0.318 to 1.27 cm, resulting in particle Reynolds numbers always less than about 0.4 based upon ball diameter and the effective viscosity of the suspension. For the moderately concentrated suspension (ϕ=0.30), the product of ΔP with the cross-sectional area A of the containing cylinder was observed to be equal to twice the drag force D on the settling sphere, in accord with theory. In the more concentrated suspension (ϕ=0.50) this product was found to be slightly, but significantly, less than twice the drag on the settling sphere. It is speculated that this lower pressure drop may result from the presence of one or more of the following phenomena: (i) migration of the falling ball off the cylinder axis ; (ii) apparent slip of the suspension at the cylinder wall; (iii) blunting of the otherwise Poiseuillian parabolic velocity profile, the latter phenomenon being known to occur during the creeping flow of concentrated suspensions through circular tubes. Incidental to the suspension experiments, for a homogeneous fluid we verify the classical theoretical formula for the off-axis pressure drop when the sphere settles at a non-concentric position in the cylinder.

This is the final part of a three-part study of the stability of vertically oriented double-diffusive interfaces having an imposed vertical stable temperature gradient. In this study, flow is forced within a fluid of infinite extent by a prescribed excess of compositionally buoyant material within a circular cylindrical interface. Compositional diffusivity is ignored while thermal diffusivity and viscosity are finite. The instability of the interface is determined by quantifying the exponential growth rate of a harmonic deflection of infinitesimal amplitude. Attention is focused on the zonal wavenumber of the fastest growing mode.

The interface is found to be unstable for some wavenumber for all values of the Prandtl number and interface radius. The zonal wavenumber of the fastest growing mode increases roughly linearly with interface radius, except for small values of the Prandtl number (<0.065). For small and moderate values of the radius, the preferred mode is either axisymmetric or has zonal wavenumber of 1, representing a helical instability. The growth rate of the fastest-growing mode is largest for interfaces having radii of from 2 to 3 salt-finger lengths.

The equation relating second- and third-order velocity structure functions was presented by Kolmogorov; Monin attempted to derive that equation on the basis of local isotropy. Recently, concerns have been raised to the effect that Kolmogorov's equation and an ancillary incompressibility condition governing the third-order structure function were proven only on the restrictive basis of isotropy and that the statistic involving pressure that appears in the derivation of Kolmogorov's equation might not vanish on the basis of local isotropy. These concerns are resolved. In so doing, results are obtained for the second- and third-order statistics on the basis of local homogeneity without use of local isotropy. These results are applicable to future studies of the approach toward local isotropy. Accuracy of Kolmogorov's equation is shown to be more sensitive to anisotropy of the third-order structure function than to anisotropy of the second-order structure function. Kolmogorov's 4/5 law for the inertial range of the third-order structure function is obtained without use of the incompressibility conditions on the second- and third-order structure functions. A generalization of Kolmogorov's 4/5 law, which applies to the inertial range of locally homogeneous turbulence at very large Reynolds numbers, is shown to also apply to the energy-containing range for the more restrictive case of stationary, homogeneous turbulence. The variety of derivations of Kolmogorov's and Monin's equations leads to a wide range of applicability to experimental conditions, including, in some cases, turbulence of moderate Reynolds number.

In most practical situations, turbulent premixed flames are ducted and, accordingly, subjected to externally imposed pressure gradients. These pressure gradients may induce strong modifications of the turbulent flame structure because of buoyancy effects between heavy cold fresh and light hot burnt gases. In the present work, the influence of a constant acceleration, inducing large pressure gradients, on a premixed turbulent flame is studied using direct numerical simulations.

A favourable pressure gradient, i.e. a pressure decrease from unburnt to burnt gases, is found to decrease the flame wrinkling, the flame brush thickness, and the turbulent flame speed. It also promotes counter-gradient turbulent transport. On the other hand, adverse pressure gradients tend to increase the flame brush thickness and turbulent flame speed, and promote classical gradient turbulent transport. As proposed by Libby (1989), the turbulent flame speed is modified by a buoyancy term linearly dependent on both the imposed pressure gradient and the integral length scale lt.

