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Volume 11, Issue 11, November 1999
 LETTERS


Oscillating crescentshaped water wave patterns
View Description Hide DescriptionA new type of persistent threedimensional crescentshaped water wave patterns was found experimentally in the laboratory. The particular features of the patterns are their oscillating character and the regular orthogonal grid the crescent crests form in the plane view. They emerge from a steep initially twodimensional paddlegenerated wave amplified by wind. The patterns are composed of the basic wave and two nonsymmetric oblique satellites with longitudinal components of wavevector equal to that of the basic wave and twice this wave number. The main mechanism of their formation was identified to be a fivewave resonant interaction process. However, the mechanism of the sharp selection of these particular harmonics was found not to be explained within the framework of existing theories.

On singularity formation in threedimensional vortex sheet evolution
View Description Hide DescriptionIt is shown that if a doublyinfinite vortex sheet has cylindrical shape and strength distributions at some initial time, then this property is retained in its subsequent evolution. It is also shown that in planes normal to the generator of the cylindrical sheet geometry, the nonlinear evolution of the sheet is the same as that of an equivalent strictly twodimensional sheet motion. These properties are used to show that when an initially flat vortex sheet is subject to a finiteamplitude, threedimensional normal mode perturbation, weak singularities develop along lines which are oblique to the undisturbed velocity jump vector at a time that can be inferred from an extension of Moore’s [Proc. R. Soc. A 365, 105 (1979)] result for twodimensional motion.

 ARTICLES


Newtonian glass fiber drawing: Chaotic variation of the crosssectional radius
View Description Hide DescriptionA model of Newtonian glass fiber drawing at fixed temperature in the unsteady range (the draw ratio is considered. In this range under steady boundary conditions, as is well known, the draw resonance instability sets in, resulting in selfsustained oscillations. These oscillations lead to a periodic variation of the crosssectional radius of the fiber. In the present work we consider the case where the spinline radius varies periodically. Such a variation may result from flow oscillations in the fiber forming channels in the directmelt process, or from the variation of the preform crosssectional radius in drawing of optical fibers. When this variation takes place in the range the selfsustained periodic oscillations of the draw resonance are replaced by quasiperiodic and periodic (modelocked) subharmonic or (under the appropriate conditions) chaotic oscillations (strange attractors). The routes to chaos found in the present work include (1) smooth period doubling bifurcation of (any) modelocked periodic solution, (2) abrupt explosions of quasiperiodic solutions. The predicted chaotic variation of the spinline radius at the winding mandrel may result in a similar variation of the crosssectional radius of the solidified fibers.

Effects of inertia on the hydrodynamics near moving contact lines
View Description Hide DescriptionWe investigate the effects of inertia on the hydrodynamics in the microscopic vicinity of moving contact lines. These hydrodynamics control the macroscopic shape and spreading of fluid bodies across solid surfaces. We perform experiments at low capillary number and negligible to moderate Reynolds number. On a microscopic scale, inertia decreases the dynamic curvature of the free surface near the contact line compared to the case with at the same Ca. On a macroscopic scale, inertia lowers the apparent contact angle of the staticlike macroscopic interface compared to the situations with the same Ca but negligible Re.

Influence of energetics on the stability of viscoelastic Taylor–Couette flow
View Description Hide DescriptionPreviously reported isothermal linear stabilityanalyses of viscoelastic Taylor–Couette flow have predicted transitions to nonaxisymmetric and timedependent secondary flows for elasticity numbers In contrast, recent experiments by Baumert and Muller using constant viscosity Boger fluids have shown that the primary flow transition leads to axisymmetric and stationary Taylortype toroidal vortices. Moreover, experimentally observed onset Deborah number is an order of magnitude lower than that predicted by isothermal linear stabilityanalyses. In this work, we explore the influence of energetics on the stability characteristics of the viscoelastic Taylor–Couette flow. Our analysis is based on a thermodynamically consistent reformulation of the OldroydB constitutive model that takes into account the influence of thermal history on polymeric stress, and an energy equation that takes into account viscous dissipation effects. Our calculations reveal that for experimentally realizable values of Peclet and Brinkman numbers, the most dangerous eigenvalue is real, corresponding to a stationary and axisymmetric mode of instability. Moreover, the critical Deborah number associated with this eigenvalue is an order of magnitude lower than those associated with the nonisothermal extensions of the most dangerous eigenvalues of the isothermal flow. Eigenfunction analysis shows stratification of perturbation hoop stress across the gap width drives a radial secondary flow. The convection of base state temperature gradients by this radial velocity perturbation leads to this new mode of instability. The influence of geometric and kinematic parameters on this instability is also investigated.

