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Volume 9, Issue 6, June 1997

A novel boundaryintegral algorithm for viscous interaction of deformable drops
View Description Hide DescriptionA new threedimensional boundaryintegral algorithm for deformable drops moving in a viscous medium at low Reynolds numbers is developed, which overcomes some familiar difficulties with boundaryintegral calculations. The algorithm is used to simulate different modes of interaction between drops or bubbles, primarily for buoyancydriven motion. The present iterative method for mean curvature calculation is found to be more robust and accurate than contour integration schemes. A novel iterative strategy based on combining biconjugate gradient and simple iterations overcomes the poor convergence of “successive substitutions” for drops in very close approach with extreme viscosity ratio. A substantially new variational method of global mesh stabilization solves the problem of mesh degradation with advantageous, soft stability constraints. A curvatureless boundaryintegral formulation is also derived and shown to provide, in principle, a more accurate description of the drop breakup than the conventional formulation. The efficiency of these techniques is demonstrated by numerical examples for two drops in gravityinduced motion with high surface resolutions. The present code successfully simulates mutual approach of slightly deformable drops to extremely small separations, as well as their rotation when in “apparent contact,” thus bridging the gap between finite deformation calculations and a recent asymptotic theory for small capillary numbers. Also provided is a 3D simulation of the experimental phenomenon of enhanced bubble coalescence, discovered by Manga and Stone [J. Fluid Mech. 256, 647 (1993); 300, 231 (1995)]. For drops of viscosity comparable to that of the surrounding fluid, it is shown in contrast that breakup is a typical result of hydrodynamic interaction in gravityinduced motion for large and even moderate capillary numbers. The code is readily applicable to any type of an ambient flow and may be adapted to more than two drops.

Forced oscillations of pendant (sessile) drops
View Description Hide DescriptionOscillations of supported liquiddrops are the subject of wide scientific interest, with applications in areas as diverse as liquid–liquid extraction, synthesis of ceramicpowders, growing of pure crystals in low gravity, and measurement of dynamic surface tension. In this study, axisymmetric forced oscillations of arbitrary amplitude of a viscousliquiddrop of fixed volume which is pendant from or sessile on a rod with a fixed contact line and surrounded by an inviscid ambient gas are induced by moving the rod in the vertical direction sinusoidally in time. This nonlinear free boundary problem is solved by a method of lines using Galerkin/finite element analysis for discretization in space and an implicit, adaptive finite difference technique for discretization in time. The variation of the drop response over a wide range of the governing parameters (Reynolds number Re, gravitational Bond number volume, and forcing frequency and amplitude) is analyzed. The results show that as the forcing frequency is increased, a sequence of oscillation modes is observed, each with its own resonance frequency at which drop response amplitude reaches a local maximum. While resonance frequencies depend strongly on drop size and on forcing amplitude, the effect of Reynolds number on is large when Re is small and diminishes when Re is large, in accord with observations for free oscillations. At high Re, a sharp increase in dropdeformation can occur for drops forced to oscillate in the vicinity of their resonance frequencies, indicating the incipience of hysteresis. The maximum observed dropdeformations increase with Re, and forcing amplitude, while the value of the dropdeformation as a function of drop size is determined by a balance between the magnitude of the viscous shear stress imposed on the dropliquid by the solid rod relative to the capillary pressure due to surface tension acting on the fluid interface. The effects of viscous dissipation are also seen in the damping of various oscillation modes and in the creation, evolution in time, and disappearance of zones of fluid recirculation within the drop.

Nusselt number for flow perpendicular to arrays of cylinders in the limit of small Reynolds and large Peclet numbers
View Description Hide DescriptionThe problem of determining the Nusselt number the nondimensional rate of heat or mass transfer, from an array of cylindrical particles to the surrounding fluid is examined in the limit of small Reynolds number and large Peclet number in this limit can be determined from the details of flow in the immediate vicinity of the particles. These are determined accurately using a method of multipole expansions for both ordered and random arrays of cylinders. The results for are presented for the complete range of the area fraction of cylinders. The results of numerical simulations for random arrays are compared with those predicted using effectivemedium approximations, and a good agreement between the two is found. A simple formula is given for relating the Nusselt number and the Darcy permeability of the arrays. Although the formula is obtained by fitting the results of numerical simulations for arrays of cylindrical particles, it is shown to yield a surprisingly accurate relationship between the two even for the arrays of spherical particles for which several known results exist in the literature suggesting thereby that this relationship may be relatively insensitive to the shape of the particles.

