ALBERT

Description: Numerical and analytical solutions to the steady compressible Euler equations corresponding to a compressible analogue of the linear Stuart vortex array are presented. These correspond to a homentropic continuation, to finite Mach number, of the Stuart solution describing a linear vortex array in an incompressible fluid. The appropriate partial differential equations describing the flow correspond to the compressible homentropic Euler equations in two dimensions, with a prescribed vorticity-density-streamfunction relationship. In order to construct a well-posed problem for this continuation, it was found, unexpectedly, to be necessary to introduce an eigenvalue into the vorticity-density-streamfunction equation. In the Rayleigh-Janzen expansion of solutions in even powers of the free-stream Mach number M(∞), this eigenvalue is determined by a solvability condition. Accurate numerical solution by both finite-difference and spectral methods are presented for the compressible Stuart vortex, over a range of M(∞), and of a parameter corresponding to a confined mass-flow rate. These also confirm the nonlinear eigenvalue character of the governing equations. All solution branches followed numerically were found to terminate when the maximum local Mach number just exceeded unity. For one such branch we present evidence for the existence of a very small range of M(∞) over which smooth transonic shock-free flow can occur.

Description: The lift force on a circular particle in plane Poiseuille flow perpendicular to gravity is studied by direct numerical simulation. The angular slip velocity Ωs = Ωp + 1/2 ̇c where - 1/2 ̇c is the angular velocity of the fluid at a point where the shear rate is ̇c and Ωp is the angular velocity of the particle, is always positive at an equilibrium position at which the hydrodynamic lift balances the buoyant weight. The particle migrates to its equilibrium position and adjusts Ωp so that Ωs 〉 0 is nearly zero because Ωp ≈ -1/2 ̇c No matter where the particle is placed, it drifts to an equilibrium position with a unique, slightly positive equilibrium angular slip velocity. The angular slip velocity discrepancy defined as the difference between the angular slip velocity of a migrating particle and the angular slip velocity at its equilibrium position is positive below the position of equilibrium and negative above it. This discrepancy is the quantity that changes sign above and below the lower equilibrium position for neutrally buoyant particles, and also above and below the lower equilibrium position for heavy particles. The existence and properties of unstable positions of equilibrium due to newly identified turning-point transitions and those near the centreline are discussed. The long particle model of Choi and Joseph (2001) that gives rise to an explicit formula for the particle velocity and the velocity profile across the channel through the centreline of the particle is modified to include the effect of the rotation of the particle. In view of the simplicity of the model, the explicit formula for Up and the velocity profile are in surprisingly good agreement with simulation values. The value of the Poiseuille flow velocity at the point at the particle's centre when the particle is absent is always larger than the particle velocity; the slip velocity is positive at steady flow.

Description: Both lighter- and hydrophobic heavier-than-liquid particles will float on liquid-air surfaces. Capillary forces cause the particles to cluster in typical situations identified here. This kind of clustering causes particles to segregate into islands and bands of high concentrations in thin liquid films rimming the inside of a slowly rotating cylinder partially filled with liquid. A second regime of particle segregation, driven by secondary motions induced by off-centre gas bubbles in a more rapidly rotating cylinder at higher filling levels, is identified. A third regime of segregation of bidisperse suspensions is found in which two layers of heavier-than-liquid particles that stratify when there is no rotation, segregate into alternate bands of particles when there is rotation.

Description: Dissipation approximations have been used to calculate the drag on bubbles and drops and the decay rate of free gravity waves on water. In these approximations, viscous effects are calculated by evaluating the viscous stresses on irrotational flows. The pressure is not involved in the dissipation integral, but it enters into the power of traction integral, which equals the dissipation. A viscous correction of the irrotational pressure is needed to resolve the discrepancy between the zero-shear-stress boundary condition at a free surface and the non-zero irrotational shear stress. Here we show that the power of the pressure correction is equal to the power of the irrotational shear stress. The viscous pressure correction on the interface can be expressed by a harmonic series. The principal mode of this series is matched to the velocity potential and its coefficient is explicitly determined. The other modes do not enter into the expression for the drag on bubbles and drops. They vanish in the case of free gravity waves. © 2004 Cambridge University Press.

Description: We report results of a series of detailed experiments designed to unveil the dynamics of a particle of radius a moving in high-frequency, low-Reynolds-number oscillatory flow. The fundamental parameters in the problem are the Strouhal (SI) and the particle Reynolds numbers (Rep), as well as the fluid-to-particle density ratio α. The experiments were designed to cover a range of S/Rep from 0.015 to 5 while keeping Rep 〈 0.5 and S1 〉 1. The primary objective of the experiments is to investigate stationary history effects associated with the Basset drag, which are maximized when the viscous time scale a2/v is of the same order of the flow time scale 9/Ω where 9 is a geometrical factor for the sphere, v is the kinematic viscosity and Ω is the angular frequency of the background flow. The theoretically determined behaviour of stationary history effects is confirmed unequivocally by the experiments, which also validate the fractional derivative behaviour (of order 1/2) of the history drag for the range of parameters under study. © 2004 Cambridge University Press.