A simple model for the turbulent flux u″c″ is also proposed, validated from simulation data and compared to existing models. It is shown that turbulent premixed flames can exhibit both gradient and counter-gradient transport and a criterion integrating the effects of pressure gradients is derived to differentiate between these regimes. In fact, counter-gradient diffusion may occur in most practical ducted flames.

The mechanism of wrap, tilt and stretch of vorticity lines around a strong thin straight vortex tube of circulation Γ starting with a vortex filament in a simple shear flow (U=SX2Xˆ1, S being a shear rate) is investigated analytically. An asymptotic expression for the vorticity field is obtained at a large Reynolds number Γ/ν[Gt ]1, ν being the kinematic viscosity of fluid, and during the initial time St[Lt ]1 of evolution as well as St[Lt ](Γ/ν)1/2. The vortex tube, which is inclined from the streamwise (X1) direction both in the vertical (X2) and spanwise (X3) directions, is tilted, stretched and diffused under the action of the uniform shear and viscosity. The simple shear vorticity is on the other hand, wrapped and stretched around the vortex tube by a swirling motion, induced by it to form double spiral vortex layers of high azimuthal vorticity of alternating sign. The magnitude of the azimuthal vorticity increases up to O((Γ/ν)1/3S) at distance r=O((Γ/ν)1/3 (νt)1/2) from the vortex tube. The spirals induce axial flows of the same spiral shape with alternate sign in adjacent spirals which in turn tilt the simple shear vorticity toward the axial direction. As a result, the vorticity lines wind helically around the vortex tube accompanied by conversion of vorticity of the simple shear to the axial direction. The axial vorticity increases in time as S2t, the direction of which is opposite to that of the vortex tube at r=O((Γ/ν)1/2 (νt)1/2) where the vorticity magnitude is strongest. In the near region r[Lt ](Γ/ν)1/3 (νt)1/2, on the other hand, a viscous cancellation takes place in tightly wrapped vorticity of alternate sign, which leads to the disappearance of the vorticity normal to the vortex tube. Only the axial component of the simple shear vorticity is left there, which is stretched by the simple shear flow itself. As a consequence, the vortex tube inclined toward the direction of the simple shear vorticity (a cyclonic vortex) is intensified, while the one oriented in the opposite direction (an anticyclonic vortex) is weakened. The growth rate of vorticity due to this effect attains a maximum (or minimum) value of ±S2/33/2 when the vortex tube is oriented in the direction of Xˆ1+Xˆ2∓ Xˆ3. The present asymptotic solutions are expected to be closely related to the flow structures around intense vortex tubes observed in various kinds of turbulence such as helical winding of vorticity lines around a vortex tube, the dominance of cyclonic vortex tubes, the appearance of opposite-signed vorticity around streamwise vortices and a zig-zag arrangement of streamwise vortices in homogeneous isotropic turbulence, homogeneous shear turbulence and near-wall turbulence.

A high-Reynolds-number asymptotic theory is developed for linear instability waves in a two-dimensional incompressible boundary layer on a flat surface coated with a thin film of a different fluid. The focus in this study is on the influence of the film flow on the lower-branch Tollmien–Schlichting waves, and also on the effect of boundary-layer/potential flow interaction on interfacial instabilities. Accordingly, the film thickness is assumed to be comparable to the thickness of a viscous sublayer in a three-tier asymptotic structure of lower-branch Tollmien–Schlichting disturbances. A fully nonlinear viscous/inviscid interaction formulation is derived, and computational and analytical solutions for small disturbances are obtained for both Tollmien–Schlichting and interfacial instabilities for a range of density and viscosity ratios of the fluids, and for various values of the surface tension coefficient and the Froude number. It is shown that the interfacial instability contains the fastest growing modes and an upper-branch neutral point within the chosen flow regime if the film viscosity is greater than the viscosity of the ambient fluid. For a less viscous film the theory predicts a lower neutral branch of shorter-scale interfacial waves. The film flow is found to have a strong effect on the Tollmien–Schlichting instability, the most dramatic outcome being a powerful destabilization of the flow due to a linear resonance between growing Tollmien–Schlichting and decaying capillary modes. Increased film viscosity also destabilizes Tollmien–Schlichting disturbances, with the maximum growth rate shifted towards shorter waves. Qualitative and quantitative comparisons are made with experimental observations by Ludwieg & Hornung (1989).

