Volume 8, Issue 4, April 1996
Index of content:

Hydrodynamics of monolayer domains at the air–water interface
View Description Hide DescriptionMolecules at the air–water interface often form inhomogeneous layers in which domains of different densities are separated by sharp interfaces. Complex interfacial pattern formation may occur through the competition of short‐ and long‐range forces acting within the monolayer. The overdamped hydrodynamics of such interfacial motion is treated here in a general manner that accounts for dissipation both within the monolayer and in the subfluid. Previous results on the linear stability of interfaces are recovered and extended, and a formulation applicable to the nonlinear regime is developed. A simplified dynamical law valid when dissipation in the monolayer itself is negligible is also proposed. Throughout the analysis, special attention is paid to the dependence of the dynamical behavior on a characteristic length scale set by the ratio of the viscosities in the monolayer and in the subphase.

Velocity fluctuations of a heavy sphere falling through a sedimenting suspension
View Description Hide DescriptionThe motion of a test sphere falling through an otherwise monodisperse suspension of different settling velocity has been examined experimentally. Marked test spheres of different sizes and densities were tracked in a monodisperse suspension of unmarked glass spheres, made optically transparent by matching the index of refraction of the suspending fluid to that of the glass spheres. The ratio, R _{ s }, between the Stokes’ velocity of the test particle and that of an isolated sphere of the suspension was varied from 1 to 13 while the particle volume fraction of the suspension was kept constant at 20%. Statistical analyses of the velocities of the test sphere yield the mean settling velocities, the velocity fluctuations and the velocity autocorrelation functions. The long time motion of the test sphere is shown to be diffusive in nature. Correlation times and self‐diffusivities are measured as a function of R _{ s }. A change in the behavior of the motion of the test particle is observed for R _{ s }≥5.

NMR imaging of velocity profiles and velocity distributions in bead packs
View Description Hide DescriptionSpatially resolved velocity profiles and spatially nonresolved velocity distributions of steady flow in a tube and bead packs were measured. Two different NMR experiments were used to measurevelocity distributions. In one, the velocity histogram was calculated from spatially resolved velocity phase encoded images acquired in a 6 mm bead pack. In the other, a Fourier flow method was used to measure the velocity distribution directly in a 0.25 mm bead pack. Axial velocity profiles in the pore space of the 6 mm bead pack at Reynolds numbers of 14.9, 29.9, and 44.8 proved to be roughly parabolic, with maxima near the pore centers. Both NMR methods yielded the same dimensionless velocity distributions that contain negative as well as positive velocity components. The velocitydistribution function derived from a bundle‐of‐tubes‐model accounts for the positive part of the velocity distribution.

Thermocapillary migration of bubbles and drops at moderate values of the Marangoni number in reduced gravity
View Description Hide DescriptionExperiments were performed on the motion of isolated drops and bubbles in a Dow‐Corning silicone oil under the action of an applied temperature gradient in a reduced gravity environment aboard the NASA Space Shuttle in orbit. Images of the interior of the test cell during these experiments were recorded on cine film and later analyzed to obtain data on the migration velocity as a function of size and the applied temperature gradient. The data are presented in scaled form. Predictions are available in the case of gas bubbles, and it is found that the scaled velocity decreases with increasing Marangoni number qualitatively as expected even though there are quantitative discrepancies. The scaled velocity also appears to approach a theoretical asymptote predicted in the limit of large values of the Marangoni number for Stokes motion. Finally, sample results from a preliminary experiment on a pair of drops are presented. They display the remarkable feature that a small drop which leads a large drop in a temperature gradient can significantly retard the motion of the large trailing drop while itself moving as though it is virtually unaffected by the presence of the large drop.

Statistical mechanics of bubbly liquids
View Description Hide DescriptionThe dynamics of bubbles at high Reynolds numbers is studied from the viewpoint of statistical mechanics. Individual bubbles are treated as dipoles in potential flow. A virtual mass matrix of the system of bubbles is introduced, which depends on the instantaneous positions of the bubbles, and is used to calculate the energy of the bubbly flow as a quadratic form of the bubbles’ velocities. The energy is shown to be the system’s Hamiltonian and is used to construct a canonical ensemble partition function, which explicitly includes the total impulse of the suspension along with its energy. The Hamiltonian is decomposed into an effective potential due to the bubbles’ collective motion and a kinetic term due to the random motion about the mean. An effective bubble temperature—ameasure of the relative importance of the bubbles’ relative to collective motion—is derived with the help of the impulse‐dependent partition function. Two effective potentials are shown to operate: one due to the mean motion of the bubbles, dominates at low bubble temperatures, where it leads to their grouping in flat clusters normal to the direction of the collective motion, while the other, temperature‐invariant, is due to the bubbles’ position‐dependent virtual mass and results in their mutual repulsion. Numerical evidence is presented for the existence of the effective potentials, the condensed and dispersed phases, and a phase transition.

