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Volume 8, Issue 6, June 1996

Direct simulation of transition in Stokes boundary layers
View Description Hide DescriptionNumerical simulations of the Stokes boundary layer over a three‐dimensional wavy wall are performed in order to investigate the role played by infinitesimal wall imperfections in triggering transition to turbulence. Our results show flow patterns qualitatively similar to those experimentally detected. In particular the laminar, disturbed‐laminar and intermittent turbulent regimes are recovered. The characteristics of the above flow regimes are analyzed.

Scaling of the fingering pattern of an impacting drop
View Description Hide DescriptionWe have studied experimentally the fingered splatter left behind after a liquiddrop impacts a solid surface at high values of the Reynolds and Weber numbers. The viscosity and surface tension of the liquid was varied by using several different fluid mixtures. The surface chosen was a thick paper sheet, on which the drop left a clear signature of the impact pattern. The maximum spreading of the fluid and the number of fingers seem to scale with an Impact Reynolds number, U(π^{2}ρD ^{3}/16σ)^{1/4}/ν^{1/2}, where U is the impact velocity, ν the kinematicviscosity of the fluid, ρ the fluid density, σ the surface tension and D the drop diameter. The number of fingers is weakly dependent on the surface tension and depends primarily on the inertial‐viscous interaction.

The instability of the separated shear layer from a bluff body
View Description Hide DescriptionIt has generally been assumed in previous studies that the normalized shear‐layer frequency (f _{SL}/f _{ K }) versus Reynolds number (Re) varies as Re^{0.5}. There has also been a great deal of scatter in quoted values of the critical Reynolds number (Re_{crit}) for the onset of the shear‐layer instability, with values ranging from 350 to 3000. In the present work, we find that Re_{crit} is strongly sensitive to the mode of primary vortex shedding and may vary between 1200–2600 depending on the angle of oblique shedding. In the case of the shear‐layer frequency, we have conducted a careful reevaluation of all previous data in the literature, and find the relationship f _{SL}/f _{ K }=0.0235×Re^{0.67}. Surprisingly, not one of the previous studies yields a relationship that is close to Re^{0.5}. We suggest an explanation for the variation Re^{0.67} based on simple physical grounds.

The three‐dimensional development of the shearing instability of convection
View Description Hide DescriptionTwo‐dimensional convection can become unstable to a mean shear flow. In three dimensions, with periodic boundary conditions in the two horizontal directions, this instability can cause the alignment of convection rolls to alternate between the x and y axes. Rolls with their axes in the y‐direction become unstable to a shear flow in the x‐direction that tilts and suppresses the rolls, but this flow does not affect rolls whose axes are aligned with it. New rolls, orthogonal to the original rolls, can grow, until they in turn become unstable to a shear flow. This behavior is illustrated through numerical simulations and low‐order models, and the sequence of local and global bifurcations is determined.

Experimental investigation of Richtmyer–Meshkov instability before and after the reflected shock compression
View Description Hide DescriptionResults of an experimental investigation on the Richtmyer–Meshkov instability of a He–CO_{2} interface are reported. A simultaneous three‐directional laser absorption technique is used to follow, at a single abscissa, the evolution of the created mixing zone before and after the interaction with the reflected shock, during the same run. CO_{2} density profiles have been determined within both the incident and the compressed mixing zones. However, near the pure CO_{2}, the wall boundary layer reflected shock interaction perturbs measurements and does not allow objective conclusions.

Stable and unstable singularities in the unforced Hele‐Shaw cell
View Description Hide DescriptionWe study singularity formation in the lubrication model for the unforced Hele‐Shaw system, describing the breaking in two of a fluid droplet confined between two narrowly spaced glass plates. By varying the initial data, we exhibit four different scenarios: (1) the droplet breaks in finite time, with two pinch points moving toward each other and merging at the singular time; (2) the droplet breaks in finite time, with two asymmetric pinch points propagating away from each other; (3) the droplet breaks in finite time, with a single symmetric pinch point; or (4) the droplet relaxes to a stable equilibrium shape without a finite time breakup. Each of the three singular scenarios has a self‐similar structure with different scaling laws; the first scenario has not been observed before in other Hele‐Shaw studies. We demonstrate instabilities of the second and third scenarios, in which the solution changes its behavior at a thickness that can be arbitrarily small depending on the initial condition. These transitions can be identified by examining the structure of the solution in the intermediate scaling region.