Selfsimilar dynamics of a viscous wedge of fluid
View Description Hide DescriptionThe 2D (two dimensional) motion of two symmetric wedges of viscous fluid is determined. The evolution of the fluid is governed by the Stokes equations plus the interfacial boundary conditions including surface tension. A similarity solution is determined and these equations are solved numerically by using a boundary integral method.Solutions for different wedge angles are found.

The development of transient fingering patterns during the spreading of surfactant coated films
View Description Hide DescriptionThe spontaneous spreading of an insoluble surfactantmonolayer on a thin liquid film produces a complex waveform whose time variant shape is strongly influenced by the surface shear stress. This Marangoni stress produces a shocklike front at the leading edge of the spreading monolayer and significant film thinning near the source. For sufficiently thin films or large initial shear stress, digitated structures appear in the wake of the advancing monolayer. These structures funnel the oncoming flow into small arteries that continuously tipsplit to produce spectacular dendritic shapes. A previous quasisteady modal analysis has predicted stable flow at asymptotically long times [Phys. Fluids A 9, 3645 (1997)]. A more recent transient analysis has revealed large amplification in the disturbance film thickness at early times [O. K. Matar and S. M. Troian, “Growth of nonmodal transient structures during the spreading of surfactant coated films,” Phys. Fluids A 10, 1234 (1998)]. In this paper, we report results of an extended sensitivity analysis which probes two aspects of the flow: the time variant character of the base state and the nonnormal character of the disturbance operators. The analysis clearly identifies Marangoni forces as the main source of digitation for both small and large wave number disturbances. Furthermore, initial conditions which increase the initial shear stress or which steepen the shape of the advancing front produce a larger transient response and deeper corrugations in the film. Disturbances applied just ahead of the deposited monolayer rapidly fall behind the advancing front eventually settling in the upstream region where their mobility is hampered. Recent findings confirm that additional forces which promote film thinning can further intensify disturbances [O. K. Matar and S. M. Troian, “Spreading of surfactantmonolayer on a thin liquid film: Onset and evolution of digitated structures,”Chaos9, 141 (1999). The transient analysis presented here corroborates our previous results for asymptotic stability but reveals a source for digitation at early times. The energy decomposition lends useful insight into the actual mechanisms preventing efficacious distribution of surfactant.

Surface equation of falling film flows with moderate Reynolds number and large but finite Weber number
View Description Hide DescriptionWaves on a thin liquid layer falling down a solid wall, either vertical or inclined, are studied by means of a reduced equation. This equation is developed by the regularized longwave expansion method, which is a combination of the Padé approximation and the longwave expansion. Its numerical solutions are compared with the calculations of the full Navier–Stokes equation, simplified Navier–Stokes equation (the “boundarylayer” equation), and the traditional longwave equations, as well as with experimental measurements. When the Reynolds numberR is as small as unity, the present equation agrees with the Navier–Stokes equation and also with the traditional longwave equations. For larger values of R, the traditional longwave equations lose their validity and make a false prediction, while the present equation agrees with the Navier–Stokes equation, as long as the rescaled Reynolds number does not exceed unity in the case of vertical films. Unlike the “boundarylayer” equation developed by previous researchers and expected to be valid at moderate and large Reynolds number, the present equation governs the surface evolution alone without postulating to resolve the velocity field. The structure of the present equation, however, has a correspondence to the depthaveraged equations, which facilitates discussing the physical mechanism of the wave dynamics. In particular, the physical origin of the instability mechanism and the wave suppression mechanism are discussed in terms of Whitham’s wave hierarchy theory. The balance of several physical effects such as drag, gravity, and inertia are also discussed in this connection. The analysis of the tail structure of permanent solitary waves predicts that the R dependence of its tail length λ exhibits two distinct regimes in the diagram. The second regime, which is not predicted by the traditional longwave equations, arises when the inertia effect becomes dominant.