When a small amount of marked solute is released into a stratified fluid, there is lateral dispersion of the marked solute and a larger lateral dispersion of any density anomaly. The method of moments is used to calculate the two dispersion coefficients. The excess dispersion for the density is shown to be proportional to the fractional density decrease from the bed to the free surface and to the cube of the water depth, and inversely proportional to the vertical mixing for lateral momentum. For weak turbulent mixing the stratification-induced lateral dispersion for the density anomaly can be several orders of magnitude greater than the lateral turbulent mixing for marked solute

In Hammerton & Kerschen (1996), the effect of the nose radius of a body on boundary-layer receptivity was analysed for the case of a symmetric mean flow past a two-dimensional body with a parabolic leading edge. A low-Mach-number two-dimensional flow was considered. The radius of curvature of the leading edge, rn, enters the theory through a Strouhal number, S=ωrn/U, where ω is the frequency of the unsteady free-stream disturbance and U is the mean flow speed. Numerical results revealed that the variation of receptivity for small S was very different for free-stream acoustic waves propagating parallel to the mean flow and those free-stream waves propagating at an angle to the mean flow. In this paper the small-S asymptotic theory is presented. For free-stream acoustic waves propagating parallel to the symmetric mean flow, the receptivity is found to vary linearly with S, giving a small increase in the amplitude of the receptivity coefficient for small S compared to the flat-plate value. In contrast, for oblique free-stream acoustic waves, the receptivity varies with S1/2, leading to a sharp decrease in the amplitude of the receptivity coefficient relative to the flat-plate value. Comparison of the asymptotic theory with numerical results obtained in the earlier paper confirms the asymptotic results but reveals that the numerical results diverge from the asymptotic result for unexpectedly small values of S.

A theoretical model is developed for the sound generated when a convected disturbance encounters a cambered airfoil at non-zero angle of attack. The model is a generalization of a previous theory for a flat-plate airfoil, and is based on a linearization of the Euler equations about the steady, subsonic flow past the airfoil. High-frequency gusts, whose wavelengths are short compared to the airfoil chord, are considered. The airfoil camber and incidence angle are restricted so that the mean flow past the airfoil is a small perturbation to a uniform flow. The singular perturbation analysis retains the asymptotic regions present in the case of a flat-plate airfoil: local regions, which scale on the gust wavelength, at the airfoil leading and trailing edges; a ‘transition’ region behind the airfoil which is similar to the transition zone between illuminated and shadow regions in optical problems; and an outer region, far away from the airfoil edges and wake, in which the solution has a geometric-acoustics form. For the cambered airfoil, an additional asymptotic region in the form of an acoustic boundary layer adjacent to the airfoil surface is required in order to account for surface curvature effects. Parametric calculations are presented which illustrate that, like incidence angle, moderate amounts of airfoil camber can significantly affect the sound field produced by airfoil–gust interactions. Most importantly, the amount of radiated sound power is found to correlate very well with a single aerodynamic loading parameter, αeff, which is an effective mean-flow incidence angle for the airfoil leading edge.