Rheology of dense bubble suspensions
View Description Hide DescriptionThe rheological behavior of rapidly sheared bubble suspensions is examined through numerical simulations and kinetic theory. The limiting case of spherical bubbles at large Reynolds number and small Weber number is examined in detail. Here, and, being the bubble radius, the imposed shear, the interfacial tension, and and , respectively, the viscosity and density of the liquid. The bubbles are assumed to undergo elastic bounces when they come into contact; coalescence can be prevented in practice by addition of salt or surfaceactive impurities. The numerical simulations account for the interactions among bubbles which are assumed to be dominated by the potential flow of the liquid caused by the motion of the bubbles and the shearinduced collision of the bubbles. A kinetic theory based on Grad’s moment method is used to predict the distribution function for the bubble velocities and the stress in the suspension. The hydrodynamic interactions are incorporated in this theory only through their influence on the virtual mass and viscous dissipation in the suspension. It is shown that this theory provides reasonable predictions for the bubblephase pressure and viscosity determined from simulations including the detailed potential flow interactions. A striking result of this study is that the variance of the bubble velocity can become large compared with in the limit of large Reynolds number. This implies that the dispersephase pressure and viscosity associated with the fluctuating motion of the bubbles is quite significant. To determine whether this prediction is reasonable even in the presence of nonlinear drag forces induced by bubble deformation, we perform simulations in which the bubbles are subject to an empirical drag law and show that the bubble velocity variance can be as large as .

A numerical calculation of the hydraulic permeability of threedimensional disordered fibrous media
View Description Hide DescriptionHydraulic permeabilities of polymeric membranes and gels are of interest both for calculating fluid flow rates and hindered diffusion coefficients. We have calculated hydraulic permeabilities for monomodal and bimodal, periodic and random fibrous media. Hydrodynamic interactions between fibers are calculated by applying a numerical version of slender body theory to a collection of fibers in a cubic cell many Brinkman screening lengths in dimension. Results for random media are obtained by averaging over many ensembles of fibers. To account for the surrounding medium, the line distribution of point forces along the fiber axes are replicated throughout space by using the Ewald summation technique. Results for periodic media agree with previous theoretical results up to a fiber volume fraction of 50% for parallel flow and 40% for transverse flow. Hydraulic permeabilities calculated for threedimensional, disordered media with monomodal and bimodal distributions of fiber radius are compared with existing theories and with experimentally determined hydraulic permeabilities for a range of fiber volume fractions. Specific calculations are performed for agarose and collagen/proteoglycan gel systems, which are well described as bimodal fibrous media and are relevant to bioseparations and physiological systems, respectively.

Breakdown of scaling in droplet fission at high Reynolds number
View Description Hide DescriptionIn this paper we address the shape of a lowviscosity fluid interface near the breaking point. Experiments show that the shape varies dramatically as a function of fluid viscosity. At low viscosities, the interface develops a region with an extremely sharp slope, with the steepness of the slope diverging with vanishing viscosity. Numerical simulations demonstrate that this tip forms as a result of a convective instability in the fluid; in the absence of viscosity this instability results in a finite time singularity of the interface far before rupture (in which the interfacial curvature diverges). The dynamics before the instability roughly follow the scaling laws consistent with predictions based on dimensional analysis, though these scaling laws are violated at the instability. Since the dynamics after rupture is completely determined by the shape at the breaking point, the time dependences of recoiling do not follow a simple scaling law. In the process of demonstrating these results, we present detailed comparisons between numerical simulations and experimental drop shapes with excellent agreement.

