ALBERT

Notes: The linear stability analysis of a simple model of a swirling jet illuminates the competition and interaction of centrifugal and Kelvin–Helmholtz instabilities. By employing potential theory, analytical expressions are derived for the growth rate and propagation velocity of both axisymmetric and helical waves. The results show that centrifugally stable flows become destabilized by sufficiently short Kelvin–Helmholtz waves. The asymptotic limits demonstrate that for long axisymmetric waves the centrifugal instability dominates, while long helical waves approach the situation of a Kelvin–Helmholtz instability in the azimuthal direction, modulated by a stable or unstable centrifugal stratification. Both short axisymmetric and short helical waves converge to the limit of a plane Kelvin–Helmholtz instability feeding on the azimuthal vorticity.

Notes: The dispersion and settling of small, heavy, spherical particles in a temporally evolving two-dimensional mixing layer under gravity is investigated. The dilute limit is assumed, in which both the effect of the particles on the fluid flow and the interaction among the particles is negligible. The particle dynamics is quantified as a function of the dimensionless Stokes and Froude numbers, St and Fr, which express the ratios of the three time scales related to (i) the fluid flow, (ii) the particles' inertia, and (iii) their settling velocity, respectively. For horizontal flow in which the upper stream is the seeded one, the mixing layer accelerates the settling of particles with small St, whereas particles with large St are slowed down in their settling motion. At intermediate St and for moderate settling velocities, root-mean-square (RMS) data for the particle concentration field demonstrate the generation of strong inhomogeneities by the mixing layer. These regions of high particle concentration have the form of bands in the initially unseeded stream. Scaling laws for their angles and the distance between them are given. Furthermore, analytical results for linearized flow fields are derived that demonstrate the optimal efficiency of the dispersion and settling process at intermediate St. The numerical simulations show the existence of different parameter regimes, in which the particle motion is dominated by the coherent vortices and by gravity, respectively. Scaling laws are derived for the particle dispersion and settling for both of these regimes, which show reasonable quantitative agreement with the simulation data. Flows that exhibit a vortex pairing process show a reduced tendency of the particles toward suspension. For vertically upward flow in which the faster stream is seeded, is observed a sharp maximum in the particle dispersion measures for intermediate St and settling velocities equal to one-half the difference between the free-stream velocities. Under these conditions, the cross-stream fluid velocity components become optimally efficient in ejecting particles into the unseeded stream. © 1995 American Institute of Physics.

Notes: This paper presents detailed computational results for the dispersion of heavy particles in transitional mixing layers forced at both the fundamental and subharmonic frequencies. The results confirm earlier observations of particle streaks forming in the braid region between successive vortices. A scaling argument based on the idealization of the spatially periodic mixing layer as a row of point vortices shows that the formation of these concentrated particle streaks proceeds with optimum efficiency for St(approximately-equal-to)1. It thereby provides a quantitative basis for experimental and numerical observations of preferential particle dispersion at Stokes numbers of order unity. Both the model and full simulation furthermore exhibit oscillatory particle motion, as well as the formation of two bands of high particle concentrations, for larger Stokes numbers. The particle dispersion as a function of time and the Stokes number is quantified by means of two different integral scales. These show that the number of dispersed particles does not reach a maximum for intermediate Stokes number. However, when the distance is weighted, optimum dispersion is observed for Stokes numbers around unity. By tracing the dispersed particles backwards in time, they are found to originate in inclined, narrow bands that initially stretch from the braid region into the seeded free stream. This suggests that particle dispersion can be optimized by phase coupling the injection device with the forcing signal for the continuous phase. In the presence of a subharmonic perturbation, enhanced particle dispersion is observed as a result of the motion of the vortices, whereby a larger part of the flow field is swept out.

Notes: Direct numerical simulations are used to analyze the evolution of a temporally growing two-dimensional shear layer seeded with dilute concentrations of bubbles under gravity. The bubble concentrations are dilute enough so that bubble–bubble interactions can be neglected, but are large enough for cumulative effects of bubbles to influence the flow. The evolution of the bubble field is determined by tracking many individual bubbles, and the flow field is advanced by using the Navier–Stokes equations with a coupling term representing the effect of the bubbles on the flow. The results are interpreted in terms of the vorticity, density, and pressure fields relative to the one-way coupled or passive case. For the coupled case, a reduction in the magnitude of the vorticity and pressure gradients near the vortex center is observed. In addition to modification of the flow, it is observed that the accumulation of bubbles is smaller and the location of the equilibrium points are shifted farther from the vortex center as a result of the coupling. It is explored how these changes are modified by different Froude numbers and bubble sizes. The differences between passive and coupled cases usually increase due to larger accumulations as larger bubbles are considered. However, for certain Froude numbers an optimum coupling is observed at intermediate bubble sizes due to the absence of equilibrium points for large bubbles.

Notes: The interface region between two fluids of different densities and viscosities in a porous medium in which gravity is directed at various angles to the interface is analyzed. Under these conditions, base states exist that involve both tangential and normal velocity components. These base states support traveling waves. In the presence of a normal displacement velocity, the amplitude of these waves grows according to the viscous fingering instability. For the immiscible case, it can easily be shown that the growth rate is not affected by the tangential velocities, while surface tension results in the usual stabilization. For the case of two miscible fluids, the stability of the base states using the quasi-steady-state approximation is investigated. The resulting equations are solved analytically for time t=0 and a criterion for instability is formulated. The stability of the flow for times t(approximately-greater-than)0 is investigated numerically using a spectral collocation method. It is found that the interaction of pressure forces and viscous forces is modified by tangential shear as compared to the classical problem, resulting in a stabilizing effect of the tangential shear. The key to understanding the physical mechanism behind this stabilization lies in the vorticity equation. While the classical problem gives rise to a dipole structure of the vorticity field, tangential shear leads to a quadrupole structure of the perturbation vorticity field, which is less unstable. This quadrupole structure is due to the finite thickness of the tangential base state velocity profile, i.e., the finite thickness of the dispersively spreading front, and hence cannot emerge on the sharp front maintained in immiscible displacements.