Description: In this paper we study the lift-off to equilibrium of a single circular particle in Newtonian and viscoelastic fluids by direct numerical simulation. A particle heavier than the fluid is driven forward on the bottom of a channel by a plane Poiseuille flow. After a certain critical Reynolds number, the particle rises from the wall to an equilibrium height at which the buoyant weight just balances the upward thrust from the hydrodynamic force. The aim of the calculation is the determination of the critical lift-off condition and the evolution of the height, velocity and angular velocity of the particle as a function of the pressure gradient and material and geometric parameters. The critical Reynolds number for lift-off is found to be larger for a heavier particle whereas it is lower for a particle in a viscoelastic fluid. A correlation for the critical shear Reynolds number for lift-off is obtained. The equilibrium height increases with the Reynolds number, the fluid elasticity and the slip angular velocity of the particle. Simulations of single particle lift-off at higher Reynolds numbers in a Newtonian fluid by Choi and Joseph (2001) but reported here show multiple steady states and hysteresis loops. This is shown here to be due to the presence of two turning points of the equilibrium solution.

Description: In this paper we study the sedimentation of several thousand circular particles in two dimensions using the method of distributed Lagrange multipliers for solid-liquid flow. The simulation gives rise to fingering which resembles Rayleigh-Taylor instabilities. The waves have a well-defined wavelength and growth rate which can be modelled as a conventional Rayleigh-Taylor instability of heavy fluid above light. The heavy fluid is modelled as a composite solid-liquid fluid with an effective composite density and viscosity. Surface tension cannot enter this problem and the characteristic shortwave instability is regularized by the viscosity of the solid-liquid dispersion. The dynamics of the Rayleigh-Taylor instability are studied using viscous potential flow, generalizing work of Joseph, Belanger and Beavers (1999) to a rectangular domain bounded by solid walls; an exact solution is obtained.

Description: In recent experiments on the growth of localized disturbances in a Blasius boundary layer, Medeiros & Gaster (1999a, b) observed that the development of nonlinear effects depends markedly on the initial phase of their imposed disturbance. Here, a simple explanation of this phenomenon is proposed. Because the disturbance is localized in space and time, it has a spread of wavenumbers and frequencies: among these are components which can initiate a pair of resonant subharmonic waves with well-determined phase, which are then amplified by the familiar three-wave resonance mechanism. The amplitude attained after some time is strongly phase-dependent, consistent with the experimental observations.

Description: The structure of energy-containing turbulence in the outer region of a zero-pressure- gradient boundary layer has been studied using particle image velocimetry (PIV) to measure the instantaneous velocity fields in a streamwise-wall-normal plane. Experiments performed at three Reynolds numbers in the range 930 〈 Reθ 〈 6845 show that the boundary layer is densely populated by velocity fields associated with hairpin vortices. (The term ‘hairpin’ is here taken to represent cane, hairpin, horseshoe, or omega-shaped vortices and deformed versions thereof, recognizing these structures are variations of a common basic flow structure at different stages of evolution and with varying size, age, aspect ratio, and symmetry.) The signature pattern of the hairpin consists of a spanwise vortex core located above a region of strong second-quadrant fluctuations (u 〈 0 and v 〉 0) that occur on a locus inclined at 30–60° to the wall.In the outer layer, hairpin vortices occur in streamwise-aligned packets that propagate with small velocity dispersion. Packets that begin in or slightly above the buffer layer are very similar to the packets created by the autogeneration mechanism (Zhou, Adrian & Balachandar 1996). Individual packets grow upwards in the streamwise direction at a mean angle of approximately 12°, and the hairpins in packets are typically spaced several hundred viscous lengthscales apart in the streamwise direction. Within the interior of the envelope the spatial coherence between the velocity fields induced by the individual vortices leads to strongly retarded streamwise momentum, explaining the zones of uniform momentum observed by Meinhart & Adrian (1995). The packets are an important type of organized structure in the wall layer in which relatively small structural units in the form of three-dimensional vortical structures are arranged coherently, i.e. with correlated spatial relationships, to form much longer structures. The formation of packets explains the occurrence of multiple VITA events in turbulent ‘bursts’, and the creation of Townsend's (1958) large-scale inactive motions. These packets share many features of the hairpin models proposed by Smith (1984) and co-workers for the near-wall layer, and by Bandyopadhyay (1980), but they are shown to occur in a hierarchy of scales across most of the boundary layer.In the logarithmic layer, the coherent vortex packets that originate close to the wall frequently occur within larger, faster moving zones of uniform momentum, which may extend up to the middle of the boundary layer. These larger zones are the induced interior flow of older packets of coherent hairpin vortices that originate upstream and over-run the younger, more recently generated packets. The occurence of small hairpin packets in the environment of larger hairpin packets is a prominent feature of the logarithmic layer. With increasing Reynolds number, the number of hairpins in a packet increases.