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Volume 8, Issue 2, February 1996

Pulsed gradient NMR measurements and numerical simulation of flow velocity distribution in sphere packings
View Description Hide DescriptionThe displacement of water molecules associated with the flow of water inside a nonconsolidated packing of 800 μm OD glass spheres has been measured by a pulsed gradient NMR technique. Using a stimulated spin‐echo sequence, mean displacements of up to 300 μm corresponding to measurement times of up to 200 ms can be analyzed. The measurement can be quantitatively calibrated using the pure molecular self‐diffusion of water at zero flow conditions. For molecular displacements much smaller than the pore size, the distribution of the flow velocity component along the mean flow direction is determined at Reynolds numbers high enough so that longitudinal molecular diffusion is negligible. An exponential decay of the probability distribution of the displacements is observed at large distances. The results are very similar to those obtained by numerical solution of the Stokes equation in random sphere packings. At longer displacement distances, a secondary peak of the displacement distribution is observed: It is interpreted as the first step toward the transition toward classical dispersion at displacements much larger than the pore size. The influence of molecular diffusion and of the heterogeneities of the magnetic permeability also are discussed.

Investigation of wetting hydrodynamics using numerical simulations
View Description Hide DescriptionMeniscus shapes from a simulation of a plate immersing into an infinitely deep liquid bath, for a range of outer length scales, have been obtained numerically. These have been compared with the leading‐order prediction from a three‐region asymptotic analysis done in the double limit, Capillary number, Ca→0, L _{ S }/L _{ C }→0, with Ca ln(L _{ C }/L _{ S }) of O(1), where L _{ S } and L _{ C } represent the slip length and an outer macroscopic length, respectively. For Ca<0.01, the numerically computed and the perturbationsolutions show excellent agreement. Within this range of Ca, the meniscus slope at a distance 10L _{ S } from the dynamic contact line is geometry independent, that is, does not vary with changes in the outer length L _{ C }. The interface slope at this point can serve as an appropriate materialboundary condition for the outer problem. For 0.01<Ca<0.1, the intermediate region solution continues to closely fit the numerically generated solution, while the match in the outer region begins to degrade. By monitoring the pressure difference between the surrounding inviscid gas phase and arbitrarily chosen point in the liquid, we attribute this breakdown to infiltration of viscous effects into the outer region, so that static capillarity does not adequately describe meniscus shapes in this regime. For Ca≳0.1, there is no match between the numerical and perturbationsolutions in both the intermediate and outer regions, indicating that higher‐order contributions must be accounted for in the perturbationsolutions.

The effect of surfactant on the rise of a spherical bubble at high Reynolds and Peclet numbers
View Description Hide DescriptionExperiments and numerical simulations of rising spherical bubbles in quiescent surfactantsolutions are presented. The rise velocities versus the concentration in the bulk are measured using three surfactants,Triton X_{100}, Brij_{30} and SDS for different bubble sizes, between 0.4 and 1 mm equivalent radius. We also present a brief description of the finite‐difference numerical method developed to solve the full Navier‐Stokes equations around the contaminated bubble for Reynolds numbers ranging from 50 to 200. The distributions of the tangential velocity, the vorticity, the pressure and the surfactant concentration on the bubble surface are calculated. In the case of high Peclet numbers surfactant molecules, which adsorb on the surface are convected and collected at the rear part of the bubble forming a stagnant cap where the no‐slip condition holds. The concentration on the bubble interface is obtained for surfactants having a desorption rate much slower than the convective rate. The sudden increase of the shear stress and pressure at the leading edge of the cap contributes mainly to decrease the rise velocity. This rapid slowdown of the bubble occurs when nearly half of the bubble surface is covered by the surfactant layer, and this is due to the particularly high values obtained for the shear stress and the pressure at the leading edge of this cap‐angle. Measured and calculated rise velocities for bubbles of 0.4 mm equivalent radius show good agreement when the sorption kinetics controls the surfactant exchange between the bulk and the surface. Calculated critical concentrations needed to cover completely the bubble agree with the measurements even for larger bubbles.