A free surface flow with compositional‐dependent interfacial properties
View Description Hide DescriptionThe effect of adsorbed surfactants on the flow that takes place in the film forming zone of the slot coater is examined. The numerical technique employed in this work, that is based on the finite element method, can suitably handle viscousfree surfaces with locally varying interfacial properties. The complete information produced by the computational code allows to relate the interfacial dynamics to the mechanism involved in the mass transport process: convection and adsorption/desorption. The results presented show that the effect produced by both interfacial viscosity and interfacial elasticity are rather similar; however, the operating conditions that make them more noticeable are different.

Exploitation of Brownian motions for the optimal control of fiber orientation distributions
View Description Hide DescriptionThe dynamical behavior of suspensions of rod‐like particles is explored in unsteady elementary flows characterized by the following special property: while there may be arbitrarily large excursions from an initial state by the carrier fluid, the fluid returns to the initial state at the end of the time interval. In such a closed flow, it is shown that the orientation distribution of a dilute non‐Brownian suspension will also return to its initial state, after arbitrarily large excursions driven by the flow of the carrier fluid. In contrast, the orientation distribution in a Brownian suspension will not return to its initial state at the end of the interval in a closed flow. These effects are exploited in a constrained nonlinear optimal control problem for the orientation distribution of a Brownian suspension in an example closed flow. The result of the control problem is a family of protocols that are most effective at achieving various degrees of anisotropy of the suspension, within the class of flows under consideration.

The nature of particle contacts in sedimentation
View Description Hide DescriptionAn experimental study of the contact of a falling sphere with a neutrally buoyant sphere in a viscous fluid is described. The experiments used a mixture of polyalklene glycol and tetrabromoethane as the Newtonian fluid, together with a pair of equal‐sized Teflon^{■} and nylon spheres of 6.355 mm diameter. The spheres have microscopic surface roughness, which allows them to make physical contact when in close approach. This contact breaks the symmetry of the relative trajectory of the two spheres and affects the rate at which the heavy sphere moves past the neutrally buoyant one. The experimental observations verify the roll/slip model of Davis [Phys. Fluids A 4, 2607 (1992)] for the interaction of the two spheres in contact. This model assumes that contact prevents the nominal surfaces from approaching closer than a minimum separation equal to the effective roughness height, and that the tangential component of the contact force is described by solid friction theory. A friction coefficient of 0.28±0.02 provides the best agreement between theory and experiment, with some variation observed when the experiments were repeated for the same and different pairs of spheres.

Terminal structure of unsteady classical and interacting boundary layers
View Description Hide DescriptionThe terminal structure of the unsteady boundary‐layer equations is investigated for the problem of impulsive flow past a circular cylinder. It is known that both the classical and interacting boundary‐layer equations for this problem are singular and the singular structure for the interacting boundary layer on a circular cylinder has not been properly resolved using the classical finite difference approach within an Eulerian framework. The objective of this paper is to resolve the singular structure for both the classical and interacting boundary layers for the impulsively started flow past a cylinder. An adaptive‐grid scheme, coupled to a panel method for the interacting case, is employed to resolve the extremely small time and length scales associated with a terminal structure defined by the emergence of a singularity. Calculations performed in an Eulerian coordinate system show that interacting boundary‐layer calculations terminate sooner than the classical boundary‐layer calculation, and as the Reynolds number decreases, the singularity occurs earlier in time and closer to the rear stagnation point of the circular cylinder. Comparisons with previous Lagrangian and fixed‐grid Eulerian results are discussed.

