ALBERT

Description: Cross-stream migration and stable orientations of elliptic particles falling in an Oldroyd-B fluid in a channel are studied. We show that the normal component of the extra stress on a rigid body vanishes; lateral forces and torques are determined by the pressure. Inertia turns the long side of the ellipse across the stream and elasticity turns it along the stream; tilted off-centre falling is unstable. There are two critical numbers: the elasticity and Mach numbers. When the elasticity number is smaller than critical the fluid is essentially Newtonian with broadside-on falling at the centreline of the channel. For larger elasticity numbers the settling turns the long side of the particle along the stream in the channel centre for all velocities below a critical one, identified with a critical Mach number of order one. For larger Mach numbers the ellipse flips into broadside-on falling again. The critical numbers are functions of the channel blockage ratio, the particle aspect ratio and the retardation/relaxation time ratio of the fluid. Two ellipses falling near to each other, attract, line-up vertically and straighten-out with long sides vertical. Stable, off-centre tilting is found for ellipses falling in shear-thinning fluids and for cylinders with flat ends in which particles tend to align their longest diameter with gravity.

Description: We consider steady compressible Euler flow corresponding to the compressible analogue of the well-known incompressible Hill's spherical vortex (HSV). We first derive appropriate compressible Euler equations for steady homentropic flow and show how these may be used to define a continuation of the HSV to finite Mach number M∞ = U∞/C∞, where U∞, C∞ are the fluid velocity and speed of sound at infinity respectively. This is referred to as the compressible Hill's spherical vortex (CHSV). It corresponds to axisymmetric compressible Euler flow in which, within a vortical bubble, the azimuthal vorticity divided by the product of the density and the distance to the axis remains constant along streamlines, with irrotational flow outside the bubble. The equations are first solved numerically using a fourth-order finite-difference method, and then using a Rayleigh-Janzen expansion in powers of M∞ 2 to order M∞ 4. When M∞ 〉 0, the vortical bubble is no longer spherical and its detailed shape must be determined by matching conditions consisting of continuity of the fluid velocity at the bubble boundary. For subsonic compressible flow the bubble boundary takes an approximately prolate spheroidal shape with major axis aligned along the flow direction. There is good agreement between the perturbation solution and Richardson extrapolation of the finite difference solutions for the bubble boundary shape up to M∞ equal to 0.5. The numerical solutions indicate that the flow first becomes locally sonic near or at the bubble centre when M∞ ≈ 0.598 and a singularity appears to form at the sonic point. We were unable to find shock-free steady CHSVs containing regions of locally supersonic flow and their existence for the present continuation of the HSV remains an open question.

Description: The modulation of isotropic turbulence by particles has been investigated using direct numerical simulation (DNS). The particular focus of the present work is on the class of dilute flows in which particle volume fractions and inter-particle collisions are negligible. Gravitational settling is also neglected and particle motion is assumed to be governed by drag with particle relaxation times ranging from the Kolmogorov scale to the Eulerian time scale of the turbulence and particle mass loadings up to 1. The velocity field was made statistically stationary by forcing the low wavenumbers of the flow. The calculations were performed using 963 collocation points and the Taylor-scale Reynolds number for the stationary flow was 62. The effect of particles on the turbulence was included in the Navier-Stokes equations using the point-force approximation in which 963 particles were used in the calculations. DNS results show that particles increasingly dissipate fluid kinetic energy with increased loading, with the reduction in kinetic energy being relatively independent of the particle relaxation time. Viscous dissipation in the fluid decreases with increased loading and is larger for particles with smaller relaxation times. Fluid energy spectra show that there is a non-uniform distortion of the turbulence with a relative increase in small-scale energy. The non-uniform distortion significantly affects the transport of the dissipation rate, with the production and destruction of dissipation exhibiting completely different behaviours. The spectrum of the fluid-particle energy exchange rate shows that the fluid drags particles at low wavenumbers while the converse is true at high wavenumbers for small particles. A spectral analysis shows that the increase of the high-wavenumber portion of the fluid energy spectrum can be attributed to transfer of the fluid particle covariance by the fluid turbulence. This in turn explains the relative increase of small-scale energy caused by small particles observed in the present simulations as well as those of Squires & Eaton (1990) and Elghobashi & Truesdell (1993).

Description: Unsteady boundary-layer development over moving walls in the limit of infinite Reynolds number is investigated using both the Eulerian and Lagrangian formulations. To illustrate general trends, two model problems are considered, namely the translating and rotating circular cylinder and a vortex convected in a uniform flow above an infinite flat plate. To enhance computational speed and accuracy for the Lagrangian formulation, a remeshing algorithm is developed. The calculated results show that unsteady separation is delayed with increasing wall speed and is eventually suppressed when the speed of the separation singularity approaches that of the local mainstream velocity. This suppression is also described analytically. Only 'upstream-slipping' separation is found to occur in the model problems. The changes in the topological features of the flow just prior to the separation that occur with increasing wall speed are discussed.

