Volume 11, Issue 1, January 1999
 LETTERS


Tumbling cards
View Description Hide DescriptionWhen a stiff rectangular card is dropped in still air with its long axis horizontal, it often settles into a regular mode of motion; while revolving around its long axis it descends along a path that is inclined to the vertical at a nearly constant angle. We show experimentally that the tumbling frequency of a card of length width w and thickness d scales as consistent with a simple dimensional argument that balances the drag against gravity.

Bluffbody propulsion produced by combined rotary and translational oscillation
View Description Hide DescriptionFlows produced by combined oscillatory translation and rotation of a circular cylinder in quiescent fluid have been studied using a twodimensional direct numerical simulation technique. Results are presented for one set of the five dimensionless groups which characterize these flows, for which it is found that a streaming flow normal to the axis of translation is generated. The consequent reaction force is then used to propel the cylinder in the direction opposite to the jet, thus demonstrating a novel propulsion mechanism for bluff bodies.

Entropylayer instabilities over a blunted flat plate in supersonic flow
View Description Hide DescriptionEntropylayer instabilities in the supersonic flow over a blunted flat plate with a nose radius of at a Mach number and a unit Reynolds number are investigated. A type of entropylayer instability most amplified at zero wave angle is found. Its streamwise wave number and the wallnormal phase distribution of the numerically obtained density eigenfunctions agree with flowfield visualization using the Schlieren technique. The frequencies of this entropylayer instability made nondimensional on the basis of an appropriate entropylayer length scale matches documented experimental observations at a blunted cone.
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 ARTICLES


Shock formation within sonoluminescence bubbles
View Description Hide DescriptionA strong case has been made by several authors that sharp, spherically symmetric shocks converging on the center of a spherical bubble driven by a strong acoustic field give rise to rapid compression and heating that produces the brief flash of light known as sonoluminescence. The formation of such shocks is considered. It is found that, although at the main collapse the bubble wall does indeed launch an inwardlytraveling compression wave, and although the subsequent reflection of the wave at the bubble center produces a very rapid temperature peak, the wave is prevented from steepening into a sharp shock by an adverse gradient in the sound speed caused by heat transfer. It is shown that the mathematical characteristics of the flow can be prevented from accumulating into a shock front by this adverse sound speed gradient. A range of results is presented for a variety of bubble ambient radii and sound field amplitudes suggested by experiments. The time scale of the peak temperature in the bubble is set by the dynamics of the compression wave: this is typically in the range 100–300 ps (FWHM) in concert with recent measurements of the sonoluminescence pulse width.

Numerical investigation of thermocapillary flow around a bubble
View Description Hide DescriptionThe thermocapillary flow induced by a gas bubble in a Newtonian liquid layer subjected to a stable temperature stratification is investigated. This flow is analyzed for a special configuration when the surface tension and buoyant forces oppose one another. The driving mechanism is the surface tension gradient related to the Marangoni number whereas the stabilizing effects are the viscous and buoyant forces related to the Prandtl and the Rayleigh numbers. In a previous work, this flow has been investigated experimentally for a few combinations of these three parameters. In order to make a more systematic study of the influence of these parameters, numerical simulations are used as a decisive tool. Indeed, it allows the contribution of the different mechanisms to be evaluated. To validate the finite element model, developed for this purpose, the numerical results are first compared to experimental ones. Then, the influence of these three dimensionless parameters on the flow pattern and the magnitude of the flow is analyzed. This sensitivity study is supplemented by a convergence study. It appears that the Rayleigh number modifies the flow pattern but has little influence on the strength of the primary vortex induced by the bubble. On the other hand, the Marangoni and Prandtl numbers induce little change in the flow pattern but they respectively enhance and reduce the strength of the primary vortex. This systematic analysis leads us to propose an empirical relationship for the strength of the flow.

Spin coating and airjet blowing of thin viscous drops
View Description Hide DescriptionUsing the lubrication approximation we investigate the spreading of a finitesized thin drop of incompressible Newtonian fluid on a planar substrate subjected to a jet of air blowing normally to the substrate. Three specific problems are studied in detail: a jet of air acting normally to the substrate when gravity effects are negligible, a jet of air directed vertically downward onto a sessile drop on a horizontal substrate, and a jet of air directed vertically upward onto a pendent drop on a horizontal substrate. The dynamics of the moving contact line are modeled by means of a generalized “Tanner Law” relating the contact angle to the speed of the contact line. Both symmetric twodimensional and axisymmetric threedimensional geometries are considered; the latter case is equivalent to that of an axisymmetric drop spreading on a substrate rotating with constant angular velocity (the simplest model for spin coating). Quasistatic solutions for the drop profile are obtained in the limit of small capillary number and the stability of equilibrium solutions to small perturbations is determined. The analysis is then extended to include drops with a dry patch at their center which are found to be unconditionally unstable in all the cases investigated.