Boundary layer separation in a rotating container
View Description Hide DescriptionThe flow field induced in a cylindrical container by the rotation of the container and/or the end wall(s) about the axis of symmetry is highly vortical. If the container and the end walls are rotated at different angular velocities, a meridional flow develops due to the tilting and stretching of the axial vorticity created by the rotation of the bounding surfaces. In the present study, the flow field induced by the differential rotation of the container and one end wall while the other end wall remains fixed is numerically studied for a range of Reynolds numbers and ratios of the container to the end wall rotation rate. It is shown that the production of azimuthal vorticity by the tilting and stretching of axial vorticity causes the container side wall boundary layer to separate for a range of Reynolds numbers and ratios of the container to the end wall rotation rate.

Linear and nonlinear evolution of boundary layer instabilities generated by acoustic‐receptivity mechanisms
View Description Hide DescriptionExperimental results are presented concerning the evolution of instabilitiesgenerated by the interaction between low‐level broad‐band acoustic waves and small two‐dimensional roughness elements. Streamwise perturbation velocity spectra are measured and it is found that on the smooth plate, naturally‐occurring Tollmien‐Schlicting (T‐S) waves grow in a manner consistent with a resonant subharmonic wave interaction. However, in the presence of small two‐dimensional roughness elements, a strong primary T‐S mode is forced by an interaction with the background acoustic field. This leads to a K‐type of nonlinear interaction characterized by the generation of harmonics (up to six harmonics are observed) at the expense of the subharmonic. The scaling of the T‐S waves and their harmonics with the number and amplitude of the wall roughness is also considered.

Two‐dimensional disturbance growth of linearly stable viscous shear flows
View Description Hide DescriptionInstability of fluid flow has long been linked with the eigenvalues of the Orr‐Sommerfeld equation. Recently, it has been shown that even though all eigenvalues may be stable, it is still possible to have disturbance growth. This is because the Orr‐Sommerfeld operator is non‐normal. We identify which eigenmodes are important in two‐dimensional disturbance growth. We find a relationship between maximum growth and growth caused by the adjoint of the leading eigenmode for both plane Poiseuille flow and plane Couette flow. The work points to a connection between the occurrence of the first degeneracy of the Orr‐Sommerfeld operator and the first appearance of disturbance growth.

Buoyant‐thermocapillary instabilities of differentially heated liquid layers
View Description Hide DescriptionThe stability of buoyant‐thermocapillary‐driven flows in a fluid layer subjected to a horizontal temperature gradient is analyzed. Our purpose is the modelization of recent experimental results obtained for a fluid of Prandtl number Pr=7, by Daviaud and Vince [Phys. Rev. E 48, 4432 (1993)], who observed a transition between traveling waves and stationary rolls when the height of fluid is increased. Our model takes into account several effects that were examined separately in previous studies. The relative importance of buoyancy and thermocapillarity is controlled by the ratio W of Marangoni number to Rayleigh number. The fluid layer is bounded below by a rigid plane whose temperature varies linearly along the direction of the thermal gradient. A Biot number is introduced to describe heat transfer at the top free surface. Our stability analysis establishes the existence of a transition between stationary and oscillatory modes when W exceeds a value W _{0} which is a function of the Biot number. Moreover, two types of oscillatory modes have been identified which differ by the range of variation of their critical parameters (wave number, frequency, angle of propagation) versus W.

Symmetry and instability in rotating hydrodynamic and magnetohydrodynamic flows
View Description Hide DescriptionThe effect of symmetries on instabilitites in rotating hydrodynamic and magnetohydrodynamicflows is considered. Examples are described in which models of such flows or special cases have O(2) symmetry instead of the expected SO(2) symmetry. This change in symmetry has a profound effect on the nature of any instabilities and on their subsequent evolution. Both primary and secondary instabilities in systems with O(2) symmetry are discussed, and the effects of restoring SO(2) symmetry are analyzed. Implications of the results for convection in a rotating annulus, magnetohydrodynamicinstabilities in accretion disks and a two‐layer baroclinic instabilitymodel are described.