Notes: The three-dimensional development of the wake behind a flat plate separating two laminar streams of equal velocity, subjected to perturbations in its initial conditions, is studied experimentally and numerically at moderate Reynolds numbers. The effect of single waves as well as subharmonic streamwise forcing is discussed. For the case of a single wave, streamwise forcing combined with a sinusoidal spanwise perturbation, the two distinct three-dimensional vorticity modes found in an early study [J. Fluid Mech. 190, 1 (1988)], are further studied and their symmetry properties analyzed. Depending on the orientation of the spanwise perturbation, the counter-rotating pairs of streamwise vortex tubes forming in the braids are shown to induce undulations in the cores of the Karman-like vortices, resulting in either an in-phase or a varicose configuration. Furthermore, when the wake is subjected to subharmonic streamwise forcing, additional modes of the topology of the vorticity field are shown to exist. The evolution of these subharmonic three-dimensional modes is analyzed as a function of the initial conditions.

Notes: The transport in vortical and stagnation point flow fields is analyzed for particles across the entire range of density ratios, based on the Maxey–Riley equation [Phys. Fluids 26, 883 (1983)] without history effects. For these elementary flow fields, the governing equations simplify substantially, so that analytical progress can be made towards quantifying ejection/entrapment trends and accumulation behavior. For a solid body vortex, the analysis shows that optimal ejection or entrapment occurs for all density ratios, as the difference between inward and outward forces reaches a maximum for intermediate values of the Stokes number. The optimal Stokes number value is provided as a function of the density ratio. Gravity is shown to shift accumulation regions, without affecting the entrapment or ejection rates. For a point vortex flow, the existence of up to three different regimes is demonstrated, which are characterized by different force balances and ejection rates. For this flow, optimal accumulation is demonstrated for intermediate Stokes numbers. The stagnation point flow gives rise to optimal accumulation for heavy particles, whereas light particles do not exhibit optimal behavior. The analysis furthermore indicates that nonvanishing density ratios give rise to a finite Stokes number regime in which the particle motion is oscillatory. Above and below this regime, the motion is overdamped. © 1997 American Institute of Physics.

Notes: The motion of small, spherical noninteracting bubbles in two-dimensional vortical flows by means of numerical simulations is investigated. After a discussion concerning the various bubble equations, bubble trajectories are calculated in a solid-body vortex, where it is found that the bubble motion can be described in terms of the location where the bubbles accumulate, or equilibrium points, and the rate of entrapment into these equilibrium points. Of importance here is that the rate of entrapment into the vortex has an optimum value for some value of the inertia parameter, or inverse Stokes number. The bubble motion in a temporally evolving shear layer is investigated, where it is found that the solid-body vortex model predicts the trends in the growth in concentration about the vortex center for the case without gravity. For the case with gravity, not all bubbles are captured by the vortex, and the percentage of bubbles captured increases with decreasing inertia parameter. Also discussed is how these factors affect the generation of the interface between regions seeded and not seeded with bubbles.

Notes: The nonlinear evolution of the interface between two miscible fluids of different densities and viscosities is simulated numerically for flow in a two-dimensional porous medium in which gravity is directed at various angles to the interface. Global velocities tangential to the interface are included in the analysis in addition to a normal displacing velocity. In unstable configurations, the viscous fingers that result translate as they amplify when nonzero tangential velocities are present. The increased stabilization by tangential shearing velocities reported in [A. Rogerson and E. Meiburg, Phys. Fluids A 5, 1344 (1993)] affects the growth and wavelength selection of the emerging fingers. Tangential shearing also breaks the symmetry in the shape and concentration distribution of emerging fingers. In addition to the fingering mechanisms reported in previous studies, new mechanisms of diagonal fingering, trailing-lobe detachment, and secondary side-finger instability, resulting from the presence of gravity and tangential velocities, have been identified. These phenomena are reflected in one-dimensional averaged profiles of the concentration field. Also, how different density–concentration relations influence the interfacial evolution is investigated. When the dependence of viscosity and density on the concentration has different functional forms, the region of instability may be localized. The nature of the interfacial development is altered by varying the density relation and thereby changing the region of instability, suggesting that careful modeling of the density and viscosity relations is warranted.

Notes: The steady thermocapillary motion in a square cavity with a top free surface in the absence of gravitational forces is considered. The cavity is heated from the side with the vertical boundaries isothermal while the horizontal boundaries are adiabatic. The relative change in the surface tension is very small, i.e., an appropriate capillary number tends to zero, so that the free surface is assumed to remain flat at leading order. A finite-difference method is employed to compute the flow field. Numerically accurate solutions are obtained for a range of Prandtl numbers and for Reynolds numbers Re as high as 5×104. Surface deflections are computed as a domain perturbation for small capillary number. In addition, asymptotic methods are used to infer the boundary layer structure in the cavity, in the limit of large values of the Reynolds and Marangoni numbers. For a fixed Prandtl number Pr, it is shown that the Nusselt number, liquid circulation, and maximum vorticity are asymptotic to Re1/3, Re−1/3, and Re2/3, respectively. These results are in agreement with the computed solutions. The leading-order solution for the free-surface deformation is sensitive to the value of Pr. With Pr〉1, the depression near the hot corner may exceed the elevation near the cold corner, while a secondary elevation may be induced near the hot corner when Pr〈1.