Spreading and imbibition of viscous liquid on a porous base
View Description Hide DescriptionThe competition between the viscous spreading of liquid on a substrate and the absorption by the substrate is studied using several models. The local behaviors near the contact lines, the time scales of droplet spreading and disappearance, and the dependencies on the physical factors that enter are discussed.

Sand ripples in an oscillating annular sand–water cell
View Description Hide DescriptionSand ripples, familiar from ocean floors, are studied by oscillating an annular sand–water cell. Empirical rules for the onset and disappearance of ripples are uncovered and compared with simple dimensional arguments, and with prior experimental and theoretical work. In addition the dynamical behavior of sand ripples are investigated, and the existence of a secondary instability where small and large wavelengths coexist in space is shown. Finally, by visualizing the fluid flow above the sand ripples the paper describes vortices which are an integral part of the ripple formation and show that there is a strong coupling between two different types of vortices and the sand ripple pattern itself.

Effects of horizontal vibration on hopper flows of granular materials
View Description Hide DescriptionThe current experiments investigate the discharge of glass spheres in a planar wedgeshaped hopper (45° sidewalls) that is vibrated horizontally. When the hopper is discharged without vibration, the discharge occurs as a funnel flow, with the material exiting the central region of the hopper and stagnant material along the sides. With horizontal vibration, the discharge rate increases with the velocity of vibration as compared with the discharge rate without vibration. For a certain range of acceleration parameters (20–35 Hz and accelerations greater than about 1 g), the discharge of the material occurs in an invertedfunnel pattern, with the material along the sides exiting first, followed by the material in the core; the free surface shows a peak at the center of the hopper with the free surface particles avalanching from the center toward the sides. During the deceleration phase of a vibration cycle, particles all along the trailing or lowpressure wall separate from the surface and fall under gravity for a short period before reconnecting the hopper. For lower frequencies (5 and 10 Hz), the free surface remains horizontal and the material appears to discharge uniformly from the hopper.

Microscopic motion of particles flowing through a porous medium
View Description Hide DescriptionStokesian dynamics simulations are used to study the microscopic motion of particles suspended in fluids passing through porous media.Modelporous media with fixed spherical particles are constructed, and mobile ones move through this fixed bed under the action of an ambient velocity field. The pore scale motion of individual suspended particles at pore junctions are first considered. The relative particle flux into different possible directions exiting from a single pore, for two and threedimensional modelporous media is found to approximately equal the corresponding fractional channel width or area. Next the waiting time distribution for particles which are delayed in a junction due to a stagnation point caused by a flow bifurcation is considered. The waiting times are found to be controlled by twoparticle interactions, and the distributions take the same form in modelporous media as in twoparticle systems. A simple theoretical estimate of the waiting time is consistent with the simulations. It is found that perturbing such a slowmoving particle by another nearby one leads to rather complicated behavior. Finally, the stability of geometrically trapped particles is studied. For simple model traps, it is found that particles passing nearby can “relaunch” the trapped particle through its hydrodynamic interaction, although the conditions for relaunching depend sensitively on the details of the trap and its surroundings.

Influence of the geometry on the structure of the flow between a pair of corotating disks
View Description Hide DescriptionConstantproperty laminar flow in the space between two coaxial disks corotating in a fixed cylindrical enclosure is investigated numerically over the Reynolds number range for interdisk spacing to disk radius aspect ratios in the range Most of the 3D calculations correspond to regions in the (Re, S) map slightly above the 2D–3D transition, where the 2D (axisymmetric) steady flow can bifurcate into two different families of unsteady 3D flow; (a) at sufficiently large values of S or Re, into a flow that is asymmetric with respect to the interdisk midplane; (b) at intermediate values of S and Re, into a flow that displays shiftandreflect symmetry with respect to the interdisk midplane. Qualitative and quantitative comparisons with previous experimental and numerical results are established. The structure of the 3D flow is analyzed as well as the jumps in wave frequency when it experiences a change in its circumferential wave length. It is shown that each family of 3D flows is characterized by a distinct behavior in the frequency/wave length transitions. Present calculations and analysis provide the necessary additional evidence about the duality of this class of flows to reinterpret the numerical and experimental results previously reported by Humphrey et al. [Phys. Fluids 7, 1225 (1995)] and Abrahamson et al. [Phys. Fluids A 1, 241 (1989)].