Elliptical vortices in shear: Hamiltonian moment formulation and Melnikov analysis
View Description Hide DescriptionThe equations of motion for interacting elliptical vortices in a background shear flow are derived from a Hamiltonian moment formulation. The equations reduce to the sixth order system of Melander et al. [J. Fluid Mech. 167, 95 (1986)] when a pair of vortices is considered and shear is neglected. The equations for a pair of identical vortices are analyzed using a number of methods, with particular emphasis on the implications for vortex merger. The splitting distance between the stable and unstable manifolds connecting the hyperbolic fixed points of the intercentroidal motion—the separatrix splitting—is estimated with a Melnikov analysis. This analysis differs from the standard time‐periodic Melnikov analysis on two counts: (a) the ‘‘periodic’’ perturbation arises from a second degree of freedom in the system which is not wholly independent of the first degree of freedom, the intercentroidal motion; (b) this perturbation has a faster time scale than the intercentroidal motion. The resulting Melnikov integral appears to be exponentially small in the perturbation as the latter goes to zero. Numerical simulations, notably Poincaré sections, provide a global view of the dynamics and indicate that, as observed in previous studies, there are essentially two modes of merger. The effect of the shear on chaotic motion is also discussed.

Study of the von Kármán flow between coaxial corotating disks
View Description Hide DescriptionWe report an experimental study of the swirling flow generated in the gap between two coaxial corotating disks. We use a free geometry, i.e., unshrouded disks in air, with moderate to high Reynolds numbers. When the relative rotation rate is varied, transitions in the flow can be observed by global power measurement and are related to the geometry of the external recirculating flow. The mean flow is studied in details with hot‐wire measurements using a boxcar‐type averaging technique. It involves a single turbulent vortex undergoing a slow precession motion. We show that statistical properties of the turbulent fluctuations are affected by the dynamics of the mean flow, which also displays a correlation with the global power fluctuations.

Solutions of uniform, open‐channel, liquid metal flow in a strong, oblique magnetic field
View Description Hide DescriptionThe flow of an electrically conducting fluid in an open channel in the presence of a strong magnetic field of oblique incidence to both the channel walls and the force of gravity is explored. This type of flow has possible applications to the protection of high heat flux surfaces in magnetic confinementfusion reactors. The governing equations of fully‐developed flow are derived retaining all viscous terms. They are then solved in the strong field limit in an asymptotic, iterative fashion, carrying the first two terms in the expansion with powers of the effective Hartmann number. The asymptotic solutions for the velocity, induced magnetic field and the flow rate are compared with a numerical solution of the complete governing equations. Good agreement is seen between the asymptotic and numerical predictions of velocity and electric current distribution when the core regions are dominated by magnetic forces. One novel feature of open channel flow of this type is the existence, predicted by the asymptotic analysis and confirmed by the numerical integration, of a second‐order Hartmann layer on the free surface. Its presence is required to ensure the condition of no shear stress on this boundary. Also seen is the presence of strong discontinuities across free shear layers, which form along the field lines that extend from the free surface/sidewall corner.

Potentiostatic electroconvection in a layer and in a loop
View Description Hide DescriptionThis paper concerns electroconvectional stability of a conduction state in an electrolyte layer flanked by cation‐permselective walls (electrodialysis membranes, electrodes) under potentiostatic conditions. It is shown through a numerical finite difference solution of the linear stability problem that above a certain voltage threshold the basic conduction state becomes electroconvectionally unstable. Marginal stability curves in the voltage/wave number plane are calculated and the dependence of the critical threshold characteristics on the system’s parameters (ionic diffusivities ratio, electroconventional Péclet number) studied. Electroconvectional instability is shown to occur for an arbitrary ionic diffusivities ratio. A model problem of electroconvection in a loop under potentiostatic conditions is solved explicitly for a steady state. It is shown that above a certain voltage threshold, the quiescent conduction in the loop bifurcates into a pair of electroconvectional steady state circulations.