On pressure and velocity boundary conditions for the lattice Boltzmann BGK model
View Description Hide DescriptionPressure (density) and velocity boundary conditions are studied for 2D and 3D lattice BoltzmannBGK models (LBGK) and a new method to specify these conditions is proposed. These conditions are constructed in consistency with the wall boundary condition, based on the idea of bounceback of the nonequilibrium distribution. When these conditions are used together with the incompressible LBGK model [J. Stat. Phys. 81, 35 (1995)] the simulation results recover the analytical solution of the plane Poiseuille flow driven by a pressure (density) difference. The halfway wall bounceback boundary condition is also used with the pressure (density) inlet/outlet conditions proposed in this paper and in Phys. Fluids 8, 2527 (1996) to study 2D Poiseuille flow and 3D square duct flow. The numerical results are approximately secondorder accurate. The magnitude of the error of the halfway wall bounceback boundary condition is comparable with that of other published boundary conditions and it has better stability behavior.

Crossover between surface tension and gravitydriven instabilities of a thin fluid layer on a horizontal cylinder
View Description Hide DescriptionA thin annular layer of fluid coating a cylinder is subject to two different instabilities. One, driven by surface tension, is analogous to the Rayleigh instability of a liquid jet. The other is the Rayleigh–Taylor instability, which is driven by gravity. Measurements of the wavelength and growth rate of periodic patterns of droplets which develop as a result of the instability of such a fluid layer are reported for cylinders with radius in the range cm. For small the wavelength and growth rate of the pattern are in agreement with theoretical predictions for the surfacetensiondriven instability. For large , the Rayleigh–Taylor instability is observed. At intermediate there is a region of crossover between the two limiting cases.

The instability of sand ripples under partially standing surface waves
View Description Hide DescriptionWe extend recent works on oscillatory flows over rigid ripples and the instability of sand ripples under such flows, by considering the instability of sand ripples under partially standing surface waves over a finite water depth. The variation of unstable ripples within a wavelength of the standing waves and inferences on the ripple distribution over a sand bar are examined. The steady circulation due to the combined effects of waves and ripples are also discussed.

Influence of thermal boundary conditions on the stability of thermocapillarydriven convection at low Prandtl numbers
View Description Hide DescriptionWe analyze the effect of various thermal boundary conditions on the linear stability of surfacetensiondriven flow in an unbounded liquid layer subject to a longitudinal temperature gradient. An original approach is devised to estimate the critical instability parameters. The order of magnitude estimates are used to solve the problem asymptotically for small Prandtl numbers. The instability is shown to be essentially determined by the thermal boundary conditions. For insulating boundaries the critical wavenumber scales as meaning that the most unstable wave is considerably longer than the depth of the layer. When the bottom is well conducting, the critical wavelength is comparable to the depth of the layer. For the case of insulating bottom and nonadiabaticfree surface the critical wavenumber depends on the Biot number as Even a weak thermal coupling between the free surface and the ambient medium such that can significantly influence the instability threshold.

The Rayleigh–Taylor instability of viscous fluid layers
View Description Hide DescriptionThe effects of viscosity and surface tension on the nonlinear evolution of Rayleigh–Taylor instability of plane fluid layers are investigated. Full twodimensional incompressible Navier–Stokes equations and exact boundary equations are solved simultaneously for a precise prediction of this phenomenon. An accurate flux line segment model (FLAIR) for fluid surface advection is employed for the interface reconstruction. The instability is characterized by three stages of development, which are defined based on the competition of the bubble and spike growth. This competition is responsible for the development of different spike and bubble morphologies and is decided based on geometrical factors, mainly the amplitude and wavelength of the initial perturbation, and on the fluid properties, mainly viscosity and surface tension. It is addressed and explained why the spike sometimes grows faster than the bubble, and vice versa. The cutoff and the most unstable wave numbers are identified numerically based on the Weber number. The effect of Weber and Reynolds numbers on the growth rate of instability and the role of viscosity in dragging the development of instability are also investigated.

