Index of content:
Volume 7, Issue 8, August 1995

Lubrication theory for reactive spreading of a thin drop
View Description Hide DescriptionSolder drops spreading on metallic substrates are a reactive form of the wetting problem. A metallic component may diffuse in the liquid toward a metal substrate, where it is consumed by a reaction that forms a solid intermetallic phase. The resulting spatial variation in the composition of the drop may cause composition gradients along the free surface of the drop. Together with any thermal gradients along the free surface,Marangoni effects may, in turn, modify the bulk transport in the spreading drop. Motivated by this situation, we extend lubricationtheory for the spreading of thin drops in the presence of gravity and thermocapillarity to include mass transport and solutocapillarity. We use an approximate solute profile in the drop to derive coupled evolution equations for the free surface shape and concentration field. Numerical solutions for the nonreactive (single component) drop agree well with previous theory. In the reactive case, we are only able to compute results for parameters outside of the range for solder materials. Including reactive effects in the model impacts the flow patterns and spreading rates at relatively early times; but by the end of the spreading, solutal effects have died out in the model.

Observations of fiber orientation in suspensions subjected to planar extensional flows
View Description Hide DescriptionFibers in a planar extensional flow tend to align parallel to the axis of extension, with a small orientational dispersion resulting from hydrodynamic, fiber–fiber interactions. We have visualized the orientation of an opaque, tracer fiber in the midst of an index of refraction and density matched fiber suspension in a four‐roll mill and determined the mean squares of the components of the fiber orientation in the compressional and neutral directions. The observed mean‐square orientations in suspensions in Newtonian fluids with nL ^{3}=5–17 are qualitatively consistent with the dilute limit of the theory of Shaqfeh and Koch [Phys. Fluids A 31, 728 (1988)]. Here n is the number of fibers per unit volume and L is the fiber length. It was not possible to perform the experiment for higher fiber concentrations, because of difficulties in keeping the tracer fiber at the stagnation point, which were attributed to translational, hydrodynamicdiffusion.Measurements of orientational dispersion in polyacrylamide solutions indicated that the non‐Newtonian stresses increase fiber alignment, as predicted by Harlen and Koch [Phys Fluids A 4, 5 (1992)].

Macroscopic behavior of vibrating beds of smooth inelastic spheres
View Description Hide DescriptionThree‐dimensional granular dynamics simulations are carried out to investigate macroscopic behavior of granular materials subjected to vibrations. Particles, idealized as smooth inelastic, uniform spheres, are gravitationally loaded into a rectangular periodic cell having an open top and plane floor. Vibrations to the bed are subsequently imposed through the sinusoidally oscillated floor. Significant differences in the character of the bed are found, depending on the strength of the applied floor accelerations Γ=aω^{2}, even if the boundary input energy is fixed. At high acceleration values, a dense upper region is supported on a fluidized low‐density region near the floor. The temperature is maximum at the floor and monotonically attenuates upward, while the solids fraction profile peaks at some intermediate depth. When lower accelerations are applied, the granular temperature no longer decreases monotonically from the bottom to the top and the solids fraction depth profile bulges at approximately three diameters from the floor. The surface of the bed appears chaotic and fluidized, where a low solids fraction and high temperature occurs. The bed height, which remains almost constant below 1.2g, undergoes a pronounced expansion when 1.2g≤Γ≤2.0g, and subsequently flattens out at Γ≂2.8g. Computed granular temperature and solids fraction depth profiles are in good agreement with recent kinetic theory predictions when the acceleration is large enough, while bed expansion at lower accelerations is quantitatively consistent with existing experimental data.

Nonlinear stability and rupture of ultrathin free films
View Description Hide DescriptionThe nonlinear long‐wave stability and lifetimes of thin free films subjected to the excess Lifshitz–van der Waals (LW) forces are studied based on numerical solutions, and a weakly nonlinear theory (WNT), which neglects mode interactions. The WNT works best for the fastest growing (dominant) disturbances of small initial amplitudes, and also for relatively thick films. For such cases, the nonlinear viscous effects (stabilizing) and inertia (destabilizing) are usually less significant than the LW force (destabilizing), surface tension force, and the unsteady effects (both stabilizing). For large initial amplitudes, linearly stable disturbances can engender strong subcritical instabilities and film rupture due to the greatly enhanced LW forces, inertia and mode interactions.