Shock waves in a liquid containing small gas bubbles
View Description Hide DescriptionNumerical and experimental studies of the transient shock wave phenomena in a liquid containing non‐condensable gas bubbles are presented. In the numerical analysis, individual bubbles are tracked to estimate the effect of volume oscillations on the wave phenomena. Thermal processes inside each bubble, which have significant influence on the volume oscillation, are calculated directly using full equations for mass, momentum and energy conservation, and those results are combined with the averaged conservation equations of the bubbly mixture to simulate the propagation of the shock wave. A silicone oil/nitrogen bubble mixture, in which the initial bubble radius is about 0.6 mm and the gas volume fraction is 0.15% – 0.4%, is used in the shock tube experiments. The inner diameter of the shock tube is chosen to be 18 mm and 52 mm in order to investigate the multidimensional effects on the wave phenomena. In a fairly uniform bubbly mixture, the experimental results agree well with the numerical ones computed using a uniform spatial distribution of bubbles. On the other hand, in all the other experiments, the bubbles in the shock tubes are not distributed uniformly, being relatively concentrated along the axis of the tube. This non‐uniformity substantially alters the profile of the shock waves. The numerical predictions where such a distribution is taken into account agree well with those experimental data.

A numerical study of three‐dimensional bubble merger in the Rayleigh–Taylor instability
View Description Hide DescriptionThe Rayleigh–Taylor instability arises when a heavy fluid adjacent to a light fluid is accelerated in a direction against the density gradient. Under this unstable configuration, a perturbation mode of small amplitude grows into bubbles of the light fluid and spikes of the heavy fluid. Taylor discovered the steady state motion with constant velocity for a single bubble or periodic bubbles in the Rayleigh–Taylor instability. Read and Youngs studied the motion of a randomly perturbed fluid interface in the Rayleigh–Taylor instability. They reported constant acceleration for the overall bubble envelope. Bubble merger is believed to cause the transition from constant velocity to constant acceleration. In this paper, we present a numerical study of this important physical phenomenon. It analyzes the physical process of bubble merger and the relationship between the horizontal bubble expansion and the vertical interface acceleration. A dynamic bubble velocity, beyond Taylor’s steady state value, is observed during the merger process. It is believed that this velocity is due to the superposition of the bubble velocity with a secondary subharmonic unstable mode. The numerical results are compared with experiments.

Singularity formation in Hele–Shaw bubbles
View Description Hide DescriptionWe provide numerical and analytic evidence for the formation of a singularity driven only by surface tension in the mathematical model describing a two‐dimensional Hele–Shaw cell with no air injection. Constantin and Pugh have proved that no such singularity is possible if the initial shape is close to a circle; thus we show that their result is not true in general. Our evidence takes the form of direct numerical simulation of the full problem, including a careful assessment of the effects of limited spatial resolution, and comparison of the full problem with the lubrication approximation.

Wetting front instability in randomly stratified soils
View Description Hide DescriptionA probabilistic criterion is derived for the onset of wetting front instability during surface waterinfiltration into a randomly stratified soil. It is based on the common assumption that the natural log hydraulic conductivity of the soil is a random, multivariate Gaussian function of space. Whereas the mean (expectation) of this function may exhibit a drift, its fluctuations about the mean are statistically homogeneous with constant variance and autocorrelation scale. The wetting front is taken to form a sharp boundary. Closed‐form expressions for the probability of instability, and for the mean critical wave number, are obtained either directly or via a first‐order reliability method. Monte Carlo simulations are used to verify these analytical solutions as well as to determine the mean maximum rate of incipient finger growth and corresponding mean wave number. The effects of applied pressure gradient, capillary pressure head at the wetting front, and statistical parameters of the hydraulic conductivity field on instability and incipient finger growth are investigated for a wide range of these variables.