Large growth rate instabilities in threelayer flow down an incline in the limit of zero Reynolds number
View Description Hide DescriptionIn this paper, we examine the effect of viscosity stratification on wave propagation in threelayer flow down an inclined plane at vanishingly small Reynolds number and at finite wavelengths, for cases of negligible liquid–liquid interfacial tensions. We have found that the longwavelength interface mode inertialess instability of Weinstein and Kurz [Phys. Fluids A 3, 2680 (1991)] persists into the finite wavelength domain in the form of nearly complex conjugate wave speed pairs; in certain limits, the interface modes are precisely complex conjugates. As in the case of Weinstein and Kurz, the physical configuration necessary to achieve inertialess instability is a low viscosity and thin internal layer with respect to the other layers in the film. The largest growth rate of the inertialess instability is found at finite wavelengths on the order of the total thickness of the film, and is orders of magnitude larger than the maximum growth rates identified by Lowenhurz and Lawrence [Phys. Fluids A 1, 1686 (1989)] for twolayer flows. We have also found an additional configuration exhibiting extremely large growth rates, also characterized by nearly complex conjugate behavior, that is not accessible via a long or short wavelength asymptotic limit; these threelayer structures have thin, high viscosity internal layers. The characteristic wavelengths associated with the largest growth rates are on the order of ten times smaller than those for the low viscosity internal layer cases. The influence of the deformable free surface on the growth rates of these interface modes is studied and found to be significant.

The inhomogeneous structure of a bidisperse sedimenting gas–solid suspension
View Description Hide DescriptionWe consider a model of a bidisperse gas–solid suspension in which the particles are subject to gravitational and Stokes drag forces and undergo elastic solidbody collisions. Dynamic simulations of many interacting particles in a unit cell with periodic boundary conditions indicate that the suspension has an inhomogeneous structure on the length scale of the cell. A linear stability analysis of averaged equations of motion for the particulate phase is used to predict the values of the Stokes number, particle volume fraction, and unit cell length for which the homogeneous suspension is unstable and these results are compared with the numerical simulations. The suspension is subject to long horizontal wave instabilities at sufficiently high particle volume fractions and low Stokes numbers. The mechanism of instability involves a coupling between the shear flow induced by particle volume fraction variations and the collisional exchange of momentum between the particles. Solutions of the averaged equations successfully capture the particle velocity fields induced by the inhomogeneous structure in the unstable suspensions. These velocity fields are characterized by the mean and variance of the particle velocity and by momentumdensity correlation functions. When the total particle volume fraction is small, the simulated suspensions are stable but still exhibit longrange structure. This structure may be attributed to a pair probability, corresponding to an excess of neighbors of the same species, and a deficit of neighbors of the other species, which decays like with radial distance r.

Threedimensional bubbles in Rayleigh–Taylor instability
View Description Hide DescriptionWe study the highly nonlinear stages of the Rayleigh–Taylor instability(RTI) for threedimensional flow. The proposed numerical and analytical methods are original approaches to the problem. They validate each other and the obtained results agree well.

Laminar flow past a rotating circular cylinder
View Description Hide DescriptionThe present study numerically investigates twodimensional laminar flow past a circular cylinder rotating with a constant angular velocity, for the purpose of controlling vortex shedding and understanding the underlying flow mechanism. Numerical simulations are performed for flows with Re=60, 100, and 160 in the range of 0⩽α⩽2.5, where α is the circumferential speed at the cylinder surface normalized by the freestream velocity. Results show that the rotation of a cylinder can suppress vortex shedding effectively. Vortex shedding exists at low rotational speeds and completely disappears at where is the critical rotational speed which shows a logarithmic dependence on Re. The Strouhal number remains nearly constant regardless of α while vortex shedding exists. With increasing α, the mean lift increases linearly and the mean drag decreases, which differ significantly from those predicted by the potential flowtheory. On the other hand, the amplitude of lift fluctuation stays nearly constant with increasing α while that of drag fluctuation increases. Further studies from the instantaneous flow fields demonstrate again that the rotation of a cylinder makes a substantial effect on the flow pattern.