Effect of three‐dimensionality on the lift and drag of nominally two‐dimensional cylinders
View Description Hide DescriptionIt has been known for some time that two‐dimensional numerical simulations of flow over nominally two‐dimensional bluff bodies at Reynolds numbers for which the flow is intrinsically three dimensional, lead to inaccurate prediction of the lift and drag forces. In particular, for flow past a normal flat plate (International Symposium on Nonsteady Fluid Dynamics, edited by J. A. Miller and D. P. Telionis, 1990, pp. 455–464) and circular cylinders [J. Wind Eng. Indus. Aerodyn. 35, 275 (1990)], it has been noted that the drag coefficient computed from two‐dimensional simulations is significantly higher than what is obtained from experiments. Furthermore, it has been found that three‐dimensional simulations of flows lead to accurate prediction of drag [J. Wind Eng. Indus. Aerodyn. 35, 275 (1990)]. The underlying cause for this discrepancy is that the surface pressure distribution obtained from two‐dimensional simulations does not match up with that obtained from experiments and three‐dimensional simulations and a number of reasons have been put forward to explain this discrepancy. However, the details of the physical mechanisms that ultimately lead to the inaccurate prediction of surface pressure and consequently the lift and drag, are still not clear. In the present study, results of two‐dimensional and three‐dimensional simulations of flow past elliptic and circular cylinders have been systematically compared in an effort to pinpoint the exact cause for the inaccurate prediction of the lift and drag by two‐dimensional simulations. The overprediction of mean drag force in two‐dimensional simulations is directly traced to higher Reynolds stresses in the wake. It is also found that the discrepancy in the drag between two‐dimensional and three‐dimensional simulations is more pronounced for bluffer cylinders. Finally, the current study also provides a detailed view of how the fluctuation, which are associated with the Kármán vortex shedding in the wake, affect the mean pressure distribution and the aerodynamic forces on the body.

Kinetic alpha effect in viscosity stratified creeping flows
View Description Hide DescriptionIt is shown that viscosity stratified plane Poiseuille flow may exhibit a long‐wavelength instability of a purely kinetic nature formally resembling the so‐called alpha effect known in magnetohydrodynamics or in anisotropic three‐dimensional flows of homogeneous fluids. In the absence of the alpha effect, the system may display a peculiar type of long‐wavelength instability, where the latter is controlled by the surface tension. The weakly nonlinear equation for the evolving interfaces is derived and solved numerically.

Capillary wave scattering from a surfactant domain
View Description Hide DescriptionThe study of capillary wavescattering by a circular region with different interfacial properties from the rest of an otherwise homogeneous interface is motivated by experiments on wave attenuation at a monolayer‐covered air–water interface where domains of one surface phase are dispersed in a second surface phase. Here the scattering function is calculated for an incident wave of frequency ω (wavevector k _{0}) scattering from an isolated circular domain of radius a with surface tension σ_{1} which is imbedded in an otherwise infinite interface of surface tension σ_{0}. The underlying fluid is treated as irrotational and the three‐dimensional flow problem coupling the heterogeneous surface to the underlying liquid is reduced to a set of dual integral equations, which are solved numerically. With this solution the scattering amplitudes and the total scattering cross sections are calculated as a function of the surface tension ratio σ_{0}/σ_{1} and incident wavenumber k _{0} a. The analogous problem of a discontinuous change in bending rigidity is also considered and the solution to the complete viscous problem is outlined in the Appendix. Experimental implications of these results are discussed.

Nonlinear evolution of waves on falling films at high Reynolds numbers
View Description Hide DescriptionA new model describing the dynamics of large‐amplitude waves on laminar falling wavy films at high Reynolds numbers (Re≳300) is presented. The model is based on second‐order boundary layer theory and includes the pressure variation across the film as well as higher‐order viscous terms. The consistency and accuracy of the model is verified by comparing the linear stability results with Kapitza’s classical boundary layermodel and Orr–Sommerfeld studies of the two‐dimensional Navier–Stokes equations. Numerical integration of a traveling wave simplification of the model predicts the existence of chaotic large‐amplitude, nonperiodic waves, as observed in the experiments. The computed wave statistics such as wave celerities, root‐mean‐square (RMS) values of film thickness, probability density function (PDF), and film thickness power spectrum using the present model are in reasonable agreement with those measured on naturally excited fully developed flows at Re≳300. The present model also overcomes the main deficiency of the classical boundary layermodel (namely, negative wall shear stress) predicts large‐amplitude waves (with peak to substrate ratios of 3 to 4) and gives better agreement with data.