Determination of surface tension from the shape oscillations of an electromagnetically levitated droplet
View Description Hide DescriptionIn the fundamental (l=2) mode, the frequency spectrum of a magnetically levitated inviscid droplet exhibits three distinct peaks. If the modes that correspond to each of these peaks is known, the surface tension of the droplet may be calculated. In experiments that make use of this principle, there is no unambiguous method of assigning mode numbers to these peaks. The dynamics of the oscillating droplet depend on the magnetic pressure on the dropletsurface. Consequently, the order of the peaks in the l=2 mode oscillations is determined by the magnetic pressure distribution. In this paper, the magnetic pressure distribution on the surface of the droplet is calculated as a function of the parameters that govern the external magnetic field. The frequencies of the droplet oscillation and its static shape deformation are also expressed in terms of these same parameters. The frequencies of oscillation are used to determine the surface tension of the liquiddroplet. Finally, the magnetic pressure distribution on the droplet is shown to yield the well‐known ‘‘pear‐like’’ shape that is assumed by liquid metaldroplets in a conical levitator.

Starting and steady quadrupolar flow
View Description Hide DescriptionPlanar flow induced in a viscous fluid by a small cylinder oscillating in the direction normal to its axis is modeled theoretically and reproduced experimentally. In the model, a line force dipole (force doublet) was used as the source of motion. In an initially quiescent unbounded fluid this source produces zero net momentum and generates symmetrical quadrupolar flow consisting of two dipolar vorticity fronts propagating in opposite directions from the source. For starting flow at low Reynolds numbers, a second‐order unsteady solution is obtained in terms of a power series of the Reynolds number, Re=Q/4πν^{2}, where Q is the forcing amplitude and ν is the kinematicviscosity. This solution demonstrates that, as time t→∞, the flow in the vicinity of the source becomes steady and radial. To describe this steady asymptote, the Jeffery–Hamel nonlinear solution for radial flow is used. A particular solution is derived using the nondimensional intensity Re of the force dipole as a governing parameter. It is shown that the problem permits a similarity solution for all values of Re when a mass sink of prescribed intensity q=q(Re) is added to the flow. This steady asymptote is reproduced experimentally, using a vertical porous cylinder that oscillates horizontally in the shallow upper layer of a two‐layer fluid and sucks fluid through its porous walls.

Internal capillary‐gravity waves of a two‐layer fluid with free surface over an obstruction—Forced extended KdV equation
View Description Hide DescriptionIn this paper we study steady capillary‐gravity waves in a two‐layer fluid bounded above by a free surface and below by a horizontal rigid boundary with a small obstruction. Two critical speeds for the waves are obtained. Near the smaller critical speed, the derivation of the usual forced KdV equation (FKdV) fails when the coefficient of the nonlinear term in the FKdV vanishes. To overcome this difficulty, a new equation, called a forced extended KdV equation (FEKdV) governing interfacial wave forms, is obtained by a refined asymptotic method. Various solutions and numerical results of this equation are presented.

Experimental study of incompressible Richtmyer–Meshkov instability
View Description Hide DescriptionThe Richtmyer–Meshkov instability of a two‐liquid system is investigated experimentally. These experiments utilize a novel technique that circumvents many of the experimental difficulties that have previously limited the study of Richtmyer–Meshkov instability. The instability is generated by vertically accelerating a tank containing two stratified liquids by bouncing it off of a fixed coil spring. A controlled two‐dimensional sinusoidal initial shape is given to the interface by oscillating the container in the horizontal direction to produce standing waves. The motion of the interface is recorded during the experiments using standard video photography. Instability growth rates are measured and compared with existing linear theory. Disagreement between measured growth rates and the theory are accredited to the finite bounce length. When the linear stability theory is modified to account for an acceleration pulse of finite duration, much better agreement is attained. Late time growth curves of many different experiments seem to collapse to a single curve when correlated with the circulation deposited by the impulsive acceleration. A theory based on modeling the late time evolution of the instability using a row of vortices is developed. The growth curve given by this model has similar shape to those measured, but underestimates the late‐time growth rate.