Magnetic damping of buoyant convection during semiconductor crystal growth in microgravity. Continuous random g‐jitters
View Description Hide DescriptionThis paper treats the buoyant convection of a liquid metal in a circular cylinder with a uniform, steady, axial magnetic field and with the random residual accelerations encountered on earth‐orbiting vehicles. The objective is to model the magnetic damping of the melt motion during semiconductorcrystal growth by the Bridgman process in space. For a typical process with a magnetic flux density of 0.2 T, the convective heat transfer and the nonlinear inertial effects are negligible, so that the governing equations are linear. Therefore, for residual accelerations or ‘‘g‐jitters’’ whose directions are random functions of time, the buoyant convection is given by a superposition of the convections for two unidirectional accelerations: an axial acceleration which is parallel to the cylinder’s axis and a transverse acceleration which is perpendicular to this axis. Similarly, the response to accelerations whose amplitudes are random functions of time is given by a Fourier‐transform superposition of the buoyant convections for sinusoidally periodic accelerations for all frequencies. Since the temperature gradient in the Bridgman process is primarily axial, the magnitude of the three‐dimensional buoyant convection for the transverse acceleration is much larger than the magnitude of the axisymmetric convection for the axial acceleration. At a low frequency, only the magnetic damping limits the magnitude of the buoyant convection. As the frequency is increased, linear inertial effects augment the magnetic damping, so that the magnitude of the convection decreases, and its phase shifts to a quarter‐period lag after the acceleration.

On the forced flow of salty water in a loop
View Description Hide DescriptionThe buoyancy‐driven flow of salty water in a loop is computed. This problem belongs to the general class of the convective behavior of solutal fluids, a specific example of which is the oceanic thermohaline circulation. The two cases of freshwater flux forcing and so‐called virtual salt flux forcing are compared and contrasted. The former is an exact statement of the saline forcing of the ocean by the atmosphere, while the latter is an approximation used in many climate models. Analytical solutions appropriate to both cases are presented for broad parameter ranges and ultimately encapsulated in the form of bifurcation maps. These allow for comparisons between the behaviors predicted for the two cases. Furthermore, the solutions are supported by means of numerical experimentation. It is found that a simple loop model, forced by a steady flux, can possess multiple solutions, either stationary solutions and limit cycles or distinctly different limit cycles. This result is closely related to climate models. In addition, this study transcends climate applications and applies to the more general classical problem of convection in a loop.
The novel aspect here is the application of freshwater flux to a salty fluid. The effect on density of this forcing is different from that due to the application of heat to a thermally sensitive fluid. Surprising and counter‐intuitive behaviors have been found which reflect these differences. As an example, in the limit where diffusion is weak relative to freshwater flux, a δ‐function‐like salinity profile appears if freshwater flux conditions are used. Models using a virtual salt flux approximation, or a relaxation condition, yield a low mode solution for these parameters. In contrast, the virtual salt flux equations can be obtained from the freshwater‐forced equations by a systematic expansion in one limit where diffusion dominates freshwater flux. Numerical experiments are used to examine the comparisons between virtual salt flux and freshwater‐forced solutions, with the result that virtual salt flux generally yields accurate results when diffusion is strong.

Convective and absolute instabilities of a buoyancy‐induced flow in a thermally stratified medium
View Description Hide DescriptionConvective and absolute instabilities of the buoyancy‐induced boundary layer adjacent to an isothermal vertical flat plate immersed in a linear ambient thermal stratification are investigated. Because of the ambient stratification, the mean temperature and velocity similarity profiles possess a region of temperature deficit and flow reversal. Linear stability analysis reveals the existence of convective and absolute instabilities, which are studied in terms of the governing nondimensional parameters of the problem, namely, the Prandtl number (Pr) and a modified Grashof number (G). The critical value of G=G _{ c } at which transition to convective instability occurs has a maximum between Pr=2 and 3; for larger values, G _{ c } is a weak function of Pr. At higher G=G _{ a }, the flow becomes absolutely unstable. It is found that G _{ a } increases with Pr. This tendency is related to the existence of the reverse flow due to the ambient stratification. It is shown that as the reverse flow becomes relatively more dominant, the flow is more susceptible to absolute instability.