Nonlinear transformation of waves in finite water depth
View Description Hide DescriptionThe formulation of a nonlinear frequency domain parabolic mild‐slope model is detailed. The resulting model describes two‐dimensional wave transformation and nonlinear coupling between frequency components. Linear dispersion and transformation characteristics are dictated by fully‐dispersive linear theory, an improvement over weakly‐dispersive Boussinesq theory. Both the present model and a weakly‐dispersive nonlinear frequency domain model are compared to laboratory data for both two‐dimensional wave transformation and pure shoaling. It is found that, in general, data‐model comparisons are enhanced by the present model, particularly in instances where the wave condition is outside the shallow water range.

Resonant generation of internal waves on the soft sea bed by a surface water wave
View Description Hide DescriptionThe nonlinear response of an initially flat sea bed to a monochromatic surface progressive wave was studied using the multiple scale perturbation method. Two opposite‐traveling subliminal internal ‘‘mud’’ waves are selectively excited and form a resonant triad with the surface wave. The amplitudes of the internal waves grow on a time scale much longer than the period of the surface wave. It was found that the sea bed response is critically dependent on the density ratio of water and soil, depth of water, and depth and viscosity of the saturated soil. The result of instability analysis is in qualitative agreement with the result of a wave flume experiment.

On the mutual interaction of two gas bubbles in a sound field
View Description Hide DescriptionThe expression for the mutual interaction force between two pulsating gas bubbles immersed in an incompressible and inviscid fluid is derived assuming that the distance between the bubbles is comparable to their sizes. The results of numerical calculations for air bubbles in water are presented. They show that at small distances between the bubbles the interaction force is substantially different from that given by Bjerknes’ theory [Fields of Force (Columbia University Press, New York, 1906)] in which the separation distance between the bubbles is assumed to be large in comparison with the bubble sizes. The discrepancy seems to result from the effect of multiple scattering which is no longer negligible when the bubbles come close to each other. The present study enables one to understand the mechanism of the formation of the stable bubble clusters (‘‘bubble grapes’’) observed experimentally by Kobelev et al. [Pis’ma Zh. Eksp. Teor. Fiz. 30, 423 (1979)].

Fractal properties of propagating fronts in a strongly stirred fluid
View Description Hide DescriptionThe fractalproperties of propagating aqueous autocatalytic chemical reaction fronts are measured in a capillary‐wave (CW) flow at values of the ratio of the RMS intensity of the fluid velocity fluctuation (u’) to the laminar propagation rate of the front (S _{ L }) up to 220. The images of the fronts are found to exhibit fractal behavior with a fractal dimension (d) of 1.31±0.06, which is very similar to some measurements in gaseous flame fronts, as well as isoscalar contours of passive dyes in CW and other randomly stirred flows. These results suggest that u’/S _{ L }, thermal expansion, variations of viscosity and diffusivity across the flame front, and the turbulence spectrum do not significantly affect d in randomly stirred flows.

Study of tracer dispersion in self‐affine fractures using lattice–gas automata
View Description Hide DescriptionThis paper studies the problem of hydrodynamic dispersion of a tracer in a fluid flowing through a two‐dimensional rough channel bounded by self‐affine surfaces. Changing the surface roughness exponent H, rough walls having different microstructure are obtained. In order to simulate hydrodynamics, a lattice–gas automata modified to introduce two different species of particles is used. In the studied range of Péclet numbers (20–50), the concentration profiles along the channel are well described by Gaussian‐type dispersion. A clear enhancement of the dispersion due to roughness is observed. For the studied regime of Péclet numbers, a simple approach is proposed which allows us to interpret the dispersion enhancement in terms of surface roughness. It is shown that the dispersion enhancement in the rough channel is due to the presence of two characteristic lengths, the hydraulic diameter δ_{H} which determines the velocity in the channel and the average aperture δ_{av} which determines the transverse diffusion length; next shown is that the dispersion in the rough channel varies as D _{∥}∼(δ_{av}/δ_{H})^{2}. The values of δ_{H} obtained from the dispersion results are compared with those obtained from permeability measures and a good agreement is observed. In the studied domain of Péclet numbers, the roughness exponent H has only a weak influence on the dispersion.