A Lagrangian for water waves
View Description Hide DescriptionA Lagrangian for strongly nonlinear unsteady water waves (including overturning waves) is obtained. It is shown that the system of quadratic equations for the Stokes coefficients, which determine the shape of a steady wave (discovered by Longuet‐Higgins 100 years after Stokes derived his system of cubic equations) directly follows from the canonical system of Lagrange equations. Applications to the investigation of the stability of water waves and to the construction of numerical schemes are pointed out.

Interaction of laminar far wake with a free surface
View Description Hide DescriptionWave disturbances caused by the uniform translatory motion of a submerged body on or beneath the free surface of a viscous fluid are investigated analytically. The submerged body is idealized as an Oseenlet or an Oseen doublet, and exact solutions in closed integral forms are obtained. Based on these exact solutions, asymptotic representations of the wave amplitude for large Reynolds numbers based on the deep‐water wavelength at large distances downstream of the body are derived. The results obtained show explicitly the effect of the laminar wake on the amplitude and the phase of the surface waves thus created.

Internal gravity wave radiation into weakly stratified fluid
View Description Hide DescriptionIt is shown by way of nonlinear numerical simulations of flow restricted to two dimensions that a compact wavepacket of large‐amplitude internal gravity waves incident upon a weakly stratified region in which the buoyancy frequency is less than the frequency of the wavepacket may partially transmit energy into this region through the generation of a wavepacket of lower frequency. In part, the transmission of waves occurs due to the transient nature of the forcing by the incident wavepacket, but if the amplitude of the wavepacket is moderately large, weakly nonlinear effects may act to significantly increase the proportion of the wavepacket that is transmitted. For a range of simulations initialized with wavepackets of different amplitude and vertical extent, the characteristics of the reflected and transmitted waves are analyzed and reflection coefficients are calculated. An explanation for how the nonlinear transmission mechanism operates is given by demonstrating that the wave induced mean‐flow, which is shown to be approximately equal to the horizontal wave pseudomomentum expressed in Eulerian variables, acts to adjust the frequency of the incident waves.

A geometrical interpretation of force on a translating body in rotational flow
View Description Hide DescriptionSome recent results for the force on a translating rigid three‐dimensional body in incompressible flow, in which the integration is over the vorticity field rather than surface pressure, are interpreted from a point of view that distinguishes changes of fluid impulse directly attributable to the vorticity field from those due to its image system in the body. An expression is first derived geometrically for a sphere in inviscid fluid; the flow is taken to consist of discrete vortex loops whose change in impulse, and that of the image system in the sphere, are calculated via their projected areas. As an example, the force on a sphere due to an infinite line vortex is calculated exactly. To generalize the geometrical derivation to bodies of any shape, a reciprocal theorem is proved concerning the impulse of the image system of a dipole. This yields the inviscid form of a result derived mathematically by Howe [J. Fluid Mech. 206, 131 (1989)]. Physical interpretations of the various terms in Howe’s expression are offered, and the relationship to a very similar independent result by Chang [Proc. R. Soc. London Ser. A 437, 517 (1992)] is discussed.

Görtler vortices in boundary layers with streamwise pressure gradient: Linear theory
View Description Hide DescriptionLinear theory is used to analyze the stability of two‐dimensional boundary layer flows to stationary Görtler vortices. The basic flow profiles in the boundary layer are described by the Falkner–Skan similarity solutions. We approach the problem both with local linear theory (with the streamwise position held fixed) and with a streamwise marching technique (to represent the evolution of the inlet disturbance). Comparisons of solutions obtained by the two methods are presented: The results are consistent in showing that adverse pressure gradients are destabilizing, as in the case of Tollmien–Schlichting waves. This is at odds with recent findings by Otto and Denier and underscores the sensitivity of the results to initial conditions.