The effects of viscosity on centre modes in the incompressible stability of a trailing line vortex
View Description Hide DescriptionThe effects of viscosity on centre‐modes of instability/stability are considered, with particular emphasis on the trailing line vortex. These effects are shown to be particularly significant when the axial wavenumber is within O(Re ^{−1/(2+λ)}) of the critical (inviscid) value, where Re is the Reynolds number, and λ is a (positive) parameter defined explicitly in the paper. Also examined are the critical (singular) layers that arise within the flow, which are shown to have thickness O(Re ^{−2λ/3(4+2λ)}). A number of numerical results are presented, and these show that viscosity seems to generally play a stabilising role on the stability of the flow. In the Appendix, wavenumbers somewhat closer to the critical inviscid value are examined, and the effects of viscosity are shown to substantially permeate outside of the critical layers, and lead to a system similar to that discussed by Stewartson, Ng and Brown [Philos. Trans. R. Soc. London Ser. A 324, 473 (1988)] in the context of swirling Poiseuille flow.

Internal waves in xenon near the critical point
View Description Hide DescriptionJust above the liquid‐vapor critical point, a fluid’s large compressibility causes a stable stratification in which the density varies by as much as 10% in 1 cm. This stratification supports internal gravity waves which we observed with an oscillator immersed in a near‐critical xenon sample. We found the number and frequencies of the observable modes depended on the sample cell’s orientation, with only two modes seen in the horizontal cell. The frequencies of the two modes had different temperature dependences: with decreasing temperature, the higher frequency increased monotonically from 0.7 to 2.8 Hz, but the lower frequency varied nonmonotonically, with a maximum of 1.0 Hz at 20 mK above the critical temperature. These temperature dependences continued to 20 mK below the critical temperature, where the xenon was separated into liquid and vapor phases. We calculated these two frequencies by solving the eigenvalue problem of internal waves in a box containing a stratified fluid. The fluid’s density profile was obtained from xenon’s equation of state. The calculated and measured frequencies agree to within 15%. Analytical calculations based on simple approximations of the density profile provide insight into the observed temperature dependences.

Three‐dimensional subharmonic waves during transition in the near‐wake region of a cylinder
View Description Hide DescriptionDetailed quantitative measurements of three‐dimensional subharmonic waves have been obtained with a scanning laser anemometer in the near‐wake region of a circular cylinder. A spatial decomposition technique has been used to separate the secondary three‐dimensional flow from the primary Strouhal vortexflow. The appearance of subharmonic frequencies in the spectrum of the secondary instability field at Reynolds numbers of 300 and above provides experimental evidence consistent with the scenario of a period‐doubling route to turbulence. The basic flow structure of the secondary field responsible for producing the subharmonics is shown to be pairs of counter‐rotating streamwise vortices, which alternate the direction of rotation every Strouhal period.

The excitation of ducted modes by passing internal waves
View Description Hide DescriptionA local peak in the atmospheric wind or stability profile supports ducted (vertically trapped) modes. If the duct is rather high in the atmosphere it cannot be directly excited by the major atmospheric sources such as convection, but an internal gravity wave radiating out of such sources transfers energy to the duct via nonlinear processes related to the resonance triad interaction. The amplitude of such a transfer is computed. The modal excitations remain after the internal wave has left the region, and while each passing wave adds only a small amount of activity to the duct such contributions are cumulative and drive the modal activity up to a level where it is limited by local dissipation. This suggests the possibility of monitoring the mean flux of internal waves through the region by measuring activity within elevated ducts, a procedure that offers several advantages over monitoring the wave flux directly.

Weakly nonlocal gravity–capillary solitary waves
View Description Hide DescriptionThere is evidence from previous analytical work based on a modelequation and from numerical computations that gravity–capillary solitary waves in a liquid layer are nonlocal—they feature oscillatory tails of constant amplitude—when the Bond number τ is less than 1/3. Here, the full gravity–capillary wave problem is examined and these tails are calculated asymptotically in the weakly nonlinear regime. For given values of τ (0<τ<1/3) and Froude number F slightly greater than 1, there exists a one‐parameter family of weakly nonlocal solitary waves characterized by the phase shift of the tails relative to the main peak. The tail amplitude depends on the phase shift and is exponentially small with respect to the wave peak amplitude. Predictions of the asymptotic theory are confirmed by numerical computations using a spectral method.