Bifurcating jets generated with stepped and sawtooth nozzles
View Description Hide DescriptionA simple round jet is perturbed by attaching either a stepped or a sawtooth trailing edge that acts as a form of passive control. Flow visualization, laser Doppler anemometer, and hot‐wire measurements are used to document the flow behavior in both water and air for Reynolds numbers of 7600 and 22 000, respectively. When the jet is forced, the flow bifurcates. The bifurcation requires a low forcing frequency that encourages the formation of tilted primary vortex rings with significant vorticity and axial spacing. In the step nozzle flow, the rings result from closed vortex loops. In the sawtooth nozzle flow, they develop from sections of a helix. A powerful secondary jet initiates in straining regions between primary ring cores, where closed‐loop or helical structures with opposite tilt are present. The radial momentum in this jet is large enough to divert the path of the primary jet in the opposite direction. The bifurcating effect was found to be strongest when the forcing frequency was less than or equal to one‐third of the natural roll‐up frequency.

The oscillatory instability of plane variable‐density jets
View Description Hide DescriptionThe results of an experimental investigation of the instability of variable‐density plane jets issuing into ambient air are reported. When the jet to ambient fluid density ratio S(S=ρ_{ j }/ρ_{∞}) is less than a critical value, an intense oscillating instability is observed. This instability is characterized by sharp peaks in the power spectral density measured in the near field of the jet. The effects of the control parameters S, Re, and H/θ (jet width to exit momentum thickness) on the instability regime are determined. It is shown that Re is a better scaling parameter than H/θ. The Strouhal number of the dominant mode St_{ H } increases with S and Re up to a constant value of 0.25, which is in rather good agreement with the theory and the experiments of Yu and Monkewitz [Phys. Fluids A 2, 1175 (1990); J. Fluid Mech. 255, 323 (1993)]. In the present experiments the critical value S _{ c } above which the oscillating regime disappears is an increasing function of Re and S _{ c } seems to reach a limiting value in the neighborhood of 0.7, which does not agree well either with the theory or with the experiments of Yu and Monkewitz [Phys. Fluids A 2, 1175 (1990); J. Fluid Mech. 255, 323 (1993)]. This difference is in qualitative agreement with the results of linear stability computations, also reported in the paper, which take into account differences in shape and relative positions of the density and velocity profiles.

On the stability of an axisymmetric rotating flow in a pipe
View Description Hide DescriptionThe linear stability of an inviscid, axisymmetric and rotating columnar flow in a finite length pipe is studied. A well posed model of the unsteady motion of swirling flows with compatible boundary conditions that may reflect the physical situation is formulated. A linearized set of equations for the development of infinitesimal axially‐symmetric disturbances imposed on a base rotating columnar flow is derived. Then, a general mode of axisymmetric disturbances, that is not limited to the axial‐Fourier mode, is introduced and an eigenvalue problem is obtained. Benjamin’s critical state concept is extended to the case of a rotating flow in a finite length pipe. It is found that a neutral mode of disturbance exists at the critical state. In the case of a solid body rotating flow with a uniform axial velocity component, analytical solution of the eigenvalue problem is found. It is demonstrated that the flow changes its stability characteristics as the swirl changes around the critical level. When the flow is supercritical an asymptotically stable mode is found, and when the flow is subcritical, an unstable mode of disturbance may develop. This result cannot be predicted by Rayleigh’s classical stability criterion. In the case of a general columnar swirling flow in a pipe, the asymptotic solution of the eigenvalue problem around the critical state is also studied. It is shown that the critical swirl ratio is a point of exchange of stability for any swirling flow in a finite length pipe. This result reveals an unknown instability mechanism of swirling flows that cannot be detected by previous stabilityanalyses and sheds new light on the relation between stability of vortexflows and the vortex breakdown phenomenon.

On the stability of non‐columnar swirling flows
View Description Hide DescriptionThe linear stability of an inviscid, axisymmetric and non‐columnar swirling flow in a finite length pipe is studied. A novel linearized set of equations for the development of infinitesimal axially‐symmetric disturbances imposed on a base non‐columnar rotating flow is derived. Then, a general mode of an axisymmetric disturbance, that is not limited to the axially‐periodic mode, is introduced and an eigenvalue problem is obtained. A neutral mode of disturbance exists at the critical state. The asymptotic behavior of the branches of non‐columnar solutions that bifurcate at the critical state is given. Using asymptotic techniques, it is shown that the critical state is a point of exchange of stability for these branches of solutions. This result, together with a previous result of Wang and Rusak [Phys. Fluids 8, 1007 (1996)] on the stability of columnar vortex flows, completes the investigation on the stability of all branches of solutions near the critical state. Results reveal the important relation between stability of vortex flows and the physical mechanism leading to the axisymmetric vortex breakdown phenomenon.