Symmetry breaking bifurcation in finite disk flow
View Description Hide DescriptionIn the Reynolds number range 8 771⩽Re⩽13 300, we find three distinctly different solutions for flow between finite, corotating disks. The basic flow, present for all Reynolds numbers, displays symmetry with respect to midplane, while the other two solutions are asymmetric, exist only within the specified Reynolds number range, and are mirrorimages of one another. These solutions were obtained in mixed formulation of the steady state problem. To circumvent the Babushka–Brezzi stability criteria yet solve for steady state, we follow Zienkiewicz and Woo and adjoin the time asymptotic form of the equation of mass conservation, in artificial compressibility form, to the steady state Navier–Stokes equations. The system so obtained is nonsingular and yields to easy solution by Galerkin’s method. To follow particular solution branches, we employ parametric continuation.

The growth of wind waves at the crests and troughs of a low amplitude swell
View Description Hide DescriptionIf the interface between the fluids is initially planar, air blowing over water is subject to an inviscid instability, which acts to generate wind waves. Conditions in nature motivate studying the influence of a longer wave (a swell) on the instability. In the absence of waves, the wind and its associated winddrift current are taken to be plane, parallel flows. Linear theory is used to produce a first approximation to a wind and current modified by the presence of a low amplitude swell, and the stability of the modified wind and current to much shorter wavelength disturbances (the wind waves) is examined. In the neighborhood of a crest or trough of the swell, where the fluid flow is nearly parallel, the linearized equations are closely related to those obtained in the absence of a swell. The amplitude of the swell appears as a parameter, and because of the large separation in wavelength assumed between the swell and the wind waves, the phase of the swell may be approximated as a parameter. It is found that, at both crests and troughs, the growth rates of the wind waves ultimately decrease rapidly with increasing swell amplitude, indicating that the swell has a stabilizing influence. This result is consistent with the experimentally observed suppression of wind waves by a swell propagating in the direction of the wind.

Measurements of a longitudinal vortex generated by a rectangular jet in a turbulent boundary layer
View Description Hide DescriptionMeasurements were made on the mean flowfield created through the interaction between a twodimensional flat plate turbulent boundary layer on a flat plate and an inclined jet. The jet was generated by a nozzle of rectangular exit and was pitched and skewed to the oncoming flow. A total of three pitch angles and two jet velocity ratios were tested. The measurements were performed with a threecomponent laser Doppler anemometer system in a low speed wind tunnel. The dominant feature of the flow physics is a single longitudinal vortex produced as a result of jet/boundary layer interaction, with additional induced secondary features in the nearwall area in the spanwise direction. A salient feature of the flow is its maximum vorticity position which is located underneath the center of the vortex. Within the measuredflow parameter range, the vortex development differs also from that of a round jet, differences in the velocity distribution are observed. The study provides contributions to flow physics.

Analysis and classification of reactiondriven stationary convective patterns in a porous medium
View Description Hide DescriptionA twodimensional model consisting of continuity, momentum, species, and energy balances is considered to analyze the convective patterns that arise due to exothermic reactions occurring in a porous medium. First, the onedimensional conduction states of the system are classified using singularity theory and the shooting technique. It is observed that there can be either one or three conduction states when the reacting fluid is a gas. Next, we use linear stability analysis to determine the boundary of the parameter values at which the conduction state loses stability leading to convective flows. Pure and mixedmode convective solutions are then analyzed using local bifurcation theory. The formulas to evaluate the coefficients appearing in the amplitude equations are developed and used to obtain the classification (phase) diagram of the convective flows in the parameter space. The classification is presented in the unique conduction solution region in the presence and absence of mode interactions. The phase diagrams are used to identify the region of parameter values where convection has a detrimental effect on the stability of the system. It is found that the Lewis number (Le), which represents the ratio of thermal to mass diffusivity, has a profound influence on the stability boundaries. For the convective solutions may bifurcate subcritically and introduce an ignition point.