Measurements of turbulent velocity fluctuations in a planar Couette cell
View Description Hide DescriptionThe anisotropy of turbulent velocityfluctuations in a planar Couette cell has been investigated by using homodynephotoncorrelation spectroscopy (HCS). We find that 〈δv _{ q }(l)〉, the mean velocity difference between two points separated by a distance l, is consistent with the scaling behavior 〈δv _{ q }(l)〉∝l ^{0.55}. These measurements were made at a Reynolds number Re much greater than Re_{ c }, where the subcritical transition first occurs. The HCS technique was also used to measure Re_{ c } itself.

Mean‐velocity scaling in and around a mild, turbulent separation bubble
View Description Hide DescriptionThis paper examines the question of the scaling of mean‐velocity profiles in adverse‐pressure‐gradient flows. In these flows, the mean velocity scaling must be different than in zero‐pressure‐gradient flows, because the friction velocity used in the latter case can become vanishingly small in the former. Two decades ago, Perry and Schofield [Phys. Fluids 16, 2068 (1973)] proposed a new outer‐region scaling law to be used when the boundary layer approaches separation. Since that time, a number of sets of experimental data close to separation have been shown to fall on a universal curve when the profiles are plotted in Perry–Schofield coordinates, and the profile shape was given by Dengel and Fernholz [J. Fluid Mech. 212, 615 (1990)]. Recently, however, a new set of scaling laws has been proposed by Durbin and Belcher [J. Fluid Mech. 238, 699 (1992)] as a result of their asymptotic analysis, in which they assumed the appropriate near‐wall velocity scale to be based on the local strength of the pressure gradient. The resulting scaling laws are different than Perry and Schofield’s scaling and, in fact, predict a three‐layered rather than a two‐layered boundary‐layer structure. Here, experimental results are shown for an adverse‐pressure‐gradient boundary layer which separates from and then reattaches to a smooth surface. These data provide a wide range of flow conditions for comparing the conflicting scaling laws mentioned above, under conditions of both decreasing and increasing skin friction, with and without instantaneous reverse flow.
It is found that the Perry–Schofield coordinates provide better collapse, over a wider range of streamwise positions and over a larger fraction of the boundary layer, than the scaling laws of Durbin and Belcher. Other proposed scaling laws are also evaluated. Yaglom’s half‐power law is shown to hold for a subset of the profiles which fall on Dengel and Fernholz’s universal profile. And the data provide a test of the range of validity of the (zero‐pressure‐gradient) logarithmic law of the wall. The law is violated here when instantaneous reverse flow exists in the boundary layer and/or when the local pressure gradient is strong enough, as is consistent with earlier work. However, after reattachment these criteria are insufficient to indicate the return to the log law, and several bubble lengths are required after reattachment before the universal log law is satisfied. The wake region responds to reattachment more slowly and does not appear fully recovered six bubble lengths (twenty boundary‐layer thicknesses) after reattachment.

The self‐similar topology of passive interfaces advected by two‐dimensional turbulent‐like flows
View Description Hide DescriptionWe study the topology, and in particular the self‐similar and space‐filling properties of the topology of line‐interfaces passively advected by five different 2‐D turbulent‐like velocity fields. Special attention is given to three fundamental aspects of the flow: the time unsteadiness, the classification of local spatial flow structure in terms of hyperbolic and elliptic points borrowed from the study of phase spaces in dynamical systems and a classification of flow structure in wavenumber space derived from the studies of Weierstrass and related functions. The methods of analysis are based on a classification of interfacial scaling topologies in terms of K‐ and H‐ fractals, and on two interfacial scaling exponents, the Kolmogorov capacity D _{ K } and the dimension D introduced by Fung and Vassilicos [Phys. Fluids 11, 2725 (1991)] who conjectured that D≳1 implies that the interface is H‐fractal. An argument is presented (in the Appendix) to show that D≳1 is a necessary condition for the evolving interface to be H‐fractal through the action of the flow, and that D≳1 is also sufficient provided that no isolated regions exist where the flow velocity is either unbounded or undefined in finite time. D is interpreted to be a degree of H‐fractality and is different from the Hausdorff dimension D _{ H }. In all our flows, steady and unsteady, interfaces in particular realisations of the flow reach a non‐space‐filling steady self‐similar state where D and D _{ K } are both constant in time even though the interface continues to be advected and deformed by the flow.
It is found that D is equal to 1 in 2‐D steady flows and always increases with unsteadiness, that D _{ K } generally decreases with unsteadiness where the interfacial topology is dominated by spirals, and that D _{ K } increases with unsteadiness where the interfacial topology is dominated by tendrils. In those flows with larger number of modes, D _{ K } is a non‐increasing function of unsteadiness and a decreasing function of the exponent p of the flow’s self‐similar energy spectrum E(k)∼k ^{−p }. D _{ K }’s decreasing dependences on unsteadiness and the exponent p can be explained by the presence of spirals in the eddy regions of the flow. The values of D and D _{ K } and their dependence on unsteadiness can change significantly only by changing the distribution of wavenumbers in wavenumber space while keeping the phases and energy spectrum constant.