Nonlinear equilibrium solutions in stratified plane Poiseuille flow
View Description Hide DescriptionBifurcationcharacteristics of the primary and secondary instabilities of thermally stratified plane Poiseuille flow are discussed based on nonlinear equilibrium solutions. For lowPrandtl number fluid under strongly stable stratification, the bifurcation diagram of the primary instability around the turning point of the linear neutral curve exhibits nonlinear degeneracy. As a result, the early transition process can be accurately predicted by the linear critical Reynolds number and the associated supercritical bifurcation surface. Secondary instability of the steady basic flow under the influence of thermal stratification is further analyzed using Floquet theory. The stratification effect on the threedimensional breakdown of Ktype has been established by studying the amplification rates and threshold amplitudes. In unstably stratified flow,Ktype breakdown leads to nonlinear equilibrium solution identical to the mixed mode solution reported by Fujimura and Kelly [Phys. Fluid7, 68–79 (1995)] in a weakly nonlinear study of mode interaction. As a result, deriving amplitude equations for the physical mechanism of secondary instability can be formulated rigorously under unstable stratification.

Experimental and numerical study of anomalous thermocapillary convection in liquid gallium
View Description Hide DescriptionThermocapillary Marangoni convection of liquid gallium was studied experimentally and numerically. A specially designed experimental setup ensured an oxidefree surface of the liquid gallium for a very long time. The convective flow at the free surface was found to be directed opposite to both buoyancydriven and ordinary thermocapillary convection. The anomalous direction of the thermocapillary flow was explained by the presence of a small amount of a surfaceactive contaminant—lead adsorbed at the free surface. Two different approaches were used to describe the observed phenomenon. First, the flow was treated as a pure thermocapillary convection with a modified dependence of the surface tension on temperature so that to reproduce the measuredvelocity distribution. Second, a novel physical model was devised for the flow driven by the gradient of the surface tension induced by the temperature dependence of the concentration of the adsorbed layer of contaminant. In contrast to the ordinary thermocapillary convection in lowPrandtlnumber liquids, there is a strong coupling between the flow and the driving force in the proposed model resulting in velocity profiles very similar to those observed in the experiment.

An experimental study of boundarylayer transition over a rotating, compliant disk
View Description Hide DescriptionAn experimental study is described which investigates the laminarturbulent transition of the boundary layer over rigid and compliant disks rotating under water. Hotfilm data are presented and analyzed to produce neutralstability curves. It appears to be the first time that such data has become available for a compliant disk. Our experiments employing a rigid disk essentially confirm the results of previous authors. For the flow over the compliant disk the turbulence levels in the transitional and fully turbulent flow regimes are found to be considerably lower than the corresponding levels for the rigid disk. The analysis of our experimental data suggests that wall compliance has a stabilizing influence in the frequency range associated with the Type I crossflow instability. Nevertheless, compliance is found to have an overall destabilizing effect on the boundarylayer flow. This results in a substantially lower transitional Reynolds number compared to the case of the rigid disk. The experimental observations are in qualitative agreement with our theoretical predictions. It is argued that the reduced transitional Reynolds number for the compliant disk might indicate that transition for such a disk results from a convectiveinstability mechanism and not from an absoluteinstability mechanism as has recently been suggested in the literature to be the case for a rigid disk.

Computational simulations of vorticity enhanced diffusion
View Description Hide DescriptionComputer simulations are used to investigate a phenomenon of vorticity enhanced diffusion (VED), a net transport and mixing of a passive scalar across a prescribed vortexflow field driven by a background gradient in the scalar quantity. The central issue under study here is the increase in scalar flux down the gradient and across the vortex field. The numerical scheme uses cylindrical coordinates centered with the vortexflow which allows an exact advective solution and 1D or 2D diffusion using simple numerical methods. In the results, the ratio of transport across a localized vortex region in the presence of the vortexflow over that expected for diffusion alone is evaluated as a measure of VED. This ratio is seen to increase dramatically while the absolute flux across the vortex decreases slowly as the diffusion coefficient is decreased. Similar results are found and compared for varying diffusion coefficient, D, or vortex rotation time, for a constant background gradient in the transported scalar vs an interface in the transported quantity, and for vortexflow fields constant in time vs flow which evolves in time from an initial state and with a Schmidt number of order unity. A simple analysis shows that for a small diffusion coefficient, the flux ratio measure of VED scales as the vortex radius over the thickness for mass diffusion in a viscous shear layer within the vortex characterized by The phenomenon is linear as investigated here and suggests that a significant enhancement of mixing in fluids may be a relatively simple linear process. Discussion touches on how this vorticity enhanced diffusion may be related to mixing in nonlinear turbulent flows.