Numerical prediction of laminar, transitional and turbulent flows in shrouded rotorstator systems
View Description Hide DescriptionThe paper deals with numerical prediction of laminar, transitional and turbulent regimes in confined flow between rotating and stationary discs. For the laminar and transitional flows, a spectral tauChebyshev method associated with a multistep time scheme is used. This approach allows accurate prediction of the two laminar regimes mentioned by Daily and Nece (1960) in their experimental studies. For the geometry under consideration (1/11 aspect ratio), the transition to unsteady motion occurs abruptly without any oscillatory behavior. Thus, the instabilities develop in a region localized near the external shroud, primarily along the stator side, according with experimental findings. For calculating turbulent flow regimes, one point secondorder transportmodeling has been implemented in a finite volume code. The superiority of advanced Reynolds stresstransportmodels over the classical model is decisive for predicting such a complex flow. This is particularly important in order to get a precise delineation of the adjacent turbulent and relaminarized regions within the cavity. This level of closure was crucial to produce numerical results in good agreement with experimental data.

Selfsimilarity and mixing characteristics of turbulent mixing layers starting from laminar initial conditions
View Description Hide DescriptionDirect numerical simulations of two turbulent mixing layers starting from low and highgradient laminar profiles have been performed, to assess the transitional and long time characteristics of these flows. The simulations include a passive scalar with Schmidt number of 0.7, and were conducted using a grid resolution of implemented in parallel on 128 nodes of the Intel Paragon supercomputer. The two mixing layers at their final turbulent condition achieved Reynolds number of . It is found that although the selfsimilarity appears to have been achieved early in the evolution of these flows, long time development is necessary for the mixingcharacteristics and the statistical content of these flows to become similar to the mixing layers that start from turbulent boundary layers.

A numerical study of selfsimilarity in a turbulent plane wake using largeeddy simulation
View Description Hide DescriptionTurbulent wakes are known to develop selfsimilarly sufficiently far downstream from obstacles that generate them. It has long been assumed that the spreading rate of the wake in the selfsimilar regime is independent of the details of the body generating the wake, being dependent only on the total drag (or momentum deficit). This assumption seems to be in contradiction with some recent experiments. In this study we attempt to complement these experimental investigations through a numerical study of a timedeveloping wake. A numerical study has the advantage of eliminating many of the uncontrolled factors present in experiments and allowing precise control of initial conditions. Largeeddy simulations employing the recently developed dynamic localization model are used to extend previous results from direct numerical simulations. The largeeddy simulation results are compared to the direct numerical simulation database, wherever such comparisons are feasible, as a check of the method. Like the experiments, the largeeddy simulations suggest that nonunique selfsimilar states, characterized by different spreading rates and turbulent statistics, are possible and that they can be maintained for significant time periods. The study also demonstrates the predictive capability of the dynamic localization subgrid model.

Application of neural networks to turbulence control for drag reduction
View Description Hide DescriptionA new adaptive controller based on a neural network was constructed and applied to turbulent channel flow for drag reduction. A simple control network, which employs blowing and suction at the wall based only on the wallshear stresses in the spanwise direction, was shown to reduce the skin friction by as much as 20% in direct numerical simulations of a lowReynolds number turbulent channel flow. Also, a stable pattern was observed in the distribution of weights associated with the neural network. This allowed us to derive a simple control scheme that produced the same amount of drag reduction. This simple control scheme generates optimum wall blowing and suction proportional to a local sum of the wallshear stress in the spanwise direction. The distribution of corresponding weights is simple and localized, thus making real implementation relatively easy. Turbulence characteristics and relevant practical issues are also discussed.

Derivation of a PDF model for turbulent flows based on principles from statistical physics
View Description Hide DescriptionClassical ideas from statistical physics are used to derive a PDF model for turbulent flows. The model is built by adopting a Lagrangian point of view and by considering separately the statistical effects of the viscous and of the pressure gradient forces which act on a fluid particle. Closures are developed alternatively in terms of the pdf itself and of the trajectories of the stochastic process. The viscous force is shown to manifest itself as an antidiffusion in phase space while modeling of the fluctuating part of the pressure gradient force is based on linear laws for nonequilibrium thermodynamics along Onsager’s regressiontoequilibrium hypothesis. The final expression is identical to a Langevin equation proposed by Pope which is thus seen to be obtained from underlying principles.