Linear two‐dimensional stability of a Burgers vortex layer
View Description Hide DescriptionA linear stabilityanalysis is presented for a stationary Burgers vortex layer in irrotational straining flow, to normal mode disturbances invariant in the direction of main flow vorticity. The whole neutral curve is calculated by combining numerical and asymptotic analysis. It is similar to that for free mixing layers which are always unstable, except that there is unconditional stability below a critical Reynolds number, in agreement with the long‐wave asymptotic result by Neu [J. Fluid Mech. 143, 253 (1984)]. The Reynolds number compares shear flowvorticity versus stretching rate and diffusion, so both latter factors are stabilizing if strong enough. Neutral disturbances represent standing waves.

Nonlinear Tollmien–Schlichting waves for a Blasius flow over compliant coatings
View Description Hide DescriptionIn a previous work, we considered the influence of a compliant wall on finite‐amplitude Tollmien–Schlichting (TS) waves in plane Poiseuille flow. In the present investigation we extend this analysis to the more relevant Blasius boundary‐layer profile. Similarly to W. Koch [J. Fluid Mech. 243, 319 (1992)], who studied finite‐amplitude TS waves over a rigid wall, we use a parallel flow assumption and expand the flow quantities in Fourier and Chebyshev modes. Our coating is of Kramer’s type. We present a model that takes into account the nonlinear fluid/structure interaction all over the flow field. The problem to be numerically feasible, we mainly focus on moderately flexible walls, being stable (at finite Reynolds numbers) with respect to flow‐induced surface instability (FISI) waves. Our computations of nonlinear TS instabilitywaves show that for a certain range of wall parameters two‐dimensional finite‐amplitude traveling waves exist well below critical Reynolds numbers predicted by linear theory, typical for a subcriticalbifurcation behavior. Computations of the flow structure and skin friction of the finite‐amplitude TS waves are provided as well.

Pair dispersion over an inertial range spanning many decades
View Description Hide DescriptionNew numerical results on scalar pair dispersion through an inertial range spanning many decades are presented here. These results are achieved through a new Monte Carlo algorithm for synthetic turbulent velocity fields, which has been developed and validated recently by the authors [J. Comput. Phys. 117, 146 (1995)]; this algorithm is capable of accurate simulation of a Gaussian incompressible random field with the Kolmogoroff spectrum over 12–15 decades of scaling behavior with low variance. The numerical results for pair dispersion reported here are within the context of random velocity fields satisfying Taylor’s hypothesis for two‐dimensional incompressible flow fields. For the Kolmogoroff spectrum, Richardson’s t ^{3} scaling law is confirmed over a range of pair separation distances spanning eight decades with a Richardson constant with the value 0.031±0.004 over nearly eight decades of pair separation, provided that the longitudinal component of the velocity structure tensor is normalized to unity. Remarkably, in appropriate units this constant agrees with the one calculated by Tatarski’s experiment from 1960 within the stated error bars. Other effects on pair dispersion of varying the energy spectrum of the velocity field and the degree of isotropy, as well as the importance of rare events in pair separation statistics, are also developed here within the context of synthetic turbulence satisfying Taylor’s hypothesis.

Hierarchical order in wall‐bounded shear turbulence
View Description Hide DescriptionSince turbulence at realistic Reynolds numbers, such as those occurring in the atmosphere or in the ocean, involve a high number of modes that cannot be resolved computationally in the foreseeable future, there is a strong motivation for finding techniques which drastically decrease the number of such required modes, particularly under inhomogeneous conditions. The significance of this work is to show that wall‐bounded shear turbulence, in its strongly inhomogeneous direction (normal to the wall), can be decomposed into one (or a few) space–time mother mode(s), with each mother generating a whole family of modes by stretching symmetry. In other words, the generated modes are similar, dilated copies of their mother. In addition, we show that the nature of all previous modes strongly depends on the symmetry itself. These findings constitute the first scaling theory of inhomogeneous turbulence.