Absolute and convective nature of the Eckhaus and zigzag instability with throughflow
View Description Hide DescriptionThe nature of the Eckhaus and of the zigzag instability is investigated for a periodic basic “flow” (a yperiodic Stokes solution) in the presence of a transverse or a longitudinal mean flow using the twodimensional extension of the absolute instability criterion. For each flow orientation, stability diagrams are obtained numerically and analytically for a simple amplitudeequation model considering both the Eckhaus and the zigzag instability. Analytical results extend and correct a previous analysis by Müller and Tveitereid. In particular, for a longitudinal flow, the Eckhaus instability is convective near its instability threshold and the absolute destabilization occurs at a finite wave number. Similar results hold for the zigzag instability for a transverse throughflow which is convective near threshold. In the presence of an arbitrarily oriented mean flow, the absolute threshold for the Eckhaus instability is also numerically determined. Implications of these results for real experiments are discussed.

On the weakly nonlinear development of the elliptic instability
View Description Hide DescriptionResults are presented for the outcome of the elliptic instability, investigated by methods of dynamicalsystems theory. Finitedimensional nonlinear systems are obtained through Galerkin truncation using a systematic truncation criterion that exactly captures any fluid behavior that can also be captured via amplitude expansions. Six different regions of parameter space are explored, corresponding to linear instabilities of different symmetries and temporal types (“steady” or “oscillatory”). Four different kinds of bifurcation behavior are found among the six cases considered. One of these equilibrates at small amplitude, and the others do not, but depart significantly from the unstable equilibrium solution.

Vortex dynamics investigation using an acoustic technique
View Description Hide DescriptionA new acoustic technique using the double timereversal mirrors system and based on geometrical acoustics, is used to study a vortical flow. The interaction between the sound and a vortex is described in details. This technique has been applied to the study of a stretched vortex. This vortex is generated by stretching the vorticity of a boundary layer in a low velocity water channel. It is shown that the velocity field can be reconstructed from the phase distortion of an ultrasonic wave. The technique gives access to a complete characterization of the vortex dynamics, such as the temporal evolution of its size, its circulation and its position.

Experimental investigation of turbulence generated by breaking waves in water of intermediate depth
View Description Hide DescriptionThis paper reports a set of laboratory data for breaking waves in the water of intermediate depth. A monochromatic wave train with a wave height of 14.5 cm and a wavelength of 121 cm was generated in a water depth, of 20 cm. The wave train breaks consistently at a distance of about 2 from the wave generator. The instantaneous velocity fields under the breaking waves on a twodimensional vertical plane were measured by using the particle imagevelocimetry(PIV) technique. By repeating the same experiments twenty times and performing the ensemble average, mean velocity, mean vorticity,turbulence intensity, and other flow properties such as the Reynolds stress and the mean strain rate were calculated. Outside the aerated region, where the density of air bubbles is high, the experimental data show that the mean vorticity was of the same order of magnitude as with being the phase speed. The maximum turbulence intensity outside the aerated region was in the order of magnitude of 0.1 cm/s). The timeaveraged (over one wave period) turbulence intensity under the wave trough level was one order of magnitude smaller, i.e., it was about 0.04 cm/s). Based the experimental data, the transport equation for turbulent kinetic energy was further examined. The turbulence dissipation rate and its time scale were also estimated. Under the trough level at the measurement section, which was about 3 downstream from the breaking point, the turbulence production, and dissipation were of the same order of magnitude, but not identical. The turbulence advection, production, and dissipation were equally important, while the turbulencediffusion was almost negligible.