Experiments were conducted to measure the collisional particle pressure in both cocurrent and countercurrent flows of liquid–solid mixtures. The collisional particle pressure, or granular pressure, is the additional pressure exerted on the containing walls of a particulate system due to the particle collisions. The present experiments involve both a liquid-fluidized bed using glass, plastic or steel spheres and a vertical gravity-driven flow using glass spheres. The particle pressure was measured using a high-frequency-response flush-mounted pressure transducer. Detailed recordings were made of many different particle collisions with the active face of this transducer. The solids fraction of the flowing mixtures was measured using an impedance volume fraction meter. Results show that the magnitude of the measured particle pressure increases from low concentrations (<10% solid volume fraction), reaches a maximum for intermediate values of solid fraction (30–40%), and decreases again for more concentrated mixtures (>40%). The measured collisional particle pressure appears to scale with the particle dynamic pressure based on the particle density and terminal velocity. Results were obtained and compared for a range of particle sizes, as well as for two different test section diameters.

In addition, a detailed analysis of the collisions was performed that included the probability density functions for the collision duration and collision impulse. Two distinct contributions to the collisional particle pressure were identified: one contribution from direct contact of particles with the pressure transducer, and the second one resulting from particle collisions in the bulk that are transmitted through the liquid to the pressure transducer.

This paper is concerned with the problem of viscous flow in an elastic tube. Elastic tubes collapse (buckle non-axisymmetrically) when the transmural pressure (internal minus external pressure) falls below a critical value. The tube's large deformation during the buckling leads to a strong interaction between the fluid and solid mechanics.

In this study, the steady three-dimensional Stokes equations are used to analyse the slow viscous flow in such a tube whose deformation is described by geometrically nonlinear shell theory. Finite element methods are used to solve the large-displacement fluid–structure interaction problem. Typical wall deformations and flow fields in the strongly collapsed tube are shown. Extensive parameter studies illustrate the tube's flow characteristics (e.g. volume flux as a function of the applied pressure drop through the tube) for boundary conditions corresponding to the four fundamental experimental setups. It is shown that lubrication theory provides an excellent approximation of the fluid traction while being computationally much less expensive than the solution of the full Stokes equations. Finally, the computational predictions for the flow characteristics and the wall deformation are compared to the results obtained from an experiment.

The so-called Crocco integral establishes a relation between the velocity and temperature distributions in steady boundary layer flow. It corresponds to an exact solution of the flow equations in the case of unity Prandtl number and an adiabatic wall, where it reduces to the condition that the total enthalpy remains constant throughout the boundary layer, irrespective of pressure gradient and compressibility. The effect of Prandtl number is usually incorporated by assuming a constant recovery factor across the entire boundary layer. Strictly, however, this modification is in conflict with the conservation-of-energy principle. In search of a more complete expression for the Crocco integral the present study applies an asymptotic solution approach to the energy equation in constant-property flow. The analysis of self-similar boundary layer solutions results in a formulation of the Crocco integral which correctly incorporates the effect of Prandtl number to first order, and that is complete in the sense that it satisfies the energy conservation requirement. Furthermore, the result is found to be applicable not only to self-similar boundary layers, but also to provide a solution to the laminar flow equations in general as well. The effect of varying properties is considered with regard to the extension of the expression to more general flow conditions. In addition to the asymptotic expression for the Crocco integral, asymptotic solutions are also obtained for the recovery factor for various classes of flows.

We examine the inviscid flow generated around a body moving impulsively from rest with a constant velocity U in a constant density gradient, ∇ρ0, which is assumed to be weak in the sense ε=a[mid ]∇ρ0[mid ] /ρ0[Lt ]1, where a is the length scale of the body. In the absence of a density gradient (ε=0), the flow is irrotational and no force acts on the body. When 0<ε[Lt ]1, vorticity is generated by a baroclinic torque and vortex stretching, which introduce a rotational component into the flow. The aim is to calculate both the flow around the body and the force acting on it.