Stability of Newtonian and viscoelastic dynamic contact lines
View Description Hide DescriptionThe stability of the moving contact line is examined for both Newtonian and viscoelastic fluids. Two methods for relieving the contact line singularity are chosen: matching the free surface profile to a precursor film of thickness b, and introducing slip at the solid substrate. The linear stability of the Newtonian capillary ridge with the precursor film model was first examined by Troian et al. [Europhys. Lett. 10, 25 (1989)]. Using energy analysis, we show that in this case the stability of the advancing capillary ridge is governed by rearrangement of fluid in the flow direction, whereby thicker regions develop that advance more rapidly under the influence of a body force. In addition, we solve the Newtonian linear stability problem for the slip model and obtain results very similar to those from the precursor film model. Interestingly, stability results for the two models compare quantitatively when the precursor film thickness b is numerically equal to the slip parameter α. With the slip model, it is possible to examine the effect of contact angle on the stability of the advancing front, which, for small contact angles, was found to be independent of the contact angle. The stability of an Oldroyd‐B fluid was examined via perturbation theory in Weissenberg number. It is found that elastic effects tend to stabilize the capillary ridge for the precursor film model, and this effect is more pronounced as the precursor film thickness is reduced. The perturbation result was examined in detail, indicating that viscoelastic stabilization arises primarily due to changes of momentum transfer in the flow direction, while elasticity has little effect on the response of the fluid to flow in the spanwise direction.

Control of laminar vortex shedding behind a circular cylinder using splitter plates
View Description Hide DescriptionLaminar vortex shedding behind a circular cylinder and its control using splitter plates attached to the cylinder are simulated. The vortex shedding behind a circular cylinder completely disappears when the length of the splitter plate is larger than a critical length, and this critical length is found to be proportional to the Reynolds number. The Strouhal number of the vortex shedding is rapidly decreasing with the increased plate length until the plate length (l) is nearly the same as the cylinder diameter (d). On the other hand, at 1<l/d<2, the control shows two different behaviors for the Reynolds numbers investigated. The net drag is significantly reduced by the splitter plate, and there exists an optimum length of the plate for minimum drag at a given Reynolds number. From an examination of the instantaneous flow fields, it is found that the Strouhal number modification by the splitter plate is closely related to the size of the primary vortex behind the cylinder and the length of the plate.

Stability of periodic arrays of vortices
View Description Hide DescriptionThe stability of periodic arrays of Mallier–Maslowe or Kelvin–Stuart vortices is discussed. We derive with the energy‐Casimir stability method the nonlinear stability of this solution in the inviscid case as a function of the solution parameters and of the domain size. We exhibit the maximum size of the domain for which the vortex street is stable. By adapting a numerical time‐stepping code, we calculate the linear stability of the Mallier–Maslowe solution in the presence of viscosity and compensating forcing. Finally, the results are discussed and compared to a recent experiment in fluids performed by Tabeling et al. [Europhy. Lett. 3, 459 (1987)]. Electromagnetically driven counter‐rotating vortices are unstable above a critical electric current, and give way to co‐rotating vortices. The importance of the friction at the bottom of the experimental apparatus is also discussed.

Experimental study of rotating disk instability. I. Natural flow
View Description Hide DescriptionThis article is devoted to the study of the rotating disk flow instability. This inflectional‐type instability, called a cross‐flow instability, exemplary of the transition to turbulence in three‐dimensional boundary layers. We first present the experimental marginal stability curve of unstable waves obtained by hot‐film probe measurements and compare it with the theoretical results available in the literature. The experiment is in accordance with different theoretical determinations of the linear threshold, but we note a difference between experimental and theoretical critical wave‐number values. The unstable wave dynamics is then investigated by means of experimental dispersion curves (linking frequencies to the wave‐number vector components) determined by two‐probe measurements. The results show, in particular, the existence of traveling dispersive waves in the boundary layer of the rotating disk. Finally, we show that the emergence of nonlinear effects occurs very early in the system, far from the transition point.