Description: Marangoni convection in a differentially heated binary mixture is studied numerically by continuation. The fluid is subject to the Soret effect and is contained in a two-dimensional small-aspect-ratio rectangular cavity with one undeformable free surface. Either or both of the temperature and concentration gradients may be destabilizing; all three possibilities are considered. A spectral-element time-stepping code is adapted to calculate bifurcation points and solution branches via Newton's method. Linear thresholds are compared to those obtained for a pure fluid. It is found that for large enough Soret coefficient, convection is initiated predominantly by solutal effects and leads to a single large roll. Computed bifurcation diagrams show a marked transition from a weakly convective Soret regime to a strongly convective Marangoni regime when the threshold for pure fluid thermal convection is passed. The presence of many secondary bifurcations means that the mode of convection at the onset of instability is often observed only over a small range of Marangoni number. In particular, two-roll states with up-flow at the centre succeed one-roll states via a well-defined sequence of bifurcations. When convection is oscillatory at onset, the limit cycle is quickly destroyed by a global (infinite-period) bifurcation leading to subcritical steady convection.

Description: X-ray lithography and micro-machining have been used to study gas-assisted liquid atomization in which a liquid film was impinged by a large number of sonic micro-gas jets. Three distinct breakup regimes were demonstrated. Two of these regimes share characteristics with previously observed atomization processes: a bubble bursting at a free surface (Newitt et al. 1954; Boulton-Stone & Blake 1993) and liquid sheet disintegration in a high gas/liquid relative velocity environment (Dombrowski & Johns 1963). The present work shows that suitable control of the gas/liquid interface creates a third regime, a new primary atomization mechanism, in which single liquid droplets are ejected directly from the liquid film without experiencing an intermediate ligament formation stage. The interaction produces a stretched liquid sheet directly above each gas orifice. This effectively pre-films the liquid prior to its breakup. Following this, surface tension contracts the stretched film of liquid into a sphere which subsequently detaches from the liquid sheet and is entrained by the gas jet that momentarily pierces the film. After droplet ejection, the stretched liquid film collapses, covering the gas orifice, and the process repeats. This new mechanism is capable of the efficient creation of finely atomized sprays at low droplet ejection velocities (e.g. 20 μm Sauter mean diameter methanol sprays using air at 239 kPa, with air-to-liquid mass ratios below 1.0, and droplet velocities lower than 2.0 m s-1). Independent control of the gas and the liquid flows allows the droplet creation process to be effectively de-coupled from the initial droplet momentum, a characteristic not observed with standard gas-assisted atomization mechanisms.

Description: A generalized similarity formulation extending the work of Terrill (1967) for Couette-Poiseuille flow in the annulus between concentric cylinders of infinite extent is given. Boundary conditions compatible with the formulation allow a study of the effects of inner and outer cylinder transpiration, rotation, translation, stretching and twisting, in addition to that of an externally imposed constant axial pressure gradient. The problem is governed by η, the ratio of inner to outer radii, a Poiseuille number, and nine Reynolds numbers. Single-cylinder and planar problems can be recovered in the limits η → 0 and η → 1, respectively. Two coupled primary nonlinear equations govern the meridional motion generated by uniform mass flux through the porous walls and the azimuthal motion generated by torsional movement of the cylinders; subsidiary equations linearly slaved to the primary flow govern the effects of cylinder translation, cylinder rotation, and an external pressure gradient. Steady solutions of the primary equations for uniform source/sink flow of strength F through the inner cylinder are reported for 0 ≤ η ≤ 1. Asymptotic results corroborating the numerical solutions are found in different limiting cases. For F 〈 0 fluid emitted through the inner cylinder fills the gap and flows uniaxially down the annulus; an asymptotic analysis leads to a scaling that removes the effect of η in the pressure parameter β, namely β = π2R*2, where R* = F(1 - η)/(1 + η). The case of sink flow for F 〉 0 is more complex in that unique solutions are found at low Reynolds numbers, a region of triple solutions exists at moderate Reynolds numbers, and a two-cell solution prevails at large Reynolds numbers. The subsidiary linear equations are solved at η = 0.5 to exhibit the effects of cylinder translation, rotation, and an axial pressure gradient on the source/sink flows.

Description: A previously unreported shock feature associated with the scouring of a horizontal granular bed by a dam-break wave is discussed. Near the wave centre, the present study shows, the free surface breaks backward and a hydraulic jump forms. This behaviour is described from the standpoint of shallow-water theory, suitably extended to deal with non-equilibrium sediment transport. The shock formation involves a particularly strong coupling between flow free-surface evolution and bed morphodynamics. Support for our conclusions is sought through experimental and numerical approaches. In order to magnify the observed phenomena, measurements were performed for the case of light bed particles moving in sheet and debris flow modes. A detailed picture of the transient two-phase flow is presented, based on whole field acquisition of the grain motions by particle tracking techniques. Corresponding shallow-water solutions are constructed numerically using a shock capturing scheme. Finally, an interpretation of the jump formation is proposed based on the theory of characteristics.

Description: The relaxation of a smooth two-dimensional vortex to axisymmetry, also known as 'axisymmetrization', is studied asymptotically and numerically. The vortex is perturbed at t = 0 and differential rotation leads to the wind-up of vorticity fluctuations to form a spiral. It is shown that for infinite Reynolds number and in the linear approximation, the vorticity distribution tends to axisymmetry in a weak or coarse-grained sense: when the vorticity field is integrated against a smooth test function the result decays asymptotically as t-λ with λ = 1 + (n2 + 8)1/2, where n is the azimuthal wavenumber of the perturbation and n ≥ 1. The far-field stream function of the perturbation decays with the same exponent. To obtain these results the paper develops a complete asymptotic picture of the linear evolution of vorticity fluctuations for large times t, which is based on that of Lundgren (1982).