Axial instability of a freesurface front in a partially filled horizontal rotating cylinder
View Description Hide DescriptionWe investigate the axial instability of the freesurface front of a viscous fluid in a horizontal cylinder rotating about its longitudinal axis. A simplified model equation for the evolution of the free surface is derived and includes the effects of gravity, capillarity, inertia, and viscosity. This equation is solved numerically to determine the base state with no axial variation, and a numerical linear stability analysis is carried out to examine the onset of unstable axial modes. Various computational results are presented for the wavelength of the axial instability. Inertia is found to play an important role in the onset of the instability and the wavelength of the instability λ satisfies the power law where γ is surface tension. Finally some numerical simulations of the simplified evolution equation are presented to show that they can capture the steady sharkteeth patterns observed in recent experiments [R. E. Johnson, in Engineering Science, Fluid Dynamics: A Symposium to Honor T. Y. Wu (World Scientific, Singapore, 1990), pp. 435–449; S. T. Thoroddsen and L. Mahadevan, “Experimental studies of the instabilities in a partially filled horizontal rotating cylinder,” Exp. Fluids 23, 1 (1997)].

Finiteamplitude topographic Rossby waves in a channel
View Description Hide DescriptionThis paper describes Rossby waves propagating along a channel with a discontinuity in potential vorticity. A finiteamplitude, longwave equation giving the displacement of the material interface separating two constant vorticity regions is derived and solved for both steady and unsteady finiteamplitude waves. Finiteamplitude Rossby waves that propagate in the opposite direction to infinitesimal waves are identified, and the mechanism for these waves is elucidated. The full parameter space of possible flows is described quantitatively. The weakly nonlinear limit of this finiteamplitude equation is also presented, including the novel case of kink solitons in an equation with only quadratic nonlinearity.

Twodimensional Navier–Stokes simulation of breaking waves
View Description Hide DescriptionNumerical simulations describing plunging breakers including the splashup phenomenon are presented. The motion is governed by the classical, incompressible, twodimensional Navier–Stokes equation. The numerical modeling of this twophase flow is based on a piecewise linear version of the volume of fluid method. Capillary effects are taken into account such as a nonisotropic stress tensor concentrated near the interface. Results concerning the time evolution of liquid–gas interface and velocity field are given for short waves, showing how an initial steep wave undergoes breaking and successive splashup cycles. Breaking processes including overturning, splashup and gas entrainment, and breaking induced vortexlike motion beneath the surface and energy dissipation, are presented and discussed. It is found that strong vorticities are generated during the breaking process, and that more than of the total prebreaking wave energy is dissipated within three wave periods. The numerical results are compared with some laboratory measurements, and a favorable agreement is found.

Optimal disturbances and bypass transition in boundary layers
View Description Hide DescriptionStreamwise streaks are ubiquitous in transitional boundary layers, particularly when subjected to high levels of freestream turbulence. Using the steady boundarylayer approximation, the upstream disturbances experiencing maximum spatial energy growth are numerically calculated. The calculations use techniques commonly employed when solving optimalcontrol problems for distributed parameter systems. The calculated optimal disturbances consist of streamwise vortices developing into streamwise streaks. The maximum spatial energy growth was found to scale linearly with the distance from the leading edge. Based on these results, a simple model for prediction of transition location is proposed. Available experiments have been used to correlate the single constant appearing in the model.

Spanwise structure of wall pressure on a cylinder in axial flow
View Description Hide DescriptionThe spanwise structure of wall pressurefluctuations was measured in an axisymmetric turbulent boundary layer on a cylinder parallel to the mean flow at a momentum thickness Reynolds number of 2530 and a boundary layer thickness to cylinder radius ratio of 4.81. The measurements were made using miniature hearing aid type condenser microphones with spanwise separations of 0°, 10°, 20°, 30°, 60°, and 90°. An improved wall pressure power spectrum was obtained at low frequencies by utilizing a twopoint subtraction method to remove low frequency acoustic background noise of the wind tunnel. The spanwise correlations indicate that the spanwise coherent length of the wall pressure is 30° or 0.11δ). The spanwise coherence is weak and concentrated in a frequency band that is substantially lower than the most energetic frequency band of the wall pressure spectrum. A mode number–frequency decomposition of the wall pressure spectrum indicates that the greatest quantity of energy is in the circumferential modes nearest zero. Modes −4 to 4 contain most of the wall pressure energy. Conditional sampling by pressure peak and VITA detection schemes (where VITA was applied to wall pressure to detect strong pressure gradient events) indicate that the spanwise extent of the high pressure peaks and high wall pressure gradients is 60° or 0.22δ).