When a two-dimensional body moves perpendicularly to the density gradient U·∇ρ0=0, the density and velocity field are both steady in the body's frame of reference and the vorticity field decays with distance from the body. When a three-dimensional body moves perpendicularly to the density gradient, the vorticity field is regular in the main flow region, [Dscr ]M, but is singular in a thin inner region [Dscr ]I located adjacent to the body and to the downstream-attached streamline, and the flow is characterized by trailing horseshoe vortices. When the body moves parallel to the density gradient U×∇ρ0=0, the density field is unsteady in the body's frame of reference; however to leading order the flow is steady in the region [Dscr ]M moving with the body for Ut/a[Gt ]1. In the thin region [Dscr ]I of thickness O(aε), the density gradient and vorticity are singular. When U×∇ρ0=0 this singularity leads to a downstream ‘jet’ with velocities of O(−(U·∇ρ0) Ua/(ρ0U)) on the downstream attached streamline(s). In the far field the flow is characterized by a sink of strength CM[Vscr ] (U·∇ρ0) /2ρ0, located at the origin, where CM is the added-mass coefficient of the body and [Vscr ] is the body's volume.

The forces acting on a body moving steadily in a weak density gradient are calculated by considering the steady relative velocity field in region [Dscr ]M and evaluating the momentum flux far from the body. When U·∇ρ0=0, a lift force, CL[Vscr ] (U·∇ρ0)×U, pushes the body towards the denser fluid, where the lift coefficient is CL=CM/2 for a three-dimensional body, that is axisymmetric about U, and is CL=(CM+1)/2 for a two-dimensional body. The direction of the lift force is unchanged when U is reversed. A general expression for the forces on bodies moving in a weak shear and perpendicularly to a density gradient is calculated. When U×∇ρ0=0, a drag force −CD[Vscr ] (U·∇ρ0)U retards the body as it moves into denser fluid, where the drag coefficient is CD=CM/2, for both two- and three-dimensional axisymmetric bodies. The direction of the drag force changes sign when U is reversed. There are two contributions to the drag calculation from the far field; the first is from the wake ‘jet’ on the attached streamline(s) caused by the rotational component of the flow and this leads to an accelerating force. The second and larger contribution arises from a downstream density variation, caused by the distortion of the isopycnal surfaces by the primary irrotational flow, and this leads to a drag force.

When cylinders or spheres move with a velocity U at arbitrary orientation to the density gradient, it is shown that they are acted on by a linear combination of lift and drag forces. Calculations of their trajectories show that they initially slow down or accelerate on a length scale of order ρ0/[mid ]∇ρ0[mid ] (independent of [Vscr ] and U) as they move into regions of increasing or decreasing density, but in general they turn and ultimately move parallel to the density gradient in the direction of increasing density gradient.

An analysis of the three-dimensional instability of two-dimensional viscoelastic elliptical flows is presented, extending the inviscid analysis of Bayly (1986) to include both viscous and elastic effects. The problem is governed by three parameters: E, a geometric parameter related to the ellipticity; Re, a wavenumber-based Reynolds number; and De, the Deborah number based on the period of the base flow. New modes and mechanisms of instability are discovered. The flow is generally susceptible to instabilities in the form of propagating plane waves with a rotating wavevector, the tip of which traces an ellipse of the same eccentricity as the flow, but with the major and minor axes interchanged. Whereas a necessary condition for purely inertial instability is that the wavevector has a non-vanishing component along the vortex axis, the viscoelastic modes of instability are most prominent when their wavevectors do vanish along this axis. Our analytical and numerical results delineate the region of parameter space of (E, ReDe) for which the new instability exists. A simple model oscillator equation of the Mathieu type is developed and shown to embody the essential qualitative and quantitative features of the secular viscoelastic instability. The cause of the instability is a buckling of the ‘compressed’ polymers as they are perturbed transversely during a particular phase of the passage of the rotating plane wave.