Scaling properties of differential molecular diffusion effects in turbulence
View Description Hide DescriptionScaling properties of a normalized concentration difference in a turbulent flow containing two scalars of unequal diffusivity are determined by similarity analysis and numerical simulation. Similarity hypotheses applied to the power spectrum of the normalized concentration difference, termed the differential diffusion, yield predicted dependences of the variance of the differential diffusion on the turbulenceReynolds number (Re) and on the Schmidt numbers (Sc) of the scalars. In particular, the variance is found to be proportional to Re^{−1/2}. This and other predictions are supported by numerical simulations of multiple scalar mixing using a one‐dimensional stochastic mixing model. The analysis and numerical results indicate fundamental distinctions between the physical mechanisms governing the scalar spectral cascade and those governing spectral transfer of the differential diffusion. The relationships of predicted scalings to passive mixing measurements that have been reported and to behaviors expected in reacting flow are noted.

Intense vortical structures in grid‐generated turbulence
View Description Hide DescriptionThis paper presents a set of experiments aimed at investigating the features and the statistical frequency of intense vortical structures (sometimes called ‘‘filaments’’, or ‘‘worms’’) as manifested by a migrating bubble technique in a mean shear free, homogeneous, isotropic, stationary turbulence generated by oscillating grids in a water tank for R _{λ} reaching up to 300. It is found that the nucleation of filaments at the surface of the walls of the tank, where boundary layers are liable to destabilize is much more frequent than in the homogeneous bulk of the tank where one filament is typically detected each hundred large scale turnover time. This distinction between the wall surface and the bulk activity, supplemented with the fact that the size of the filaments and their lifetime compare with the length and time‐scales of the largest structures of the flow leads us to formulate an elementary model explaining the origin and the geometrical features of these intense vortical structures in turbulent flows for arbitrary Reynolds numbers.

Influence of a strong adverse pressure gradient on the turbulent structure in a boundary layer
View Description Hide DescriptionA comparison between the turbulent structures found in a zero pressure gradient boundary layer and a boundary layer subjected to a strong adverse pressure gradient is presented. The pressure gradient reverses the direction of the dominant turbulent diffusion, resulting in considerable turbulenttransport towards the wall. Two‐point space–time correlations and the invariants show that this reduces the anisotropy in the near wall region and indicate an important reflection of the turbulent motion from the wall back into the outer layer. This is verified by a quadrant analysis [Lu and Willmarth, J. Fluid Mech. 60, 481 (1973)] which demonstrates that the strong events near the wall are totally dominated by motions in the first and fourth quadrants.

Approximation of subgrid‐scale energy transfer based on the dynamics of resolved scales of turbulence
View Description Hide DescriptionPreviously established properties of subgrid‐scale nonlinear interactions in turbulent flows at low Reynolds numbers are employed to approximate the subgrid‐scale energy transfer in terms of the resolved scales for statistically stationary channel flow. The approximation is obtained by focusing on the dynamics of resolved scales in the vicinity of the cutoff wave number. Predicted plane‐averaged transfer is in a good agreement with the exactly computed quantity. The modeled forward transfer and inverse transfer are comparable, similarly to the case of the exact transfer, but each is overestimated by the model. The exact and the modeled transfers exhibit only moderate pointwise correlation, which increases if conditional probabilities of forward and inverse transfers are considered. Probability distribution functions for modeled and exact transfers are in a good qualitative agreement. In all cases the results provided by the approximation are at least comparable to, and generally better than those provided by the classical Smagorinsky model.