Numerical investigation of the entrainment and mixing processes in neutral and stablystratified mixing layers
View Description Hide DescriptionThe direct numerical simulation (DNS) of a temporallygrowing mixing layer has been carried out, for a variety of initial conditions at various Richardson and Prandtl numbers, by means of a pseudospectral technique; the main objective being to elucidate how the entrainment and mixing processes in mixinglayer turbulence are altered under the combined influence of stable stratification and thermal conductivity. Stratification is seen to significantly modify the way by which entrainment and mixing occur by introducing highlylocalized, convective instabilities, which in turn cause a substantially different threedimensionalization of the flow compared to the unstratified situation. Fluid which was able to cross the braid region mainly undisturbed (unmixed) in the unstratified case, pumped by the action of rib pairs and giving rise to wellformed mushroom structures, is not available with stratified flow. This is because of the large number of ribs which efficiently mix the fluid crossing the braid region. More efficient entrainment and mixing has been noticed for high Prandtl number computations, where vorticity is significantly reinforced by the baroclinic torque. In liquid sodium, however, for which the Prandtl number is very low, the generation of vorticity is very effectively suppressed by the large thermal conduction, since only small temperature gradients, and thus negligible baroclinic vorticity reinforcement, are then available to counterbalance the effects of buoyancy. This is then reflected in less efficient entrainment and mixing. The influence of the stratification and the thermal conductivity can also be clearly identified from the calculated entrainment coefficients and turbulent Prandtl numbers, which were seen to accurately match experimental data. The turbulent Prandtl number increases rapidly with increasing stratification in liquid sodium, whereas for air and water the stratification effect is less significant. A general law for the entrainment coefficient as a function of the Richardson and Prandtl numbers is proposed, and critically assessed against experimental data.

Experimental observations of the mixing transition in a shockaccelerated gas curtain
View Description Hide DescriptionRichtmyer–Meshkov instability of a thin curtain of heavy gas embedded in air and accelerated by a planar shock wave(Mach 1.2) leads to the growth of interfacial perturbations in the curtain and to mixing. Our experiments produce a phenomenological description of the mixing transition and incipient turbulence during the first millisecond after the shock interaction. Growth of scales both larger and smaller than that of initial perturbations is visually observed and quantified by applying a wavelet transform to lasersheet images of the evolving gas curtain. Histogram and wavelet analyses show an abrupt mixing transition for a multimode initial perturbation that is not apparent for singlemode perturbations.

Planar visualizations of largescale turbulent structures in axisymmetric supersonic separated flows
View Description Hide DescriptionThe spatial evolution of largescale turbulent structures in the shear layer of an axisymmetric, supersonic separated flow has been investigated. The experimental diagnostic used was planar visualization of condensed ethanoldroplets that were suspended in the supersonic free stream. Spatial correlation analyses of large ensembles of images show that the mean sideview structure is highly strained and elliptical in shape and is inclined toward the local free stream direction. It is also shown that the effect of lateral streamline convergence for this axisymmetric case causes a reduction in sideview structure size and eccentricity at the reattachment point as compared to the planar case. Endview structures are wedge shaped, wider on the freestream side than on the recirculation region or developing wake side. It is concluded that the wedge shape is caused by the axisymmetric confinement of the shear layer as it approaches the wake centerline. The average number of structures present in the endview plane decreases significantly from 10–14 at recompression to 4–5 in the developing wake region. Evidence of an amalgamation of endview structures in the images at the reattachment point illustrates one of the mechanisms responsible for this reduction.

Modeling of Richtmyer–Meshkov instabilityinduced turbulent mixing in shocktube experiments
View Description Hide DescriptionA turbulencetransportmodel for the analysis of shock interface interactioninduced turbulent mixing is presented. Results given by the onedimensional (1D) version of this model are compared with data obtained in shocktube experiments.Calibrations are made from an air/He interface destabilized by a 1.3 incident shock waveMach number, taking into account the successive interactions with the different reshocks on the shock tube end wall. Then, using the same set of model constants, different gas pairs with both various Atwood and incident shock waveMach numbers are considered, in order to point out the influence of these main parameters on the results. Mixing zone thickness time evolutions and 1D density profiles are presented and directly compared with experimental results. Profiles of other variables such as the space integral of the turbulent kinetic energy translation energy and the ratio given by the computations, are also shown. Using two different initializations, in particular, to better describe the first phases of the phenomenon under study (i.e., taking into account the initial membrane in a horizontal shock tube configuration), we have found good agreement between calculations and experiments.
